MathsGee Answers - Recent questions in Mathematics
https://mathsgee.com/questions/mathematics
Powered by Question2AnswerSketch the function $f(x)=\left\{\begin{array}{cc}|x| & \text { if }|x| \leq 1 \\ 1 & \text { if }|x| \geq 1\end{array}\right.$ and write down its domain and range.
https://mathsgee.com/27966/sketch-function-begin-array-array-right-write-domain-range
Sketch the function $f(x)=\left\{\begin{array}{cc}|x| & \text { if }|x| \leq 1 \\ 1 & \text { if }|x| \geq 1\end{array}\right.$ and write down its domain and range.Mathematicshttps://mathsgee.com/27966/sketch-function-begin-array-array-right-write-domain-rangeMon, 10 May 2021 08:36:40 +0000Sketch the function $f(x)=-\frac{1}{2} \sqrt{x+1}$ and write down its domain and range.
https://mathsgee.com/27964/sketch-the-function-frac-sqrt-and-write-down-its-domain-range
Sketch the function $f(x)=-\frac{1}{2} \sqrt{x+1}$ and write down its domain and range.Mathematicshttps://mathsgee.com/27964/sketch-the-function-frac-sqrt-and-write-down-its-domain-rangeMon, 10 May 2021 08:33:57 +0000Evaluate $\lim _{x \rightarrow 1}\left(\frac{y-4 \sqrt{y}+3}{y^{2}-1}\right)$
https://mathsgee.com/27962/evaluate-lim-x-rightarrow-1-left-frac-y-4-sqrt-y-3-y-2-1-right
Evaluate $\lim _{x \rightarrow 1}\left(\frac{y-4 \sqrt{y}+3}{y^{2}-1}\right)$Mathematicshttps://mathsgee.com/27962/evaluate-lim-x-rightarrow-1-left-frac-y-4-sqrt-y-3-y-2-1-rightMon, 10 May 2021 08:28:46 +0000List the Compound Angle Identities in trigonometry
https://mathsgee.com/27960/list-the-compound-angle-identities-in-trigonometry
List the Compound Angle Identities in trigonometryMathematicshttps://mathsgee.com/27960/list-the-compound-angle-identities-in-trigonometryMon, 10 May 2021 08:26:32 +0000List the Double Angle Identities in trigonometry
https://mathsgee.com/27958/list-the-double-angle-identities-in-trigonometry
List the Double Angle Identities in trigonometryMathematicshttps://mathsgee.com/27958/list-the-double-angle-identities-in-trigonometryMon, 10 May 2021 08:24:36 +0000There are 14 people at a party. Every pair of people shakes hands exactly once. How many handshakes occur?
https://mathsgee.com/27956/there-people-people-shakes-hands-exactly-handshakes-occur
There are 14 people at a party. Every pair of people shakes hands exactly once. How many handshakes occur?Mathematicshttps://mathsgee.com/27956/there-people-people-shakes-hands-exactly-handshakes-occurMon, 10 May 2021 08:18:47 +0000Find the $83^{\text {rd }}$ term in the sequence, $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6, \ldots$
https://mathsgee.com/27954/find-the-83-text-rd-term-in-the-sequence-1-4-4-5-5-5-5-5-6-6-ldots
Find the $83^{\text {rd }}$ term in the sequence, $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6, \ldots$<br />
Here the number 1 is repeated once, 2 twice, 3 three times. Later 10 is repeated ten times, 11 eleven times and so on.Mathematicshttps://mathsgee.com/27954/find-the-83-text-rd-term-in-the-sequence-1-4-4-5-5-5-5-5-6-6-ldotsMon, 10 May 2021 08:17:18 +0000Which fraction is the smallest?
https://mathsgee.com/27952/which-fraction-is-the-smallest
Which fraction is the smallest?<br />
A. $\frac{1}{3}$<br />
B. $\frac{3}{10}$<br />
C. $\frac{6}{9}$<br />
D. $\frac{2}{7}$<br />
E. $\frac{5}{13}$Mathematicshttps://mathsgee.com/27952/which-fraction-is-the-smallestMon, 10 May 2021 08:15:04 +0000Find $726 \times 32+726 \times 68$.
