Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. (i) $\sqrt{\frac{\sec \theta-1}{\sec \theta+1}}+\sqrt{\frac{\sec \theta+1}{\sec \theta-1}}=2 \operatorname{cosec} \theta$ (ii) $\sqrt{\frac{1+\sin \theta}{1-\sin \theta}}+\sqrt{\frac{1-\sin \theta}{1+\sin \theta}}=2 \sec \theta$ (iii) $\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}+\sqrt{\frac{1-\cos \theta}{1+\cos \theta}}=2 \operatorname{cosec} \theta$ (iv) $\frac{\sec \theta-1}{\sec \theta+1}=\left(\frac{\sin \theta}{1+\cos \theta}\right)^{2...
Read More →A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is
Question: A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\frac{1}{100} .$ What is the probability that he will in a prize (a) at least once (b) exactly once (c) at least twice? Solution: Let X represent the number of winning prizes in 50 lotteries. The trials are Bernoulli trials. Clearly, $\mathrm{X}$ has a binomial distribution with $n=50$ and $p=\frac{1}{100}$ $\therefore q=1-p=1-\frac{1}{100}=\frac{99}{100}$ $\therefore \mathrm{P}(\mathrm{X...
Read More →Write the number of solutions of the equation tan
Question: Write the number of solutions of the equation tanx+ secx= 2 cosxin the interval [0, 2]. Solution: Given: tanx+ secx= 2 cosx $\Rightarrow \frac{\sin x}{\cos x}+\frac{1}{\cos x}=2 \cos x$ $\Rightarrow \frac{\sin x+1}{\cos x}=2 \cos x$ $\Rightarrow \sin x+1=2 \cos ^{2} x$ $\Rightarrow \sin x=2 \cos ^{2} x-1$ $\Rightarrow 2\left(1-\sin ^{2} x\right)-1=\sin x$ $\Rightarrow 2-2 \sin ^{2} x-1=\sin x$ $\Rightarrow 1-2 \sin ^{2} x=\sin x$ $\Rightarrow 2 \sin ^{2} x+\sin x-1=0$ $\Rightarrow 2 \s...
Read More →Write the number of solutions of the equation tan
Question: Write the number of solutions of the equation tanx+ secx= 2 cosxin the interval [0, 2]. Solution: Given: tanx+ secx= 2 cosx $\Rightarrow \frac{\sin x}{\cos x}+\frac{1}{\cos x}=2 \cos x$ $\Rightarrow \frac{\sin x+1}{\cos x}=2 \cos x$ $\Rightarrow \sin x+1=2 \cos ^{2} x$ $\Rightarrow \sin x=2 \cos ^{2} x-1$ $\Rightarrow 2\left(1-\sin ^{2} x\right)-1=\sin x$ $\Rightarrow 2-2 \sin ^{2} x-1=\sin x$ $\Rightarrow 1-2 \sin ^{2} x=\sin x$ $\Rightarrow 2 \sin ^{2} x+\sin x-1=0$ $\Rightarrow 2 \s...
Read More →Find the value of x in each of the following.
Question: Find the value ofxin each of the following. (i) $\sqrt[5]{5 x+2}=2$ (ii) $\sqrt[3]{3 x-2}=4$ (iii) $\left(\frac{3}{4}\right)^{3}\left(\frac{4}{3}\right)^{-7}=\left(\frac{3}{4}\right)^{2 x}$ (iv) $5^{x-3} \times 3^{2 x-8}=225$ (v) $\frac{3^{3 x} \cdot 3^{2 x}}{3^{x}}=\sqrt[4]{3^{20}}$ Solution: (i) $\sqrt[5]{5 x+2}=2$ $\Rightarrow(5 x+2)^{\frac{1}{5}}=2$ $\Rightarrow\left[(5 x+2)^{\frac{1}{5}}\right]^{5}=(2)^{5}$ $\Rightarrow(5 x+2)=32$ $\Rightarrow 5 x=32-2$ $\Rightarrow 5 x=30$ $\Righ...
