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Check Divisibility

Variable Divide

What is a factor?

Factors of a number are numbers that divide the given number exactly without leaving any remainder. That is, they are exact divisors of the given numbers. For example, the number 12 is divisible by 2, 3, 4, and 6.

What is HCF?

The greatest common factor (GCF or GCD or HCF) of a set of whole numbers is the largest positive integer that divides evenly all the given numbers with zero remainders. HCF stands for Highest Common Factor. HCF is also known as GCF (Greatest Common Factor) or GCD (Greatest Common Divisor).

The Highest Common Factor of two or more numbers is the highest number by which all the given numbers are divisible without leaving any remainders. Basically, it is the largest number that divides all the given numbers.

Let’s understand the concept of HCF with an example:

Consider two numbers: 12 and 18

Factors of 12: 1, 2, 3, 4, 6, and 12

Factors of 18: 1, 2, 3, 6, 9, and 18

Both 12 and 18 have 1, 2, 3, and 6 as common factors, of which 6 is the highest. So, 6 is the HCF of 12 and 18.

What is LCM?

LCM stands for Lowest or Least Common Multiple. The LCM of two or more numbers is the smallest positive integer that is divisible by all the given numbers.

Let’s understand the concept of LCM with an example:

Consider two numbers: 8 and 12.

The multiples of 8 are:

8 X 1 = 8,
8 X 2 = 16,
8 X 3 = 24,
8 X 4 = 32, and so on…

The multiples of 12 are:

12 X 1 = 12,
12 X 2 = 24,
12 X 3 = 36,
12 X 4 = 48, and so on…

Of all these multiples of 8 and 12, 24 is the lowest and common multiple of both. So, 24 is the LCM of 8 and 12.

LCM And HCF Formula

The relation between HCF and LCM for two numbers is defined by the HCF and LCM formula given by:

Product of two numbers = (HCF of the two numbers) x (LCM of the two numbers)

CAUCHY INTEGRAL,LIOUVILLE AND ROUCHE THEOREM



Cauchy Integral Theorem


If f(z) is analytic in some simply connected region R, then

\[\int f(z)\,dz\space =\space 0\]

for any closed contour gamma completely contained in R. Writing z as

\[z\space =\space x+iy\]

For a multiply connected region,

\[\int_{C_1} f(z)\,dz\space =\space \int_{C_2} f(z)\,dz\]


Liouville Theorem


In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f(z)|<= M for all z in C is constant. Equivalently, non-constant holomorphic functions on C have unbounded images.

\[{\displaystyle |a_{k}|\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {|f(\zeta )|}{|\zeta |^{k+1}}}\,|d\zeta |\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {M}{r^{k+1}}}\,|d\zeta |={\frac {M}{2\pi r^{k+1}}}\oint _{C_{r}}|d\zeta |={\frac {M}{2\pi r^{k+1}}}2\pi r={\frac {M}{r^{k}}},}\]

A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if f and g are entire, and |f| ≤ |g| everywhere, then f = α·g for some complex number α. Consider that for g = 0 the theorem is trivial so we assume. Consider the function h = f/g. It is enough to prove that h can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of h is clear except at points in g−1(0). But since h is bounded and all the zeroes of g are isolated, any singularities must be removable. Thus h can be extended to an entire bounded function which by Liouville's theorem implies it is constant.


Rouche Theorem


Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K with closed contour K, if |g(z)| < |f(z)| on partial K, then f and f + g have the same number of zeros inside K, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour K is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.

Let C be a closed, simple curve (i.e., not self-intersecting). Let h(z) = f(z) + g(z). If f and g are both holomorphic on the interior of C, then h must also be holomorphic on the interior of C. Then, with the conditions imposed above, the Rouche's theorem in its original (and not symmetric) form says that

If |f(z)| > |h(z) − f(z)|, for every z in C, then f and h have the same number of zeros in the interior of C.

What is Fourier Series?

Fourier Series is an infinite series representation of Periodic function in terms of trignometric functions sine and cosine .
Fourier series make use of the orthogonality relationships of the sine and cosine functions.



The Fourier seriesof the function is given by

\[{f\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {\left\{ {{a_n}\cos nx + {b_n}\sin nx} \right\}} ,}\]

where the coefficients are defined by integrals :

\[{{a_0} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)dx} ,\;\;\;}\kern-0.3pt{{a_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\cos nx dx} ,\;\;\;}\kern-0.3pt{{b_n} = \frac{1}{\pi }\int\limits_{ – \pi }^\pi {f\left( x \right)\sin nx dx} .}\]


Fourier Series of Even and Odd Functions


Fourier Series of even function

The Fourier series expansion of an even function f(x) with the period of 2π
does not involve the terms with sines and has the form:

\[{f\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {{a_n}\cos nx} ,}\]

where the Fourier coefficients are given by the formulas

\[{{a_0} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)dx} ,\;\;\;}\kern-0.3pt{{a_n} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\cos nxdx} .}\]

Fourier Series of odd function

Accordingly, the Fourier series expansion of an odd 2π-periodic
function f(x) consists of sine terms only and has the form:

\[f\left( x \right) = \sum\limits_{n = 1}^\infty {{b_n}\sin nx} ,\]

where the coefficients are given by the formula

\[{b_n} = \frac{2}{\pi }\int\limits_0^\pi {f\left( x \right)\sin nxdx} .\]

Difference between Statement and Sentence in Logic

\[ A\, statement\, is\, a\, sentence\, which\, is\, either\, true\, or\, false,\]

\[ but\, not\, both\, true\, and\, false\, simultaneously.\]

\[ A \,sentence\, will\, not\, be\, considered\, to\, be\, a\, Statement\, if:\]

\[ It\, is\, an\, exclamation,\, order,\, request,\, question,\, depicts\, time,\, places,\, pronouns.\]


Simple Statement

\[If\, a\, statement\, cannot\, be\, further\, broken\, down\, into\, various\, statements,\]

\[ or\, in\, simpler\, words\, if\, it\, is\, concrete\, by\, itself,\, it\, is\, called\, a\, Simple\, Statement.\]


Compound Statement

\[If\, a\, statement\, can\, further\, be\, broken\, down\, into\, simpler\, statements\, so\, that\,from\,a\,main\]

\[ statement,\, we\, can\, yield\, more\, than\, one\, statement,\, then\, it \,is\, called\, a\, Compound \,Statement.\]


Conjunction, Disjunction and Negation

CONJUNCTION

\[The\, statement\, p^q\, has\, the\, truth\, value\, T(true)\, if\, both\, p\, and\, q\, have\, the\, truth\, value\, T.\]

\[Similarly,\, the\, statement\, p^q\, has\, the\, truth\, value\, F(false)\, if\, either\, p\, or\, q \,have\, the\]

\[truth \,value \,F \,or\, both\, have\, the\, truth \,value\, F.\]


DISJUNCTION

\[The\, statement\, pvq\, has\, the\, truth\, value\, F(false)\, if\, both\, p\, and \,q \,have \,the\, truth\, value\, F.\]

\[Similarly,\, the\, statement\, pvq\, has\, the \,truth\, value \,T(true)\]

\[ if\, either\, p \,or\, q\, have\, the\, truth\, value\, T\, or\, both\, have\, the\, truth\, value\, T.\]


NEGATION

\[~p\, has\, truth \,value\, T\, whenever\, p\, has\, truth\, value\, F.\]

\[~p\, has\, truth\, value\, F\, whenever\, p\, has\, truth\, value\, T.\]


CONVERSE OF A CONDITIONAL STATEMENT

\[If\, p=>q\, is\, a\, conditional \,statement;\, then\, its\, converse\, is\, q=>p.\]


CONTRAPOSITIVE OF A CONDITIONAL STATEMENT

\[If\, a\, connective\, “if\, then”\, is\, used\, to\, make\, the\, compound\, statement\, if \,p\, then\, q,\]

\[ it\, is\, a \,conditional\, statement.\,it\, is\, also\, called\, as\, an\, implication\]

\[and\, is\, written \,as\, p => q.\]

\[ CONTRAPOSITIVE\, of\, a\, Conditional\, Statement\, is:\, ~q => ~p.\]


Truth Table For Logical Operations

AND operation:

p q p ∧ q
T T T
T F F
F T F
F F F

OR operation:

p q p ∨ q
T T T
T F T
F T T
F F F

Negation Operation:

p ∼ p
T F
T F
F T
F T

Conditional Operation

p q p ⇒ q
T T T
T F F
F T T
F F T

Biconditional Operation:

p q p ⇔ q
T T T
T F F
F T F
F F T

Tautology and Fallacy

\[A\, tautology\, asserts \,that \,every \,possible\, interpretation\, has \,only\, one\, output, \,namely\, true.\]

\[On \,the\, other\, hand,\, fallacy\, implies\, an \,assertion\, of\, false\, in\, every\, possible\, interpretation\]


p q ⇒q ⇒p (p⇒q)∨(q⇒p) ∼{(p⇒q) ∨(q⇒p)}
T T T T T F
T F F T T F
F T T F T F
F F T T T F

Integration Formulae

\[1)\space\int x^n \,dx\space=\frac{x^{(n+1)}}{n+1}+c\]

\[2)\space\int\frac{1}{x}\,dx\space=ln\,|x|+c\]

\[3)\space\int e^x\,dx\space=e^x+c\]

\[4)\space\int a^x\,dx\space=\frac{a^x}{ln\space a}+c\]

\[5)\space\int ln\space x\,dx\space=x\,ln\space x-x+c\]

\[6)\space\int sin\space x\,dx\space=-\,cos\space x+c\]

\[7)\space\int cos\space x\,dx\space=sin\space x+c\]

\[8)\space\int sec^2\space x\,dx\space=tan\space x+c\]

\[9)\space\int cosec^2\space x\,dx\space=-\,cot\space x+c\]

\[10)\space\int sec\space x\space tan\space x\,dx=sec\space x+c\]

\[11)\space\int cosec\space x\space cot\space x\,dx=-\,cosec\space x+c\]

\[12)\space\int\tan\space x\,dx=ln\,|sec\space x|+c\]

\[13)\space\int\cot\space x\,dx=ln\,|sin\space x|=-\,ln\,|cosec\space x|+c\]

\[14)\space\int\sec\space x\,dx=ln\,|tan\space x\,+\,sec\space x|=-\,ln\,|sec\space x\,-\,tan\space x|+c\]

\[15)\space\int\cosec\space x\,dx=ln\,|cosec\space x\,-\,cot\space x|=ln\,|tan\,\frac{x}{2}|+c\]

\[16)\space\int\frac{1}{x^2+a^2}\,dx=\frac{1}{a}\,tan^{-1}\lparen\frac{x}{a}\rparen+c\]

\[17)\space\int\frac{1}{x^2-a^2}\,dx=\frac{1}{2a}\,ln\,|\frac{x-a}{x+a}|+c\]

\[18)\space\int\frac{1}{a^2-x^2}\,dx=\frac{1}{2a}\,ln\,|\frac{x+a}{x-a}|+c\]

\[19)\space\int\frac{1}{\sqrt{a^2-x^2}}\,dx=sin^{-1}(\frac{x}{a})+c\]

\[20)\space\int\frac{1}{\sqrt{a^2+x^2}}\,dx=ln\,|x+\sqrt{x^2+a^2}|+c\]

\[21)\space\int\frac{1}{\sqrt{x^2-a^2}}\,dx=ln\,|x+\sqrt{x^2-a^2}|+c\]

\[22)\space\int\sqrt{a^2-x^2}\,dx=\frac{x}{2}\sqrt{a^2-x^2}\,+\,\frac{a^2}{2}sin^{-1}(\frac{x}{a})+c\]

\[23)\space\int\sqrt{x^2-a^2}\,dx=\frac{x}{2}\sqrt{x^2-a^2}\,-\,\frac{a^2}{2}ln\,|x+\sqrt{x^2-a^2}|+c\]

\[24)\space\int\sqrt{x^2+a^2}\,dx=\frac{x}{2}\sqrt{x^2+a^2}\,+\,\frac{a^2}{2}ln\,|x+\sqrt{x^2+a^2}|+c\]

\[25)\space\int\frac{1}{|x|\,\sqrt{x^2-a^2}}\,dx=\frac{1}{a}\,sec^{-1}\lparen\frac{x}{a}\rparen+c\]

\[26)\space\int f(ax+b) \,dx\space=\frac{1}{a}\,f(ax+b)\,+c\]

\[27)\space\int \frac{f'(x)}{f(x)} \,dx\space=ln|f(x)|\,+c\]

\[28)\space\int \frac{f'(x)}{\sqrt{f(x)}} \,dx\space=2\sqrt{f(x)}\,+c\]

\[29)\space\int (f(x))^{n}\,f'(x)\,dx\space=\frac{(f(x))^{(n+1)}}{n+1}+c\]

\[30)\space\int e^{x}\,(f(x)\,+\,f'(x)\,)\,dx\space=e^{x}\,f(x)\,+c\]

\[31)\space\int e^{ax}sinbx \,dx\space=\frac{e^{ax}}{a^2+b^2}(asinbx-bcosbx)+c\]

\[32)\space\int e^{ax}cosbx \,dx\space=\frac{e^{ax}}{a^2+b^2}(acosbx+bsinbx)+c\]

Distance Formula: To Calculate Distance Between Two Points

\[Let \,the\, two \,points\, be \,A\, and \,B,\, having\, coordinates\, to\]

\[ be\, (x1,y1) \,and \,(x2,y2)\, respectively.\]

\[Thus,\, the \,distance\, between\, two\, points \,is\, given\, as-\]

\[\large d = \sqrt{(x_{2} -x_{1})^{2} + (y_{2} – y_{1})^{2}}\]




Midpoint Theorem: To Find Mid-point of a Line Connecting Two Points

\[Consider\, the\, same\, points \,A\, and \,B, \,having\, coordinates\, to\]

\[ be\, (x1,y1) \,and\, (x2,y2) \,respectively.\, Let\, M(x,y)\, be\, the\]

\[ midpoint\, of\, lying\, on \,the\, line\, connecting\, these\, two\]

\[ points \,A\, and\, B.\, The \,coordinates\, of\, the\, point\, M \,is\, given\, as-\]

\[\large M (x,y)= \left ( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right )\]




Angle Formula: To Find The Angle Between Two Lines

\[Consider\, two \,lines A\, and\, B,\, having \,their \,slopes\, to\, be\, m1\, and\, m2\, respectively.\]

\[Let\, “θ” \,be\, the\, angle\, between \,these \,two\, lines,\, then\, the\, angle \,between \,them \,can \,be\, represented\, as\]

\[\large \tan \theta = \frac{m_{1} – m_{2}}{1 + m_{1} m_{2}}\]


\[SPECIAL\, \, CASES:\]


\[i) \space \,Case \,1\,:\, When\, the\, two \,lines \,are\, parallel\, to \,each\, other,\]

\[\large m_{1} = m_{2}=m\]

\[Substituting\, the\, value \,in \,the\, equation\, above,\]

\[\large \tan \theta = \frac{m – m }{1 + m^{2}} = 0\]

\[\large \Rightarrow \theta = 0\]


\[ii) \space \,Case\, 2\,:\, When \,the\, two\, lines \,are\, perpendicular \,to \,each\, other,\]

\[m1 . m2 = -1\]

\[Substituting\, the\, value\, in \,the\, original \,equation,\]

\[\large \tan \theta = \frac{m_{1} – m_{2}}{1 + (-1)} = \frac{m_{1} – m_{2}}{0}\,\, which\, is\, undefined.\]

\[⇒ \theta = 90°\]




Section Formula: To Find a Point Which Divides a Line into m:n Ratio

\[Consider \, a \, line \, A \, and \, B \, having \, coordinates \, (x_1,y_1) \, and \, (x_2,y_2) \, respectively. \]

\[Let \,P\, be\, a\, point\, that \,which\, divides\, the \,line\, in \,the\, ratio\, m:n, \]

\[then \,the\, coordinates\,of\, the \,point\, P \,is \,given\, as-\]

\[i) \space When\, the\, ratio\, m:n \,is\, internal\,:\]

\[\large \left (\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n} \right )\]

\[ii) \space When \,the\, ratio \,m:n \,is\, external\,:\]

\[\large \left (\frac{mx_{2} – nx_{1}}{m – n}, \frac{my_{2} – ny_{1}}{m – n} \right )\]




Area of a Triangle in Cartesian Plane

\[The\, area \,of\, a\, triangle\, In \,coordinate\, geometry\,whose \,vertices\, are\, (x_1,y_1),\,(x_2,y_2) \,and\, (x_3,y_3)\, is\]

\[\frac{1}{2}|x_1 (y_2~ -~ y_3)~ + ~x_2(y_3~ – ~y_1)~ +~ x_3(y_1~ – ~y_2)|\]

Distance


Coordinate 1 = ( , )
Coordinate 2 = ( , )





Midpoint


Coordinate A = ( , )
Coordinate B = ( , )





Intersection Point of Two Lines


Equation 1 : x + y + = 0

Equation 2 : x + y + = 0





Angle between two lines


Equation 1 : x + y + = 0

Equation 2 : x + y + = 0





Distance between point and a line


Line Equation : x + y + = 0

Point = ( , )



Parallel Line Checker


Coordinate 1 = ( , )
Coordinate 2 = ( , )
Coordinate 3 = ( , )
Coordinate 4 = ( , )





Perpendicular Line Checker


Coordinate 1 = ( , )
Coordinate 2 = ( , )
Coordinate 3 = ( , )
Coordinate 4 = ( , )





Section Formula


Coordinate 1 = ( , )

Coordinate 2 = ( , )

Ratio = ( , )





Collinear Points Checker


Coordinate 1 = ( , )

Coordinate 2 = ( , )

Coordinate 3 = ( , )


Line Perpendicular To Given Line and Pass through the Point


Line Equation : x + y =

Point = ( , )



Line Parallel To Given Line and Pass through the Point


Line Equation : x + y =

Point = ( , )



Slope of the line with 2 points known

 

Required Coordinates: , , ,

 

Magnitude of a Vector

\[The\, magnitude\, of\, a\, vector\, is \,shown\, by\, vertical\, lines\, on \,both\, the\, sides\, of\, the \,given\, vector\, “|a|”\]

\[ It \,represents\, the \,length\, of\, the\, vector.\, Mathematically, \,the \,magnitude\]

\[of \,a \,vector\, is\, calculated\, by\, the \,help\, of \,“Pythagoras\, Theorem,”\, i.e.\]

\[|a|= √(x2+y2)\]



Unit Vector

\[A \,unit\, vector \,has \,a \,length \,(or\, magnitude)\, equal \,to\, one,\, which \,is\, basically\, used \,to\, show\, the\]

\[ direction \,of\, any \,vector.\, A\, unit \,vector\, is \,equal \,to\, the \,ratio\, of \,a\, vector\, and\, its\, magnitude.\]

\[ Symbolically,\, it\, is\, represented \,by\, a\, cap\, or \,hat (^). \]

\[\hat{\mathbf{a}}=\frac{\mathbf{a}}{\|\mathbf{a}\|}\]



Zero Vector

\[A \,vector\, with\, zero \,magnitudes \,is\, called \,a \,zero\, vector.\, The \,coordinates\, of\, zero\, vector\]

\[ are\, given\, by\, (0,0,0) \,and \,it\, is\, usually \,represented \,by\, 0\, with\, an\, arrow (→) \,at \,the\, top\, or\, just\, 0\]



Characteristics of Vector Math Addition

\[* \,Commutative\, Law-\, the\, order \,of\, addition\, does\, not \,matter,\, i.e,\, a + b = b + a\]

\[* \,Associative\, law-\, the\, sum \,of \,three\, vectors\, has\, nothing \,to\, do\, with\]

\[ which\, pair\, of\, the\, vectors\, are \,added \,at\, the \,beginning.\]



Scalar Multiplication

\[Only \,the\, magnitude \,of\, a\, vector\, is \,changed\, not \,the\, direction.\]

\[* \, \, S(a+b) = Sa + Sb\]

\[* \, \, a.1 = a\]

\[* \, \, a.0 = 0\]

\[* \, \, a.(-1) = -a\]



Scalar Triple Product

\[The \,scalar\,triple\, product,\, also\, called\, as\, box\, product \,or\, mixed \,triple\]

\[ product,\, of \,three \,vectors,\, say \,a,\, b\, and \,c\, is\, given\, by \,(a×b)⋅c.\]

\[ It\, is\, also\, denoted \,by\, (a\, b\, c). \]

\[(a \,b\, c)\, = (a×b)⋅c\]

\[\begin{aligned} (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &=\left|\begin{array}{ll} a_{2} & a_{3} \\ b_{2} & b_{3} \end{array}\right| c_{1}-\left|\begin{array}{ll} a_{1} & a_{3} \\ b_{1} & b_{3} \end{array}\right| c_{2}+\left|\begin{array}{ll} a_{1} & a_{2} \\ b_{1} & b_{2} \end{array}\right| c_{3} \\ &=\left|\begin{array}{lll} c_{1} & c_{2} & c_{3} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array}\right| \end{aligned}\]



Cross Product


\[The \,cross\, product\, of \,two\, vectors\, results \,in\, a\, vector\, quantity.\]

\[ It\, is \,represented\, by\, a \,cross\, sign \,between\, two\, vectors.\]

\[a x b = | a | | b | sin θ\]


Properties of Cross Product


\[1)\space \overrightarrow{a}\,\times\, \overrightarrow{b}\,=\,-\, \overrightarrow{b}\,\times\, \overrightarrow{a} \]

\[2)\space \overrightarrow{a}\,\times\, \overrightarrow{a}\,=\,0 \]

\[3)\space \overrightarrow{a}\,\times\,(\,\overrightarrow{b}\,+\,\overrightarrow{c}\,)\,=\, \overrightarrow{a}\,\times\, \overrightarrow{b}\,+\,\overrightarrow{a}\,\times\,\overrightarrow{c} \]

\[4)\space \hat{i}\,\times\, \hat{i}\,=\, \hat{j}\,\times\, \hat{j}\,=\,\hat{k}\,\times\,\hat{k}\,=\,0\,\space and \space\,\hat{i}\,\times\,\hat{j}\,=\,\hat{k}\,,\,\hat{j}\,\times\,\hat{k}\,=\,\hat{i}\,,\,\hat{k}\,\times\,\hat{i}\,=\,\hat{j} \]

\[5)\space Two\,non {\text -} zero\,vectors\, \overrightarrow{a}\,and\, \overrightarrow{b}\,are\,collinear\,if\,and\,only\,if\, \overrightarrow{a}\,\times\, \overrightarrow{b}\,=\,0 \]

\[6)\space If\, \overrightarrow{a}\,=\,a_1\,\hat{i}\,+\,a_2\,\hat{j}\,+\,a_3\,\hat{k}\,and\,\overrightarrow{b}\,=\,b_1\,\hat{i}\,+\,b_2\,\hat{j}\,+\,b_3\,\hat{k}\,,\,then\\ \\ \overrightarrow{a}\,\times\, \overrightarrow{b}\,=\,\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\,=\,(\,a_2b_3\,-\,a_3b_2\,)\,\hat{i}\,+\,(\,a_3b_1\,-\,a_1b_3\,)\,\hat{j}\,+\,(\,a_1b_2\,-\,a_2b_1\,)\,\hat{k} \]


DOT PRODUCT

\[The\, dot \,product\, of \,two \,vectors \,always\, results\, in\, scalar\, quantity,\]

\[ i.e.\, it \,has \,only \,magnitude\, and \,no \,direction.\, It \,is\, represented\]

\[ by\, a\, dot (.)\, in \,between\, two \,vectors.\]

\[a . b = | a | | b | cos θ\]


Properties of Dot Product

\[1)\space \overrightarrow{a}\,.\, \overrightarrow{a}\,=\,|\,\overrightarrow{a}\,|\,|\,\overrightarrow{a}\,|\,cos\,0\,=\,|\,\overrightarrow{a}\,|^2\,=\,a^2 \]

\[2)\space \hat{i}\,.\, \hat{i}\,=\,\hat{j}\,.\, \hat{j}\,=\,\hat{k}\,.\, \hat{k}\,=\,1 \]

\[3)\space \overrightarrow{a}\,.\, \overrightarrow{b}\,=\, \overrightarrow{b}\,.\, \overrightarrow{a}\,(\,commutative\,) \]

\[4)\space \overrightarrow{a}\,.\,(\, \overrightarrow{b}\,+\,\overrightarrow{c}\,)\,=\, \overrightarrow{a}\,.\, \overrightarrow{b}\,+\, \overrightarrow{a}\,.\, \overrightarrow{c}\,(\,distributive\,) \]

\[5)\space (\,l\,\overrightarrow{a}\,)\,.\,(\,m\,\overrightarrow{b}\,)\,=\,lm\,(\,\overrightarrow{a}\,.\,\overrightarrow{b}\,)\,,\,where\,l\,and\,m\,are\,scalars \]

\[6)\space If\, \overrightarrow{a}\,and\, \overrightarrow{b}\,are\,two\,non {\text -} zero\,vectors\,,\,then\, \overrightarrow{a}\,.\, \overrightarrow{b}\,=\,0 \]


Components of Vectors (Horizontal & Vertical)

\[ Vector \,“a”\, can\, be\, broken \,down \,into \,two\, components\, i.e. \,a_x\, and\, a_y\]

\[The \,component \,a_x \,is\, called\, a \,“Horizontal\, component”\, whose \,value\, is\, a\, cos θ.\]

\[The\, component\, a_y \,is\, called\, a \,“Vertical\, component”\, whose\, value \,is \,a\, sin θ.\]