https://mathsgee.com/27950/find-726-times-32-726-times-68
Find $726 \times 32+726 \times 68$.Mathematicshttps://mathsgee.com/27950/find-726-times-32-726-times-68Mon, 10 May 2021 08:11:11 +0000My friend is 4 years older than me. The sum of our ages is 24 . How old am I?
https://mathsgee.com/27948/my-friend-years-older-than-me-the-sum-of-our-ages-is-24-how-old-am
My friend is 4 years older than me. The sum of our ages is 24 . How old am I?Mathematicshttps://mathsgee.com/27948/my-friend-years-older-than-me-the-sum-of-our-ages-is-24-how-old-amMon, 10 May 2021 08:09:48 +0000What is 30 percent of 20 percent of 50 percent of 7000 ?
https://mathsgee.com/27946/what-is-30-percent-of-20-percent-of-50-percent-of-7000
What is 30 percent of 20 percent of 50 percent of 7000 ?Mathematicshttps://mathsgee.com/27946/what-is-30-percent-of-20-percent-of-50-percent-of-7000Mon, 10 May 2021 08:07:31 +0000Which of the following numbers is closest to $0,000246 \times 7982413 ?$
https://mathsgee.com/27944/which-of-the-following-numbers-closest-000246-times-7982413
Which of the following numbers is closest to $0,000246 \times 7982413 ?$<br />
A. 2<br />
B. 20<br />
C. 200<br />
D. 2000<br />
E. 20000Mathematicshttps://mathsgee.com/27944/which-of-the-following-numbers-closest-000246-times-7982413Mon, 10 May 2021 08:04:06 +0000Some students in a Grade five class line up. Thabo is $10^{t h}$ from the front and $12^{t h}$ from the back. How many students are in the class?
https://mathsgee.com/27942/students-grade-class-thabo-from-front-back-many-students-class
Some students in a Grade five class line up. Thabo is $10^{t h}$ from the front and $12^{t h}$ from the back. How many students are in the class?Mathematicshttps://mathsgee.com/27942/students-grade-class-thabo-from-front-back-many-students-classMon, 10 May 2021 08:03:05 +0000Find the value of $0,2019-0,02019$.
https://mathsgee.com/27940/find-the-value-of-0-2019-0-02019
Find the value of $0,2019-0,02019$.Mathematicshttps://mathsgee.com/27940/find-the-value-of-0-2019-0-02019Mon, 10 May 2021 08:02:02 +0000Find the value of $1+2+3+4+5+6+7+8+9+10$.
https://mathsgee.com/27938/find-the-value-of-1-2-3-4-5-6-7-8-9-10
Find the value of $1+2+3+4+5+6+7+8+9+10$.Mathematicshttps://mathsgee.com/27938/find-the-value-of-1-2-3-4-5-6-7-8-9-10Mon, 10 May 2021 08:00:28 +0000Nkosi played soccer with some friends. They started at 5:20 pm and finished at 6:05 pm. How long was their game?
https://mathsgee.com/27936/nkosi-played-soccer-friends-started-finished-long-their-game
Nkosi played soccer with some friends. They started at 5:20 pm and finished at 6:05 pm. How long was their game?Mathematicshttps://mathsgee.com/27936/nkosi-played-soccer-friends-started-finished-long-their-gameMon, 10 May 2021 07:59:20 +0000Mr Smith has a sweet shop. The graph below shows his sales for each day of a week, from Monday to Sunday. On which day were sales highest?
https://mathsgee.com/27934/smith-sweet-graph-below-shows-sales-monday-sunday-which-highest
<p>Mr Smith has a sweet shop. The graph below shows his sales for each day of a week, from Monday to Sunday. On which day were sales highest?</p>
<p><img alt="" src="https://mathsgee.com/?qa=blob&qa_blobid=7057141151932863797" style="height:318px; width:600px"></p>
<p> </p>Mathematicshttps://mathsgee.com/27934/smith-sweet-graph-below-shows-sales-monday-sunday-which-highestMon, 10 May 2021 07:58:14 +0000Find the value of $20 \times 19$.
https://mathsgee.com/27932/find-the-value-of-20-times-19
Find the value of $20 \times 19$.Mathematicshttps://mathsgee.com/27932/find-the-value-of-20-times-19Mon, 10 May 2021 07:56:03 +0000Let $f(z)$ be the principal branch of $\sqrt[3]{z}$.