Read More →If
Question: If $\frac{\tan 3 x-1}{\tan 3 x+1}=\sqrt{3}$, then $x=$ Solution: $\frac{\tan 3 x-1}{\tan 3 x+1}=\sqrt{3}$ i. e $\tan 3 x-1=\sqrt{3} \tan 3 x+\sqrt{3}$ $\tan 3 x-1=\sqrt{3} \tan 3 x+\sqrt{3}$ i.e $-1-\sqrt{3}=\tan 3 x(\sqrt{3}-1)$ i. e $\tan 3 x=-\frac{(1+\sqrt{3})}{\sqrt{3}-1}$ $\tan 3 x=-\tan 75^{\circ}$ $=\tan \left(180-75^{\circ}\right) \quad[\because \tan (180-\theta)=-\tan \theta]$ $\tan 3 x=\tan \frac{7 \pi}{12}$ i. e $3 x=n \pi+\frac{7 \pi}{12}$ i. e $x=\frac{n \pi}{3}+\frac{7 \...
Read More →On a multiple choice examination with three possible answers for each of the five questions,
Question: On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing? Solution: The repeated guessing of correct answers from multiple choice questions are Bernoulli trials. Let X represent the number of correct answers by guessing in the set of 5 multiple choice questions. Probability of getting a correct answer is, $p=\frac{1}{3}$ $\therefore q=1-p=1-\frac{1}{3}=\f...
Read More →The set of values of x satisfying the equation
Question: The set of values of $x$ satisfying the equation $\frac{\tan 3 x-\tan 2 x}{1+\tan 3 x \tan 2 x}=1$ is __________________ Solution: $\frac{\tan 3 x-\tan 2 x}{1+\tan 3 x \tan 2 x}=1$ i.e $\tan (3 x-2 x)=1 \quad$ (using identity :- $\left.\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \tan y}\right)$ i.e $\tan x=1$ i.e $x=\tan ^{-1} \frac{\pi}{4}$ i.e $x=n \pi+\frac{\pi}{4} \quad ; n \in Z$...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. if $\cos A+\cos ^{2} A=1$, prove that $\sin ^{2} A+\sin ^{4} A=1$ Solution: Given: $\cos A+\cos ^{2} A=1$ We have to prove $\sin ^{2} A+\sin ^{4} A=1$ Now, $\cos A+\cos ^{2} A=1$ $\Rightarrow \quad \cos A=1-\cos ^{2} A$ $\Rightarrow \quad \cos A=\sin ^{2} A$ $\Rightarrow \quad \sin ^{2} A=\cos A$ Therefore, we have $\sin ^{2} A+\sin ^{4} A=\cos A+(\cos A)^{2}$ $=\cos A+\cos ^{2} A$ $=1$ Hence proved....
Read More →The number of values of x ∈ [0, 2π]
Question: The number of values ofx [0, 2] satisfying the equation 2 sin2x= 4 + 3 cosxis______________. Solution: Forx [0, 2] 2sin2x= 4 + 3cosx i.e 2(1 cos2x) = 4 + 3cosx i.e 2 2cos2x= 4 + 3cosx ⇒ 2cos2x+ 3cosx+ 2 = 0 ⇒ 2cos2x+ 4cosx cosx+ 2 = 0 i.e No solution is possible No value ofxsatisfies 2sin2x= 4 + 3cosx...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $\operatorname{cosec} \theta+\cot \theta=m$ and $\operatorname{cosec} \theta-\cot \theta=n$, prove that $m n=1$ Solution: Given: $\operatorname{cosec} \theta+\cot \theta=m$ $\operatorname{cosec} \theta-\cot \theta=n$ We have to prove $m n=1$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Multiplying the two equations, we have $(\operatorname{cosec} \theta+\cot \theta)(\operatorname{cosec} \theta-\cot \theta)=m n$ $\Rightarrow\left(\...
Read More →If cos mx = cos nx, m ≠ n,
Question: If cosmx= cosnx,mn, thenx=______________. Solution: If cosmx= cosnx ;mn i.e mx= 2r nx;rZ i.ex(m n) = 2r i.e $x=\frac{2 r \pi}{m \pm n}$...