Straight Line

\[1)\space Centroid\,with\,given\,three \,coordinates{(x_1,y_1),(x_2,y_2),(x_3,y_3)}:\,\]

\[\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\]


\[2)\space Relation\,between\,two\,lines\]

\[Let \,L1 \,and\, L2 \,be \,the \,two\, lines \,as\]

\[ L_1\, :\, a_1x + b_1y + c_1 = 0 \space \space \space \, \, \, L_2 \, :\, a_2x + b_2y + c_2 = 0\]

\[a)\space For\, Parallel\, Lines:\]

\[\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\]

\[b)\space For\, Intersecting\, Lines:\]

\[\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\]

\[c)\space For\, Coincident\, Lines:\]

\[\frac{a_1}{a_2}=\frac{b_1}{b_2} = \frac{c_1}{c_2}\]


\[3) \space Angle \,between\, Straight\, lines\]

\[Let\,L_1\,≡y=m_1 x+c_1 \, \,and \,\,\, {{L}_{2}}\,\,\,\equiv \,\,\,y={{m}_{2}}x+{{c}_{2}}\]

\[Angle = \theta ={{\tan }^{-1}}|( \frac{{{m}_{2}}-{{m}_{1}}}{1+{{m}_{1}}{{m}_{2}}})​\]

\[Special Cases:\]

\[i) \space m_2=m_1\,→\,lines\,are\,parallel\]

\[ii) \space m_1 m_2=−1\,→\,lines \,are\,perpendicular\,to\,each\,other\]


\[4) \space Length\, of\, Perpendicular \,from \,a\, Point\, on\, a\, Line\]

\[The\, length\, of\, the\, perpendicular \,from\]

\[ P(x1, y1) \,on\, ax + by + c = 0 \,is\]

\[\]

\[l=\,|\frac{|ax_1+by_1+c}{\sqrt{a^2+b^2}}|\]

\[B (x, y)\, is\, foot\, of\, perpendicular\, is\, given \,by:\]

\[\frac{x-x_1}{a}=\frac{y-y_1}{b}=-\frac{(ax_1+by_1+c)}{(a^2+b^2)}\]

\[A’(h, k)\, is\, mirror\, image, \,given\, by:\]

\[\frac{h-x_1}{a}=\frac{k-y_1}{b}=-2\,\frac{(ax_1+by_1+c)}{(a^2+b^2)}\]


\[5) \space Angular\, Bisector\, of \,Straight \,lines\]

\[An \,angle\, bisector \,has \,equal \,perpendicular\, distance\]

\[ from \,the \,two\, given\, lines\]

\[The\, equation\, of\, line\, L \,can\, be\, given\, as:\]

\[\frac{(a_1 x+b_1 y+c_1)}{\sqrt{(a_1^2+b_1^2)}}\,=\,± \frac{(a_2 x+b_2 y+c_2)}{\sqrt{(a_2^2+b_2^2)}}\]


\[6) \space Distance\, of\, a \,Point\, From\, a\, Line\]

\[Let\, the\, perpendicular\, distance (d)\, of \,a \,line \,Ax + By+ C = 0\, from \,a \,point (x1, y1)\, is\, defined \,by\]

\[d=\frac{\left | Ax_{1}+By_{1}+C \right |}{\sqrt{A^{2}+B^{2}}}\]


\[7) \space Distance \,Between \,Two\, Parallel\, Lines\]

\[The \,distance \,d\, between\, two \,parallel\, lines \,say, \,y= m x+ c1\, and\, y = m x + c2 \,is \,given\, by:\]

\[d=\frac{\left | C_{1}-C_{2} \right |}{\sqrt{1+m^{2}}}\]

\[Consider\, the \,general\, form\, of \,the\, line,\, i.e.,\, Ax + By + C1= 0 \,and\, Ax + By + C2=0\]

\[d=\frac{\left | c_{1}-c_{2} \right |}{\sqrt{A^{2}+B^{2}}}\]


\[8) \space Concurrency \,of \,Three \,Lines\]

\[Let \,the\, lines \,be:\]

\[{{L}_{1}}\equiv {{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}\]

\[ {{L}_{2}}\equiv {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}\]

\[ {{L}_{3}}\equiv {{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}\]

\[\begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3} \end{vmatrix} = 0\]

Lagrange’s Mean Value Theorem

\[If \,a \,function\, f\, is\, defined\, on \,the\, closed \,interval\, [a,b]\, satisfying\, the \,following\, conditions –\]

\[i)\, \, The \,function \,f\, is \,continuous\, on\, the\, closed \,interval \,[a, b]\]

\[ii)\, \, The \,function \,f\, is \,differentiable \,on\, the \,open\, interval\, (a, b)\]

\[Then \,there \,exists\, a\, value\, x = c\, in\, such\, a\, way\, that\, f'(c) = \,[f(b) – f(a)]/(b-a)\]


Geometrical Interpretation of Lagrange’s Mean Value Theorem



Rolle’s Theorem

\[A\, special\, case\, of\, Lagrange’s\, mean\, value \,theorem \,is\, Rolle’s\, Theorem\, which \,states\, that:\]

\[If\, a \,function\, f \, is\, defined\, in\, the\, closed \,interval\, [a, b] \,in \,such \,a \,way \,that\, it \,satisfies\, the \,following:\]

\[i) \, The\, function\, f \,is\, continuous\, on \,the\, closed\, interval\, [a, b]\]

\[ii)\, The \,function\, f \,is\, differentiable \,on \,the\, open\, interval\, (a, b)\]

\[iii) \, Now\, if\, f (a) = f (b) ,\, then\, there\, exists\, at \,least \,one\, value \,of\, x,\]

\[ let's \,assume\, this\, value\, to\, be \,c,\, which \,lies \,between\, a \,and\, b\, i.e. \,(a < c < b )\, such \,that\, f‘(c) = 0 .\]

\[Precisely,\, if \,a\, function\, is\, continuous \,on\, the \,closed \,interval \,[a, b]\, and \,differentiable\]

\[ on\, the\, open\, interval\, (a, b) \,then\, there\, exists\, a \,point\, x = c \, in\, (a, b) \,such\, that\, f'(c) = 0\]


Geometric interpretation of Rolle’s Theorem



Cauchy's Theorem

\[Let \,the\, functions \,f ( x ) \,and\, g ( x ) \,be\, continuous\, on \,an\, interval \,[ a , b ],\]

\[ differentiable \,on \,(a , b ) ,\, and\, g ′ ( x ) ≠ 0\, for\, all \,x ∈ ( a , b ) .\]

\[ Then\, there \,is\, a\, point\, x = c \,in \,this\, interval\, such \,that\]

\[{\frac{{f\left( b \right) – f\left( a \right)}}{{g\left( b \right) – g\left( a \right)}}} = {\frac{{f’\left( c \right)}}{{g’\left( c \right)}}.}\]



Critical Point

\[A \,function\, f \,which \,is\, continuous\, with \,x\, in \,its\, domain \,contains \,a \,critical \,point \,at \,point\, x\]

\[ if \,the\, following\, conditions \,hold \,good.\]

\[* \,f ’(x) = 0 \]

\[* \,f ’(x) \,is\, undefined \]

\[A \,point \,of\, differentiable\, function\, f\, at \,which \,derivative \,is\, zero \,can \,be \,termed \,as\,critical \,point.\]


\[The\, types \,of\, critical\, points \,are\, as \,follows:\]

\[* \, A\, critical \,point\, is \,a\, local \,maximum \,if\, the \,function \,changes\]

\[ from\, increasing\, to \,decreasing\, at \,that \,point, \,whereas \,it\, is\, called\]

\[ a\, local\, minimum\, if \,the\, function\, changes \,from\, decreasing\, to\, increasing\, at\, that\, point.\]

\[* \, A \,critical\, point \,is\, an\, inflexion\, point \,if\, the \,concavity\, of\, the\]

\[ function\, changes\, at \,that\, point.\]

\[* \,If\, a\, critical\, point \,is\, neither\, of\, the\, above, \,then\, it\, signifies \,a\]

\[ vertical\, tangent\, in\, the\, graph\, of\, a\, function.\]

Application of Derivative


Rate of Change of a Quantity

\[ This\, is \,the\, general\, and\, most\, important\, application \,of\, derivative.\]

\[ For\, example, \,to\, check \,the \,rate \,of \,change \,of \,the\, volume \,of\, a\, cube\]

\[ with\, respect\, to\, its \,decreasing \,sides,\, we\, can \,use \,the \,derivative\, form\, as \,dy/dx.\]

\[ Where \,dy \,represents\, the\, rate\, of\, change\, of\, volume\, of \,cube\, and\]

\[ dx \,represents\, the\, change\, of \,sides\, of \,the\, cube.\]


Increasing and Decreasing Functions

\[To\, find \,that\, a\, given\, function \,is \,increasing\, or \,decreasing\, or\]

\[ constant,\, say\, in\, a\, graph,\, we\, use\, derivatives.\, If \,f\, is\, a\, function\]

\[ which\, is\, continuous\, in \,[p, q] \,and \,differentiable\, in\, the \,open\, interval\, (p, q),\, then,\]

\[* f\, is\, increasing \,at\, [p, q]\, if\, f'(x) > 0 \,for\, each \, x ∈ (p, q)\]

\[* f\, is\, decreasing\, at\, [p, q]\, if\, f'(x) < 0\, for \,each \,x ∈ (p, q)\]

\[* f\, is\, constant\, function\, in\, [p, q], if\, f'(x)=0 \,for \,each\, x ∈ (p, q)\]


Monotonicity At a Point

\[Functions \,are \,said \,to\, be\, monotonic\, if\, they\]

\[are\, either\, increasing\, or \,decreasing\, in\, their \,entire \,domain. \]

\[(a) \space A \space function \space f(x) \space is \space called \space an \space increasing \space function \space at \space point \space x \space = \space a, \space if \, in \, a \]\[sufficiently \, small \, neighbourhood \, of \, x=a \, ; \, f(a-h) < f(a) < f(a+h)\]

\[(b) \, A \,function\, is \,called \,a \,decreasing\, function\, at\, at\, point \, x = a\, if\, in \,a\]\[sufficiently \, small \, neighbourhood \, of \, x=a \, ; \, f(a-h) > f(a)>f(a+h)\]

\[(c) f'(x)\, will\, be \,zero \,when \,the\, function\, is \,at\, its \,maxima\, or\, minima\]


Point of Inflection

\[For\, continuous\, function\, f(x), \,if \,f'(x_0) = 0\, or\, f’”(x_0)\, does\, not\, exist\]

\[ at\, points\, where\, f'(x_0)\, exists\, and \,if\, f”(x)\, changes\, sign \,when \,passing\, through\]

\[ x = x_0 \,then\, x_0\, is\, called\, the\, point \,of\, inflection.\]

\[(a) \,If \,f”(x) < 0,\, x ∈ (a, b) \,then \,the \,curve \,y = f(x)\, is \,concave \,downward\]

\[(b) \,If \,f” (x) > 0, \,x ∈ (a, b)\, then\, the \,curve\, y = f(x)\, is \,concave\, upwards\]

\[ in (a, b)\]


Special Points

\[(a) Critical \space Points \space : \space The \space points \space of \space domain \space for \space which \space f`(x) \space is \space equal \space to \space zero \space or \space doesn't \space exist \]\[\space are \space called \space critical \space points. \]

\[(b) Stationary \space Points \space : \space The \space stationary \space points \space are \space the \space points \space of \space domain \space where \space f`(x) \space = \space 0 .\]\[ Every \space stationary \space point \space is \space a \space critical \space point.\]



Equation of a plane


Point A = ( , , )

Point B = ( , , )

Point C = ( , , )




Distance from point to plane calculator


Plane Equation : x + y + z + = 0

Point M = ( , , )




Distance between perpendicular line and plane




Distance between two planes calculator


Plane Equation 1: x + y + z + = 0

Plane Equation 2: x + y + z + = 0




Angle between planes calculator


Plane Equation 1: x + y + z + = 0

Plane Equation 2: x + y + z + = 0



3-D Geometry


\[1)\space Distance\, of\,P(x,y,z)\, from\, origin(0,0,0)= \sqrt {(x^2+y^2+z^2)}\]


\[2) \space Distance\, between \,2 \,points\, P\left( x_1,y_1,z_1 \right)\, and \,Q\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\]

\[=\sqrt{ (x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]


\[3) \space Let\, P\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)\, and \, Q\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\,be\, 2\, points.\]

\[ R\, divides\, the\, line \,segment\, PQ\, in\, ratio\, internally.\, Then\, R\, has\, coordinate:\]

\[(\frac {mx_2+nx_1}{m+n},\,\frac {my_2+ny_1}{m+n},\,\frac {mz_2+nz_1}{m+n})\]


\[4) \space Direction\, cosines\, and \,direction \,Ratios\, of\, a \,Line \,in \,Cartesian \,Plane\]

\[Cosines\, of \,the \,angles\, a \,line \,makes\, with \,the \,positive\, x,\, y\, and \, z \,axis\, respectively,\]

\[ are \,called \,direction \,cosines\, of \,that \,line.\]

\[So\, if\, those \,angles\, are\, \alpha ,\,\beta ,\,\gamma\, then\, \cos \alpha ,\,cos\beta ,\,cos\gamma\, are \,the\, direction\, cosines\, of\, the\]

\[ line. \,They \,are\, denoted\, by\, l,m,n.\]

\[l^2+m^2+n^2=1\]

\[Any\, 3 \,number\, a,\,b,\,c\,which\, are\, proportional\, to\, direction\, cosines\]

\[ are \,called \,direction\, ratios.\]

\[Hence \frac{l}{a}=\frac{m}{b}=\frac{n}{c}=\frac{\sqrt{{{l}^{2}}+{{m}^{2}}+{{n}^{2}}}}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}=\frac{1}{\sqrt{{{a}^{2}}{{b}^{2}}+{{c}^{2}}}} ​\]


\[5) \space Let \,P(x,y,z) \,and\, Q\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)\, be \,2 \,points. \]

\[Then\, direction \,cosines\, will \,be\]

\[ l=\frac{{{x}_{2}}-{{x}_{1}}}{\left| PQ \right|},m=\frac{{{y}_{2}}-{{y}_{1}}}{\left| PQ \right|},n=\frac{{{z}_{2}}-{{z}_{1}}}{\left| PQ \right|}\]


\[6) \space Projection\, of\, PQ,\,(P(x_1,y_1,z_1),\, Q(x_2,y_2,z_2)) \,on \,a \,line\, whose\]

\[direction \,cosines\, are \,l,m,n is \,l\left( {{x}_{2}}-{{x}_{1}} \right)+m\left( {{y}_{2}}-{{y}_{1}} \right)+n\left( {{z}_{2}}-{{z}_{1}} \right)\]


\[7) \space 2\, lines \,having\, direction\, cosines\, \left( {{l}_{1}},{{m}_{1}},{{n}_{1}} \right)\, and\, \left( {{l}_{2}},{{m}_{2}},{{n}_{2}} \right).\]

\[ Then \,angle\, between\, them\, is\, \theta ={{\cos }^{-1}}\left( {{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}} \right)\]


\[8) \space Let\, \bar{A} \,be\, the\, vector\, area. \]

\[If\, its\, direction\, cosines \,are \, \cos \alpha ,\cos \beta ,\cos \gamma.\]

\[ Then\, projections \,are\, {{A}_{1}}=A\cos \alpha ,{{A}_{2}}=A\cos \beta ,{{A}_{3}}=A\cos \gamma .\]

\[∴ {{A}^{2}}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}\]


\[9) \space Area \,of\, a\, triangle:\]

\[\alpha = \frac 12\left| \begin{matrix} x_1 & y_1 & 1\cr x_2 & y_2 & 1 \cr x_3 & y_3 & 1 \cr \end{matrix} \right|\]


\[10) \space A \,first\, degree\, equation\, in\, x,\, y, \,z \, \, represents\, a\, plane\, in\, 3D\]

\[ ax+by+cz=0,{{z}^{2}}{{b}^{2}}+{{c}^{2}} \ne \,represents\, a \,plane.\]


\[11) \space Let\, P\, be\, the\, length\, of\, the\, normal\, from\, the \,origin\, to\, the\, plane\]

\[ and \,\, l,m,n \,be\, the \,direction\, cosines\, of\, that \,normal. \]

\[Then\, the\, equation \,of \,the\, plane\, is\, given\, by \, lx+my+nz=P.\]


\[12) \space Let\, a\, plane\, cuts\, length\, a,b,c \,from\, the\, coordinate\, axis.\]

\[ Then\, equation\, of \,the \,plane \,is \, \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1.\]


\[13) \space Plane\, passing\, through \, \left( {{x}_{1}}{{y}_{1}}{{z}_{1}} \right), \, \left( {{x}_{2}}{{y}_{2}}{{z}_{2}} \right), \, \left( {{x}_{3}}{{y}_{3}}{{z}_{3}} \right) \, is\]

\[ \left| \begin{matrix} x & y & z & 1 \\ {{x}_{1}} & {{y}_{1}} & {{z}_{1}} & 1 \\ {{x}_{2}} & {{y}_{2}} & {{z}_{2}} & 1 \\ {{x}_{3}} & {{y}_{3}} & {{z}_{3}} & 1 \\ \end{matrix} \right|=0\]


\[14) \space Angle\, between\, 2\, planes\]

\[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0 \, and \, {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}=0 \, is \, given \, by:\]

\[ \cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]


\[15) \space Two\, sides \,of\, a\, plane\]

\[2\, points\, A\left( {{x}_{1}}{{y}_{1}}{{z}_{1}} \right) \,and\, B\left( {{x}_{2}}{{y}_{2}}{{z}_{2}} \right)\, lie\, on\, the\, same\, side\, or\]

\[ opposite \,sides\, of\, a\, plane\, ax+by+cz+d=0 \,accordingly\, as\, a{{x}_{1}}+b{{y}_{1}}+c{{z}_{1}}+d \]

\[and\, a{{x}_{2}}+b{{y}_{2}}+c{{z}_{2}}+d \,are\, of\, same\, sign \,or\, opposite\, sign.\]


\[16) \space Distance\, from\, a\, point\, to\, a \,plane\]

\[Distance \,of\,{{x}_{1}}{{y}_{1}}{{z}_{1}} \,from\, ax+by+cz+d\, is\]

\[ | \frac{a{{x}_{1}}+b{{y}_{1}}+c{{z}_{1}}+d}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\]


\[17) \space Equation\, of \,the\, planes\, bisecting\, the\, angle\, between \,2\, planes\]

\[Let\, {{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+d=0 \, and \, {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+d=0 \, be \, 2 \, planes. \]

\[The\, planes \,bisecting\, the\, angles \,between \,them \,are \]

\[ \frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=\pm \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]


\[18) \space Two\, Intersecting\, plane\]

\[If\, U=0\, and\, V=0 \,be\, 2\, planes\, then \,the\, plane\]

\[ passing\, through \,the\, line \,of \,their\, intersection\, is\]

\[ U+\lambda V=0\lambda \,to\, be \,determined\, from\, given\, condition.\]

Properties of Logarithm



Principle Properties of Logarithm

\[Let\, m\, and\, n\, be\, arbitrary\, positive\, numbers\, ,\, a\,>\,0\, ,\, a\, \mathrel{\mathtt{!=}}\, 1\, , \, b\,>\,0\, ,\,b\mathrel{\mathtt{!=}}\,1\,and\space\,α\,,\,β\] \[\,be\,any\,real\,numbers\,,\,then\]

\[1)\space\,log_a\,(mn)\space=\space\,log_a\,m\,+\,log_a\,n\]

\[2)\space\,log_a\,(\,\frac{m}{n}\,)\,\space=\space\,log_a\,m\,-\,log_a\,n\]

\[3)\space\,log_a\,m^α\space=\space\,α\,log_a\,m\]

\[4)\space\,log_{a\space^β}\,m\space=\space\frac{1}{β}\,log_a\,m\]

\[5)\space\,log_b\,m\space=\space\frac{log_a\,m}{log_a\,b}\]

Extra properties of Logarithm

\[1)\space\,a^{{log_b}\,x}\space=\space\,x^{{log_b}\,a}\,,\,b\,\mathrel{\mathtt{!=}}\,1\,,\,a\,,\,b\,,\,x\,are\,positive\,numbers\]

\[2)\space\,a^{{log_a}\,x}\space=\space\,x\,,\,a\,>\,0\,,\,a\,\mathrel{\mathtt{!=}}\,1\,,\,x\,>\,0\]

\[3)\space\,log_{a\space^k}\,x\space=\space\frac{1}{k}\,log_a\,x\,,\,a\,>\,0\,,\,a\,\mathrel{\mathtt{!=}}\,1\,,\,x\,>\,0\]

\[4)\space\,log_a\,x^{2k}\space=\space\,2k\,log_a\,|x|\,,\,a\,>\,0\,,\,a\,\mathrel{\mathtt{!=}}\,1\,,\,k\,\in\,I\]

\[5)\space\,log_{a\space^{2k}}\,x\space=\space\frac{1}{2k}\,log_{|a|}\,x\,,\,x\,>\,0\,,\,a\,>\,0\,,\,a\,\mathrel{\mathtt{!=}}\,±1\space\,and\space\,k\,\in\,I\,\sim\{\,0\}\]

\[6)\space\,log_{a\space^α}\,x^β\space=\space\frac{β}{α}\,log_a\,x\,,\,x\,>\,0\,,\,a\,>\,0\,,\,a\,\mathrel{\mathtt{!=}}\,1\,,\,α\,\mathrel{\mathtt{!=}}\,0\]

\[7)\space\,a^{{log_b}\,a}\space=\space\sqrt{a}\,,\,if\,b\,=\,a^2\,,\,a\,>\,0\,,\,b\,>\,0\,,\,b\,\mathrel{\mathtt{!=}}\,1\]

\[8)\space\,a^{{log_b}\,a}\space=\space\,a^2\,,\,if\,b\,=\sqrt{a}\,,\,a\,>\,0\,,\,b\,>\,0\,,\,b\,\mathrel{\mathtt{!=}}\,1\]

Binomial Theorem



General Formula

\[(x+y)^{n}\,=\,nC_r x^{n-r} . y^{r}\,+ nC_r x^{n-r}.y^{r}+\,..................\,+nC_{n-1} x . y^{n-1} \, + nC_n y^{n}\]\[where,\,nC_r = \, \frac{n!}{(n-r)!r!}\]

Some Useful Expansions



\[1)\space\,(x+y)^{n} + (x-y)^{n} =\, 2[nC_0 x^{n} + nC_2 x^{n-1} y^{2} + nC_4 x^{n-4}y^{4} + ...............]\]

\[2)\space\,(x+y)^{n} - (x-y)^{n} =\, 2[nC_1 x^{n-1} y + nC_3 x^{n-3} y^{3} + nC_5 x^{n-5} y^{5} + ..................]\]

\[3)\space\,(1+x)^{n} = [nC_0 + nC_1 x + nC_2 x^{2} + ....... +nC_n x^{n} ]\]

Properties of Binomial Coefficients

\[nC_0,nC_1,nC_2,\space\ ... ...\space\ nC_{n}\,are\, called\, Binomial\, Coefficients\, and\, also\, represented\, by\, C_0,C_1,C_2,....,C_n \]

\[1)\space\,C_0\,+\,C_1\,+,C_2\,......\,C_n\,=\,2^{n}\]

\[2)\space\,C_0\,+\,C_2\,+\,C_4\,......\,=\,C_1\,+\,C_3\,+\,C_5\,......\,=\,2^{n-1}\]

\[3)\space\,C_0\,-\,C_1\,+\,C_2\,-\,C_3\, .......\,+\, (-1)^{n}. nC_n\,=\,0\]

\[4)\space\,C_1\,+\,2C_2\,+\,3C_3\,+\,......\,+\,n.C_n\,=\,n.2^{n-1}\]

\[5)\space\,C_1\,-\,2C_2\,+\,3C_3\,-\,4C_4\,+\,....\,+\,(-1)^{n-1} n.C_n\,=\,0\,for\,n\,>\,1)\]

\[6)\space\,C_0^{2}\,+\,C_1^{2}\,+\,C_2^{2}\,+\,.....\,+\,C_n^{2}\,=\,[\frac{(2n)!}{(n!)^2}]\]

Probability Properties



General Formula

\[P(E)\,=\,\frac{Number\space of\space favourable\space outcomes}{Total\space Number\space of\space outcomes}\]

General form of Addition Theorem of Probability

\[P(A_1\,\cup\,A_2\,\cup\,....\,\cup\,A_n)\,=[\,\sum_{i=1}^{n} P(A_i) \,-\,\sum_{i < j} P(A_i \,\cap \, A_j) \, + \,\sum_{i < j < k} P(A_i \,\cap \, A_j \, \cap \, A_k) \, - \\ ...\,+\,(-1)^{n-1}\,P(A_1 \,\cap\, A_2\,\cap\,...\,\cap \, A_n) ] \]

Conditional Probability

\[The\,probability\,of\,occurrence\,of\,an\,event\,E_1\, , \,given\,that\,E_2\,has\,already\,occured\\is\,called\,the\,Conditional\,Probabilty\, , \,it\,is\,denoted\,by\,P(\frac{E_1}{E_2}) \]

\[P(\frac{E_1}{E_2})\,=\,\frac{P(E_1\,\cap\,E_2)}{P(E_2)}\,=\,\frac{n(E_1\,\cap\,E_2)}{n(E_2)}\]

Total Probability Theorem

\[Let\,E_1\, , \, E_2 \, ,\, ... \,E_n\,be\,n\,mutually\,exclusive\,and\,exhaustive\,events\,i.e.,\\E_i\, \cap \, E_j\,=\,\phi\,for\,i\,!= \,j\,and\,\bigcup_{i=1}^{n}\,E_i\,=\,S\,.\,Suppose\,that\, , \\ P(E_i)\,>\,0\, ,\,\forall\,1\,\leq\,i\,\leq\,n\,Then\,for\,any\,event\,E \]

\[P(E)\,=\,\sum_{i=1}^{n}\,P(E_i)\,.\,P(\frac{E}{E_i})\]