https://mathsgee.com/27930/let-f-z-be-the-principal-branch-of-sqrt-3-z
Let $f(z)$ be the principal branch of $\sqrt[3]{z}$. Find $f(-i)$.Mathematicshttps://mathsgee.com/27930/let-f-z-be-the-principal-branch-of-sqrt-3-zMon, 10 May 2021 07:46:49 +0000Find $i^{i}$ and its principal value.
https://mathsgee.com/27928/find-i-i-and-its-principal-value
Find $i^{i}$ and its principal value.Mathematicshttps://mathsgee.com/27928/find-i-i-and-its-principal-valueMon, 10 May 2021 07:42:47 +0000Compute $\cos \left(\frac{\pi}{3}+i\right)$
https://mathsgee.com/27926/compute-cos-left-frac-pi-3-i-right
Compute $\cos \left(\frac{\pi}{3}+i\right)$Mathematicshttps://mathsgee.com/27926/compute-cos-left-frac-pi-3-i-rightMon, 10 May 2021 07:41:40 +0000Calculate $\sin \left(\frac{\pi}{4}+i\right)$.
https://mathsgee.com/27924/calculate-sin-left-frac-pi-4-i-right
Calculate $\sin \left(\frac{\pi}{4}+i\right)$.Mathematicshttps://mathsgee.com/27924/calculate-sin-left-frac-pi-4-i-rightMon, 10 May 2021 07:40:03 +0000Find all the complex roots of the equation $\cos z=3$.
https://mathsgee.com/27922/find-all-the-complex-roots-of-the-equation-cos-z-3
Find all the complex roots of the equation $\cos z=3$.Mathematicshttps://mathsgee.com/27922/find-all-the-complex-roots-of-the-equation-cos-z-3Mon, 10 May 2021 07:38:44 +0000Show that $$ \tan \left(z_{1}+z_{2}\right)=\frac{\tan z_{1}+\tan z_{2}}{1-\left(\tan z_{1}\right)\left(\tan z_{2}\right)} $$
https://mathsgee.com/27920/show-that-left-right-frac-tan-tan-left-tan-right-left-tan-right
Show that<br />
$$<br />
\tan \left(z_{1}+z_{2}\right)=\frac{\tan z_{1}+\tan z_{2}}{1-\left(\tan z_{1}\right)\left(\tan z_{2}\right)}<br />
$$<br />
for all complex numbers $z_{1}$ and $z_{2}$ satisfying $z_{1}, z_{2}, z_{1}+z_{2} \neq n \pi+\pi / 2$ for any integer<br />
$n$Mathematicshttps://mathsgee.com/27920/show-that-left-right-frac-tan-tan-left-tan-right-left-tan-rightMon, 10 May 2021 07:35:57 +0000Show that $$ |\cos (z)|^{2}=(\cos x)^{2}+(\sinh y)^{2} $$ for all $z \in \mathbb{C}$, where $x=\operatorname{Re}(z)$ and $y=\operatorname{Im}(z)$
https://mathsgee.com/27918/show-that-cos-sinh-mathbb-where-operatorname-operatorname
Show that<br />
$$<br />
|\cos (z)|^{2}=(\cos x)^{2}+(\sinh y)^{2}<br />
$$<br />
for all $z \in \mathbb{C}$, where $x=\operatorname{Re}(z)$ and $y=\operatorname{Im}(z)$Mathematicshttps://mathsgee.com/27918/show-that-cos-sinh-mathbb-where-operatorname-operatornameMon, 10 May 2021 07:34:11 +0000Show that $$ |\sin z|^{2}=(\sin x)^{2}+(\sinh y)^{2} $$ for all complex numbers $z=x+y i$.
https://mathsgee.com/27916/show-that-sin-2-sin-x-2-sinh-2-for-all-complex-numbers-z-i
Show that<br />
$$<br />
|\sin z|^{2}=(\sin x)^{2}+(\sinh y)^{2}<br />
$$<br />
for all complex numbers $z=x+y i$.Mathematicshttps://mathsgee.com/27916/show-that-sin-2-sin-x-2-sinh-2-for-all-complex-numbers-z-iMon, 10 May 2021 07:32:50 +0000Sketch the following sets in the complex plane $\mathbb{C}$ and determine whether they are open, closed, or neither; bounded; connected. Briefly state your reason.