Read More →The general value of x satisfying tan x tan 2x = 1 is
Question: The general value ofxsatisfying tanxtan 2x= 1 is ______________. Solution: tanxtan 2x= 1 i.e 1 tanxtan 2x= 0 ...(1) Since $\tan 3 x=\tan (x+2 x)$ $=\frac{\tan x+\tan 2 x}{1-\tan x \tan 2 x}$ i.e $\tan 3 x=\infty$ .......(from 1) i.e $3 x=n \pi+\frac{\pi}{2}$ i. e $x=\frac{n \pi}{3}+\frac{\pi}{6}$...
Read More →Suppose X has a binomial distribution
Question: Suppose $X$ has a binomial distribution $B\left(6, \frac{1}{2}\right)$. Show that $X=3$ is the most likely outcome. (Hint: P(X = 3) is the maximum among all P (xi),xi= 0, 1, 2, 3, 4, 5, 6) Solution: $X$ is the random variable whose binomial distribution is $B\left(6, \frac{1}{2}\right)$. Therefore, $n=6$ and $p=\frac{1}{2}$ $\therefore q=1-p=1-\frac{1}{2}=\frac{1}{2}$ Then, $\mathrm{P}(\mathrm{X}=x)={ }^{n} \mathrm{C}_{x} q^{n-x} p^{x}$ $={ }^{6} \mathrm{C}_{x}\left(\frac{1}{2}\right)^...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $a \cos \theta+b \sin \theta=m$ and $a \sin \theta-b \cos \theta=n$, prove that $a^{2}+b^{2}=m^{2}+n^{2}$ Solution: Given: $a \cos \theta+b \sin \theta=m$ $a \sin \theta-b \cos \theta=n$ We have to prove $a^{2}+b^{2}=m^{2}+n^{2}$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Now, squaring and adding the two equations, we get $(a \cos \theta+b \sin \theta)^{2}+(a \sin \theta-b \cos \theta)^{2}=m^{2}+n^{2}$ $\Rightarrow\left(a^{2} \c...
Read More →The number of values of x ∈ (–π, π) satisfying 2 tan
Question: The number of values ofx (, ) satisfying 2 tan2x= sec2xis ______________. Solution: Forx∊ (, ) 2tan2x= sec2x i. e $\frac{2 \sin ^{2} x}{\cos ^{2} x}=\frac{1}{\cos ^{2} x} \quad$ i. e $x \notin(2 n+1) \frac{\pi}{2}$ i. e $\sin ^{2} x=\frac{1}{2}$ i. e $\sin x=\pm \frac{1}{\sqrt{2}}$ i. e $x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$ Number of value ofxin (, ) Satisfying2tan2x= sec2xis 4....
Read More →The number of values of x ∈ (–π, π) satisfying 2 tan
Question: The number of values ofx (, ) satisfying 2 tan2x= sec2xis ______________. Solution: Forx∊ (, ) 2tan2x= sec2x i. e $\frac{2 \sin ^{2} x}{\cos ^{2} x}=\frac{1}{\cos ^{2} x} \quad$ i. e $x \notin(2 n+1) \frac{\pi}{2}$ i. e $\sin ^{2} x=\frac{1}{2}$ i. e $\sin x=\pm \frac{1}{\sqrt{2}}$ i. e $x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$ Number of value ofxin (, ) Satisfying2tan2x= sec2xis 4....
Read More →The most general value of θ satisfying 2sin
Question: The most general value ofsatisfying 2sin2 1 = 0 is ______________. Solution: $2 \sin ^{2} \theta-1=0$ $\Rightarrow \sin ^{2} \theta=\frac{1}{2}$ i. e $\sin \theta=\pm \frac{1}{\sqrt{2}} \quad$ i. e $\theta=n \pi \pm \frac{\pi}{6} ; n \in \mathbb{Z}$ ie $\theta=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$...