Baye's Theorem

\[If\,an\,event\, E \,can\, occur \,only\,with\,one\,of\,the\,n\,mutually\,exclusive\,and\,exhaustive\,events\\E_1\, ,\, E_2 \, , \, E_3 \, ... \, E_n\,and\,the\,probabilities\,P(E/E_1)\, ,\,P(E/E_2)\, ,\, ...\, , \, P(E/E_n)\,are\,known\, , \,then \]

\[P(\frac{E_k}{E})\,=\,\frac{P(E_k)\,.\,\frac{P(E)}{P(E_k)}}{\sum_{k=1}^{n}P(E_k)\,.\,\frac{P(E)}{P(E_k)}}\]

Binomial Theorem on Probability

\[Suppose\,a\,binomial\,experiment\,has\,probability\,of\,success\,p\,and\,that\,of\,failure\,q\,.\\If\,E\,be\,an\,event\,and\,let\,X\,=\,number\,of\,times\,event\,E\,occurs\,in\,n\,trials\\Then\,the\,probability\,of\,occurence\,of\,event\,E\,exactly\,r\,times\,in\,n\,trials\,is\,denoted\,by \]

\[P(r)\,=\,nC_r\,p^{r}\,q^{n-r}\\where\, , \,r\,=\,0\, ,\,1\, , \,2\, , \,3\, , \, ... \, ,\,n\, . \]

\[Statistics\space Calculator\]


Statistics Formula Sheet

\[Mean\] \[\bar{x}=\frac{\sum x}{n}\]

\[x\,=\, Observations \,given\]

\[n\, =\, Total\, number\, of\]

\[observations\]

\[Median\]

\[If\, n\, is\, odd,\, then\]

\[M\,=(\frac{n+1}{2})\,term\]

\[If\, n\, is\, even,\, then\]

\[M \,=\,\frac{(\frac{n}{2})^{th}term+(\frac{n}{2}+1)^{th}term}{2}\]

\[n \,= \,Total\, number \,of\]

\[observations\]

\[Mode\]

\[The\, value\, which\, occurs\]

\[most\, frequently\]

\[-\]
\[Variance\] \[\sigma ^{2}\,=\,\frac{\sum (x-\bar{x})^{2}}{n}\]

\[x\, =\, Observations\, given\]

\[x¯\, =\, Mean\]

\[n \,= \,Total\, number \,of\]

\[observations\]

\[Standard \,Deviation\] \[S = \sigma = \sqrt{\frac{\sum (x-\bar{x})^{2}}{n}}\]

\[x\, =\, Observations\, given\]

\[x¯\, =\, Mean\]

\[n \,= \,Total\, number \,of\]

\[observations\]

\[Range\] \[L-S\]

\[L\, =\, Largest\, value\]

\[S\, =\, Smallest\, value\]

\[Coeff.\,of\,Range\] \[\frac{L-S}{L+S}\]

\[L\, =\, Largest\, value\]

\[S\, =\, Smallest\, value\]

\[Coeff.\,of\,Variation\] \[\frac{\sigma}{\bar{x}}*100\]

\[\sigma=Standard\, Deviation\]

\[x¯\, =\, Mean\]

\[Combined \,Mean\] \[\frac{m\bar{x}\,+\,n\bar{y}}{m+n}\]

\[x¯\,/y¯\, =\, Mean\, of\, two\]

\[distributions\]

\[m\,/n\,=no.\,of\,elements\,in\,\]

\[each\,distributions\]

\[Weighted\,Mean\] \[\frac{\sum Wi\,xi}{\sum Wi}\]

\[Wi=Weight \,of\, each\]

\[distributions\]

\[xi=ith\,observation\]

\[Mean\,Deviation\]

\[about\,mean\]

\[\bar{X}\]

\[\frac{\sum|xi-\bar{x}|}{n}\]

\[n \,= \,Total\, number \,of\]

\[observations\]

\[x¯\, =\, Mean\]

\[xi=ith\,observation\]

\[Mean\,Deviation\]

\[about\,median\]

\[M\]

\[\frac{\sum|xi-M|}{n}\]

\[n \,= \,Total\, number \,of\]

\[observations\]

\[M\, =\, Median\]

\[xi=ith\,observation\]

\[Mean\,Deviation\]

\[about\,mode\]

\[Z\]

\[\frac{\sum|xi-Z|}{n}\]

\[n \,= \,Total\, number \,of\]

\[observations\]

\[Z\, =\, Mode\]

\[xi=ith\,observation\]

\[Mean\,Deviation\]

\[about\,any\,no.\,'a'\]

\[\frac{\sum|xi-a|}{n}\]

\[n \,= \,Total\, number \,of\]

\[observations\]

\[xi=ith\,observation\]

\[For \,symmetrical \]

\[Distribution:\]

\[Mode\,=\,3Median\,-\,2Mean\] \[-\]

Sets Formulas

\[1)\space If A\space and\space B\space are\space overlapping\space sets\space n(AUB)\space =n(A) + n(B) - n(A⋂B)\]

\[2)\space If A\space and\space B\space are\space disjoint\space sets\space n(AUB)\space = n(A) + n(B)\]

\[3)\space n(A)\space = n(AUB) + n(A⋂B) - n(B)\]

\[4)\space n(A⋂B)\space = n(A) + n(B) - n(AUB)\]

\[5)\space n(B)\space = n(AUB) + n(A⋂B) - n(A)\]

\[6)\space n(U)\space = n(A) + n(B) - n(A⋂B) + n((AUB)^c)\]

\[7)\space n((AUB)^c)\space = n(U) + n(A⋂B) - n(A) - n(B)\]

\[8)\space n(AUB)\space = n(A-B) + n(B-A) + n(A⋂B)\]

\[9)\space n(A-B)\space = n(AUB) - n(B) \]

\[10)\space n(A-B)\space = n(A) - n(A⋂B) \]

\[11)\space n(A^c)\space = n(U) - n(A) \]

Log Calculator
Base
Number

General Formula

\[1)\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{b}{f(t)dt}\]

\[2)\int\limits_{a}^{b}{f(x)dx=}-\int\limits_{b}^{a}{f(x)dx}\]

\[3)\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{c}{f(x)dx}+\int\limits_{c}^{b}{f(x)dx}\]

\[4)\int\limits_{-a}^{a}{f(x)dx=\int\limits_{0}^{a}[f(x)+f(-x)]=\left\{ \begin{matrix} 0 & if\,\,f\left( x \right)\,is\,odd\,function \\ 2\int\limits_{0}^{a}{f\left( x \right)dx} & if\,\,f\left( x \right)\,is\,even\,function \\ \end{matrix} \right.\,\,\,\,\,\,}\]

Shifting

\[1)\space\int_{a}^b\,f(x)\,dx\space=\int_{a-c}^{b-c}\,f(x+c)\,dx\]

\[2)\space\int_{a}^b\,f(x)\,dx\space=\int_{a+c}^{b+c}\,f(x-c)\,dx\]

\[3)\space\int_{a}^b\,f(x)\,dx\space=\int_{ak}^{bk}\,\frac{1}{k}\,f(\frac{x}{k})\,dx\]

\[4)\space\int_{a}^b\,f(x)\,dx\space=\int_{0}^{b-a}\,f(x+a)\,dx\]

Periodicity

\[f(x)\,is\,a\,periodic\,function\,with\,period\,\,T\,and\,m\,and\,n\,are\,integers\]

\[1)\space\int_{0}^{nT}\,f(x)\,dx\space=n\,\int_{0}^T\,f(x)\,dx\]

\[2)\space\int_{nT}^{mT}\,f(x)\,dx\space=(m-n)\,\int_{0}^T\,f(x)\,dx\]

\[3)\space\int_{a}^{a+nT}\,f(x)\,dx\space=n\,\int_{0}^T\,f(x)\,dx\]

\[4)\space\int_{a+nT}^{a+mT}\,f(x)\,dx\space=(m-n)\,\int_{0}^T\,f(x)\,dx\]

\[5)\space\int_{a+nT}^{b+nT}\,f(x)\,dx\space=\int_{a}^b\,f(x)\,dx\]

King's Rule

\[1)\space\int_{a}^b\,f(x)\,dx\space=\int_{a}^b\,f(a+b-x)\,dx\]

\[2)\space\int_{0}^a\,f(x)\,dx\space=\int_{0}^a\,f(a-x)\,dx\]

\[3)\space\int_{-a}^a\,f(x)\,dx\space=\int_{-a}^a\,f(-x)\,dx\]

\[4)\int\limits_{0}^{2a}{f(x)dx=\int\limits_{0}^{a}f(x) + \int_{0}^a f(2a-x)dx \space = \left\{ \begin{matrix} 0 & if\,\,f\left( 2a-x \right)\,-f(x) \\ 2\int\limits_{0}^{a}{f\left( x \right)dx} & if\,\,f\left( 2a-x \right)\,=f(x) \\ \end{matrix} \right.\,\,\,\,\,\,}\]

Integration By Part

\[\int_{a}^{b} du(\frac{dv}{dx})dx=[uv]_{a}^{b}-\int_{a}^{b} v(\frac{du}{dx})dx\]

Leibnitz's rule

\[For\,an\,integral\,of\,the\,form\,\displaystyle{ \int_{a(x)}^{b(x)} f(x,t)\,dt, }\]

\[where\,\displaystyle{ -\infty \lt a(x), b(x) \lt \infty },\,the\,derivative\,of\,this\,integral\,is\,expressible\,as\]

\[\displaystyle{ \frac{d}{dx} \left (\int_{a(x)}^{b(x)} f(x,t)\,dt \right )= f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) - f\big(x,a(x)\big)\cdot \frac{d}{dx} a(x) + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x} f(x,t) \,dt, }\]

\[where\,the\,partial\,derivative\,indicates\,that\,inside\,the\,integral,\]

\[only\,the\,variation\,of\,f(x, t)\,with\,x\,is\,considered\,in\,taking\,the\,derivative.\]

Wallis Integrals

\[If\,\displaystyle{ W_n = \int_0^{\frac{\pi}{2}} \sin^n x \,dx }\,or\,\displaystyle{ W_n = \int_0^{\frac{\pi}{2}} \cos^n x \,dx. }\]

\[then,\,\,\,If\,n\,is\,even,\,\,\displaystyle{ W_{2p}=\frac{2p-1}{2p} \cdot \frac{2p-3}{2p-2} \cdots \frac{1}{2} W_0 = \frac{(2p-1)!!}{(2p)!!} \cdot \frac{\pi}{2} = \frac{(2p)!}{2^{2p} (p!)^2} \cdot \frac{\pi}{2}, }\]

\[If\,n\,is\,odd,\,\,\displaystyle{ W_{2p}=\frac{2p-1}{2p} \cdot \frac{2p-3}{2p-2} \cdots \frac{1}{2} W_0 = \frac{(2p-1)!!}{(2p)!!} \cdot \frac{\pi}{2} = \frac{(2p)!}{2^{2p} (p!)^2} \cdot \frac{\pi}{2}. }\]

\[If\,n\,is\,odd,\,\,\displaystyle{ W_{2p}=\frac{2p-1}{2p} \cdot \frac{2p-3}{2p-2} \cdots \frac{1}{2} W_0 = \frac{(2p-1)!!}{(2p)!!} \cdot \frac{\pi}{2} = \frac{(2p)!}{2^{2p} (p!)^2} \cdot \frac{\pi}{2}} \]

Inequalities

\[1)\,\,If\,at\,every\,point\,x\,of\,an\,interval\,[a,b]\,the\,inequalities\,g(x)\,≤\,f(x)\,≤\,h(x)\,are\,fulfilled\,,\,then\]

\[\space\int_{a}^b\,g(x)\,dx\space≤\int_{a}^{b}\,f(x)\,dx\space≤\int_{a}^{b}\,h(x)\,dx\]

\[2)\,\,If\,m\,is\,the\,least\,value\,and\,M\,is\,the\,greatest\,value\,of\,function\,f(x)\,on\,the\,interval\,[a,b]\,,\,then\]

\[\space\,m\,(b-a)\space≤\int_{a}^b\,f(x)\,dx\space≤\,M\,(b-a)\]

\[3)\space\,|\int_{a}^b\,f(x)\,dx\,|\space≤\int_{a}^b\,|\,f(x)\,|\,dx\]

Some Standard Results

\[1)\space\int\limits_{0}^{\frac{\pi}{2}}logsinx \space dx \space = -\frac{\pi}{2}log2 \space=\int\limits_{0}^{\frac{\pi}{2}}logcosx \space dx\]

\[2)\space\int\limits_{0}^{\frac{\pi}{2}}logsecx \space dx \space = \frac{\pi}{2}log2 \space=\int\limits_{0}^{\frac{\pi}{2}}logcosecx \space dx\]

\[3)\space\int\limits_{0}^{\frac{\pi}{2}}logtanx \space dx \space = \,0 \space=\int\limits_{0}^{\frac{\pi}{2}}logcotx \space dx\]

\[4)\space\int\limits_{a}^{b}{x}dx \space=\frac{b-a}{2} \space a,b \in I\]

\[5)\space\int \limits_{a}^{b}\frac{|x|}{x}dx \space = |b|-|a|\]

Basic Derivative Rules

\[1)Constant \space Rule:\frac{d}{dx}(c)=0\]

\[2)Constant \space Multiple\space Rule:\frac{d}{dx}(cf(x))=cf'(x)\]

\[3)Power \space Rule:\frac{d}{dx}(x^n)=nx^{n-1}\]

\[4)Sum \space Rule:\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)\]

\[5)Difference \space Rule:\frac{d}{dx}[f(x)-g(x)]=f'(x)-g'(x)\]

\[6)Product \space Rule:\frac{d}{dx}[f(x)g(x)]=f(x)g'(x)+g(x)f'(x)\]

\[7)Quotient \space Rule:\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}\]

\[8)Chain \space Rule:\frac{d}{dx}[f(g(x)]=f'(g(x))g'(x)\]

Exponential Functions

\[\frac{d}{dx}(e^x)=e^x\]

\[\frac{d}{dx}(a^x)=a^xln \space a\]

\[\frac{d}{dx}(e^{g(x)})=e^{g(x)}g'(x)\]

\[\frac{d}{dx}(a^{g(x)})=ln(a)a^{g(x)}g'(x)\]

Logarithmic Functions

\[\frac{d}{dx}(ln \space x)=\frac{1}{x}\]

\[\frac{d}{dx}(ln(g(x)))=\frac{g'(x)}{g(x)}\]

\[\frac{d}{dx}(log_{a}x)=\frac{1}{xlna}, \space x>0 \]

\[\frac{d}{dx}(log_{a}g(x))=\frac{g'(x)}{g(x)lna}\]

Trignometric Functions

\[\frac{d}{dx}(sinx)=cosx\]

\[\frac{d}{dx}(cosx)=-sinx\]

\[\frac{d}{dx}(tanx)=sec^2x\]

\[\frac{d}{dx}(cosecx)=-cosecx \space cotx\]

\[\frac{d}{dx}(secx)=secx \space tanx\]

\[\frac{d}{dx}(cotx)=-cosec^2x\]

Inverse Trignometric Functions

\[\frac{d}{dx}(sin^{-1}x)=\frac{1}{\sqrt{1+x^2}}\]

\[\frac{d}{dx}(cos^{-1}x)=\frac{-1}{\sqrt{1-x^2}}\]

\[\frac{d}{dx}(tan^{-1}x)=\frac{1}{\sqrt{1+x^2}}\]

\[\frac{d}{dx}(cot^{-1}x)=\frac{-1}{\sqrt{1+x^2}}\]

\[\frac{d}{dx}(sec^{-1}x)=\frac{1}{x\sqrt{x^2-1}}, \space x !=±1,0\]

\[\frac{d}{dx}(cosec^{-1}x)=\frac{-1}{x\sqrt{x^2-1}},\space x !=±1,0\]

Differential Equations



1. Variable Seperable


It can be used when,
All the y terms (including dy) can be moved to one side of the equation, and
All the x terms (including dx) to the other side.

\[\frac{dx}{dy} = 4xy\ which\space can\space be\space written\space as, \\ \frac{dx}{x} = 4ydy\]


2. Equation reducible to variable seperable


\[ The\space equation\space \frac{dy}{dx}-f(ax+by+c)\space can\space be\space written\space as \\ \frac{dy}{dx}-f(t)\space keeping\space ax+by+c = t\space then\space solve \]


3. Homogenous Equations


A differential equation, M dx + N dy = 0, is homogeneous if replacement of x and y by λx and λy results in the original function multiplied by some power of λ, where the power of λ is called the degree of the original function. If so, follow these steps.

\[ Let\space y=vx\space so\space \frac{dy}{dx} = x\frac{dv}{dx}+v\]

\[Now\space put\space value\space of\space \frac{dy}{dx} in\space the\space equation\space and\space solve\]


4. First Order Linear Differential Equations


When the equation is written in form

\[\frac{dy}{dx} + Py = Q,\space where\space P\space and\space Q\space are\space functions\space of\space x\space or , \\ \frac{dx}{dy} + Px = Q,\space where\space P\space and\space Q\space are\space functions\space of\space y\]

Then these can be solved by -

\[Let\space Integrating\space Factor(I)\space =\space e^{\int P\,dx} (for\space first\space equation\space) or, \\ I = e^{\int P \,dy}(for\space second\space equation)\]

Put in \[yI = \int {QI} + c\] and integrate to get the answer


5. Formulae (Exact Form)


\[a)\space xdy \,+\,ydx\,=\,d(xy)\]

\[b)\space dx \,\plusmn\,ydx\,=\,d(x \plusmn y)\]

\[c)\space xdx \,+\,ydy\,=\,\frac{1}{2}d(x^2+y^2)\]

\[d)\space \frac{xdy \,-\,ydx}{x^2}\,=\,d(\frac{y}{x})\]

\[e)\space \frac{xdy \,-\,ydx}{y^2}\,=\,-d(\frac{x}{y})\]

\[f)\space \frac{2xydx \,-\,x^2dy}{y^2}\,=\,d(\frac{x^2}{y})\]

\[g)\space \frac{xdy \,-\,ydx}{x^2+y^2}\,=\,d(tan^{-1}\frac{y}{x})\]

\[h)\space \frac{xdx \,+\,ydy}{\sqrt(x^2+y^2)}\,=\,\frac{1}{2}d(\frac{x^2+y^2}{\sqrt(x^2+y^2)})\]

\[i)\space \frac{ye^{x}dx\,-\,e^{x}dy}{y^2}\,=\,d(\frac{e^x}{y})\]


6. Linear Differential Equation with constant coeffiecients

Definition

A diff. equation,

\[\frac{d^n}{dx^{n}}y+a_1\frac{d^{n-1}}{dx^{n-1}}y+......ay\space =\space Q\]

Where a1 , a2 ,............., an are constants
While Q is a function of x.

is called Linear differential equation of nth order first degree.

Above equation can be written as -

\[D^n y+ a_1 D^{n-1}y + a_2 D^{n-2}y+....a_n y\space =\space Q\]

General Solution = Complementary Function(C.F.) + Particular Integral(P.I.)


How to find C.F.?

Put D = m and equate LHS to 0 to get values of m

Case-1 : If m is real and distinct

\[C.F\space =\space C_1 e^{m_1 x} + C_2 e^{m_2 x} + .... \]

Case-2 : If m is real and equal

\[C.F\space =\space (C_1 + C_2 x + C_3 x^2 + ...) e^{mx}\]

Case-3 : If m is imaginary and distinct

\[C.F\space =\space e^{ax} (C_1 cos(bx)\space +\space C_2 sin(bx))\]


How to find P.I.?

To find P.I. check the nature of Q

\[\frac{1}{(LHS \spaceremoving\space y)}*Q\]

1. If Q = eax

Put D = a to get the desired P.I.

2. If Q = sin(ax)/cos(ax)

Put D2 = -a2 and use identities of trigo to solve

3. If Q = xm

take out least degree term and use binomial expansion to solve

4. If Q = e^{ax}.V

Take out eax by putting D to D+a and then solve


Homogenous Differential equations


A linear differential equation of the form,

\[x^{n} \frac{d^{n}y}{dx^{n}} + a_1x^{n-1}\frac{d^{n-1}y}{dx^{n-1}}+a_2 x^{n-2}\frac{d^{n-2}y}{dx^{n-2}}+.....+a_n y\space =\space Q\]

with a0, a1, . . . , an constants is called the homogeneous Euler-Cauchy equation of order n.

We introduce a new variable z where,

\[z\space =\space ln(x)\space or\space x\space =\space e^{z}\]

Then we change differentiation terms to

\[x\frac{d}{dx}\space =\space D\space,\space where\space D = \frac{d}{dz}\]

\[x^{2}\frac{d^{2}}{dx^{2}}\space =\space D(D-1)\space\space and\space\space x^{3}\frac{d^{3}}{dx^{3}}\space =\space D(D-1)(D-2) and\space so\space on\]

Now it reduces to linear differntial eqn with constant coefficients,find CF and PI to get solution and put back z = ln(x) to get final answer

Legendre's Differential eqn

A linear differential equation of the form,

\[(ax+b)^{n} \frac{d^{n}y}{dx^{n}} + a_1(ax+b)^{n-1}\frac{d^{n-1}y}{dx^{n-1}}+a_2 (ax+b)^{n-2}\frac{d^{n-2}y}{dx^{n-2}}+.....+a_n y\space =\space Q\]

Put ax+b = ez and solve line homogenous equations

\[(ax+b)\frac{dy}{dx}\space = \space bD\space (ax+b)^{2}\frac{d^{2}}{dx^{2}}\space =\space b^{2}D(D-1)\space and\space so\space on\]


Solving Simultaneous Linear Differential Equation


Each equation in a system will be assumed to have constant coefficients and be of the general form
f(D)x + g(D)y + ....... + h(D)u = G(t)
where x, y, ...... , u are the dependent variables, t is the independent variable.


A system of n equations in the n dependent variables x, y, ..... , u and independent variable t will have a solution consisting of n functions

x = x(t)
y = y(t)
.............
.............
u = u(t)

The method used to solve a system of n equations in n variables is analogous to the procedure used to solve a system of n linear equations in n unknowns in algebra. In algebra, we solve a system of n equations in n unknowns by eliminating unknowns between equations until we obtain an equation containing a single unknown, from which we deduce the value of the unknown. Then we substitute the value of that unknown into other equations to obtain the values of other unknowns. In solving systems of linear differential equations we go through the same type process to obtain an equation containing a single dependent variable. The equation in this single dependent variable will be a linear differential equation with constant coefficients. We then solve this equation, using methods for solving such equations, to obtain an expression for that dependent variable. We then substitute the expression for that variable into another equation to obtain an expression for another variable. As with the algebraic problem, we can also employ determinants.

Theorem. The number of constants in the general solution of a system of equations must equal the sum of the orders of the equations.


Linear Differential equation of second order


NOTE - The coefficient of second order derivative should be 1

Method-1 - Part of C.F. is known

Steps -

\[1.\space Compare\space the\space differential\space eqn\space with\space \frac{d^{2}y}{dx{2}}+P\frac{dy}{dx}+Qy\space =\space R\space and\space find\space value\space of\space P,Q,R\]

\[2.\space Apply\space conditions\space as\space per\space P,Q\space and\space find\space part\space of\space C.F.\space=\space u\]

\[3.\space Let\space y=uv\space be\space complete\space solution\space find\space \frac{dy}{dx},\frac{d^{2}y}{dx^{2}}\]

\[4.\space Put\space in\space original\space question\]

\[5.\space Put\space P=\frac{dv}{dx} and\space\space solve\space using\space first\space order\space eqn\]

\[6.\space Integrate\space P\space to\space get\space v and\space find\space complete\space solution \]


Method-2 - Removal of First Derivative

Steps -

\[1.\space Find\space u\space = e^{-\frac{1}{2}\int P\,dx}\space I=Q-\frac{1}{2}\frac{dP}{dx}-\frac{P^{2}}{4}\space and S=\frac{R}{u}\]

\[2.\space If\space I\space is\space a constant\space or \frac{constant}{x^{2}}\space then this method is applicable\]

\[3. \space Put\space values\space in\space \frac{d^{2}v}{dx^{2}}+Iv = S\space and\space it\space becomes\space differential\space eqn\space with\space constant\space coefficients\]


Method-3 - Changing the independent variable

Steps -

\[1.\space Choose z such as (\frac{dz}{dx})^{2}\space = \space Q\]

\[2.\space Find \frac{d^{2}z}{dx^{2}}\]

\[3.\space Find\space P_1,Q_1,R_1\space\]

\[P_1\space =\space \frac{\frac{d^{2}z}{dx^{2}}+P\frac{dz}{dx}}{(\frac{dz}{dx})^{2}}\space,Q_1\space =\space \frac{Q}{(\frac{dz}{dx})^{2}}\space and\space R_1\space =\space \frac{R}{(\frac{dz}{dx})^{2}}\]

\[4.\space Put\space in\space \frac{d^{2}y}{dz^{2}}\space +\space P_1\frac{dy}{dx}\space +Q_1 y\space =\space R_1\space and\space solve\space using\space differential\space eqn\space with\space constant\space coeffiecients\]


Method-4 - Variation of Parameters

Steps -

\[1.\space Convert\space to\space homogenous\space eqns\space by\space putting\space x\space =\space e^{z}\space and\space z\space =\space logx\]

\[2.\space Put\space x^{2}\frac{d^{2}y}{dx^{2}}\space =\space D(D-1)\space and\space x\frac{dy}{dx}\space =\space D,\space where\space D\space is\space \frac{d}{dz}\]

\[3.\space Solve\space for\space C.F.\space and\space let\space it\space be\space y\space =\space u+v\]

\[4.\space Find\space out\space u v_1 - v u_1\space where\space u_1\space =\space \frac{du}{dx}\]

\[5.\space Let\space the\space complete\space solution\space be\space y\space =\space Au+Bv\]

\[A\space =\space -\int\frac{Rv}{u v_1 - v u_1}\,dx\space +\space C_1\]

\[B\space =\space \int\frac{Ru}{u v_1 - v u_1}\,dx\space +\space C_2\]

\[6.\space Put\space in\space equation\space to\space get\space the\space answer\]

Power Series

A power series about a, or just power series, is any series that can be written in the form,

\[\sum_{n=0}^{\infty} c_n (x-a)^{n}\]

where anand cnn are numbers. The cn ’s are often called the coefficients of the series. The first thing to notice about a power series is that it is a function of x . That is different from any other kind of series that we’ve looked at to this point. In all the prior sections we’ve only allowed numbers in the series and now we are allowing variables to be in the series as well. This will not change how things work however. Everything that we know about series still holds.