https://mathsgee.com/27914/following-complex-determine-whether-bounded-connected-briefly
Sketch the following sets in the complex plane $\mathbb{C}$ and determine whether they are open, closed, or neither; bounded; connected. Briefly state your reason.<br />
(a) $|z+3|<1$;<br />
(b) $|\operatorname{Im}(z)| \geq 1$;<br />
(c) $1 \leq|z+3|<2$.Mathematicshttps://mathsgee.com/27914/following-complex-determine-whether-bounded-connected-brieflyMon, 10 May 2021 07:24:41 +0000Let $$ T(z)=\frac{z}{z+1} $$ Find the inverse image of the disk $|z|<1 / 2$ under $T$ and sketch it.
https://mathsgee.com/27912/let-frac-find-the-inverse-image-the-disk-under-t-and-sketch
Let<br />
$$<br />
T(z)=\frac{z}{z+1}<br />
$$<br />
Find the inverse image of the disk $|z|<1 / 2$ under $T$ and sketch it.Mathematicshttps://mathsgee.com/27912/let-frac-find-the-inverse-image-the-disk-under-t-and-sketchMon, 10 May 2021 07:15:54 +0000Suppose $p(z)$ is a polynomial with real coefficients. Prove that
https://mathsgee.com/27910/suppose-p-is-polynomial-with-real-coefficients-prove-that
Suppose $p(z)$ is a polynomial with real coefficients. Prove that<br />
(a) $\overline{p(z)}=p(\bar{z})$;<br />
(b) $p(z)=0$ if and only if $p(\bar{z})=0$;<br />
(c) the roots of $p(z)=0$ appear in conjugate pairs, i.e., if $z_{0}$ is a root of $p(z)=0$, so is $\bar{z}_{0}$.Mathematicshttps://mathsgee.com/27910/suppose-p-is-polynomial-with-real-coefficients-prove-thatMon, 10 May 2021 07:13:35 +0000Suppose that $f(z)=x^{2}-y^{2}-2 y+i(2 x-2 x y)$, where $z=x+i y$.
https://mathsgee.com/27908/suppose-that-f-z-x-2-y-2-2-y-i-2-x-2-x-y-where-z-x-i-y
Suppose that $f(z)=x^{2}-y^{2}-2 y+i(2 x-2 x y)$, where $z=x+i y$. Use the expressions<br />
$$<br />
x=\frac{z+\bar{z}}{2} \quad \text { and } \quad y=\frac{z-\bar{z}}{2 i}<br />
$$<br />
to write $f(z)$ in terms of $z$ and simplify the result.Mathematicshttps://mathsgee.com/27908/suppose-that-f-z-x-2-y-2-2-y-i-2-x-2-x-y-where-z-x-i-yMon, 10 May 2021 07:04:36 +0000Write the following functions $f(z)$ in the forms $f(z)=u(x, y)+i v(x, y)$ under Cartesian coordinates with $u(x, y)=\operatorname{Re}(f(z))$ and $v(x, y)=\operatorname{Im}(f(z))$ :
https://mathsgee.com/27906/following-functions-coordinates-operatorname-operatorname
Write the following functions $f(z)$ in the forms $f(z)=u(x, y)+i v(x, y)$ under Cartesian coordinates with $u(x, y)=\operatorname{Re}(f(z))$ and $v(x, y)=\operatorname{Im}(f(z))$ :<br />
(a) $f(z)=z^{3}+z+1$<br />
(b) $f(z)=z^{3}-z$;<br />
(c) $f(z)=\frac{1}{i-z}$;<br />
(d) $f(z)=\overline{\exp \left(z^{2}\right)}$Mathematicshttps://mathsgee.com/27906/following-functions-coordinates-operatorname-operatornameMon, 10 May 2021 06:56:26 +0000Use complex numbers to prove the Law of Cosine: Let $\triangle A B C$ be a triangle with
https://mathsgee.com/27904/use-complex-numbers-prove-law-cosine-triangle-triangle-with
Use complex numbers to prove the Law of Cosine: Let $\triangle A B C$ be a triangle with<br />
$$<br />
\begin{aligned}<br />
|B C|=& a,|C A|=b,|A B|=c \text { and } \angle B C A=\theta . \text { Then } \\<br />
& a^{2}+b^{2}-2 a b \cos \theta=c^{2}<br />
\end{aligned}<br />
$$<br />
Hint: Place $C$ at the origin, $B$ at $z_{1}$ and $A$ at $z_{2}$. Prove that<br />
$$<br />
z_{1} \bar{z}_{2}+z_{2} \bar{z}_{1}=2\left|z_{1} z_{2}\right| \cos \theta<br />
$$Mathematicshttps://mathsgee.com/27904/use-complex-numbers-prove-law-cosine-triangle-triangle-withMon, 10 May 2021 06:53:47 +0000Establish the identity and then use it to derive Lagrange's trigonometric identity