Read More →The number of values of θ∈
Question: The number of values of $\theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ satisfying $\frac{1-\cos 2 \theta}{1+\cos 2 \theta}=3$ is ____________________ Solution: for $\theta \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$ $\frac{1-\cos 2 \theta}{1+\cos 2 \theta}=3$ i.e $\frac{2 \sin ^{2} \theta}{2 \cos ^{2} \theta}=3 \quad$ (using identity :- $1-\cos ^{2} \theta=2 \sin ^{2} \theta, 1+\cos 2 \theta=2 \cos ^{2} \theta$ ) i.e $\tan ^{2} \theta=3$ i.e $\tan \theta=\pm \sqrt{3}$ i.e $\...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $3 \sin \theta+5 \cos \theta=5$, prove that $5 \sin \theta-3 \cos \theta=\pm 3$. Solution: Given: $3 \sin \theta+5 \cos \theta=5$ We have to prove that $5 \sin \theta-3 \cos \theta=\pm 3$. We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Squaring the given equation, we have $(3 \sin \theta+5 \cos \theta)^{2}=(5)^{2}$ $\Rightarrow \quad 9 \sin ^{2} \theta+2 \times 3 \sin \theta \times 5 \cos \theta+25 \cos ^{2} \theta=25$ $\Rightarrow ...
Read More →The values of θ∈
Question: The values of $\theta \in\left[0, \frac{\pi}{4}\right]$ satisfying $\tan 5 \theta=\cot 2 \theta$ are ___________________ Solution: for $\theta \in\left[0, \frac{\pi}{4}\right]$ $\tan 5 \theta=\cot 2 \theta$ i.e $\tan 5 \theta=\frac{1}{\tan 2 \theta}$ $\Rightarrow \tan 5 \theta \tan 2 \theta=1 \quad \ldots(1)$ Since $\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}$ i.e $\tan (5 \theta+2 \theta)=\frac{\tan 5 \theta+\tan 2 \theta}{1-\tan 5 \theta \tan 2 \theta}$ i.e $\tan 7 \theta=\frac{...
Read More →The values of θ∈
Question: The values of $\theta \in\left[0, \frac{\pi}{4}\right]$ satisfying $\tan 5 \theta=\cot 2 \theta$ are ___________________ Solution: for $\theta \in\left[0, \frac{\pi}{4}\right]$ $\tan 5 \theta=\cot 2 \theta$ i.e $\tan 5 \theta=\frac{1}{\tan 2 \theta}$ $\Rightarrow \tan 5 \theta \tan 2 \theta=1 \quad \ldots(1)$ Since $\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}$ i.e $\tan (5 \theta+2 \theta)=\frac{\tan 5 \theta+\tan 2 \theta}{1-\tan 5 \theta \tan 2 \theta}$ i.e $\tan 7 \theta=\frac{...
Read More →In an examination, 20 questions of true-false type are asked.
Question: In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers true; if it falls tails, he answers false. Find the probability that he answers at least 12 questions correctly. Solution: Let X represent the number of correctly answered questions out of 20 questions. The repeated tosses of a coin are Bernoulli trails. Since head on a coin represents the true answer and tail r...
Read More →Simplify
Question: Simplify (i) $\left(\frac{15^{\frac{1}{3}}}{9^{\frac{1}{4}}}\right)^{-6}$ (ii) $\left(\frac{12^{\frac{1}{5}}}{27^{\frac{1}{5}}}\right)^{\frac{5}{2}}$ (iii) $\left(\frac{15^{\frac{1}{4}}}{3^{\frac{1}{2}}}\right)^{-2}$ Solution: (i) $\left(\frac{15^{\frac{1}{3}}}{9^{\frac{1}{4}}}\right)^{-6}$ $\left(\frac{15^{\frac{1}{3}}}{9^{\frac{1}{4}}}\right)^{-6}=\left(\frac{9^{\frac{1}{4}}}{15^{\frac{1}{3}}}\right)^{6}$ $=\frac{9 \frac{6}{4}}{15^{\frac{6}{3}}}$ $=\frac{9^{\frac{3}{2}}}{15^{2}}$ $=\...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $x=a \cos ^{3} \theta, y=b \sin ^{3} \theta$, prove that $\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=1$ Solution: Given: $x=a \cos ^{3} \theta$ $\Rightarrow \frac{x}{a}=\cos ^{3} \theta$ $x=b \sin ^{3} \theta$ $\Rightarrow \frac{y}{b}=\sin ^{3} \theta$ We have to prove $\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=1$ We know that $\sin ^{2} \theta+\cos ^{2} \theta=1$ So, we have $\left(\frac{x}...
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