The interval of convergence must then contain the interval a−R<x<a+R since we know that the power series will converge for these values. We also know that the interval of convergence can’t contain x’s in the ranges x<a−R and x>a+R since we know the power series diverges for these value of x . Therefore, to completely identify the interval of convergence all that we have to do is determine if the power series will converge for x=a−R or x=a+R. If the power series converges for one or both of these values then we’ll need to include those in the interval of convergence.


Analytic Function

A function f(x) containing the point x=x0 is called analytic at x0 if its Taylor's Series exists and converges to f(x) for all x

All polynomial functions are analytic everywhere


It is important to note that no matter what else is happening in the power series we are guaranteed to get convergence for x = a. The series may not converge for any other value of x, but it will always converge for x = a.

Ordinary Point

At point x=x0 is called an ordinary point

\[\frac{d^{2}y}{dx^{2}}+P(x)\frac{dy}{dx}+Q(x)y=0\]

If both functions are analytic at x = x0


Regular and Irregular Singular Points

If a point is not ordinary then the point is singular

A singular point x = x0 is called regular singular point if both (x-x0) and (x-x0)2 are analytic at x = x0 else it is irregular


Legendre's Differential Equation


The differential equation of form,

\[(1-x^{2})\frac{d^{2}y}{dx^{2}}-2x\frac{dy}{dx}+n(n+1)y\space =\space 0\]

\[\frac{d}{dx}((1-x^{2})\frac{dy}{dx})+n(n+1)y\space =\space 0\]

is called Legendre's Differential Equation,where n is a real number


Bessel's Differential Equation


The differential equation of form,

\[x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+(x^{2}-n^{2})y\space =\space 0\]

is called Bessel's Differential Equation of order n, where n is a positive constant


Chebyshev's Differential Equation


The differential equation of form,

\[(1-x^{2})\frac{d^{2}y}{dx^{2}}-x\frac{dy}{dx}+n^{2}y\space =\space 0\]

is called Chebyshev's Differential Equation


Frobenius Method


Steps-

1. Assume y = a0xm+a1xm+1+...

2. Substitute from 1 for y,dydx and d2xdx2

3. Equate to 0 the coefficient of lowest power of x. This gives a qudratic eqn in m

4. Equate to 0 other powers of x to find a1,a2,a3....

5. Substitute in eqn


Case 1 - If roots are distinct and dont differ by an integer

Let m1,m2 be roots,then complete solution is

\[y\space =\space c_1(y)_{m_1} + c_2(y)_{m_2}\]


Case 2 - If roots are equal

\[y\space =\space c_1(y)_{m_1}+c_2(\frac{\delta y}{\delta x})_{m_1}\]


Case 3 - If roots are distinct , differ by integer and making a coefficient of y infinite

\[y\space =\space c_1(y)_{m_1} + c_2(\frac{\delta y}{\delta x})_{m_2}\]


Case 4 - If roots are distinct,differ by an integer and making one or more coefficients indeterminate

Let the roots be m1,m2. If one of the coefficients becomes indeterminate when m=m2, the complete solution is given by putting m=m2 in y

Maxima and Minima of Functions

A local maximum point on a function is a point (x,y) on the graph of the function whose y coordinate is larger than all other y coordinates on the graph at points "close to'' (x,y). More precisely, (x,f(x)) is a local maximum if there is an interval (a,b) with (a < x < b) and f(x)≥f(z) for every z in (a,b). Similarly, (x,y) is a local minimum point if it has locally the smallest y coordinate. Again being more precise: (x,f(x)) is a local minimum if there is an interval (a,b) with a < x < b and f(x)≤f(z) for every z in (a,b). A local extremum is either a local minimum or a local maximum.



First Order Derivative Test

  1. Find out first derivative of f(x)
  2. Equate to 0 and find out values of x
  3. Plot a wavy curve
  4. If at a point curve goes from -ve slope to +ve,Minima at that point and viceversa for Maxima


Second Order Derivative Test

  1. Equate to 0 first order derivative and find values of x
  2. Find out second derivative
  3. Put values of x in seoond derivative
  4. If answer is +ve, Minima and -ve,Maxima

Tangent & Normal



Things To Remember


(a) The value of the derivative at P(x1,y1) gives the slope of the tangent of the curve at P. Symbolically

\[\space f'(x_{1}) \space = \frac{dy}{dx}\Big]_{(x_{1},y_{1})} = Slope \space of \space tangent \space at \space P(x_{1},y_{1}) \space = m \]


(b) Equation of tangent at (x1,y1) is:

\[ y \space - \space y_{1} = \frac{dy}{dx}\Big]_{(x_{1},y_{1})} \space (x-x_{1}) \]


(c) Equation of normal at (x1,y1) is :

\[ y \space - \space y_{1} \space = -\frac{1}{\frac{dy}{dx}\Big]_{(x_{1},y_{1})}} \space (x-x_{1}) \]

(d) If the Angle of intersection between two curves is 90° then
they are called ORTHOGONAL curves


Length of Tangent, Subtangent, Normal & Subnormal.

Tangent-Normal

\[Length \space of \space the \space tangent(PT) \space = \space \frac{y_{1}\sqrt{1+[f'(x_{1})]^2}}{f'(x_{1})}\]

\[Length \space of \space Subtangent \space (MT) \space = \space \frac{y_{1}}{f'(x_{1})} \]

\[Length \space of \space Normal \space (PN) \space = y_{1}\sqrt{1+[f'(x_{1})]^2}\]

\[Length \space of \space Subnormal \space (MN) \space = \space y_{1}f'(x_{1})\]

Inverse,Periodicity of Functions



What is a Period?

The time interval between two waves is known as a Period whereas a function that repeats its values at regular intervals or periods is known as a Periodic Function. In other words, a periodic function is a function that repeats its values after every particular interval.

\[f(x+a)\space = \space f(x) for\space a>0\]


How to find a period?

  • If a function repeats over at a constant period we say that is a periodic function.
  • It is represented like f(x) = f(x + p), p is the real number and this is the period of the function.
  • Period means the time interval between the two occurrences of the wave.


\[If\space for\space f(x)\space period\space =\space T,\space then\space period\space for\space f(ax)\space =\space \frac{T}{a}\]

\[Rest\space for\space all\space other\space cases\space of\space functions,\space the\space period\space remains\space same\]


What are inverse functions?

An inverse function or an anti function is defined as a function, which can reverse into another function. In simple words, if any function “f” takes x to y then, the inverse of “f” will take y to x.

If the function is denoted by ‘f’ or ‘F’, then the inverse function is denoted by f-1- or F-1.


How to find inverse?

The graph of the inverse of a function reflects two things, one is the function and second is the inverse of the function, over the line y = x.

\[The\space inverse\space function\space returns\space the\space original\space value\space for\space which\space a\space function\space gave\space the\space output.\]

Transformation of Functions



If function f(x) changes to f(x) + b or f(x) - b -
If f(x) changes to |f(x)| -

The graph of f(x) and |f(x)| would coincide if f(x) > 0 and the portions where f(x) < 0 would get inverted in the upwards direction.

If f(x) changes to f(|x|) -

Graphs of f(|x|) and f(x) would be identical in the first and the fourth quadrants (as x > 0) and as such the graph of f(|x|) would be symmetrical about the y-axis (as (|x|) is even).

If y = f(x) changes to |y| = f(x)

Graph of |y| = f(x) would exist only in the regions where f(x) is non-negative and will be reflected about the x-axis only in those regions.

If f(x) changes to af(x) -

No change in period of function, The maxima and minima gets increased

If f(x) changes to f(ax) -

The period of function changes from P to P/|a|


Check Leap Year




Beta and Gamma Functions



What are Beta functions?

The beta function is a unique function where it is classified as the first kind of Euler’s integrals. The beta function is defined in the domains of real numbers. The notation to represent the beta function is “β”. The beta function is meant by B(p, q), where the parameters p and q should be real numbers.

\[B(p,q)\space =\space \int_{0}^{1} t^{p-1} (1-t)^{q-1}\,dt \\ Where\space p,q\space greater\space than\space 0\]

Properties of Beta Functions

  • This function is symmetric which means that the value of beta function is irrespective to the order of its parameters, i.e B(p, q) = B(q, p)
  • B(p, q) = B(p, q+1) + B(p+1, q)
  • B(p, q+1) = B(p, q). [q/(p+q)]
  • B(p+1, q) = B(p, q). [p/(p+q)]
  • B (p, q). B (p+q, 1-q) = π/ p sin (πq)

\[B(p,q)\space =\space \int_{0}^{\infty} \frac{t^{p-1}}{(1+t) ^{p+q}} \,dt\]

\[B(p,q)\space =\space 2\int_{0}^{\frac{\pi}{2}} sin^{2p-1}\theta\ cos^{2q-1}\theta \]

\[B(p,q)\space =\space \int_{0}^{1}x^{m-1}(1-x)^{n-1}\,dx\]

\[B(p,q) = B(q,p)\]

What are Gamma functions?

\[\gamma(x)\space =\space \int_{0}^{\infty} t^{x-1}e^{-t}\,dt\]

\[\gamma(n)\space =\space (n-1)!\]

\[\gamma (\frac{1}{2})\space =\space\frac{\sqrt{\pi}}{2} \]

\[\gamma(\frac{n}{2})\space =\space \gamma(\frac{n-2}{2})\frac{n}{2}\]

\[\int_{0}^{\frac{\pi}{2}}sin^{m}\theta cos^{n}\theta \,d\theta\space =\space \frac{\gamma\frac{m+1}{2}\gamma\frac{n+1}{2}}{2\gamma\frac{m+n+2}{2}} \]

\[\gamma n \gamma(1-n)\space =\space \frac{\pi}{sin \space n\pi}\]

\[\gamma(m)\gamma(m+\frac{1}{2})\space =\space \frac{\sqrt{\pi}}{2^{2m-1}}\gamma(2m)\]

\[\gamma(\frac{1}{2})\space =\space \sqrt{\pi}\]

Relation between beta and gamma functions

\[B(p,q)\space =\space \frac{\gamma(p)\space\gamma(q)}{\gamma(p+q)}\]

\[F(s)=\mathcal{L}\{f(t)\}=\int_{0}^{\infty}e^{-st} f(t).dt\]

\[\mathcal{L}^{−1}(F(s))\]

Table of Laplace Transforms

\[f(t)=\mathcal{L}^{-1}\{F(s)\}\] \[F(s)=\mathcal{L}\{f(t)\}\] \[f(t)=\mathcal{L}^{-1}\{F(s)\}\] \[F(s)=\mathcal{L}\{f(t)\}\]
\[1\] \[\frac{1}{s}\] \[e^{at}\] \[\frac{1}{s-a}\]
\[t^n, \space n=1,2,3...\] \[\frac{n!}{s^{n+1}}\] \[t^p, \space p>-1\] \[\frac{p+1}{s^{p+1}}\]
\[\sqrt{t}\] \[\frac{\sqrt{\pi}}{2s^{\frac{1}{2}}}\] \[t^{n-\frac{1}{2}}, \space n=1,2,3,...\] \[\frac{1.3.5....(2n-1)\sqrt{\pi}}{2^{n}s^{n+\frac{1}{2}}}\]
\[sin(at)\] \[\frac{a}{s^2+a^2}\] \[cos(at)\] \[\frac{s}{s^2+a^2}\]
\[tsin(at)\] \[\frac{2as}{(s^2+a^2)^2}\] \[tcos(at)\] \[\frac{s^2-a^2}{(s^2+a^2)^2}\]
\[sin(at)-atcos(at)\] \[\frac{2a^3}{(s^2+a^2)^2}\] \[sin(at)+atcos(at)\] \[\frac{2as^2}{(s^2+a^2)^2}\]
\[cos(at)-atsin(at)\] \[\frac{s(a^2-a^2)}{(s^2+a^2)^2}\] \[cos(at)+atsin(at)\] \[\frac{s(s^2+3a^2)}{(s^2+a^2)^2}\]
\[sin(at+b)\] \[\frac{ssin(b)+acos(b)}{s^2+a^2}\] \[cos(at+b)\] \[\frac{scos(b)-asin(b)}{s^2+a^2}\]
\[sinh(at)\] \[\frac{a}{s^2-a^2}\] \[cosh(at)\] \[\frac{s}{s^2-a^2}\]
\[e^{at}sin(bt)\] \[\frac{b}{(s-a)^2+b^2}\] \[e^{at}cos(bt)\] \[\frac{s-a}{(s-a)^2+b^2}\]
\[e^{at}sinh(bt)\] \[\frac{b}{(s-a)^2-b^2}\] \[e^{at}cosh(bt)\] \[\frac{s-a}{(s-a)^2-b^2}\]
\[t^ne^{at}, \space n=1,2,3,...\] \[\frac{n!}{(s-a)^{n+1}}\] \[f(ct)\] \[\frac{1}{c}F\Big(\frac{s}{c}\Big)\]

Properties of Laplace Transforms

\[1) \space Shifting \space Property : \space If \space \mathcal{L}\{f(t)\} = \bar{f}(s)\]\[then \space \mathcal{L}\{e^{at}f(t)\} = \space \bar{f}(s-a)\]

\[2) \space Division \space Property : \space If \space \mathcal{L}\{f(t)\} = \bar{f}(s)\]\[then \space \mathcal{L}\{\frac{1}{t}f(t)\} \space = \space \int\limits_{s}^{\infty}\bar{f}(s)ds\]

\[3) \space Transforms \space of \space integral : \space If \space \mathcal{L}\{f(t)\} = \bar{f}(s)\]\[then \space \mathcal{L}\{\int\limits_{0}^{t}f(t)dt\} \space = \space \frac{1}{s} \bar{f}(s)\]

\[4) \space Multiplication \space Property : \space If \space \mathcal{L}\{f(t)\} = \bar{f}(s)\]\[then \space \mathcal{L}\{t^nf(t)\} \space = \space (-1)^n \frac{d^n}{ds^n}[\bar {f}(s)] \]

\[5) \space Laplace \space transform \space of \space a \space Derivative : \space If \space \mathcal{L}\{f(t)\} = \bar{f}(s)\]\[then \space \mathcal{L}\{f^n(t)\} \space = \space s^n \mathcal{L}\{f(t)\} - s^{n-1}f(0) - s^{n-2}f'(0) - s^{n-3}f''(0) - s^{n-4}f'''(0)-...\]

Table of Inverse Laplace Transforms

\[F(s)\] \[\mathcal{L}^{-1}\{F(s)\}\] \[F(s)\] \[\mathcal{L}^{-1}\{F(s)\}\]
\[\frac{1}{s}\] \[1\] \[\frac{1}{s^n}, \space n=1,2,3...\] \[\frac{t^{n-1}}{(n-1)!}\]
\[\frac{1}{s+a}\] \[e^{-at}\] \[\frac{1}{s-a}\] \[e^{at}\]
\[\frac{1}{s^2+a^2}\] \[\frac{sin(at)}{a}\] \[\frac{s}{s^2+a^2}\] \[cos(at)\]
\[\frac{1}{s^2-a^2}\] \[\frac{sinh(at)}{a}\] \[\frac{s}{s^2-a^2}\] \[cosh(at)\]
\[\frac{1}{(s^2+a^2)^2}\] \[\frac{sin(at)-atcos(at)}{2a^3}\] \[\frac{s}{(s^2+a^2)^2}\] \[\frac{tsin(at)}{2a}\]
\[\frac{s^2}{(s^2+a^2)^2}\] \[\frac{sin(at)+atcos(at)}{2a}\] \[\frac{s^3}{(s^2+a^2)^2}\] \[cos(at) - \frac{1}{2}atsin(at)\]
\[\frac{s^2-a^2}{(s^2+a^2)^2}\] \[tcos(at)\] \[\frac{1}{(s^2-a^2)^2}\] \[\frac{atcosh(at)-sinh(at)}{2a^3}\]
\[\frac{s}{(s^2-a^2)^2}\] \[\frac{tsinh(at)}{2a}\] \[\frac{s^2}{(s^2-a^2)^2}\] \[\frac{atcosh(at)+sinh(at)}{2a}\]
\[\frac{s^3}{(s^2-a^2)^2}\] \[cosh(at) + \frac{1}{2}atsinh(at)\] \[\frac{s^2+a^2}{(s^2-a^2)^2}\] \[tcosh(at)\]

Identities:

\[\left(Z_{1} \space + \space Z_{2}\right)^2 \space = \space \left(Z_{1}\right)^2 \space + \space \left(Z_{2}\right)^2 \space + \space 2 \space Z_{1}Z_{2} \]

\[\left(Z_{1} \space - \space Z_{2}\right)^2 \space = \space \left(Z_{1}\right)^2 \space + \space \left(Z_{2}\right)^2 \space - \space 2 \space Z_{1}Z_{2} \]

\[\left(Z_{1}\right)^2 \space - \space \left(Z_{2}\right)^2 \space = \space \left(Z_{1} \space + \space Z_{2} \right)\left(Z_{1} \space - \space Z_{2} \right) \]

\[\left(Z_{1} \space + \space Z_{2}\right)^3 \space = \space \left(Z_{1}\right)^3 \space + \space 3\left(Z_{1}\right)^2Z_{2} \space + \space 3\left(Z_{2}\right)^2Z_{1} \space + \left(Z_{2}\right)^3 \]

\[\left(Z_{1} \space - \space Z_{2}\right)^3 \space = \space \left(Z_{1}\right)^3 \space - \space 3\left(Z_{1}\right)^2Z_{2} \space + \space 3\left(Z_{2}\right)^2Z_{1} \space - \left(Z_{2}\right)^3 \]

De Moivre's Theorem

For any real number x, we have

\[(cos(x)+isin(x))^{n}\space =\space cos(nx)\space +\space isin(nx)\]

Remark: Result can be shown true when n is a negative integer and even n is a rational number.


Cauchy-Reimaan Theorem


What are analytic functions?

A function f(z) is said to be analytic in a region R of the complex plane if f(z) has a derivative at each point of R and if f(z) is single valued.

Let f(x,y) = u(x,y) + iv(x,y) and the function should be analytic

Then,

\[\frac{∂v}{∂x}\space =\space -\frac{∂u}{∂y}\]

If, z = re, then Cauchy-Reimaan Theorem become-

\[\frac{1}{r}\frac{∂u}{∂θ}\space =\space -\frac{∂v}{∂r}\]

Properties of Complex Numbers:

  1. The addition of two conjugate complex numbers will result in a real number.
  2. The multiplication of two conjugate complex number will also result in a real number.
  3. If x and y are the real numbers and x+yi =0, then x =0 and y =0.
  4. If p, q, r, and s are the real numbers and p+qi = r+si, then p = r, and q=s.
  5. The complex number obeys the commutative law of addition and multiplication.
    \[Z_{1} \space + \space Z_{2} \space = \space Z_{2} \space + \space Z_{1} \\ Z_{1} \space * \space Z_{2} \space = \space Z_{2} \space * \space Z_{1} \]
  6. The complex number obeys the associative law of addition and multiplication.
    \[\left(Z_{1} \space + \space Z_{2}\right) \space + Z_{3} \space = \space Z_{1} \space + \space \left(Z_{2} \space + \space Z_{3}\right) \\ \left(Z_{1} \space * \space Z_{2}\right) \space * Z_{3} \space = \space Z_{1} \space * \space \left(Z_{2} \space * \space Z_{3}\right) \]
  7. The complex number obeys the distributive law.
    \[Z_{1} \space * \space \left(Z_{2} \space + \space Z_{3}\right) \space = \space Z_{1}*Z_{2} \space + \space Z_{1}*Z_{3} \]
  8. If the sum of two complex number is real, and also the product of two complex number is also real, then these complex numbers are conjugate to each other.
  9. For any two complex numbers, say Z1 and Z2, then \[\mid Z_{1} \space + \space Z_{2}\mid \space \le \space \mid Z_{1}\mid \space + \space \mid Z_{2}\mid \]
  10. The result of the multiplication of two complex numbers and its conjugate value should result in a complex number and it should be a positive value.

Calculations for 1 complex number

   

Polar representaion of complex number

   

Find Euler Representaion of complex number

   

Calculations for 2 complex numbers

Vector Calculus Theorems



1. Gauss Divergence Theorem

The divergence theorem states that the surface integral of the normal component of a vector point function “F” over a closed surface “S” is equal to the volume integral of the divergence of F, taken over the volume “V” enclosed by the surface S.

\[\iint_S F.n \,dS = \iiint_v divF\,dV\]

2. Stokes' Theorem

Let S be an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve C with positive orientation. Also let F be a vector field then,

\[\int_C F\,dr = \iint_S curlF\,dS\]

3. Green's Theorem

Let C be a positively oriented, piecewise smooth, simple, closed curve and let D be the region enclosed by the curve. If P and Q have continuous first order partial derivatives on D then,

\[\int_C {P\,dx + Q\,dy} = \iint_S ({\frac {∂Q}{∂x}-\frac{∂P}{∂y}})\,dA \]



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Limits Of Trigonometry Functions

\[1)\space \lim_{x→0}\space sin\space x=0\]

\[2)\space \lim_{x→0}\space cos\space x=1\]

\[3)\space \lim_{x→0}\space\frac{1-cos\space x}{x}=0\]

\[4)\space \lim_{x→0}\space \frac{sin^{-1}\space x}{x}\space =1\]

\[5)\space \lim_{x→0}\space \frac{tan^{-1}\space x}{x}\space =1\]

\[6)\space \lim_{x→0}\space\frac{sin\space x}{x}=1=\lim_{x→0}\space\frac{x}{sin\space x}\]

\[7)\space \lim_{x→0}\space\frac{tan\space x}{x}=1=\lim_{x→0}\space\frac{x}{tan\space x}\]

\[8)\space \lim_{x→a}\space sin^{-1}\space x=sin^{-1}a,\space\space |a|<=1\]

\[9)\space \lim_{x→a}\space cos^{-1}\space x=cos^{-1}a,\space\space |a|<=1\]

\[10)\space \lim_{x→a}\space tan^{-1}\space x=tan^{-1}a,\space\space -∞< a <∞ \]

Limits Of Log And Exponential Function

\[1)\space \lim_{x→0}\space e^{x}\space =1\]

\[2)\space \lim_{x→e}\space \log_{e}\space x =1\]

\[3)\space \lim_{x→0}\space \frac{e^{x}-1}{x}\space =1\]

\[4)\space \lim_{x→∞}\space (1 +\frac{1}{x})^{x}=e\]

\[5)\space \lim_{x→0}\space (1+x)^{\frac{1}{x}}\space =e\]

\[6)\space \lim_{x→∞}\space (1 +\frac{a}{x})^{x}\space =e^{a}\]

\[7)\space \lim_{x→0}\space \frac{a^{x}-1}{x}\space=\log_{e} a\]

\[8)\space \lim_{x→0}\space \frac{log(1+x)}{x}\space =1\]

Some More Important Formulas

\[1)\space \lim_{x→a}\space \frac{(x^{n}-a^{n})}{x-a}\space=n(a)^{n-1}\]

\[2)\space \lim_{x→0}\space a^{x}\space =\frac{∞ \space if a > 1}{0\space if a< 1}\] \[i.e.,\space a^{∞}\space=∞,\space if\space a>1\space and\space a^∞\space = 0,\space if\space a<1\]

\[3)\space if\space\lim_{x→a}\space f(x)\space =+∞,\space then\space\lim_{x→a}\space \frac{1}{(x)}\space =0\]

\[4)\space\lim_{x→a}\space fog(x)=\space f(\lim_{x→a}\space g(x))\space =\space f(m) \]

\[\ In particual, \]

\[a) \space \space \lim_{x→a}\space log\space f(x)\space =\space log(\lim_{x→a}\space f(x))=\space log(l)\]

\[b) \space \space \lim_{x→a}\space e^{f(x)}\space =\space e^{\lim_{x→a}\space f(x)}\space =\space e^{l} \]

Checking If Limit Exist

\[To\space check\space if\space limit\space exists\space for\space f(x)\space at\space x\space = a\]

\[We\space check\space if\]

\[Left\space Hand\space Limit\space=\space Right\space Hand\space Limit\space =\space f(a)\]

\[i.e.\space \lim_{x→a^{-}}\space f(x)=\space \lim_{x→a^{+}}\space f(x)=\space f(a)\]

L'hospital's Rule

\[If \space \lim_{x→a}\space \frac{f(x)}{g(x)}\space gives\space \frac{0}{0}\space form\]

\[Where\]

\[1)\space f(a)=0,\]

\[2)\space f(b)=0\]

\[Then,\]

\[\lim_{x→a}\space \frac{f(x)}{g(x)}=\frac{f'(a)}{g'(a)}\]

Euclid Geometry - Axioms and Postulates



Euclid's Axioms

There are 7 axioms in Euclid Geometry:

  1. Things which are equal to the same thing are equal to one another.

  2. If equals are added to equals, the wholes are equal.

  3. If equals are subtracted from equals, the remainders are equal.

  4. Things which coincide with one another are equal to one another.

  5. The whole is greater than the part.

  6. Things which are double of the same things are equal to one another.

  7. Things which are halves of the same things are equal to one another

Euclid's Postulates

There are 5 postulates in Euclid's Geometry:

  1. A straight line can be drawn from anyone point to another point


  2. A terminated line can be further produced indefinitely


  3. A circle can be drawn with any centre and any radius.

  4. All right angles are equal to one another.

  5. If a straight line falling on two other straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on the side on which the sum of angles is less than two right angles

Operations on Sets

1. Union

The union of two sets is a set containing all elements that are in A or in B (possibly both). For example,

\[\{1,2\}\cup\{2,3\} = \{1,2,3\}\]



2. Intersection

The intersection of two sets A and B, denoted by A∩B, consists of all elements that are both in A and B. For example,

\[\{1,2\}\cap\{2,3\} = \{2\}\]


The intersection of A and B is the middle part




3. Disjoint

Set A and set B are called disjoint sets if no element is common to A and B. i.e. A and B are disjoint sets then

\(A\cap B = ∅\)

For example, \[A = \{1,3,5\}, B = \{2,4,6\}\space and\space C = \{a,b,c\}\\ A\cap B\cap C = ∅\]



4. Complement

The complement of a set A, denoted by Ac is elements in Universal Set which are not in A




5. Difference

The set A−B consists of elements that are in A but not in B. For example if

\[A = \{1,2,3\}\space and\space B = \{3,5\} then,\\ A-B = \{1,2\}\]



Some conclusions from set operations

Let S be the universal set, and A,B,C,D are subsets of S. Then,

  1. A ∪ ∅ = A

    Hence ∅ i.e. empty set is identity set for 'union operation'.


