# Recent questions tagged prove

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In the right triangle the length of one leg is 5 in., and the measure of the angle opposite to that side is $30^{\circ}$. Determine which of the given statements is correct.
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In the right triangle the length of one leg is 5 in., and the measure of the angle opposite to that side is $30^{\circ}$. Determine which of the given statements is correct.In the right triangle the length of one leg is 5 in., and the measure of the angle opposite to that side is $30^{\circ}$. Determine which of the given ...
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Triangle $\triangle A B C$, right angled at $C$, is given. Height and the median from point $C$ form an angle $\varphi$. The measure of larger acute angle of $\triangle A B C$ is:Triangle $\triangle A B C$, right angled at $C$, is given. Height and the median from point $C$ form an angle $\varphi$. The measure of larger acute a ...
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Solve the equation $\frac{x-8}{\sqrt{25+(8-x)^{2}}}+\frac{x}{\sqrt{x^{2}+16}}=0$
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Solve the equation $\frac{x-8}{\sqrt{25+(8-x)^{2}}}+\frac{x}{\sqrt{x^{2}+16}}=0$Solve the equation $\frac{x-8}{\sqrt{25+(8-x)^{2}}}+\frac{x}{\sqrt{x^{2}+16}}=0$ ...
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1 answer 551 views
If the equation  $a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots .+a_{1} x=0, a_{1} \neq 0, n \geq 2$ has a positive root $x=\alpha$, then the equation $n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots .+a_{1}=0$ has a positive roots, which isIf the equation &nbsp;$a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots .+a_{1} x=0, a_{1} \neq 0, n \geq 2$ has a positive root $x=\alpha$, then the equation $n a_ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What strategy can be used to complete the square for any equation of the form$x^{2}+a x+b=0$1 answer 64 views What strategy can be used to complete the square for any equation of the form$x^{2}+a x+b=0$What strategy can be used to complete the square for any equation of the form$x^{2}+a x+b=0$... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What are the Sum-Difference identities in trigonometry? 1 answer 48 views What are the Sum-Difference identities in trigonometry?What are the Sum-Difference identities in trigonometry? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Factorize$x^{3}+4 x^{2}+3 x+12$1 answer 42 views Factorize$x^{3}+4 x^{2}+3 x+12$Factorize$x^{3}+4 x^{2}+3 x+12$... close 1 answer 479 views If the equation$a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots .+a_{1} x=0, a_{1} \neq 0, n \geq 2$, has a positive root$x=\alpha$, then the equation$n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots .+a_{1}=0$has a positive roots, which isIf the equation$a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots .+a_{1} x=0, a_{1} \neq 0, n \geq 2$, has a positive root$x=\alpha$, then the equation$n a_{n} x ...
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Prove Wilson's Theorem
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Prove Wilson's TheoremProve Wilson's Theorem ...
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Prove the following
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Prove the followingLet $n,k \in &nbsp;\mathbb{R}^+$ such that $k =(p-1)q +r, 0 \leq r \leq p$ Prove that $n^k \equiv n^r \ (mod \ p)$ ...
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The heights of tomato plants exhibited on a farmer's trade show satisfied the equation $\frac{x+1}{1+2 x}=\frac{1 x+1}{31+2 x}=\frac{1}{3}$. Find the value of $x$.
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The heights of tomato plants exhibited on a farmer's trade show satisfied the equation $\frac{x+1}{1+2 x}=\frac{1 x+1}{31+2 x}=\frac{1}{3}$. Find the value of $x$.The heights of tomato plants exhibited on a farmer's trade show satisfied the equation $\frac{x+1}{1+2 x}=\frac{1 x+1}{31+2 x}=\frac{1}{3}$. Find the ...
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Given a real value $x$ such that $0 \leq x \leq 1$, show that $$\frac{x^{2}}{x+1}+\frac{(1-x)^{2}}{2-x} \geq \frac{1}{3}$$
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Given a real value $x$ such that $0 \leq x \leq 1$, show that $$\frac{x^{2}}{x+1}+\frac{(1-x)^{2}}{2-x} \geq \frac{1}{3}$$Given a real value $x$ such that $0 \leq x \leq 1$, show that $$\frac{x^{2}}{x+1}+\frac{(1-x)^{2}}{2-x} \geq \frac{1}{3}$$ ...