https://mathsgee.com/27902/establish-identity-derive-lagranges-trigonometric-identity
Establish the identity<br />
$$<br />
1+z+z^{2}+\cdots+z^{n}=\frac{1-z^{n+1}}{1-z} \quad(z \neq 1)<br />
$$<br />
and then use it to derive Lagrange's trigonometric identity:<br />
$$<br />
1+\cos \theta+\cos 2 \theta \cdots+\cos n \theta=\frac{1}{2}+\frac{\sin \frac{(2 n+1) \theta}{2}}{2 \sin \frac{\theta}{2}} \quad(0<\theta<2 \pi)<br />
$$<br />
Hint: As for the first identity, write $S=1+z+z^{2}+\cdots+z^{n}$ and consider the difference $S-z S$. To derive the second identity, write $z=e^{i \theta}$ in the first one.Mathematicshttps://mathsgee.com/27902/establish-identity-derive-lagranges-trigonometric-identityMon, 10 May 2021 06:51:02 +0000Use exponential form to compute
https://mathsgee.com/27900/use-exponential-form-to-compute
Do the following:<br />
(a) Use exponential form to compute<br />
i. $(1+\sqrt{3} i)^{2011}$;<br />
$$<br />
\text { ii. }(1+\sqrt{3} i)^{-2011} \text { . }<br />
$$<br />
(b) Prove that<br />
$$<br />
\sum_{m=0}^{1005}\left(\begin{array}{c}<br />
2011 \\<br />
2 m<br />
\end{array}\right)(-3)^{m}=2^{2010}<br />
$$<br />
and<br />
$$<br />
\sum_{m=0}^{1005}\left(\begin{array}{c}<br />
2011 \\<br />
2 m+1<br />
\end{array}\right)(-3)^{m}=2^{2010}<br />
$$Mathematicshttps://mathsgee.com/27900/use-exponential-form-to-computeMon, 10 May 2021 06:48:15 +0000Find the four roots of the polynomial $z^{4}+16$ and use these to factor $z^{4}+16$ into two quadratic polynomials with real coefficients.
https://mathsgee.com/27898/polynomial-these-factor-quadratic-polynomials-coefficients
Find the four roots of the polynomial $z^{4}+16$ and use these to factor $z^{4}+16$ into two quadratic polynomials with real coefficients.Mathematicshttps://mathsgee.com/27898/polynomial-these-factor-quadratic-polynomials-coefficientsMon, 10 May 2021 06:46:24 +0000Find all the complex roots of the equations:
https://mathsgee.com/27896/find-all-the-complex-roots-of-the-equations
Find all the complex roots of the equations:<br />
(a) $z^{6}=-9$;<br />
(b) $z^{2}+2 z+(1-i)=0$.Mathematicshttps://mathsgee.com/27896/find-all-the-complex-roots-of-the-equationsMon, 10 May 2021 06:44:30 +0000Find the principal argument and exponential form of
https://mathsgee.com/27894/find-the-principal-argument-and-exponential-form-of
Find the principal argument and exponential form of<br />
(a) $z=\frac{i}{1+i}$;<br />
(b) $z=\sqrt{3}+i$;<br />
(c) $z=2-i$.Mathematicshttps://mathsgee.com/27894/find-the-principal-argument-and-exponential-form-ofMon, 10 May 2021 06:43:14 +0000Express the following in the form $x+i y$, with $x, y \in \mathbb{R}$ :
https://mathsgee.com/27892/express-the-following-in-the-form-x-i-y-with-x-y-in-mathbb-r
Express the following in the form $x+i y$, with $x, y \in \mathbb{R}$ :<br />
(a) $\frac{i}{1-i}+\frac{1-i}{i}$;<br />
(b) all the 3 rd roots of $-8 i$;<br />
(c) $\left(\frac{i+1}{\sqrt{2}}\right)^{1337}$Mathematicshttps://mathsgee.com/27892/express-the-following-in-the-form-x-i-y-with-x-y-in-mathbb-rMon, 10 May 2021 06:40:45 +0000Show that $$ |\log (z)| \leq|\ln | z||+\pi $$ for all $z \neq 0$
https://mathsgee.com/27890/show-that-log-z-leq-ln-z-pi-for-all-z-neq-0
Show that<br />
$$<br />
|\log (z)| \leq|\ln | z||+\pi<br />
$$<br />
for all $z \neq 0$Mathematicshttps://mathsgee.com/27890/show-that-log-z-leq-ln-z-pi-for-all-z-neq-0Mon, 10 May 2021 06:36:52 +0000Show that $$ \frac{R^{4}-R}{R^{2}+R+1} \leq\left|\frac{z^{4}+i z}{z^{2}+z+1}\right| \leq \frac{R^{4}+R}{(R-1)^{2}} $$ for all $z$ satisfying $|z|=R>1$.