  2. A ∩ S = A

    S i.e. universal set is identity set for 'intersection operation'.


  3. A ∪ S = S

    A ⊂ S and A ∪ S = S

    ⇒ if B ⊂ A then A ∪ B = A i.e. union of super set and subset is super set.


  4. A ∩ ∅ = ∅

    ∅ ⊂ A and A ∩ ∅ = ∅

    ⇒ if B ⊂ A then A ∩ B = B i.e. intersection of super set and subset is subset.


  5. If B ⊂ A, then A ∩ B = B and A ∪ B = A

  6. A ⊂ A ∪ B also B ⊂ A ∪ B

  7. A ∩ B ⊂ A also A ∩ B ⊂ B

  8. A ∩ B ⊂ A ⊂ A ∪ B and A ∩ B ⊂ B ⊂ A ∪ B i.e, A and B sets. A ∩ B is the smallest set and A ∪ B is largest set.

    A contains A ∩ B and A is contained in A ∪ B.

    Similarly B contains A ∩ B and B is contained in A ∪ B.


  9. As A ∩ B ⊂ A

    ⇒ (A ∩ B) ∪ A = A [super set]

    and (A ∩ B) ∩ A = (A ∩ B) [subset]


  10. As A ⊂ A ∪ B

    ⇒ A ∩ (A ∪ B) = A [subset]

    and A ∪ (A ∪ B) = (A ∪ B) [super set]


  11. (A - B) ∪ (A ∩ B) ∪ (B - A) = A ∪ B and (A - B), (A ∩ B), (B - A) are pairwise disjoint. i.e.

    (A - B) ∩ (A ∩ B) = ∅

    (A - B) ∩ (B - A) = ∅

    (A ∩ B) ∩ (B - A) = ∅

    Hence (A - B), (A ∩ B) and (B - A) are partitions of A ∪ B.


  12. A ∪ A' = S

    i.e. Union of A and its complement gives identity set for intersection operation.


  13. A ∩ A' = ∅

    i.e. Intersection of A and its complement gives identity set for union operation.


Remark:

  • Property 11 and 12 are very useful for Boolean algebra as in Boolean algebra these properties are formed in a way: a + a' = 1 i.e. sum of element and its complement gives multiplicative identity element and a. a' = 0

  • Multiplication of element and its complement gives additive identity element.

Sets,Relations and Functions

What are Sets?

A set is the representation of a collection of objects; distinct objects with one or more common properties.

Types of Sets
  1. Empty Set - A set with no elements. Empty sets are also called null sets or void sets and are denoted by { } or Φ.
  2. Singleton Set - A set with a single element. For example, {9}.
  3. Power Set - A set qualifies as the subset of another set if all of its elements are also the elements of that another set. A collection of all the subsets of a given set is a power set.
  4. Super Set - A super-set can be thought of as the parent set that at least contains all the elements of the subset and may or may not contain some extra elements.

What are Relations and Functions?

Relations and functions are the set operations that help to trace the relationship between the elements of two or more distinct sets or between the elements of the same set.



The relation is the subset of the Cartesian product which contains only some of the ordered pair based on the relationships defined between the first and second elements. The relation is usually denoted by R.


If every element of a set A is related with one and only one element of another set then this kind of relation qualifies as a function. A function is a special case of relation where no two ordered pairs can have the same first element.

Conditions to be a function -

A relation from a set X to a set Y is called a function if each element of X is related to exactly one element in Y. That is, given an element x in X, there is only one element in Y that x is related to.

Types of Functions -
  1. One to one function(Injective): For each element in the domain there is one and only one element in the range.
  2. Many to one function: When two or more elements from the domain are mapped to the same single elements in the range.
  3. Onto function(Surjective): When every element of the range has been mapped to an element in the domain.
  4. One-one and onto function(Bijective): A function which is both one to one and onto function.

Matrix 1
Matrix 2

Cayley-Hamilton Theorem

every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.

A square matrix can be written as |A- λI|=0 and the corresponding eqn is

\[A^m+c_{m-1}A^{m-1}+...+c_0I\space=\space0\]

The Cayley-Hamilton theorem states that an n×n matrix A is annihilated by its characteristic polynomial det(xI-A), which is monic of degree n.

Diagonalization of Matrix

Matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix--a so-called diagonal matrix--that shares the same fundamental properties of the underlying matrix. Matrix diagonalization is equivalent to transforming the underlying system of equations into a special set of coordinate axes in which the matrix takes this canonical form. Diagonalizing a matrix is also equivalent to finding the matrix's eigenvalues, which turn out to be precisely the entries of the diagonalized matrix. Similarly, the eigenvectors make up the new set of axes corresponding to the diagonal matrix.

The remarkable relationship between a diagonalized matrix, eigenvalues, and eigenvectors follows from the beautiful mathematical identity (the eigen decomposition) that a square matrix A can be decomposed into the very special form

\[A\space =\space PDP^{-1}\]

D is the diagonalized matrix and can be written as diagnol matrix containing eigen values on the major diagnol


Properties of Matrices


1. If A be any matrix then the matrix obtained by interchanging the rows and columns of A is called the Transpose of A. Transpose of the matrix A is denoted by A' or AT

2. A square matrix A is called orthogonal if A AT = I where I is identity matrix

3. If the matrix product AB is defined, then (AB)T = BTAT

4. AA-1 = I where A-1 is inverse of matrix A

5. (AB)-1 = B-1A-1

6. A matrix is said to be Involutary if A2 = I

7. A matrix is said to be Idempotent if A2 = A

8. A matrix is said to be Nilpotent if Ap = 0 , where p is any positive integer

9. A square matrix A is called Periodic if Ak+1 = A where k is a positive integer

10. A square matrix A is said to be a Singular if |A| = 0 and a square matrix A is said to be Non Singular if |A| ≠ 0

11. A matrix is said to be symmetric if A = AT and skew-symmetric if A = -AT

12. A matrix is said to be Hermitian if A = Aθ and skew-Hermitian if A = -Aθ where Aθ is transpose conjugate of matrix

13. A square matrix A is said to be Unitary if AAθ = I

14. If A be a square matrix, then AAθ and AθA are Hermitian matrix

15. If A be a square matrix of order n and C be the cofactor in A then the transpose of the matrix of cofactors of element of A is called the adjoint of A and is denoted by adj(A). Thus adj(A) = [Cij]T

16. A square matrix A(non-singular) of order n is said to be invertible, if there exists a square matrix B of same order such that
AB = In = BA then B is called the inverse of A and is denoted by A-1 . Thus A-1 = adjA/|A|

17. If A is a non singular square matrix of order n then adj(adj A) = |A|n-2A



Properties of Determinants


1. det(XY) = det(X)det(Y) where det is determinant, this is Distributive property

2. If rows and columns are interchanged then value of determinant remains same (value does not change), det(XT) = det(X)

3. Determinant of inverse of a matrix is, det(X-1) = -det(X)

4. The determinant of a matrix will be 0 if all the elements of any one column or row is 0

5. If two rows(or columns) of a determinant are identical then the values of the determinant is zero

6. If determinant Δ becomes zero on putting x = α then we say that (x-α) is a factor of Δ

7. If X is orthogonal matrix then det(X) = ±1

8. If X is a skew-symmetric matrix of odd order then det(X) = 0

9. If X is a skew-symmetric matrix of even order then det(X) is a perfect square.

10. det(kX) = kndet(X) where n is order of X and k is scalar

11. If all the elements of a determinant above or below the main diagonal consist of zeros, determinant equals to product of diagonal elements

12. A matrix A can be written as det(A- λI) = 0 , the λ are eigen values of the matrix

13. Determinant remains unaltered if its rows are changed into columns and the columns into rows, this is property of reflection.

14. If i not equals to j, subtracting t times row i from row j doesn’t change the determinant value

15. If det A is exactly 0, where det A is determinant of A, then A is singular

16. If det A not equals to 0, where det A is determinant of A, then A is invertible

17. The determinant of A is defined for all matrices A

\[Equation\] \[Graph\] \[Centre\] \[Radius\]
\[x^2+y^2=a^2\] \[(0,0)\] \[a\]
\[(x-h)^2+(y-k)^2=a^2\] \[(h,k)\] \[a\]
\[x^2+y^2+2gx+2fy+c=0\] \[(-g,-f)\] \[\sqrt{g^2+f^2-c}\]

Equations of Tangent of all Circles

\[Equations\space of\]
\[Circle\]
\[Point/Line\space of\space contact\space of \space \]
\[circle \]
\[\ m=slope \space of \space tangent\]
\[Equation\space of\space \]
\[ tangent\]
\[x^2+y^2=a^2\] \[(x1,y1)\] \[xx1 +yy1=a^2\]
\[x^2+y^2=a^2\] \[(a \space cos\theta,b \space sin\theta)\] \[x \space cos\theta + y \space sin\theta=a\]
\[x^2+y^2=a^2\] \[y=mx+c\] \[y=mx±a\sqrt{1+m^2}\]
\[x^2+y^2+2gx+2fy+c=0\] \[(x1,y1)\] \[xx1+yy1+g(x+x1)+f(y+y1)+c=0\]

Equations of Normal of all Circles

\[Equations\space of\]
\[Circle\]
\[Point/Line\space of\space contact\space of \space \]
\[circle \]
\[\ m=slope\space of \space tangent\]
\[Equation\space of\space\]
\[\ Normal\]
\[x^2+y^2=a^2\] \[(x1,y1)\] \[\frac{x}{x1}=\frac{y}{y1}\]
\[x^2+y^2=a^2\] \[(a \space cos\theta,b \space sin\theta)\] \[y=x\space tan\theta\]
\[x^2+y^2=a^2\] \[y=mx+c\] \[x+my=±a\sqrt{1+m^2}\]
\[x^2+y^2+2gx+2fy+c=0\] \[(x1,y1)\] \[\frac{y-y1}{x-x1}=\frac{y1+f}{x1+g}\]

Director Circle of all Circles

\[Equations\space of\space Circle\] \[Equation\space of\space Director\space Circle\]
\[x^2+y^2=a^2\] \[x^2+y^2=2 \space a^2\]
\[(x-h)^2+(y-k)^2=a^2\] \[(x-h)^2+(y-k)^2=2 \space a^2\]
\[x^2+y^2+2gx+2fy+c=0\] \[(x+g)^2+(y+f)^2=2(g^2+f^2-c)\]
\[Choose\space the\space calculator\space according\space to\space your\space equation\space type\]
\[For\space Equation:\space y^2=4ax\] \[For\space Equation:\space x^2=4ay\]


\[Equation\] \[Graph\] \[Focus\] \[Length\space of\space LR\] \[Equation\space of\space Directrix\] \[Equation\space of\space Axis\]
\[y^2=4ax\] \[(a,0)\] \[4a\] \[x=-a\] \[y=0\]
\[y^2=-4ax\] \[(-a,0)\] \[4a\] \[x=a\] \[y=0\]
\[x^2=4ay\] \[(0,a)\] \[4a\] \[y=-a\] \[x=0\]
\[x^2=-4ay\] \[(0,-a)\] \[4a\] \[y=a\] \[x=0\]

Equations of Tangent of all Parabolas in slope form

\[Equations\space of\]
\[Parabola\]
\[Point\space of\space contact\space in\]
\[\space terms\space of\space slope(m)\]
\[Equation\space of\space tangent\space in\]
\[\space terms\space of\space slope(m)\]
\[Condition\space of\space Tangency\]

\[y^2=4ax\] \[(\frac{a}{m^2},\frac{2a}{m})\] \[y=mx+\frac{a}{m}\] \[c=\frac{a}{m}\]
\[y^2=-4ax\] \[(-\frac{a}{m^2},-\frac{2a}{m})\] \[y=mx-\frac{a}{m}\] \[c=-\frac{a}{m}\]
\[x^2=4ay\] \[(2am,am^2)\] \[y=mx-am^2\] \[c=-am^2\]
\[x^2=-4ay\] \[(-2am,am^2)\] \[y=mx+am^2\] \[c=am^2\]

Equations of Normal of all Parabolas in slope form

\[Equations\space of\]
\[Parabola\]
\[Point\space of\space contact\space in\]
\[\space terms\space of\space slope(m)\]
\[Equation\space of\space normal\space in\]
\[\space terms\space of\space slope(m)\]
\[Condition\space of\space Normality\]

\[y^2=4ax\] \[(am^2,-2am)\] \[y=mx-2am-am^3\] \[c=-2am-am^3\]
\[y^2=-4ax\] \[(am^2,2am)\] \[y=mx+2am+am^3\] \[c=2am+am^3\]
\[x^2=4ay\] \[(-\frac{2a}{m},\frac{a}{m^2})\] \[y=mx+2a+\frac{a}{m^2}\] \[c=2a+\frac{a}{m^2}\]
\[x^2=-4ay\] \[(\frac{2a}{m},-\frac{a}{m^2})\] \[y=mx-2a-\frac{a}{m^2}\] \[c=-2a-\frac{a}{m^2}\]

Director Circle of all Parabolas

\[Equations\space of\space Parabola\] \[Equation\space of\space Director\space Circle\]
\[y^2=4ax\] \[x\,+a\space=\,0\]
\[y^2=-4ax\] \[x\,-a\space=\,0\]
\[x^2=4ay\] \[y\,+a\space=\,0\]
\[x^2=-4ay\] \[y\,-a\space=\,0\]
\[For\space Equation:\space \frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\] \[Eccentricity\space of\space ecllipse\space:\space \frac{\sqrt{a^2 - b^2}}{a}\]


\[Equation\] \[Graph\] \[Focus\] \[Length\space of\space LR\] \[Directrix\] \[Length\space of\space Major\space Axis\]
\[\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\]
\[a>b\]
\[(±ae,0)\] \[\frac{2b^2}{a}\] \[x=±\frac{a}{e}\] \[2a\]
\[\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\]
\[a < b\]
\[(0,±be)\] \[\frac{2a^2}{b}\] \[y=±\frac{b}{e}\] \[2b\]

Equations of Tangent of Ellipse

\[Equation\] \[Parametric\space Coordinates\] \[Equation\space of\space tangent\] \[Condition\space of \space Tangency\]
\[\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\]
\[a>b\]
\[(acos\theta,bsin\theta)\] \[y=mx±\sqrt{am^2+b^2}\]
\[\frac{xcos\theta}{a}+\frac{ysin\theta}{b}=1\]
\[c=±\sqrt{am^2+b^2}\]
\[\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\]
\[a < b\]
\[(bcos\theta,asin\theta)\] \[y=mx±\sqrt{bm^2+a^2}\]
\[\frac{xcos\theta}{b}+\frac{ysin\theta}{a}=1\]
\[c=±\sqrt{bm^2+a^2}\]

Equations of Normal of Ellipse

\[Equation\] \[Parametric\space Coordinates\] \[Equation\space of\space Normal\] \[Condition\space of \space Normality\]
\[\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\]
\[a>b\]
\[(acos\theta,bsin\theta)\] \[\frac{ax}{cos\theta}-\frac{by}{sin\theta}=a^2-b^2\] \[c=±\frac{m(a^2-b^2)}{\sqrt{a^2+b^2m^2}}\]
\[\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\]
\[a < b\]
\[(bcos\theta,asin\theta)\] \[\frac{bx}{cos\theta}-\frac{ay}{sin\theta}=b^2-a^2\] \[c=±\frac{m(b^2-a^2)}{\sqrt{b^2+a^2m^2}}\]

Equations of Director circle of Ellipse

\[Equation\] \[Equation\space of\space Director\space Circle\]
\[\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\]
\[a>b\]
\[x^2\,+\,y^2\space=\,a^2\,+\,b^2\]
\[\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1\]
\[a < b\]
\[x^2\,+\,y^2\space=\,a^2\,+\,b^2\]

ParaBoloid


Paraboloid, an open surface generated by rotating a parabola (q.v.) about its axis. If the axis of the surface is the z axis and the vertex is at the origin, the intersections of the surface with planes parallel to the xz and yz planes are parabolas

The surface of revolution of the parabola which is the shape used in the reflectors of automobile headlights

Its equation is given by -

\[z\space =\space b(x^{2}+y^{2})\]

The paraboloid which has radius a at height h is then given parametrically by

\[x\space =\space a\sqrt{\frac{u}{h}}cosv\]

\[y\space =\space a\sqrt{\frac{u}{h}}sinv\]

\[z\space =\space u\]

Volume of paraboloid =

\[\frac{1}{2}\pi a^{2}h\]

\[Geometric\space centroid\space = \]

\[\frac{2}{3}h\]

A circular or elliptical paraboloid surface may be used as a parabolic reflector. Applications of this property are used in automobile headlights, solar furnaces, radar, and radio relay stations.

\[Choose\space your\space Equation \space type:\]
\[Eccentricity\space of\space hyperbola\space:\space \frac{\sqrt{a^2 + b^2}}{a}\]


\[Equation\] \[Graph\] \[Focus\] \[Length\space of\space LR\] \[Directrix\] \[Length\space of\space Transverse\space Axis\]
\[\frac{x^2}{a^2}\space-\frac{y^2}{b^2}\space=1\] \[(±ae,0)\] \[\frac{2b^2}{a}\] \[x=±\frac{a}{e}\] \[2a\]
\[\frac{y^2}{b^2}\space-\frac{x^2}{a^2}\space=1\] \[(0,±be)\] \[\frac{2a^2}{b}\] \[y=±\frac{b}{e}\] \[2b\]
\[{x^2}\space-{y^2}\space={a^2}\] \[(0, ±a{\sqrt{2}} )\] \[2a\] \[x=±\frac{a}{\sqrt{2}}\] \[2a\]

Equations of Tangent of Hyperbola

\[Equation\] \[Parametric\space Coordinates\] \[Equation\space of\space tangent\] \[Condition\space of \space Tangency\]
\[\frac{x^2}{a^2}\space-\frac{y^2}{b^2}\space=1\] \[(asec\theta,btan\theta)\] \[y=mx±\sqrt{am^2-b^2}\] \[c=±\sqrt{am^2-b^2}\]
\[\frac{y^2}{b^2}\space-\frac{x^2}{a^2}\space=1\] \[(bsec\theta,atan\theta)\] \[y=mx±\sqrt{-bm^2+a^2}\] \[c=±\sqrt{-bm^2+a^2}\]
\[{x^2}\space-{y^2}\space={a^2}\] \[(asec\theta,atan\theta)\] \[y=mx±\sqrt{am^2-a^2}\] \[c=±\sqrt{am^2-a^2}\]

Equations of Normal of Hyperbola

\[Equation\] \[Parametric\space Coordinates\] \[Equation\space of\space Normal\] \[Condition\space of \space Normality\]
\[\frac{x^2}{a^2}\space-\frac{y^2}{b^2}\space=1\] \[(asec\theta,btan\theta)\] \[\frac{ax}{sec\theta}+\frac{by}{tan\theta}=a^2+b^2\] \[c=\frac{m(a^2+b^2)}{\sqrt{a^2-b^2m^2}}\]
\[\frac{y^2}{b^2}\space-\frac{x^2}{a^2}\space=1\] \[(bsec\theta,atan\theta)\] \[\frac{bx}{sec\theta}+\frac{ay}{tan\theta}=b^2+a^2\] \[c=\frac{m(b^2-a^2)}{\sqrt{a^2m^2-b^2}}\]
\[{x^2}\space-{y^2}\space={a^2}\] \[(asec\theta,atan\theta)\] \[\frac{x}{sec\theta}+\frac{y}{tan\theta}=2a\] \[c=\frac{2am}{\sqrt{1-m^2}}\]

Equations of Director Circle of Hyperbola

\[Equation\] \[Equation\space of\space Director\space Circle\]
\[\frac{x^2}{a^2}\space-\frac{y^2}{b^2}\space=1\] \[x^2\,+\,y^2\space\,=a^2\,-\,b^2\]
\[\frac{y^2}{b^2}\space-\frac{x^2}{a^2}\space=1\] \[x^2\,+\,y^2\space\,=b^2\,-\,a^2\]

Volume, CSA/LSA and TSA of 3D solids

\[Shape\] \[Volume\] \[C\,S\,A \space/\,L\,S\,A \] \[T\,S\,A\]
\[Cube\] \[a^{3}\] \[4\,a^{2}\] \[6\,a^{2}\]
\[Cuboid\] \[l\,b\,h\] \[2\,(\,l\,+\,b\,)\,h\] \[2\,(\,lb\,+\,bh\,+\,lh\,)\]
\[Cylinder\] \[ π\,r^{2}\,h\] \[2\,π\,r\,h\] \[2πr\, (\, r\, +\, h\, )\]
\[Hollow\space Cylinder\] \[ πh\,(\, R \,+ \,r\, )\, (\, R\, -\, r\, )\] \[2πh\space ( \,R\, +\, r\, )\] \[2πh\, (\, R\, +\, r\, )\]

\[+\]

\[2π\, (\, R\, +\, r\, )\, (\, R\, -\, r\, )\]

\[Cone\] \[ \frac{1}{3} \, π\,r^{2} \, h\] \[πrl\, =\, πr\,√( r^{2}\,+ \,h^{2} )\] \[πr \,(\, l\, +\, r\, )\]
\[Frustum\space Of\space Cone\] \[ ( \frac{rh}{3} )\, ( \,r^{2} \,+\, R^{2} \,+\, rR \,)\] \[π(R+r)l\, =\, π(R+r)√(h^{2}+(R-r)^{2})\] \[π(rl+Rl+r^{2}+R^{2})\]
\[Sphere\] \[\frac{4}{3}\, π\,r^{3}\] \[ 4\, π\,r^{2}\] \[4\, π\,r^{2}\]
\[Hollow\space Sphere\] \[\frac{4}{3}\, π\,(\,r_2^{3}\,-\,r_1^{3})\]

\[4\, π\,(r_2^{2}-r_1^{2})\]

\[4\, π\,(r_2^{2}-r_1^{2})\]

\[TSA\,=\,CSA\]

\[Hemisphere\] \[\frac{(2\, π\,r^{3}\,)}{3}\] \[2\, π\,r^{2}\] \[3\, π\,r^{2}\]
\[Prism\] \[ Area \,of \,base× Height \]

\[Perimeter\, of\, base\]

\[×\]

\[Height\, of\, prism\]

\[LSA + 2\,× Area \,of\, base\]
\[Rectangular\space Prism\] \[(l\,b)\,×h\] \[(\,2l\,+\,2b\,)\,×h\] \[LSA +2×(\,lb\,)\]
\[Triangular\space Prism\]

\[(\frac{1}{2}×apothem\space length\]

\[×\]

\[base\space length)\,×Height\]

\[(3\,×base\,length)\]

\[×\]

\[Height\]

\[LSA + 2×(\,\frac{1}{2}×apothem\space length\]

\[×\]

\[base\space length\,)\]

\[Pentagonal\space Prism\]

\[(\frac{5}{2}×apothem\space length\]

\[×\]

\[base\space length)\,×Height\]

\[(5\,×base\,length)\]

\[×\]

\[Height\]

\[LSA + 2×(\,\frac{5}{2}×apothem\space length\]

\[×\]

\[base\space length\,)\]

\[Hexagonal\space Prism\]

\[(3\,×apothem\space length\]

\[×\]

\[base\space length)\,×Height\]

\[(6\,×base\,length)\]

\[×\]

\[Height\]

\[LSA + 2×(\,3\,×apothem\space length\]

\[×\]

\[base\space length\,)\]

\[Pyramid\] \[\frac{1}{3}×base\,area×Height\]

\[ \frac{1}{2} × Perimeter\, of\, base\]

\[×\]

\[Slant\, height\, Total\]

\[LSA + Area\, of \,base\]
\[Square\space Pyramid\] \[\frac{1}{3}(\,base\space length\,)^{2}\,×Height\] \[\frac{1}{2} (4\,×base\, length)\]\[×\]\[slant\, height\] \[LSA + (\,base\space length\,)^{2}\]
\[Triangular\space Pyramid\]

\[\frac{1}{3}(\,\frac{1}{2}×apothem\space length\]

\[×\]

\[base\space length\,)\,×Height\]

\[\frac{1}{2}(\,3×base\,length)\]

\[×\]

\[slant\space height\]

\[LSA +(\, \frac{1}{2}×apothem\space length\]

\[×\]