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Given $n$ positive real numbers $x_{1}, x_{2}, \ldots, x_{n}$, show that $$\frac{x_{1}^{2}}{x_{2}}+\frac{x_{2}^{2}}{x_{3}}+\cdots+\frac{x_{n}^{2}}{x_{1}} \geq x_{1}+x_{2}+\cdots+x_{n} \text { . }$$Given $n$ positive real numbers $x_{1}, x_{2}, \ldots, x_{n}$, show that $$\frac{x_{1}^{2}}{x_{2}}+\frac{x_{2}^{2}}{x_{3}}+\cdots+\frac{x_{n}^{2}}{x_ ... close 1 answer 456 views If \left\{b_{1}, \ldots, b_{n}\right\} is a permutation of the sequence \left\{a_{1}, \ldots, a_{n}\right\} of positive reals, then prove that$$ \frac{a_{1}}{b_{1}}+\frac{a_{2}}{b_{2}}+\cdots+\frac{a_{n}}{b_{n}} \geq n $$If \left\{b_{1}, \ldots, b_{n}\right\} is a permutation of the sequence \left\{a_{1}, \ldots, a_{n}\right\} of positive reals, then prove that$$ ...
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Prove that if the product of $n$ positive real numbers is 1, then their sum is at least $n$
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Prove that if the product of $n$ positive real numbers is 1, then their sum is at least $n$Prove that if the product of $n$ positive real numbers is 1, then their sum is at least $n$ ...
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Find the number of real solutions to the equation: $$\left(16 x^{200}+1\right)\left(y^{200}+1\right)=16(x y)^{100}$$
1 answer 208 views
Find the number of real solutions to the equation: $$\left(16 x^{200}+1\right)\left(y^{200}+1\right)=16(x y)^{100}$$Find the number of real solutions to the equation: $$\left(16 x^{200}+1\right)\left(y^{200}+1\right)=16(x y)^{100}$$ ...
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Use the central limit theorem to provide an approximation to this probability.
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Use the central limit theorem to provide an approximation to this probability.$N$ independent trials are to be conducted, each with &quot;success&quot; probability $p .$ Let $X_{i}=1$ if trial $i$ is a success and $X_{i}=0$ if i ...
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What is the value of $t$ such that $Q(t)=4 ?$
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What is the value of $t$ such that $Q(t)=4 ?$$\begin{array}{c} \frac{d Q}{d t}=6(5-Q(t)) \\ Q(0)=0 \end{array}$$ The function$Q(t)$satisfies the differential equation shown above. What is th ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Given the following center-radius form of the equation for a circle, find the center of the circle. $$(x-3)^{2}+(y+2)^{2}=16$$ 1 answer 91 views Given the following center-radius form of the equation for a circle, find the center of the circle. $$(x-3)^{2}+(y+2)^{2}=16$$Given the following center-radius form of the equation for a circle, find the center of the circle. $$(x-3)^{2}+(y+2)^{2}=16$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Prove that a positive integer$n$is a prime number if no prime number less than or equal to$\sqrt{n}$divides$n$. 0 answers 81 views Prove that a positive integer$n$is a prime number if no prime number less than or equal to$\sqrt{n}$divides$n$.Prove that a positive integer$n$is a prime number if no prime number less than or equal to$\sqrt{n}$divides$n$. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Prove the trigonometric identity 0 answers 48 views Prove the trigonometric identityProve that$$\cos 2 \alpha=\left\{\begin{array}{l}\cos ^{2} \alpha-\sin ^{2} \alpha \\ 1-2 \sin ^{2} \alpha \\ 2 \cos ^{2} \alpha-1\end{array}\right ... close 1 answer 187 views A convergent geometric series consisting of only positive terms has first term$a$, constant ratio$r$and$n^{\text {th }}$term,$T_{n}$, such that$\sum_{n=3}^{\infty} \mathrm{T}_{n}=\frac{1}{4}$. A convergent geometric series consisting of only positive terms has first term$a$, constant ratio$r$and$n^{\text {th }}$term,$T_{n}$, such tha ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find the Maclaurin series for$\ln (1+x)$. 1 answer 53 views Find the Maclaurin series for$\ln (1+x)$.(a) Find the Maclaurin series for$\ln (1+x)$. (b) Use the alternating series estimation theorem to prove that$x-\frac{x^{2}}{2} \leq \ln (1+x) \leq ...
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By using a suitable Maclaurin series given in the text find the sum to infinity of the following infinite series:
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By using a suitable Maclaurin series given in the text find the sum to infinity of the following infinite series:By using a suitable Maclaurin series given in the text find the sum to infinity of the following infinite series: (a) $\pi-\frac{\pi^{3}}{3 !}+\frac{ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Using the axioms of probability, prove the following: 1 answer 361 views Using the axioms of probability, prove the following:Using the axioms of probability, prove the following: a. For any event$A, P\left(A^{c}\right)=1-P(A)\$. b. The probability of the empty set is zero, ...
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