https://mathsgee.com/27888/show-that-frac-left-frac-right-leq-frac-for-all-z-satisfying
Show that<br />
$$<br />
\frac{R^{4}-R}{R^{2}+R+1} \leq\left|\frac{z^{4}+i z}{z^{2}+z+1}\right| \leq \frac{R^{4}+R}{(R-1)^{2}}<br />
$$<br />
for all $z$ satisfying $|z|=R>1$.Mathematicshttps://mathsgee.com/27888/show-that-frac-left-frac-right-leq-frac-for-all-z-satisfyingMon, 10 May 2021 06:29:49 +0000Sketch the curves in the complex plane given by
https://mathsgee.com/27886/sketch-the-curves-in-the-complex-plane-given-by
Sketch the curves in the complex plane given by<br />
(a) $\operatorname{Im}(z)=-1$;<br />
(b) $|z-1|=|z+i|$;<br />
(c) $2|z|=|z-2|$.Mathematicshttps://mathsgee.com/27886/sketch-the-curves-in-the-complex-plane-given-byMon, 10 May 2021 06:27:04 +0000Verify that $\sqrt{2}|z| \geq|\operatorname{Re} z|+|\operatorname{Im} z|$.
https://mathsgee.com/27884/verify-that-sqrt-2-z-geq-operatorname-re-z-operatorname-im-z
Verify that $\sqrt{2}|z| \geq|\operatorname{Re} z|+|\operatorname{Im} z|$.<br />
Hint: Reduce this inequality to $(|x|-|y|)^{2} \geq 0$.Mathematicshttps://mathsgee.com/27884/verify-that-sqrt-2-z-geq-operatorname-re-z-operatorname-im-zMon, 10 May 2021 06:25:50 +0000Show that $$ \left|z_{1}-z_{2}\right|^{2}+\left|z_{1}+z_{2}\right|^{2}=2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right) $$ for all $z_{1}, z_{2} \in \mathbb{C}$.
https://mathsgee.com/27882/show-left-right-left-right-left-right-left-right-right-mathbb
Show that<br />
$$<br />
\left|z_{1}-z_{2}\right|^{2}+\left|z_{1}+z_{2}\right|^{2}=2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)<br />
$$<br />
for all $z_{1}, z_{2} \in \mathbb{C}$.Mathematicshttps://mathsgee.com/27882/show-left-right-left-right-left-right-left-right-right-mathbbMon, 10 May 2021 06:24:28 +0000The set $\mathbb{Q}$ adjoin $\sqrt{2}$ is defined by $\mathbb{Q}(\sqrt{2})=\{p+q \sqrt{2}: p, q \in \mathbb{Q}\}$.
https://mathsgee.com/27880/the-set-mathbb-adjoin-sqrt-is-defined-mathbb-sqrt-sqrt-mathbb
The set $\mathbb{Q}$ adjoin $\sqrt{2}$ is defined by $\mathbb{Q}(\sqrt{2})=\{p+q \sqrt{2}: p, q \in \mathbb{Q}\}$.<br />
(a) Show that $\mathbb{Q}(\sqrt{2})$ is a field.<br />
(b) Is $\sqrt{3} \in \mathbb{Q}(\sqrt{2})$ ?Mathematicshttps://mathsgee.com/27880/the-set-mathbb-adjoin-sqrt-is-defined-mathbb-sqrt-sqrt-mathbbMon, 10 May 2021 06:21:53 +0000Suppose that $z_{1}$ and $z_{2}$ are complex numbers, with $z_{1} z_{2}$ real and non-zero. Show that there exists a real number $r$ such that $z_{1}=r \bar{z}_{2}$.