\[base\space length\,)\]

\[Pentagonal\space Pyramid\]

\[\frac{1}{3}(\,\frac{5}{2}×apothem\space length\]

\[×\]

\[base\space length\,)\,×Height\]

\[\frac{1}{2}(\,5×base\space length)\]

\[×\]

\[slant\space height\]

\[LSA +(\, \frac{5}{2}×apothem\space length\]

\[×\]

\[base\space length\,)\]

\[Hexagonal\space Pyramid\]

\[\frac{1}{3}(\,3×\,apothem\space length\]

\[×\]

\[base\space length\,)\,×Height\]

\[\frac{1}{2}(\, 6×base\space length)\]

\[×\]

\[slant\space height\]

\[LSA +(\, 3×apothem\space length\]

\[×\]

\[base\space length\,)\]

\[Frustum\space Of\space Pyramid\]

\[\frac{1}{3}×(\,A_1+A_2+\,\sqrt{A_1A_2})\,×Height\]

\[^{*}A_1\,-Area\, of\, Upper\, Base\]

\[^{*}A_2\,-Area\, of\, Lower\, Base\]

\[\frac{1}{2}×(P_1+P_2)\,×slant\,height\]\[^{*}P_1\,-Perimeter\,Of\,Upper\,Base\]\[^{*}P_2\,-Perimeter\,Of\,Lower\,Base\] \[LSA\,+\,A_1\,+\,A_2\]
\[Torus\] \[(\,2π\,R\,)\,(\,π\,r^{2}\,)\]\[^{*}R\,-\,Major\,Radius\]\[^{*}r\,-\,Minor\,Radius\] \[(\,2π\,R\,)\,(\,2π\,r\,)\] \[(\,2π\,R\,)\,(\,2π\,r\,)\]
\[Angles→\]
\[Ratios↓\]
\[0^{\circ}\] \[30^{\circ}\] \[45^{\circ}\] \[60^{\circ}\] \[90^{\circ}\]
\[\sin\theta\] \[0\] \[\frac{1}{2}\] \[\frac{1}{\sqrt{2}}\] \[\frac{\sqrt{3}}{2}\] \[1\]
\[\cos\theta\] \[1\] \[\frac{\sqrt{3}}{2}\] \[\frac{1}{\sqrt{2}}\] \[\frac{1}{2}\] \[0\]
\[\tan\theta\] \[0\] \[\frac{1}{\sqrt{3}}\] \[1\] \[\sqrt{3}\] \[Not \space Defined\]
\[\cosec\theta\] \[Not \space Defined\] \[2\] \[\sqrt{2}\] \[\frac{2}{\sqrt{3}}\] \[1\]
\[\sec\theta\] \[1\] \[\frac{2}{\sqrt{3}}\] \[\sqrt{2}\] \[2\] \[Not \space Defined\]
\[\cot\theta\] \[Not \space Defined\] \[\sqrt{3}\] \[1\] \[\frac{1}{\sqrt{3}}\] \[0\]

\[Values \space of \space some \space T-Ratios \space for \space many \space angles \]

\[1) \space sin(7.5^{\circ})= \frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}{2}= cos(82.5^{\circ})= sin \frac{\pi}{24}\]

\[2) \space cos(7.5^{\circ})= \frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2}= sin(82.5^{\circ})= cos \frac{\pi}{24}\]

\[3) \space tan(7.5^{\circ})= \sqrt{6}-\sqrt{3}+\sqrt{2}-2=(\sqrt{2}-1)(\sqrt{3}-\sqrt{2})= cot(82.5^{\circ})= tan\frac{\pi}{24}\]

\[4) \space cot(7.5^{\circ})= \sqrt{6}+\sqrt{3}+\sqrt{2}+2=(\sqrt{2}+1)(\sqrt{3}+\sqrt{2})= tan(82.5^{\circ})= cot\frac{\pi}{24}\]

\[5) \space sin15^{\circ}= \frac{\sqrt{3}-1}{2\sqrt{2}}= cos75^{\circ}= sin \frac{\pi}{12}\]

\[6) \space cos15^{\circ}= \frac{\sqrt{3}+1}{2\sqrt{2}}= sin75^{\circ}= cos \frac{\pi}{12}\]

\[7) \space tan15^{\circ}= 2-\sqrt{3}= cot75^{\circ}= tan\frac{\pi}{12}\]

\[8) \space cot15^{\circ}= 2+\sqrt{3}= tan75^{\circ}= cot\frac{\pi}{12}\]

\[9) \space sin18^{\circ}= \frac{\sqrt{5}-1}{4}= \sqrt{\frac{3-\sqrt{5}}{8}} = cos72^{\circ}= sin\frac{\pi}{10} \]

\[10) \space cos18^{\circ}= \frac{\sqrt{10+2\sqrt{5}}}{4}= \sqrt{\frac{5+\sqrt{5}}{8}} = sin72^{\circ}= cos\frac{\pi}{10} \]

\[11) \space tan18^{\circ}= \sqrt{1-\frac{2\sqrt{5}}{5}}= cot72^{\circ}= tan\frac{\pi}{10}\]

\[12) \space cot18^{\circ}= \sqrt{5+2\sqrt{5}}= tan72^{\circ}= cot\frac{\pi}{10}\]

\[13) \space sin(22.5^{\circ})= \frac{\sqrt{2-\sqrt{2}}}{2}= \sqrt{\frac{4-\sqrt{8}}{8}} = cos(67.5^{\circ})= sin\frac{\pi}{8}\]

\[14) \space cos(22.5^{\circ})= \frac{\sqrt{2+\sqrt{2}}}{2}= \sqrt{\frac{4+\sqrt{8}}{8}} = sin(67.5^{\circ})= cos\frac{\pi}{8}\]

\[15) \space tan(22.5^{\circ})=\sqrt{2}-1=cot(67.5^{\circ})= tan\frac{\pi}{8}\]

\[16) \space cot(22.5^{\circ})=1+\sqrt{2}=tan(67.5^{\circ})= cot\frac{\pi}{8}\]

\[17) \space sin36^{\circ}= \frac{\sqrt{10-2\sqrt{5}}}{4}= \sqrt{\frac{5-\sqrt{5}}{8}}= cos54^{\circ}= sin\frac{\pi}{5}\]

\[18) \space cos36^{\circ}= \frac{\sqrt{5}+1}{4}= \sqrt{\frac{3+\sqrt{5}}{8}}= sin54^{\circ}=cos \frac{\pi}{5}\]

\[19) \space tan36^{\circ}= \sqrt{5-2\sqrt{5}}= cot54^{\circ}= tan \frac{\pi}{5}\]

\[20) \space cot36^{\circ}= \sqrt{\frac{5+2\sqrt{5}}{5}}= tan54^{\circ}= cot \frac{\pi}{5}\]

\[21) \space sin(37.5^{\circ})= \frac{\sqrt{2-\sqrt{2-\sqrt{3}}}}{2}= cos(52.5^{\circ})= sin \frac{5\pi}{24}\]

\[22) \space cos(37.5^{\circ})= \frac{\sqrt{2+\sqrt{2-\sqrt{3}}}}{2}= sin(52.5^{\circ})= cos \frac{5\pi}{24}\]

\[23) \space tan(37.5^{\circ})= \sqrt{6}+\sqrt{3}-\sqrt{2}-2=(\sqrt{2}+1)(\sqrt{3}-\sqrt{2})= cot(52.5^{\circ})= tan\frac{5\pi}{24}\]

\[24) \space cot(37.5^{\circ})= \sqrt{6}-\sqrt{3}-\sqrt{2}+2=(\sqrt{2}-1)(\sqrt{3}+\sqrt{2})= tan(52.5^{\circ})= cot\frac{5\pi}{24} \]

Quotient Identities

\[tan \space \theta=\frac{sin\space\theta}{cos\space\theta}\]

\[cot \space \theta=\frac{cos\space\theta}{sin\space\theta}\]

Reciprocal Identities

\[sin\space\theta=\frac{1}{cosec\space\theta}\]

\[cos\space\theta=\frac{1}{sec\space\theta}\]

\[tan\space\theta=\frac{1}{cot\space\theta}\]

\[cosec\space\theta=\frac{1}{sin\space\theta}\]

\[sec\space\theta=\frac{1}{cos\space\theta}\]

\[cot\space\theta=\frac{1}{tan\space\theta}\]

Pythagorean Identities

\[sin^2\space\theta\space+\space cos^2\space\theta=1\]

\[sec^2\space\theta\space-\space tan^2\space\theta=1\]

\[cosec^2\space\theta\space-\space cot^2\space\theta=1\]

Even/Odd Identities

\[sin\space (-\theta )=\space -sin\space\theta\]

\[tan\space (-\theta )=\space -tan\space\theta\]

\[cosec\space (-\theta )=\space -cosec\space\theta\]

\[cos\space (-\theta )=\space cos\space\theta\]

\[cot\space (-\theta )=\space -sin\space\theta\]

\[sec\space (-\theta )=\space sec\space\theta\]

Double angle Identities

\[sin\space 2 \theta=2 \space sin\space\theta\space cos\space\theta\]

\[cos\space 2 \theta= cos^2 \space\theta - sin^2 \space\theta\]

\[cos\space 2 \theta = 2\space cos^2\space\theta - 1\]

\[cos\space 2 \theta= 1- 2\space sin^2 \space\theta\]

\[tan\space 2 \theta=\frac{2\space tan \theta}{1 - tan^2 \theta}\]

Cofunction Identities

\[sin \space (\frac{π}{2} - \theta ) = cos\space\theta\]

\[tan \space (\frac{π}{2} - \theta ) = cot\space\theta\]

\[cosec \space (\frac{π}{2} - \theta ) = sec\space\theta\]

\[cos \space (\frac{π}{2} - \theta ) = sin\space\theta\]

\[cot \space (\frac{π}{2} - \theta ) = tan\space\theta\]

\[sec \space (\frac{π}{2} - \theta ) = cosec\space\theta\]

Half angle Identities

\[sin^2\space\theta = \frac{1 - cos 2\theta}{2}\]

\[cos^2\space\theta = \frac{1 + cos 2\theta}{2}\]

\[tan^2\space\theta = \frac{1 - cos 2\theta}{1 + cos 2\theta}\]

Sum/Difference Identities

\[sin (\theta ± \phi ) = sin\space\theta cos\space\phi ± cos\space\theta sin\space\phi\]

\[cos (\theta ± \phi ) = cos\space\theta cos\space\phi ∓ sin\space\theta sin\space\phi\]

\[tan (\theta ± \phi ) = \frac{tan \space\theta ± tan \space\phi}{1 ∓ tan \space\theta tan\space\phi}\]

Product-to-Sum Identities

\[2 cos \space \theta \cos \space \varphi = {{\cos(\theta - \varphi) + \cos(\theta + \varphi)}}\]

\[2 sin \space \theta \sin \space \varphi = {{\cos(\theta - \varphi) - \cos(\theta + \varphi)}}\]

\[2 sin \space \theta \cos \space \varphi = {{\sin(\theta + \varphi) + \sin(\theta - \varphi)}}\]

\[2 cos \space \theta \sin \space \varphi = {{\sin(\theta + \varphi) - \sin(\theta - \varphi)}}\]

\[2 tan \space \theta \tan \space \varphi =\frac{\cos(\theta-\varphi)-\cos(\theta+\varphi)}{\cos(\theta-\varphi)+\cos(\theta+\varphi)}\]

Sum-to-Product Identities

\[sin \space \theta \pm \sin \space \varphi = 2\sin\left( \frac{\theta \pm \varphi}{2} \right) \cos\left( \frac{\theta \mp \varphi}{2} \right)\]

\[cos \space \theta + \cos \space \varphi = 2\cos\left( \frac{\theta + \varphi} {2} \right) \cos\left( \frac{\theta - \varphi}{2} \right)\]

\[cos \space \theta - \cos \space \varphi = -2\sin\left( \frac{\theta + \varphi}{2}\right) \sin\left(\frac{\theta - \varphi}{2}\right)\]

Triple Angle Identities

\[sin\space 3 \theta=3 \space sin\space\theta - 4 \space sin^3 \space\theta\]

\[cos\space 3 \theta=4 \space cos^3\space\theta - 3 \space cos \space\theta\]

\[tan\space 3 \theta=\frac{3 \space tan\space\theta - tan^3 \space\theta}{1 - 3 \space tan^2 \space \theta}\]

Conditional Identities

\[Given:\space A\,+\,B\,+\,C\,=\,π\]

\[sin\space 2A\space+\space sin\space 2B\space+\space sin\space 2C\space=4\,sin\,A\,sin\,B\,sin\,C\]

\[sin\space A\space+\space sin\space B\space+\space sin\space C\space=4\,cos\,\frac{A}{2}\,cos\,\frac{B}{2}\,cos\,\frac{C}{2}\]

\[cos\space 2A\space+\space cos\space 2B\space+\space cos\space 2C\space=-\,1\,-\,4\,cos\,A\,cos\,B\,cos\,C\]

\[cos\space A\space+\space cos\space B\space+\space cos\space C\space=1\,+\,4\,sin\,\frac{A}{2}\,sin\,\frac{B}{2}\,sin\,\frac{C}{2}\]

\[tan\space A\space+\space tan\space B\space+\space tan\space C\space=\,tan\,A\,tan\,B\,tan\,C\]

\[tan\space \frac{A}{2}\space tan\space \frac{B}{2}\space+\space tan\space \frac{B}{2}\space tan\space \frac{C}{2}\space+\space tan\space \frac{C}{2}\space tan\space \frac{A}{2}\space=\,1\]

\[cot\space \frac{A}{2}\space+\space cot\space \frac{B}{2}\space+\space cot\space \frac{C}{2}\space=\,cot\,\frac{A}{2}\,cot\,\frac{B}{2}\,cot\,\frac{C}{2}\]

\[cot\space A\space cot\space B\space+\space cot\space B\space cot\space C\space+\space cot\space C\space cot\space A\space=\,1\]

Trigonometric Identities involving three different angles

\[1) \space sin\space (A+B+C) \space = sinAcosBcosC \space + \space sinBcosAcosC \space + \space sinCcosAcosB \space - \space sinAsinBsinC \]\[=cosAcosBcosC \space [tanA +tanB+tanc-tanAtanBtanc]\]

\[2) \space cos\space (A+B+C) \space = cosAcosBcosC \space - \space sinAsinBcosC \space \space sinAcosBsinC \space - \space cosAsinBsinC \]\[=cosAcosBcosC \space [1-tanAtanB - tanBtanC-tanCtanA]\]

\[3) \space tan\space (A+B+C) \space = \frac{tanA \space + \space tanB \space + \space tanC \space - \space tanAtanBtanC}{1-tanAtanB-tanBtanC-tanCtanA}\]


Sine and Cosine Series

\[sin\alpha + sin(\alpha + \beta) + sin(\alpha +2\beta) +...+sin(\alpha+ \overline{n-1}\beta)\]\[=\frac{sin( \alpha+(\frac{n-1}{2})\beta) sin(\frac{n\beta}{2})}{sin(\frac{\beta}{2})}\]

\[cos\alpha + cos(\alpha + \beta) + cos(\alpha +2\beta) +...+cos(\alpha+ \overline{n-1}\beta)\]\[=\frac{cos( \alpha+(\frac{n-1}{2})\beta) sin(\frac{n\beta}{2})}{sin(\frac{\beta}{2})}\]


IMPORTANT RESULTS

\[1) \space sin\theta \space sin(60^{\circ}-\theta) \space sin(60^{\circ}+\theta) \space = \space \frac{1}{4}sin3\theta\]

\[2) \space cos\theta \space cos(60^{\circ}-\theta) \space cos(60^{\circ}+\theta) \space = \space \frac{1}{4}cos3\theta\]

\[3) \space tan\theta \space tan(60^{\circ}-\theta) \space tan(60^{\circ}+\theta) \space = \space tan3\theta\]

\[4) \space cot\theta \space cot(60^{\circ}-\theta) \space cot(60^{\circ}+\theta) \space = \space cot3\theta\]

\[5) \space (i) sin^2\theta \space + \space sin^2(60^{\circ}+\theta) \space + \space sin^2(60^{\circ}-\theta) \space = \frac{3}{2}\]

\[\space \space \space \space (ii) cos^2\theta \space + \space cos^2(60^{\circ}+\theta) \space + \space cos^2(60^{\circ}-\theta) \space = \frac{3}{2}\]

\[6) \space If \space tanA+tanB+tanC \space = \space tanAtanBtanC,\]\[then \space A+B+C \space = \space n\pi, \space n \in I\]

\[7) \space If \space tanAtanB+tanBtanC+tanCtanA \space = \space 1,\]\[\space \space then \space A+B+C \space = \space (2n+1)\frac{\pi}{2}, \space n \in I\]

\[8) \space cos\theta \space cos2\theta \space cos4\theta ..... cos(2^{n-1}\theta) \space = \space \frac{sin(2^n\theta)}{2^nsin\theta}\]

\[9) \space cotA \space - \space tanA \space = \space 2cot2A\]

Inverse Hyperbolic Trigonometric Identities

The hyperbolic sine function is a one-to-one function and thus has an inverse. As usual, the graph of the inverse hyperbolic sine function sinh-1(x) also denoted by arcsinh(x) by reflecting the graph of sinh(x) about the line y=x For all inverse hyperbolic functions but the inverse hyperbolic cotangent and the inverse hyperbolic cosecant, the domain of the real function is connected.

\[sinh^{-1}(x)\space =\space ln(x+\sqrt{x^{2}+1})\]

\[cosh^{-1}(x)\space =\space ln(x+\sqrt{x^{2}-1})\]

\[tanh^{-1}(x)\space =\space \frac{1}{2}\space ln(\frac{1+x}{1-x})\]

\[coth^{-1}(x)\space =\space \frac{1}{2}\space ln(\frac{x+1}{x-1})\]

\[sech^{-1}(x)\space =\space ln(\frac{1}{x}+\sqrt{\frac{1}{x^{2}}-1})\]

\[cosech^{-1}(x)\space =\space ln(\frac{1}{x}+\sqrt{\frac{1}{x^{2}}+1})\]

Differentiation Formulaes

\[\frac{d}{dx}sinh^{-1}(x)\space =\space \frac{1}{\sqrt{x^{2}+1}}\]

\[\frac{d}{dx}cosh^{-1}(x)\space =\space \frac{1}{\sqrt{x^{2}-1}}\]

\[\frac{d}{dx}tanh^{-1}(x)\space =\space \frac{1}{1+x^{2}}\]

\[\frac{d}{dx}coth^{-1}(x)\space =\space \frac{1}{1-x^{2}}\]

\[\frac{d}{dx}sech^{-1}(x)\space =\space \frac{1}{x\sqrt{1-x^{2}}}\]

\[\frac{d}{dx}cosech^{-1}(x)\space =\space \frac{1}{x\sqrt{1+x^{2}}}\]

Hyperbolic Trigonometric Identities

The hyperbolic functions are analogs of the circular function or the trigonometric functions. The hyperbolic function occurs in the solutions of linear differential equations, calculation of distance and angles in the hyperbolic geometry, Laplace’s equations in the cartesian coordinates. Generally, the hyperbolic function takes place in the real argument called the hyperbolic angle. The basic hyperbolic functions are:


  • Hyperbolic Sine(sinh)
  • Hyperbolic Cosine(cosh)
  • Hyperbolic Tangent(tanh)

From these three basic functions, the other functions such as hyperbolic cosecant (cosech), hyperbolic secant(sech) and hyperbolic cotangent (coth) functions are derived.

\[sinh(x)\space =\space \frac{e^{x}-e^{-x}}{2}\]

\[cosh(x)\space =\space \frac{e^{x}+e^{-x}}{2}\]

\[tanh(x)\space =\space \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\]

Properties of Hyperbolic Functions

\[sinh(-x)\space =\space -sinh(x)\]

\[cosh(-x)\space =\space cosh(x)\]

\[sinh(2x)\space =\space 2sinh(x)cosh(x)\]

\[cosh(2x)\space =\space cosh^{2}x + sinh^{2}x\]

\[sinh(x)\space =\space -isinh(ix)\]

\[cosh(2x)\space –\space sinh(2x)\space =\space 1\]

\[tanh(2x)\space +\space sech(2x)\space =\space 1\]

\[coth(2x)\space –\space cosech(2x)\space =\space 1\]

Inverse Trigonometric Identities



Negative Formulaes

\[ sin^{-1}(-x) = -sin^{-1}(x)\]

\[ cos^{-1}(-x) = π-cos^{-1}(x)\]

\[ tan^{-1}(-x) = -tan^{-1}(x)\]

\[ cot^{-1}(-x) = π-cot^{-1}(x)\]

\[ sec^{-1}(-x) = π-sec^{-1}(x)\]

\[ cosec^{-1}(-x) = -cosec^{-1}(x)\]

Addition of two functions

\[sin^{-1}x + cos^{-1}x = π/2 \space,\space x ∈ [-1, 1] \]

\[tan^{-1}x + cot^{-1}x = π/2 , x ∈ R\]

\[cosec^{-1}x + sec^{-1}x = π/2 \]

Reciprocal Formulaes

\[sin^{-1}\frac{1}{x} = cosec^{-1}x \]

\[cos^{-1}\frac{1}{x} = sec^{-1}x \]

\[tan^{-1}\frac{1}{x} = cot^{-1}x, x > 0\]

Identity Formulaes

\[sin(sin^{-1}x) = x, -1≤ x ≤1\]

\[cos(cos^{-1}x) = x, -1≤ x ≤1\]

\[tan(tan^{-1}x) = x, – ∞ < x < ∞ \]

\[cot(cot^{-1}x) = x, – ∞ < x < ∞ \]

\[sec(sec^{-1}x) = x, - ∞ < x ≤ 1\space or\space 1 ≤ x < ∞\]

\[cosec(cosec^{-1}x) = x, - ∞ < x ≤ 1\space or\space 1 ≤ x < ∞\]

Addition of Same functions

\[sin^{-1}x+sin^{-1}y = sin^{-1}(x\sqrt{1-y^2} + y\sqrt{1-x^2})\]

\[sin^{-1}x-sin^{-1}y = sin^{-1}(x\sqrt{1-y^2} - y\sqrt{1-x^2})\]

\[cos^{-1}x+cos{-1}y = cos^{-1}(xy-\sqrt{(1-x^2)(1-y^2)}\]

\[cos^{-1}x-cos{-1}y = cos^{-1}(xy+\sqrt{(1-x^2)(1-y^2)}\]

\[tan^{-1}x+tan^{-1}y = tan^{-1}\frac{x+y}{1-xy}\]

\[tan^{-1}x-tan^{-1}y = tan^{-1}\frac{x-y}{1+xy}\]

Simplified Inverse Trigonometric Functions

\[Equation\] \[Conditions\] \[Graph\]
\[y=sin^{-1}(\frac{2x}{1+x^2})\] \[\space \space \space \space \space \space \space \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space |x| \leqslant 1\] \[ \pi \space - \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x>1\]\[-(\pi + 2tan^{-1}x) \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space \space x<-1\]
\[y=cos^{-1}(\frac{1-x^2}{1+x^2})\] \[2tan^{-1}x \space \space \space \space \space \space \space \space \space if \space \space \space \space \space x \geqslant 0\] \[ -2tan^{-1}x \space \space \space \space \space \space if \space \space \space \space \space x<0\]
\[y=tan^{-1}(\frac{2x}{1-x^2})\] \[\space \space \space \space \space \space \space \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space |x|<1\] \[ \pi \space + \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x<-1\]\[-(\pi - 2tan^{-1}x) \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x>1\]
\[y=sin^{-1}(3x-4x^3)\] \[-(\pi+3sin^{-1}x) \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x \leqslant -\frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 3sin^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space - \frac{1}{2} \leqslant x \leqslant \frac{1}{2}\]\[\pi - 3sin^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \space \frac{1}{2} \leqslant x \leqslant 1\]
\[y=cos^{-1}(4x^3-3x)\] \[3cos^{-1}x-2\pi \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x \leqslant -\frac{1}{2}\]\[2\pi - 3cos^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space -\frac{1}{2} \leqslant x \leqslant \frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 3cos^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \frac{1}{2} \leqslant x \leqslant 1\]
\[y=sin^{-1}(2x \sqrt{1-x^2})\] \[-(\pi+2sin^{-1}x) \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x\leqslant -\frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 2sin^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space -\frac{1}{\sqrt{2}} \leqslant x \leqslant \frac{1}{\sqrt{2}}\]\[\pi - 2sin^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \frac{1}{\sqrt{2}} \leqslant x \leqslant 1\]
\[y=cos^{-1}(2x^2-1)\] \[\space \space \space \space 2cos^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \space 0 \leqslant x \leqslant 1\]\[2\pi-2cos^{-1}x \space \space \space \space \space \space if \space \space \space \space\space -1 \leqslant x \leqslant 0\]

Domain,Range and Graph of Inverse Functions

\[Function\] \[Domain\]

\[Range \,of\, an\]

\[ Inverse\, Function\]

\[Graphs\]

\[sin^{-1}x\]

\[(arcsine\,x)\]

\[-1≤ x ≤1\] \[-\frac{π}{2}≤y≤ \frac{π}{2}\]

\[cos^{-1}x\]

\[(arccosine\,x)\]

\[-1≤ x ≤1\] \[0≤y ≤π\]

\[tan^{-1}x\]

\[(arctangent\,x)\]

\[– ∞ < x <  ∞\] \[-\frac{π}{2}<y<\frac{π}{2}\]

\[cot^{-1}x\]

\[(arccotangent\,x)\]

\[– ∞ < x <  ∞\] \[0<y<π\]

\[sec^{-1}x\]

\[(arcsecant\,x)\]

\[– ∞ ≤ x ≤-1 \]

\[or\]

\[ 1≤x≤ ∞\]

\[0≤y≤\pi,\]

\[y \neq \frac{\pi}{2}\]

\[cosec^{-1}x\]

\[(arccosecant\,x)\]
\[– ∞ ≤ x ≤-1\, or\, 1≤x≤ ∞\] \[-\frac{π}{2}≤y≤\frac{π}{2},y \neq 0\]

Domain,Range,Period and Graph of Trigonometric Functions

\[Function\] \[Domain\]

\[Range \,and\,Period\,of\, a\]

\[ Trigonometric\, Function\]

\[Graphs\]

\[sin\,x\]

\[R\]

\[-1≤ y ≤1\]


\[2π\]

\[cos\,x\]

\[R\]

\[-1≤ y ≤1\]


\[2π\]

\[tan\,x\]

\[R\,-\,(\,2n+1\,)\,\frac{π}{2}\,,\,n\,\in\,I\]

\[– ∞ < y <  ∞\]


\[π\]

\[cot\,x\]

\[R\,-\,nπ\,,\,n\,\in\,I\]

\[– ∞ < y <  ∞\]


\[π\]

\[sec\,x\]

\[R\,-\,(\,2n+1\,)\,\frac{π}{2}\,,\,n\,\in\,I\]

\[– ∞ < y ≤-1\, or\, 1≤y < ∞\]


\[2π\,,\,sec^2x\,,\,|\,secx\,|\,\in\,[\,1\,,\, ∞\,)\]

\[cosec\,x\]

\[R\,-\,nπ\,,\,n\,\in\,I\]

\[– ∞ < y ≤-1\, or\, 1≤y< ∞\]


\[2π\,,\,cosec^2x\,,\,|\,cosecx\,|\,\in\,[\,1\,,\, ∞\,)\]

General Solution of Some Trigonometric Equations

\[1)\space If \space sin\theta=0, \space then \space \theta=n\pi, \space n \in I(set \space of \space integers)\]

\[2)\space If \space cos\theta=0, \space then \space \theta=(2n+1)\frac{\pi}{2}, \space n \in I\]

\[3)\space If \space tan\theta=0, \space then \space \theta=n\pi, \space n \in I\]

\[4)\space If \space sin\theta=sin\alpha, \space then \space \theta=n\pi +(-1)^n\alpha, \space where \space \alpha \in [\frac{-\pi}{2},\frac{\pi}{2}], \space n\in I\]

\[5)\space If \space cos\theta=cos\alpha, \space then \space \theta=2n\pi ± \alpha, \space n \in I \space \alpha \in [0,\pi]\]

\[6)\space If \space tan\theta=tan\alpha, \space then \space \theta=n\pi + \alpha, \space n \in I \space \alpha \in (\frac{-\pi}{2},\frac{\pi}{2})\]

\[7)\space If \space sin\theta=1, \space \theta=2n\pi+ \frac{\pi}{2}=(4n+1)\frac{\pi}{2}, \space n \in I\]

\[8)\space If \space cos\theta=1 \space then \space \theta=2n\pi, \space n \in I\]

\[9)\space If sin^2\theta \space =sin^2\alpha \space \space or \space cos^2\theta \space = cos^2\alpha \space \space or \space tan^2\theta \space = tan^2\alpha,\]\[then \space \theta=n\pi \space ± \space \alpha, \space n \in I\]

\[10)\space For \space n \in I, \space sinn\pi=0 \space and \space cosn\pi \space =(-1)^n, \space n \in I \]\[sin(n\pi + \theta) \space = (-1)^nsin\theta\]\[cos(n\pi+ \theta) \space =(-1)^ncos\theta\]

\[11)\space cos \space n\pi \space = (-1)^n, \space n \in I\]

\[12)\space If \space n \space is \space an \space odd \space integer \space then \space sin\frac{n\pi}{2}=(-1)^{\frac{n-1}{2}}, \space cos\frac{n\pi}{2}=0\]

\[13)\space sin(\frac{n\pi}{2}+\theta) \space = (-1)^{\frac{n-1}{2}} \space cos\theta, \space cos(\frac{n\pi}{2}+\theta) \space = (-1)^{\frac{n+1}{2}}sin\theta\]

Expansion of Trigonometric Functions

\[sine\space =\space x\space - \frac{x^3}{3!}\space + \frac{x^5}{5!} - \frac{x^7}{7!}.....\]

\[cosine\space =\space 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}.....\]

\[tan\space =\space x+\frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315}.....\]

\[cot\space = \space x^{-1} - \frac{1}{3x} -\frac{1}{45}x^3 - \frac{2}{945}x^5.....\]

\[sec\space = \space 1 + \frac{1}{2x^2} + \frac{5}{24}x^4 + \frac{61}{720}x^6......\]

\[cosec\space =\space x^{-1}+\frac{1}{6x+7}360x^3 + \frac{31}{15120}x^5.....\]

Expansion of Inverse Trigonometric Functions

\[sin^{-1}\space =\space x + \frac{1}{6}x^3 +\frac{3}{40}x^5 +\frac{5}{112}x^7.....\]

\[cos^{-1}\space =\space \frac{π}{2} - (expansion\space of\space sine)\]

\[tan^{-1}\space =\space x - \frac{1}{3}x^3 + \frac{1}{5}x^5.....\]

\[cot^{-1}\space =\space \frac{π}{2} - (expansion\space of\space tan)\]

The rest can be found out using Identities

Fill atleast any Two


Solution of Triangles


Law of Cosines Calculator

\[1) \space Sine\space rule : \frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}\]

\[2) \space Cosine\space rule : cosA = \frac{(b^2+c^2-a^2)}{2bc}\]

\[3) \space Projection\space Formula : a = b cos C + c cos B \]

\[4) \space Napier's\space Analogy : tan\frac{B-C}{2} = (\frac{b-c}{b+c}) cot\frac{A}{2}\]

\[5) \space Half\space Angle\space Formula: sin\frac{A}{2} = \sqrt\frac{(s-b)(s-c)}{bc}\]

\[cos\frac{A}{2} = \sqrt\frac{(s)(s-a)}{bc}\]

\[tan\frac{A}{2} = \sqrt\frac{(s-b)(s-c)}{(s)(s-a)}\]

\[6) \space Area\space of\space \triangle : \frac{1}{2}absinC = \frac{1}{2}bcsinA = \frac{1}{2}acsinB \]

\[8) \space m-n \space Theorem \space : \space If\space D\space be\space the\space point\space on\space the\space side\space BC\space of a triangle\space ABC\space which\space divides\space the\space side\space BC\space in\space the\space ratio\space m: n, \\If\space BD:DC = m:n, then\\ (m+n) cot θ = m cot α- n cot β = n cot B – m cot C\]

\[9) \space Radius\space of\space CircumCircle : R = \frac{a}{2sinA} = \frac{b}{2sinB} = \frac{c}{sinC} = \frac{abc}{4\triangle}\]

\[10) \space Radius\space of\space InCircle : r = \frac{\triangle}{s} \space = (s-a)tan\frac{A}{2} \space = 4R sin\frac{A}{2}sin\frac{B}{2}sin\frac{C}{2}\]

\[11) \space Radius\space of\space ExCircle : r_1 = \frac{\triangle}{s-a} \space = (s)tan\frac{A}{2} \space = 4R sin\frac{A}{2}cos\frac{B}{2}cos\frac{C}{2}\]

\[12) \space Orthocentre\space and\space Pedal\space triangle : \]

\[ The\space triangle\space formed\space by\space joining\space the\space feet\space of\space the\space altitudes\space is\space called\space the\space Pedal\space Triangle.\] \[a) \space The \space sides\space are\space a cos A = R sin 2A , \space a cos B = R sin 2B \space a cos C = R sin 2C\]

\[b) \space The \space distances \space of \space the \space orthocenter \space from \space the \space angular \space points \space of \space the \triangle ABC \space are \space \]\[2RcosA, \space 2RcosB, \space and \space 2RcosC.\]

\[c) \space The \space distance \space of \space orthocenter \space from \space sides \space are\]\[ 2RcosB \space cosC, \space 2RcosCcosA \space and 2RcosCcosA.\]

\[d) \space Area \space of \space Pedal \space Triangle \space = \space 2 \triangle cosAcosBcosC \space = \frac{1}{2}R^2sin2Asin2Bsin2C \]

\[e) \space Circumradii \space of \space pedal \space triangle \space = \space \frac{R}{2}\]


\[13) \space Important Results\]

\[a)\space Circumcentre (O) : O_A\space =\space R\space\space and\space\space O_A\space =\space R cos A\]

\[b)\space Orthocentre: H_A = 2R cos A and H_A = 2R cos B cos C\]

\[c)\space Excentre (I_1): I_1A\space =\space r_1\space cosec\frac{A}{2}\space=\,4Rcos \frac{A}{2}\]

\[d)\space Centroid(G) :G_A =\sqrt[3]{2b^2+2c^2-a^2}\]

\[e)\space Incentre(I) :I =r\,cosec\frac{A}{2}\]

\[f)\space Length\,of\,Angle\,Bisector\,from\,angle\,A\,=\frac{2bc\,cos\frac{A}{2}}{b+c}\]

\[g)\space Length\, of\, Median \,from \,angle \,A\, =\frac{1}{2}\sqrt{2b^2+2c^2-a^2}\]

\[h)\space Length \,of\, Altitude\, from \,Angle\, A\,=\,\frac{2\triangle}{a}\,=c\,sinB\]

\[i)\space Relation\, between\, area \,of\, Inscribed(A)\, and \,Excribed(A_1,A_2,A_3)\, Circle\, of \,\triangle ABC:\]

\[\frac{1}{\sqrt{A}}\,=\,\frac{1}{\sqrt{A_1}}\,+\,\frac{1}{\sqrt{A_2}}\,+\,\frac{1}{\sqrt{A_3}}\]


\[j)\space Relation \,between\, median \,and \,sides \,of\, \triangle ABC:\]

\[(Length\, of\, median\, from\, angle\, A)^2\,+\,(Length\, of\, median\, from\, angle\, B)^2\,+\]

\[(Length\, of\, median\, from\, angle\, C)^2\,=\,\frac{3}{4}(a^2+b^2+c^2)\]


\[k) \space Relation\,between\, angle\, of \triangle ABC,\, Inradius,\,Circumradius:\]

\[ cosA\,+\,cosB\,+\,cosC\,=\,1+\frac{r}{R}\]


\[l)\space Incentre \,and \,Centroid \,always\,lie\,inside\, the\, \triangle\]


\[m)\space Circumcentre\, lies\, inside\, acute\, angled\, \triangle,\,outside\, obtuse\, angled\, \triangle\]

\[and\, on \,hypotenuse\, for\, right\, angled\, \triangle\]


\[n)\space Orthocentre\, lies\, inside\, acute\, angled\, \triangle,\,outside\, obtuse\, angled\, \triangle\]

\[and\, at \,B\, for\, right\, angled\, \triangle ABC \, right \,angled \,at \,B\]

STANDARD DEVIATION

 

 

 

  • Select Population if data contains all the measurable values or all of the values you are interested in.


  • Select Sample if the data is sample of large or unlimited population an you wish to make a statement about entire population.

 

 

Methods of Integration



1. Integration by Substitution


Sometimes, it is really difficult to find the integration of a function, thus we can find the integration by introducing a new independent variable. This method is called Integration By Substitution.

The given form of integral function (say ∫f(x)) can be transformed into another by changing the independent variable x to t,

Substituting x = g(t) in the function ∫f(x), we get;

\[\frac{dx}{dt}\space =\space g^{'}t\]

\[Thus,\space I\space =\space \int f(x)\,dx\space =\space f(g(t))g^{'}t\,dt\]


2. Integration by Parts


Integration by parts requires a special technique for integration of a function, where the integrand function is the multiple of two or more function.
Let us consider an integrand function to be f(x).g(x).
Mathematically, integration by parts can be represented as;

\[\int f(x)g(x)\,dx\space =\space f(x)\int g(x)\,dx\space -\space \int f^{'}(x)\int g(x)\,dx\]

For deciding the first and the second functions, one can follow the ILATE rule for integration.

\[Inverse,Logarithmic,Arithmetic,Trigonometric,Exponential\space =\space ILATE\]


3. Integration by Partial Fractions


We know that a Rational Number can be expressed in the form of p/q, where p and q are integers and q≠0. Similarly, a rational function is defined as the ratio of two polynomials which can be expressed in the form of partial fractions: P(x)/Q(x), where Q(x)≠0.

There are in general two forms of partial fraction:

Proper partial fraction: When the degree of the numerator is less than the degree of the denominator, then the partial fraction is known as a proper partial fraction.

Improper partial fraction: When the degree of the numerator is greater than the degree of denominator then the partial fraction is known as an improper partial fraction. Thus, the fraction can be simplified into simpler partial fractions, that can be easily integrated.

Algebraic Equations Formulas

\[1)\space a^2-b^2\space =(a-b)(a+b)\]

\[2)\space (a+b)^2\space =a^2+2ab+b^2\]

\[3)\space a^2+b^2\space =(a+b)^2-2ab\]

\[4)\space (a-b)^2\space =a^2-2ab+b^2\]

\[5)\space (a+b+c)^2\space =a^2+b^2+c^2+2ab+2bc+2ca\]

\[6)\space (a-b-c)^2\space =a^2+b^2+c^2-2ab+2bc-2ca\]

\[7)\space (a+b)^3\space =a^3+b^3+3a^2b+3ab^2\]

\[8)\space (a-b)^3\space =a^3-b^3-3a^2b+3ab^2\]

\[9)\space a^3-b^3\space =(a-b)(a^2+ab+b^2)\]

\[10)\space a^3+b^3\space =(a+b)(a^2-ab+b^2)\]

\[11)\space a^3+b^3+c^3-3abc\space=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\]

\[if\space a+b+c=0 \space then \space a^3+b^3+c^3\space=\space3abc\]

\[12)\space (a+b)^4\space =a^4+4a^3b+6a^2b^2+4ab^3+b^4\]

\[13)\space (a-b)^4\space =a^4-4a^3b+6a^2b^2-4ab^3+b^4\]

\[14)\space a^4-b^4\space =(a-b)(a+b)(a^2+b^2)\]

\[15)\space a^5-b^5\space =(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)\]

\[16)\space (a+b+c+...)^2\space =a^2+b^2+c^2+....+2(ab+bc+ca+...)\]

\[17)\space Considering\space n\space as\space a\space natural\space number:\] \[a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b+…+ b^{n-2}a + b^{n-1})\]

\[18)\space Considering\space n\space as\space an\space even\space number:\] \[a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b+…+ b^{n-2}a - b^{n-1})\]

\[18)\space Considering\space n\space as\space an\space odd\space number:\] \[a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b+ a^{n-3}b^2…- b^{n-2}a + b^{n-1})\]

\[19)\space Exponential\space laws:\] \[(a^m)(a^n) = a^{m+n}\] \[(ab)^m = a^mb^m \] \[(a^m)^n = a^{mn} \] \[{a^m\over a^n} = a^{m-n} \] \[a^m = {1\over a^{-m}} \] \[a^{-m} = {1\over a^m} \]

Location of Roots

\[Let\space f(x) = ax^2+bx+c = 0 where\space a>0\space then,\]


1.If both the roots are positive i.e. they lie in (0, ¥), then the sum of the roots as well as the product of the roots must be positive.

\[ ie. \space a+b = \frac{-b}{a} > 0 \space and\space ab=\frac{c}{a}>0\]

2.Similarly, if both the roots are negative i.e. they lie in (–¥, 0) then the sum of the roots will be negative and the product of the roots must be positive.

\[ie.\space a+b = \frac{-b}{a} < 0 \space and\space ab >0\]

3.Both the roots are greater than a given number k if the following three conditions are satisfied,

\[D>=0\space \frac{-b}{2a} > k\space and\space a.f(k)>0\]



4.Both the roots will lie in the given interval (k1, k2) if the following conditions are satisfied

\[D>=0\space k_1 < \frac{-b}{2a} < k_2\space and\space a.f(k_1)\space and\space a.f(k_2) are >0\]



5.Exactly one of the roots lies in the given interval (k1, k2) if,

\[f(k_1).f(k_2)\space >\space 0\]

6.A given number k will lie between the roots if a.f(k) < 0.

In particular, the roots of the equation will be of opposite signs if 0 lies between the roots a.f(0) < 0.

\[f(x)=ax^2+bx+c\]

\[a=\]

\[b=\]

\[c=\]

Nature of the roots of the Quadratic Equation ax2 + bx + c, a ≠ 0, a, b, c ∈ R

To find out the nature of the roots of a quadratic equation,

  1. We find the value of the discriminant D. If D < 0, we immediately say that equation has no solution in R.

  2. If D ≥ 0, then we examine whether it is a perfect square of a rational number or not.
    1. If D > 0 and D is not a perfect square, then roots are real and distinct.
    2. If D > 0 and it is a perfect square of a rational number and a, b, c ∈ Q, then the roots are real and rational. Roots are also distinct.

  3. If D = 0 then the roots are real and equal and if a, b, c ∈ Q the roots are equal rational numbers.

Pythagorean Triplets





Greatest Integer Function and Fractional Part



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Instructions :

Seperate Each data with ',' for both the axis
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for Y axis enter ----> a,b,c,d :
for Y axis enter ----> 10,5,6,4 :









Properties of Parallel Lines



What are parallel lines?

Parallel lines are equidistant lines (lines having equal distance from each other) that will never meet.




Conditions for Lines to be parallel

  1. the pair of alternate angles is equal, then two straight lines are parallel to each other.




  2. the pair of interior angles are on the same side of traversals is supplementary, then the two straight lines are parallel.




  3. the pair of corresponding angles is equal, then the two straight lines are parallel to each other.




  4. Two lines cut by a transversal line are parallel when the sum of the consecutive exterior angles is 180 °




  5. The sum of angles together on a line is 180 °

Coordinate Systems



1. Cartesian Coordinate System

It uses the concept of mutually perpendicular lines to denote the coordinate of a point. To locate the position of a point in a plane using two perpendicular lines, we use the cartesian coordinate system. Points are represented in the form of coordinates (x, y) in two-dimension with respect to x- and y- axes. The x-coordinate of a point is its perpendicular distance from the y-axis measured along the x-axis and it is known as Abscissa. The y-coordinate of a point is its perpendicular distance from the x-axis measured along the y-axis and it is known as Ordinate.

2. Polar Coordinate System

Here, a point is chosen as the pole and a ray from this point is taken as the polar axis. Basically, we have two parameters namely angle and radius. The angle Ɵ with the polar axis has a single line through the pole measured anti-clockwise from the axis to the line.


The point will have a unique distance from the origin (r). Thus, a point in Polar coordinate system is represented as a pair of coordinates (r, Ɵ). The pole is represented by (0, Ɵ) for any value of Ɵ, where r = 0.


The distance from the pole is called the radial coordinate, radial distance or simply radius and the angular coordinate, polar angle or azimuth.


Consider the figure below that depicts the relationship between polar and cartesian coordinates.
X = r cos Ɵ and y = r sin Ɵ
\[r =\space \sqrt{x^2 + y^2}\space and\space tan Ɵ\space =\space \frac{y}{x}\]


3. Spherical Coordinate System

The coordinate ρ is the distance from P to the origin. If the point Q is the projection of P to the xy-plane, then θ is the angle between the positive x-axis and the line segment from the origin to Q. Lastly, ϕ is the angle between the positive z-axis and the line segment from the origin to P.

x = ρsinϕcosθ
y = ρsinϕsinθ
z = ρcosϕ


4. Cylindrical Coordinate System

x=rcosθ
y=rsinθ
z=z

Its just Polar coordinates with z as height forming a cylinder

Important Graph Calculators



FInding between chord and a tangent



Ceva and Thales Theorem



Ceva's Theorem


Ceva’s theorem is a theorem regarding triangles in Euclidean Plane Geometry.

Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.


Ceva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O, to meet opposite sides at D, E and F respectively.

\[{\frac {AF}{FB}}\cdot {\frac {BD}{DC}}\cdot {\frac {CE}{EA}}=1\]

Another form of the theorem is that three concurrent lines from the polygon vertices of a triangle divide the opposite sides in such fashion that the product of three nonadjacent segments equals the product of the other three


Thales' Theorem


Basic Proportionality theorem was introduced by a famous Greek Mathematician, Thales, hence it is also called Thales Theorem. According to him, for any two equiangular triangles, the ratio of any two corresponding sides is always the same. Based on this concept, he gave theorem of basic proportionality (BPT). This concept has been introduced in similar triangles. If two triangles are similar to each other then,

1. Corresponding angles of both the triangles are equal

2. Corresponding sides of both the triangles are in proportion to each other


Thales Theorem Statement

If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio.

\[\frac{AP}{PB}\space =\space \frac{AQ}{QC}\]

The MidPoint theorem is a special case of the basic proportionality theorem. According to mid-point theorem, a line drawn joining the midpoints of the two sides of a triangle is parallel to the third side.


Converse

According to this theorem, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Properties of Circles

  1. The circles are said to be congruent if they have equal radii


  2. The diameter of a circle is the longest chord of a circle


  3. Equal chords subtend equal angles at the circumference


  4. The radius drawn perpendicular to the chord bisects the chord


  5. Circles having different radius are similar


  6. A circle can circumscribe a rectangle, trapezium, triangle, square, kite


  7. A circle can be inscribed inside a square, triangle and kite


  8. The chords that are equidistant from the centre are equal in length


  9. The distance from the centre of the circle to the longest chord (diameter) is zero


  10. The perpendicular distance from the centre of the circle decreases when the length of the chord increases


  11. If the tangents are drawn at the end of the diameter, they are parallel to each other


  12. An isosceles triangle is formed when the radii joining the ends of a chord to the centre of a circle

Properties of Quadrilaterals


  • Every quadrilateral has 4 vertices, 4 angles, and 4 sides
  • The total of its interior angles = 360 degrees

  • Different types of Quadrilaterals Are:

    Parallelogram
    Rectangle
    Square
    Rhombus
    Kite
    Trapezium
    Summary Of Quadrilaterals
    \[Property\] \[Parallelogram\] \[Rectangle\] \[Square\] \[Rhombus\] \[Kite\] \[Trapezium\]
    Sides All sides are equal ✔️ ✔️
    Opposite sides are equal ✔️ ✔️ ✔️ ✔️
    Opposite sides are parallel ✔️ ✔️ ✔️ ✔️ ✔️
    Angles All angles are equal ✔️ ✔️
    Opposite angles are equal ✔️ ✔️ ✔️ ✔️ ❌(only 1 pair equal)
    Sum of two adjacent angles is 180 ✔️ ✔️ ✔️ ✔️
    Diagonals Bisect each other ✔️ ✔️ ✔️ ✔️ ❌(only 1 diagonal bisects)
    Bisect perpendicularly ✔️ ✔️
    Intersect Perpendicularly ✔️ ✔️ ✔️
    Formulas Area l x h l x b a2 1⁄ 2 x d1 x d2 1⁄ 2 x d1 x d2 1⁄ 2 x Sum of parallel sides x h
    Perimeter 2 x (l + b) 2 x (l + b) 4a 4a 2 x (l + b) Sum of length of sides

    Curve Tracing

    For Cartesian Curves

    \[1.\space Symmetry\space - \\ If\space even\space powers\space of\space x,\space then\space symmetrical\space about\space y\space axis \\ If\space even\space powers\space of\space y,\space then\space symmetrical\space about\space x\space axis \\ If\space replacing\space y\space by\space x\space no\space change\space in\space curve,\space symmetrical\space about\space y=x\]


    \[2.\space Origin\space - \\ If\space curve\space has\space no\space constant\space term,\space Equate\space to\space 0\space the\space lowest\space degree\space term\space to\space get\space equation\space of\space tangent\space at\space origin\]


    \[3.\space Asymptotes\space - \\ For\space Asymptotes\space parallel\space to\space x\space axis,\space equate\space highest\space degree's\space coefficient\space of \space x\space to\space 0 \\ For\space Asymptotes\space parallel\space to\space y\space axis,\space equate\space highest\space degree's\space coefficient\space of \space y\space to\space 0 \]


    \[4.\space Region\space - \\ Using\space logic\space find\space region\space in\space which\space the\space curve\space will\space lie \\ Use\space wavy\space curve\space method\space to\space find\space the\space region(best\space way)\]


    \[5.\space Point\space of\space Intersection\space - \\ Put\space certain\space values\space of\space x\space to\space get\space corresponding\space values\space of\space y\space and\space plot\space them\space to\space get\space final\space answer\]

    For Polar Curves

    \[1.\space Symmetry\space -\\ To\space check\space for\space any\space angle\space \theta\space for\space symmetry\space check\space for\space (2\alpha - \theta),\space If\space curve\space remains\space same\space then\space symmetrical\]


    \[2.\space Pole\space -\\ Put\space r=0\space to\space get\space equation\space of\space tangent\space at\space pole\]


    \[3.\space Asymptotes\space -\\ Put\space r = \infty\space to\space get\space Asymptotes\]


    \[4.\space Region\space -\\ Put\space extreme\space values\space of\space cosine/sine\space to\space get\space region\space in\space which\space curve\space lie\]


    \[5.\space Point\space of\space Intersection\space \\ Put\space values\space of\space \theta\space to\space get\space specific\space values\space of\space r,\space then\space plot\]

    Kindly press the button twice for re-entering the angle

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    Congruence Properties of Triangles

    1. SSS Property (Side-Side-Side) - When all the three corresponding sides of 2 triangles are of equal size then triangles are congruent




    2. SAS Property (Side-Angle-Side) - If any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by SAS rule.




    3. ASA Property (Angle-Side-Angle) - If any two angles and the side included between the angles of one triangle are equivalent to the corresponding two angles and side included between the angles of the second triangle, then the two triangles are said to be congruent by ASA rule.