https://mathsgee.com/27878/suppose-complex-numbers-there-exists-real-number-such-that
Suppose that $z_{1}$ and $z_{2}$ are complex numbers, with $z_{1} z_{2}$ real and non-zero. Show that there exists a real number $r$ such that $z_{1}=r \bar{z}_{2}$.Mathematicshttps://mathsgee.com/27878/suppose-complex-numbers-there-exists-real-number-such-thatMon, 10 May 2021 06:20:01 +0000Find all complex solutions of the following equations:
https://mathsgee.com/27876/find-all-complex-solutions-of-the-following-equations
Find all complex solutions of the following equations:<br />
(a) $\bar{z}=z$;<br />
(b) $\bar{z}+z=0$;<br />
(c) $\bar{z}=\frac{9}{z}$.Mathematicshttps://mathsgee.com/27876/find-all-complex-solutions-of-the-following-equationsMon, 10 May 2021 06:17:55 +0000Graph the following regions in the complex plane:
https://mathsgee.com/27874/graph-the-following-regions-in-the-complex-plane
Graph the following regions in the complex plane:<br />
(a) $\{z: \operatorname{Re} z \geq 2 \operatorname{Im} z\}$<br />
(b) $\{z: \pi / 2<\operatorname{Arg} z \leq 3 \pi / 4\}$<br />
(c) $\{z:|z-4 i+2|>2\}$.Mathematicshttps://mathsgee.com/27874/graph-the-following-regions-in-the-complex-planeMon, 10 May 2021 06:14:16 +0000Use the binomial theorem to expand
https://mathsgee.com/27872/use-the-binomial-theorem-to-expand
Use binomial theorem<br />
$$<br />
\begin{aligned}<br />
(a+b)^{n} &=\left(\begin{array}{c}<br />
n \\<br />
0<br />
\end{array}\right) a^{n}+\left(\begin{array}{c}<br />
n \\<br />
1<br />
\end{array}\right) a^{n-1} b+\ldots+\left(\begin{array}{c}<br />
n \\<br />
n-1<br />
\end{array}\right) a b^{n-1}+\left(\begin{array}{l}<br />
n \\<br />
n<br />
\end{array}\right) b^{n} \\<br />
&=\sum_{k=0}^{n}\left(\begin{array}{l}<br />
n \\<br />
k<br />
\end{array}\right) a^{n-k} b^{k}<br />
\end{aligned}<br />
$$<br />
to expand<br />
(a) $(1+\sqrt{3} i)^{2011}$;<br />
$$<br />
\text { (b) }(1+\sqrt{3} i)^{-2011} \text { . }<br />
$$Mathematicshttps://mathsgee.com/27872/use-the-binomial-theorem-to-expandMon, 10 May 2021 06:12:02 +0000Show that $f\left(z_{1} z_{2}\right)=f\left(z_{1}\right) f\left(z_{2}\right)$ for all $z_{1}, z_{2} \in \mathbb{C}$.
https://mathsgee.com/27870/show-that-left-right-left-right-left-right-for-all-z-mathbb
Let $f$ be the map sending each complex number<br />
$$<br />
z=x+y i \rightarrow\left[\begin{array}{cc}<br />
x & y \\<br />
-y & x<br />
\end{array}\right]<br />
$$<br />
<br />
Show that $f\left(z_{1} z_{2}\right)=f\left(z_{1}\right) f\left(z_{2}\right)$ for all $z_{1}, z_{2} \in \mathbb{C}$.Mathematicshttps://mathsgee.com/27870/show-that-left-right-left-right-left-right-for-all-z-mathbbMon, 10 May 2021 06:10:17 +0000Compute (a) $\frac{2+i}{2-i}$;
https://mathsgee.com/27868/compute-a-frac-2-i-2-i
Compute<br />
(a) $\frac{2+i}{2-i}$;<br />
(b) $(1-2 i)^{4}$.Mathematicshttps://mathsgee.com/27868/compute-a-frac-2-i-2-iMon, 10 May 2021 06:08:57 +0000