    4. AAS Property (Angle-Angle-Side) - If any two angles and a non included side of one triangle are equivalent to the corresponding two angles and the non included side of the second triangle, then the two triangles are said to be congruent by AAS rule.




    5. RHS Property (Right-angle - Hypotenuse - Side) - If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two right triangles are said to be congruent by RHS rule.






    Similarity of Triangles


    1. AA(or AAA) Similarity - If any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other.




    2. SAS Similarity - If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar.




    3. SSS Similarity - If all the three sides of a triangle are in proportion to the three sides of another triangle, then the two triangles are similar.






    Fundamental Theorem of Proportionality


    If a line parallel to one of the sides of a triangle intersects the other two sides in distinct points, then the segments of the other two sides in one halfplane are proportional to the segments in the other halfplane.






    Using the fundamental theorem on proportionality, we can obtain some results:

    1. If a line parallel to one side of a triangle intersects the other two sides of the triangle in distinct points, the segments of the other sides of the triangle in the same halfplane of the line are proportional to the corresponding sides of the triangle.

    2. If three (or more than three) parallel lines are intercepted by two transversals, the segments cut off on the transversals between the same parallel lines are proportional.

    3. In a triangle the bisector of an angle divides the side opposite to the angle in the segments whose lengths are in the ratio of their corresponding sides.





    Converse of the fundamental theorem of proportionality


    If a line intersects two sides of a triangle such that segments of the sides in each half-plane are proportional then the line is parallel to the third side.



    0

    0

    Diffrent Centers of Triangel


    Using Edges

    X +

    Y =

    X +

    Y =

    X +

    Y =


    Using Vertex

    Centroid Calculator

    Coordinate 1 = ( , )
    Coordinate 2 = ( , )
    Coordinate 3 = ( , )




    Circumcentre Calculator


    Coordinate 1 = ( , )

    Coordinate 2 = ( , )

    Coordinate 3 = ( , )

    A = B = C=


    Incentre Calculator


    Coordinate 1 = ( , )

    Coordinate 2 = ( , )

    Coordinate 3 = ( , )

    a = b = c=


    Excentre Calculator


    Coordinate 1 = ( , )

    Coordinate 2 = ( , )

    Coordinate 3 = ( , )

    a = b = c=


    Area of Triangle


    Coordinate 1 = ( , )

    Coordinate 2 = ( , )

    Coordinate 3 = ( , )


    EFFECTIVE INTEREST RATE CALCULATOR

     

     

     

     

     

    The effective annual interest rate is the real return on a savings account or any interest-paying investment when the effects of compounding over time are taken into account.
    It also reveals the real percentage rate owed in interest on a loan, a credit card, or any other debt.
    It is also called the effective interest rate, the effective rate, or the annual equivalent rate.
    First Set
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    Result

    SPI/CGPA Convertor




    1. Percentage(%) = (SPI/CGPA - 0.5 ) *10
    2. SPI/CGPA = ( Percentage(%) / 10 ) + 0.5
    3. SPI/CGPA scaled out of 10.



    \[x^n \space = \space y\]







    Percentage Calculator


    1) What is % of ?



    2) What % of is ?



    3) out of what is % ?



    4) plus % is what?



    5) minus % is what?






    1) Sum of N terms of an Arithmetic Progression


    If first term, number of terms and common difference is given








    If first term,number of terms and last term is given








    If number of terms and the AP is given








    2) Sum of N terms of a Geometric Progression




    Sum of infinite terms of a Geometric Progression with common ratio less than 1






    3) Sum of N terms of a Harmonic Progression



    Harmonic Progression's Nth term

     

     

     

     

    A Harmonic Progression (HP) is defined as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression
    that does not contain 0. In harmonic progression, any term in the sequence is considered as the
    harmonic means of its two neighbours. For example, the sequence a, b, c, d, …is considered as an arithmetic progression;
    the harmonic progression can be written as 1/a, 1/b, 1/c, 1/d, …

    Cost and Selling Prices

    Cost Price: The price, at which an article is purchased, is called its cost price, abbreviated as C.P. Selling Price: The price, at which an article is sold, is called its selling prices, abbreviated as S.P. Profit or Gain: If S.P. is greater than C.P., the seller is said to have a profit or gain. Loss: If S.P. is less than C.P., the seller is said to have incurred a loss.

    IMPORTANT FORMULAE

    1) Selling Price: (S.P.): SP = [(100 + Gain %) x C.P] / 100
    2) Selling Price: (S.P.): Sp = [(100 - Loss %) x C.P.] / 100
    3) Cost Price: (C.P.): C.P. = (100 x S.P.)(100 + Gain %)
    4) Cost Price: (C.P.): C.P. = (100 x S.P.)(100 - Loss %)

    Enter Profit/Loss percent [if profit then +ve, if loss -ve]
    Enter Cost Price
    Enter Selling Price

    Arithmetic Mean







    Geometric Mean








    Harmonic Mean





    Convergence and Divergence of Series

    \[1.\space If\space \lim_{n\to\infty}a_n\space is\space finite\space and\space unique\space,\space the\space sequence\space is\space said\space to\space be\space convergent\]

    \[2.\space If\space \lim_{n\to\infty}a_n\space is\space =\space +/-\infty,\space then\space the\space sequence\space is\space said\space to\space be\space divergent \]

    \[3.\space If\space \lim_{n\to\infty}a_n\space is\space not\space unique,\space then\space the\space sequence\space is\space said\space to\space be\space oscillatory\]

    D'Alembert's Ratio Test

    \[Let\space u_k\space be\space a\space series\space with\space positive\space terms\space and\space suppose \\ \rho=\lim_{k\to\infty}\frac{u_{k+1}}{u_k}.\\ \\ Then,\\ 1.\space If\space \rho<1,\space the\space series\space converges.\\ 2.\space If\space \rho>1\space or\space \rho=\infty,\space the\space series\space diverges.\\ 3.\space If\space \rho=1,\space the\space series\space may\space converge\space or\space diverge.\]

    Rabbe's Test

    \[Given\space a\space series\space of\space positive\space terms\space u_i\space and\space a\space sequence\space of\space positive\space constants\space {a_i},\\ \rho = \lim_{n\to\infty}n(\frac{u_n}{u_{n+1}}-1)\\ 1. If\space \rho>1,\space the\space series\space converges.\\ 2. If\space \rho<1,\space the\space series\space diverges.>\\ 3. If\space \rho=1,\space the\space series\space may\space converge\space or\space diverge.\]

    Comparison Test

    \[Let\space b[n]\space be\space a\space second\space series.\space Require\space that\space all\space a[n]\space and\space b[n]\space are\space positive.\space If b[n] converges,\space and\space a[n]<=b[n]\\ \space for\space all\space n,\space then\space a[n]\space also\space converges.\space If\space the\space sum\space of\space b[n]\space diverges,\space and\space a[n]>=b[n]\space for\space all\space n, \\ \space then\space the\space sum\space of\space a[n]\space also\space diverges.\]

    Root's Test

    \[If\space \lim_{n\to\infty}{\sqrt{n}a_n}\ < 1\space then\space convergent\space else\space divergent\]



    Permutations,  n P r   =  

    Combinations,  n C r   =  

    Permutations

    An arrangement in sequence of elements of a set is called a permutation of the elements.

    Type I

    Let 0 ≤ r ≤ n.

    The number of ways to have an ordered sequence of n distinct elements, taken r at a time is called as an r-permutation of n elements and is denoted by P(n,r) or n P r .

    The first place in the sequence can be filled up in n-ways, the second place in (n - 1) ways and proceeding in this manner the r th place can be filled up in (n - r + 1) ways.

    Hence, P(n,r) or n P r = n (n - 1) (n - 2) ... (n - r + 1)

    or

    \[nP_r = \, \frac{n!}{(n-r)!}\]

    Type II

    A general formula for the number of ways to place r coloured balls in n number boxes, where m 1 of these are of one colour , m2 of them are of a second colour .... and m r of them are of a rth colour.

    Here the placement of the r balls is not changed by rearranging the m 1 balls of the same colour among the boxes in which they are placed or rearranging the m 2 balls of the same colour among the boxes in which they are placed, ...

    On the other hand, if the r balls were distinctly coloured, any arrangement will yield a different placement.

    It follows that each way to place the r not completely distinctly coloured balls corresponds to m 1 ! m 2 ! ... m r ! ways to place r distinctly coloured balls. Since there are P(n,r) ways to place r distinctly coloured balls in n numbered boxes, the total number of ways to place r coloured balls in n numbered boxes, where

    m 1 of these balls are of one colour,

    m 2 of these balls are of second colour,

    .

    .

    .

    m r of these balls are of a r th colour, is \[\frac{P(n,r)}{m_{1}! m_{2}! ... m_r!}\]

    i.e.

    \[Number\space of\space ways = \, \frac{nP_r}{m_{1}! m_{2}! ... m_r!}\]

    Type III

    The number of permutations of n elements, r at a time, when each element may be repeated once, twice, .... upto r times in any arrangement.

    In this case, the first place may be filled up in n ways, the second place may also be filled up in n ways, and so on .... Proceeding in this manner, the number of ways in which the r places can be filled up is n r.



    Combinations

    Selection of a set of r elements from a set of n distinct elements is called a combination.

    Let 0 ≤ r ≤ n.

    Consider a problem of choosing 5 mangoes from 12 identical mangoes. We want to know the number of ways in which 5 mangoes can be chosen out of 12 mangoes.

    If all the mangoes are identical then according to the rule of permutation, the number of ways = \[\frac{P(12,5)}{5!}\]

    In general the number of ways of choosing r mangoes out of n identical magoes is \[= \frac{n\space (n - 1)\space ...\space (n - r + 1)}{r!}\]



    \[= \frac{n!}{r!\space (n - r)!}\]

    The above quantity is denoted by C(n,r) or n C r [out of n choose r].

    Hence \[nC_r = \frac{n!}{r!\space (n - r)!}\]



    Relation between n P r and n C r

    \[\frac{nP_r}{r!} = nC_r\] or \[nP_r = r!.(nC_r) \]

    HYDROSTATIC PRESSURE CALCULATOR

     

     

    Z SCORE CALCULATOR

     

     

     

     

    The z-score, also referred to as standard score, z-value, and normal score, among other things, is a dimensionless quantity that is used to indicate the signed, fractional, number of standard deviations by which an event is above the mean value being measured.
    Values above the mean have positive z-scores, while values below the mean have negative z-scores.
    The z-score can be calculated by subtracting the population mean from the raw score, or data point in question (a test score, height, age, etc.), then dividing the difference by the population standard deviation

    Supplementary Angle Calculator

     

     

     

    Verify if two angles are supplementary

     

     

     

    FACTORIZATION CALCULATOR

     

     

    ERROR PERCNTAGE CALCULATOR

     

     

     

     

     

    Percentage error is a measurement of the discrepancy between an observed and a true, or accepted value.
    When measuring data, the result often varies from the true value.
    Error can arise due to many different reasons that are often related to human error, but can also be due to estimations and limitations of devices used in measurement.
    Regardless, in cases such as these, it can be valuable to calculate the percentage error.
    The computation of percentage error involves the use of the absolute error, which is simply the difference between the observed and the true value. The absolute error is then divided by the true value, resulting in the relative error, which is multiplied by 100 to obtain the percentage error.

    SSS Triangle's Angle Calculator

     

     

     

     

     

    "SSS" is when we know three sides of the triangle, and want to find the missing angles.
    To solve an SSS triangle:
    Use The Law of Cosines first to calculate one of the angles
    Use The Law of Cosines again to find another angle
    Finally use angles of a triangle add to 180° to find the last ang

    IMPROPER TO MIXED FRACTION CONVERTER

     

     

    Sum of Square Calculator

     

     

     

    Formula for Sum of Square :

    For two numbers : x2 + y2 = (x + y)2– 2ab ; x and y are real numbers

     

    For three numbers : x2 + y2+z2 = (x+y+z)2-2xy-2yz-2xz ; x,y and z are real numbers

     

    For n Natural Numbers : Σn2 = [n(n+1)(2n+1)]/6]

     

    For first n Even Numbers : Σ(2n)2 =[2n(n+1)(2n+1)]/3

     

    For first n Odd Numbers : Σ(2n-1)2 = [n(2n+1)(2n-1)]/3

    Number of Perfect Cubes in a Range

     

     

     

     

    Sum of Squares of N Natural Numbers

     

     

     

    Formula for Sum of Squares of N Natural numbers :

    Σn2 = [n(n+1)(2n+1)]/6]

     

    Sum of Cubes of N Natural Numbers

     

     

     

    Formula for Sum of Cubes of N Natural numbers :

    Σn2 = [n(n+1)/2]^2

     

    Squares in a Range

     

     

     


    Cubes in a Range

     

     

     

    Area of Segment Calculator

     

     

     

     

    A segment of a circle can be defined as a region bounded by a chord and a corresponding arc lying between the chord’s endpoints.
    Area of a Segment in Radians: A = (½) × r x r x (θ – Sin θ)
    Area of a Segment in Degrees: A = (½) × r x r x [(π/180) θ – sin θ]

    ARC LENGTH CALCULATOR

     

     

     

    The length of an arc depends on the radius of a circle and the central angle θ.
    L / θ = C / 2π
    As circumference C = 2πr,
    L / θ = 2πr / 2π
    L / θ = r
    We find out the arc length formula when multiplying this equation by θ:
    L = r * θ

    Sum of Square Calculator

     

     

     

    Formula for Sum of Square :

    For two numbers : x2 + y2 = (x + y)2– 2ab ; x and y are real numbers

     

    For three numbers : x2 + y2+z2 = (x+y+z)2-2xy-2yz-2xz ; x,y and z are real numbers

     

    For n Natural Numbers : Σn2 = [n(n+1)(2n+1)]/6]

     

    For first n Even Numbers : Σ(2n)2 =[2n(n+1)(2n+1)]/3

     

    For first n Odd Numbers : Σ(2n-1)2 = [n(2n+1)(2n-1)]/3

    Area of Segment Calculator

     

     

     

     

    A segment of a circle can be defined as a region bounded by a chord and a corresponding arc lying between the chord’s endpoints.
    Area of a Segment in Radians: A = (½) × r x r x (θ – Sin θ)
    Area of a Segment in Degrees: A = (½) × r x r x [(π/180) θ – sin θ]

    ARC LENGTH CALCULATOR

     

     

     

    The length of an arc depends on the radius of a circle and the central angle θ.
    L / θ = C / 2π
    As circumference C = 2πr,
    L / θ = 2πr / 2π
    L / θ = r
    We find out the arc length formula when multiplying this equation by θ:
    L = r * θ

    STRAIN CALCULATOR

     

     

    PMF of Binomial Distribution Calculator

     

     

     

    \[P.M.F = \, \frac{n!}{(n-r)!r!} p^x (1-p)^y\]

    where,
    n= no. of trials
    r= no. of successful events
    x= probability of success
    y= porobability of not being successful(1-x)

    PMF of Poisson Distribution Calculator

     

     

     

     

     

    The Poisson distribution is a discrete probability distribution which results from the Poisson experiment.
    It classifies the experiment into two different categories, such as success and failure.
    Generally, the Poisson random variable “x” defines the probability of the success of the experiment.
    From the average rate of success, the Poisson distribution probability can be easily calculated.

    STRESS CALCULATOR

     

     

     

    Boolean Algebra


    Laws of Boolean Algebra


    \[1.\,Annulment \space\, Law\\\,(\,i\,)\,A \centerdot 0\,=\,0\\\,(\,ii\,)\,A\,+\,1\,=\,1\]

    \[2.\,Identity \space\, Law\\\,(\,i\,)\,A\,+\,0\,=\,A\\\,(\,ii\,)\,A \centerdot 1\,=\,A\]

    \[3.\,Idempotent \space\, Law\\\,(\,i\,)\,A\,+\,A\,=\,A\\\,(\,ii\,)\,A \centerdot A\,=\,A\]


    \[4.\,Complement \space\, Law\\\,(\,i\,)\,A \centerdot A\,'\,=\,0\\\,(\,ii\,)\,A\,+\,A\,'\,=1\]

    \[5.\,Demorgan's \space\,Law\\ \,(\,i\,)\, ( \,A\,+\,B\,) \,'\,=\,A\,' \centerdot B\,'\\\,(\,ii\,)\, (\,A \centerdot B \,)\,'\,= \, A\,'\,+\,B\,'\]

    \[6.\,Distributive \space \, Law \\\,(\,i\,)\, A \, (\,B\,+\,C\,)\, = \,A \centerdot B \,+\,A \centerdot C \\\,(\,ii\,)\,A \,+ \,B \centerdot C\,=\,(\,A\,+\,B\,)\centerdot (\,A\,+\,C\,)\]


    \[7.\,Absorption \,/\,Redundancy \space \, Law\\\,(\,i\,)\, A \, +\,A \centerdot B\,= \,A\\\,(\,ii\,)\,A\,(\,A\,+\,B\,)\, = \,A\]

    \[8.\,Commutative \space Law \\\,(\,i\,)\, A \,+\,B\,=\,B\,+\,A\\\,(\,ii\,)\, A \centerdot B\,=\,B \centerdot A\]

    \[9.\,Associative \space Law \\ \,(\,i\,)\,A \,+\,(\,B\,+\,C\,)\,=\,(\,A\,+\,B\,)\,+\,C \\\,(\,ii\,)\, A \centerdot (\, B \centerdot C \,)\, = \, (\, A \centerdot B \,) \centerdot C\]


    \[10.\,Double \space Negation \space Law \\\,(\,i\,)\, A \,'\,'\, = \, A \\\,(\,ii\,)\, B\,'\,'\, = \, B\]

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    1st Input

    Choose Number System

    2nd Input

    Choose Number System

    Result Of Addition

    Choose Number System
    Choose Number System



    Input Of Minuend



    Input Of Subtrahend



    Result Of Subtraction

    1st Input

    Choose Number System

    2nd Input

    Choose Number System

    Result Of Addition

    Choose Number System
    Choose your numbers system
    First number
    Operation
    Second number
    Result

    One's and Two's Complement


    Enter Binary number




    Seven's and Eight's Complement


    Enter Octal number



    Nine's and Ten's Complement


    Enter Decimal number



    Fifteen's Sixteen's Complement


    Enter Hexa Decimal number

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    Residue at Pole



    Let z=a be a pole of order m of a one valued section f(z) and gamma be any circle of radius r with centre at z=a which doesnt have any singularities except at z=a, then f(z) is analytic within the annulus

    Hence it can be written in form of Laurent Series as

    \[f(z)\space =\space \sum_{n=0}^{\infty} a_n (z-a)^{n}\space +\space \sum_{n=0}^{\infty} b_n (z-a)^{-n}\]

    \[a_n \space =\space \frac{1}{2\pi i} \int \frac{f(z)}{(z-a)^{n+1}}\,dz\]

    \[b_n\space =\space \frac{1}{2\pi i} \int \frac{f(z)}{(z-a)^{-n+1}}\,dz\]

    \[ Particularly,\space b_1 \space =\space \frac{1}{2\pi i} \int f(z)\,dz\]

    The coefficient of b1 is called residue of f(z) at z=a


    Cauchy Residue Theorem


    In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem.

    Let U be a simply connected open subset of the complex plane containing a finite list of points a1, ..., an, U0 = U \ {a1, ..., an}, and a function f defined and holomorphic on U0. Let γ be a closed rectifiable curve in U0, and denote the winding number of γ around ak by I(γ, ak). The line integral of f around γ is equal to 2πi times the sum of residues of f at the points, each counted as many times as γ winds around the point:

    \[{\displaystyle \oint _{\gamma }f(z)\,dz=2\pi i\sum _{k=1}^{n}\operatorname {I} (\gamma ,a_{k})\operatorname {Res} (f,a_{k}).}\]

    If γ is a positively oriented simple closed curve, I(γ, ak) = 1 if ak is in the interior of γ, and 0 if not, therefore

    \[{\displaystyle \oint _{\gamma }f(z)\,dz=2\pi i\sum \operatorname {Res} (f,a_{k})}\]

    The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve γ must first be reduced to a set of simple closed curves {γi} whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of f dz along a Jordan curve γi with interior V. The requirement that f be holomorphic on U0 = U \ {ak} is equivalent to the statement that the exterior derivative d(f dz) = 0 on U0. Thus if two planar regions V and W of U enclose the same subset {aj} of {ak}, the regions V \ W and W \ V lie entirely in U0, and hence

    \[{\displaystyle \int _{V\smallsetminus W}d(f\,dz)-\int _{W\smallsetminus V}d(f\,dz)}\]

    Milne Thomson Method



    With the help of this method, we can directly construct f(z) in terms of z without finding v when u is given or u when v is given

    \[f(z)\space =\space \int {\phi _1(z,0)-i\phi _2(z,0)}\,dz\space +\space C\]

    Case -1 : WHen only real part of u(x,y) is given

    \[1.\space Find\space \frac{\delta u}{\delta x}\]

    \[2.\space Write\space it\space as\space equal\space to\space \phi _1 (x,y)\]

    \[3.\space Find\space \frac{\delta u}{\delta y}\]

    f(z) is obtained by the formula -

    \[f(z)\space =\space \int {\phi _1(z,0)-i\phi _2(z,0)}\,dz\space +\space C\]


    Case -2 : When only imaginary part of v(x,y) is given

    \[1.\space Find \frac{\delta v}{\delta y}\]

    \[2.\space Write it equal to \psi _1(x,y)\]

    \[3.\space Find \frac{\delta v}{\delta x}\space and\space equate\space to\space \psi _2(x,y)\]


    Case -3 : When u-v is given

    \[1.\space f(z)\space =\space u+iv\]

    \[2.\space if(z)\space =\space iu-v\]

    \[3.\space F(z)\space =\space f(z)+if(z)\]

    \[4.\space Find\space \frac{\delta U}{\delta x}\space and\space \frac{\delta U}{\delta y}\]

    \[f(z)\space =\space \int {\phi _1(z,0)-i\phi _2(z,0)}\,dz\space +\space C\]


    Case -4 : When u+v is given

    \[1.\space f(z)\space =\space u+iv\]

    \[2.\space if(z)\space =\space iu-v\]

    \[3.\space F(z)\space =\space f(z)+if(z)\]

    \[4.\space Find\space \frac{\delta V}{\delta x}\space and\space \frac{\delta V}{\delta y}\]

    \[f(z)\space =\space \int {\phi _1(z,0)-i\phi _2(z,0)}\,dz\space +\space C\]

    Representation by Power Series


    Every complex number which is analytic in domain D can be represented by a power series about a point z0 inside D. This is called Taylor Series. If f(z) is not analytic at point z0, we can still expand in both negative and positive side of z-z0. This is called Laurent's Series


    Taylor Series

    If f(z) is analytic inside a circle C with point a as centre, then for all z inside C,

    \[f(z)\space =\space f(a)+f(z-a)f^{'}(a) + \frac{(z-a)^{2}}{2}f^{''}(a)+.......\frac{(z-a)^{n}}{n}f^{n}(a)\]

    \[f(z)\space =\space \sum_{n=0}^{\infty}a_n (z_a)^{n},\space where\space a_n\space =\space \frac{f^{n}a}{n!}\]


    If a = 0, the series becomes,

    \[f(z)\space =\space f(0) + zf^{'}(0) + \frac{z^{2}}{2}f^{''}(0)+......+ \frac{z^{n}}{n!}f^{n}(0)\]

    This series is called Maclaurin Series


    Laurent's Series



    If f(z) is analytic inside and on the boundary of annular(ring-shaped) region R bounded by two concentric circled C1 and C2 of radii r1 and r2 having centre at a then for all z in R,

    \[f(z)\space =\space \sum_{n=0}^{\infty}a_n (z-a)^{n}\space +\space \sum_{n=1}^{\infty}b_n (z-a)^{-n}\]

    \[a_n\space =\space \frac{1}{2\pi i} \int\frac{f(w)}{(w-a)^{-n+1}}\,dw\]

    \[b_n\space =\space \frac{1}{2\pi i} \int\frac{f(w)}{(w-a)^{-n+1}}\]


    In case f(z) is anaytic inside C1 then bn=0 Laurent Seris reduces to Taylor Series

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    Decimal/BCD Code Converter

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    BCD Code Addition

    +

    Decimal-Excess-3 Code Converter

    Decimal
    Excess-3


    Binary-Excess-3 Code Converter

    Binary
    Excess
    Decimal
    2421

    If all the letters of the word '_____' are arranged in all possible ways and written out in alphabetical(dictionary) order, then find the rank of the given word

    Word in english characters
    Rank in Dictionary

    Binary-Grey Code Converter


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    Empirical Probability

    Enter number of favourable outcomes
    Enter number of possible outcomes

    Conditional Probability



    Probability of Event A and Event B (P(A ∩ B))
    Probability of Event A (P(A))

    Joint Probability

    The probability that two events will both occur

    Enter number of favourable outcomes in case of Event 1
    Enter number of possible outcomes in case of Event 1
    Enter number of favourable outcomes in case of Event 2
    Enter number of possible outcomes in case of Event 2

    Bayes' Theorem

    Bayes’ theorem describes the probability of an event based on prior knowledge of the conditions that might be relevant to the event.

    \[ P \, \left( \frac{A}{B}\right)\space = \space \frac{P \left( \frac{B}{A} \right) \centerdot P(A)}{P(B)} \]

    P(A|B) = probability of A given B is true

    P(B|A) = probability of B given A is true

    P(A), P(B) = the independent probabilities of A and B

    Enter number of favourable outcomes in case of Event A
    Enter number of possible outcomes in case of Event A
    Enter number of favourable outcomes in case of Event B
    Enter number of possible outcomes in case of Event B
    Enter the probability of A and B(decimal)

    C More ( ) e exp() 00 ^ 7 8 9 divide-icon 4 5 6 crossed-icon % 1 2 3 minus-icon Deci. 0 . add-icon =