There are 5% defective items in a large bulk of items.
Question: There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item? Solution: Let X denote the number of defective items in a sample of 10 items drawn successively. Since the drawing is done with replacement, the trials are Bernoulli trials. $\Rightarrow p=\frac{5}{100}=\frac{1}{20}$ $\therefore q=1-\frac{1}{20}=\frac{19}{20}$ $X$ has a binomial distribution with $n=10$ and $p=\frac{1}{20}$ $\mathrm{P}(...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\cot ^{2} A \operatorname{cosec}^{2} B-\cot ^{2} B \operatorname{cosec}^{2} A=\cot ^{2} A-\cot ^{2} B$ Solution: We have to prove $\cot ^{2} A \operatorname{cosec}^{2} B-\cot ^{2} B \operatorname{cosec}^{2} A=\cot ^{2} A-\cot ^{2} B$ We know that, $\operatorname{cosec}^{2} A-\cot ^{2} A=1$ So, $\cot ^{2} A \operatorname{cosec}^{2} B-\cot ^{2} B \operatorname{cosec}^{2} A=\cot ^{2} A\left(1+\cot ^{2} B\right)-\cot ^{2} B\left(1+\cot ^{2} A\...
Read More →Evaluate
Question: Evaluate (i) $\frac{4}{(216)^{-\frac{2}{3}}}+\frac{1}{(256)^{-\frac{3}{4}}}+\frac{2}{(243)^{-\frac{1}{5}}}$ (ii) $\left(\frac{64}{125}\right)^{-\frac{2}{3}}+\left(\frac{256}{625}\right)^{-\frac{1}{4}}+\left(\frac{3}{7}\right)^{0}$ (iii) $\left(\frac{81}{16}\right)^{-\frac{3}{4}}\left[\left(\frac{25}{9}\right)^{-\frac{3}{2}} \div\left(\frac{5}{2}\right)^{-3}\right]$ (iv) $\frac{(25)^{\frac{5}{2}} \times(729)^{\frac{1}{3}}}{(125)^{\frac{2}{3}} \times(27)^{\frac{2}{3}} \times 8^{\frac{4}{...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\tan A+\tan B}{\cot A+\cot B}=\tan A \tan B$ Solution: We have to prove $\frac{\tan A+\tan B}{\cot A+\cot B}=\tan A \tan B$ Now, $\frac{\tan A+\tan B}{\cot A+\cot B}=\frac{\tan A+\tan B}{\frac{1}{\tan A}+\frac{1}{\tan B}}$ $=\frac{\tan A+\tan B}{\frac{\tan B+\tan A}{\tan A \tan B}}$ $=\tan A \tan B$ Hence proved....
Read More →If e
Question: If $e^{\sin x}-e^{-\sin x}-4=0$, then $x=$ (a) 0 (b) $\sin ^{-1}\left\{\log _{\rho}(2-\sqrt{5})\right\}$ (c) 1 (d) none of these Solution: (d) none of these Given equation: $e^{\sin x}-e^{-\sin x}-4=0$ Let: $e^{\sin x}=y$ Now, $y-y^{-1}-4=0$ $\Rightarrow y^{2}-4 y-1=0$ $\therefore y=\frac{4 \pm \sqrt{16+4}}{2}$ $\Rightarrow y=\frac{4 \pm \sqrt{20}}{2}$ $\Rightarrow y=\frac{4 \pm 2 \sqrt{5}}{2}=2 \pm \sqrt{5}$ And, $y=e^{\sin x}$ $y=e^{\sin x}$ $\Rightarrow e^{\sin x}=2 \pm \sqrt{5}$ Ta...
Read More →If e
Question: If $e^{\sin x}-e^{-\sin x}-4=0$, then $x=$ (a) 0 (b) $\sin ^{-1}\left\{\log _{\rho}(2-\sqrt{5})\right\}$ (c) 1 (d) none of these Solution: (d) none of these Given equation: $e^{\sin x}-e^{-\sin x}-4=0$ Let: $e^{\sin x}=y$ Now, $y-y^{-1}-4=0$ $\Rightarrow y^{2}-4 y-1=0$ $\therefore y=\frac{4 \pm \sqrt{16+4}}{2}$ $\Rightarrow y=\frac{4 \pm \sqrt{20}}{2}$ $\Rightarrow y=\frac{4 \pm 2 \sqrt{5}}{2}=2 \pm \sqrt{5}$ And, $y=e^{\sin x}$ $y=e^{\sin x}$ $\Rightarrow e^{\sin x}=2 \pm \sqrt{5}$ Ta...
Read More →A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.
Question: A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes. Solution: The repeated tosses of a pair of dice are Bernoulli trials. Let X denote the number of times of getting doublets in an experiment of throwing two dice simultaneously four times. Probability of getting doublets in a single throw of the pair of dice is $p=\frac{6}{36}=\frac{1}{6}$ $\therefore q=1-p=1-\frac{1}{6}=\frac{5}{6}$ Clearly, X has the binomial distribu...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. (i) $\frac{\cot A+\tan B}{\cot B \tan A}=\cot A \tan B$ (ii) $\frac{\tan A+\tan B}{\cot A+\cot B}=\tan A \tan B$ Solution: (i) We have to prove $\frac{\cot A+\tan B}{\cot B+\tan A}=\cot A \tan B$ Now, $\frac{\cot A+\tan B}{\cot B+\tan A}=\frac{\cot A+\frac{1}{\cot B}}{\cot B+\frac{1}{\cot A}}$ $=\frac{\frac{\cot A \cot B+1}{\cot B}}{\frac{\cot A \cot B+1}{\cot A}}$ $=\frac{\cot A}{\cot B}$ $=\cot A \frac{1}{\cot B}$ $=\cot A \tan B$ Hence p...
Read More →The number of values of x in [0, 2π]
Question: The number of values of $x$ in $[0,2 \pi]$ that satisfy the equation $\sin ^{2} x-\cos x=\frac{1}{4}$ (a) 1 (b) 2 (c) 3 (d) 4 Solution: (b) 2 $\sin ^{2} x-\cos x=\frac{1}{4}$ $\Rightarrow\left(1-\cos ^{2} x\right)-\cos x=\frac{1}{4}$ $\Rightarrow 4-4 \cos ^{2} x-4 \cos x=1$ $\Rightarrow 4 \cos ^{2} x+4 \cos x-3=0$ $\Rightarrow 4 \cos ^{2} x+6 \cos x-2 \cos x-3=0$ $\Rightarrow 2 \cos x(2 \cos x+3)-1(2 \cos x+3)=0$ $\Rightarrow(2 \cos x+3)(2 \cos x-1)=0$ $\Rightarrow 2 \cos x+3=0$ or, $2...
Read More →The number of values of x in [0, 2π]
Question: The number of values of $x$ in $[0,2 \pi]$ that satisfy the equation $\sin ^{2} x-\cos x=\frac{1}{4}$ (a) 1 (b) 2 (c) 3 (d) 4 Solution: (b) 2 $\sin ^{2} x-\cos x=\frac{1}{4}$ $\Rightarrow\left(1-\cos ^{2} x\right)-\cos x=\frac{1}{4}$ $\Rightarrow 4-4 \cos ^{2} x-4 \cos x=1$ $\Rightarrow 4 \cos ^{2} x+4 \cos x-3=0$ $\Rightarrow 4 \cos ^{2} x+6 \cos x-2 \cos x-3=0$ $\Rightarrow 2 \cos x(2 \cos x+3)-1(2 \cos x+3)=0$ $\Rightarrow(2 \cos x+3)(2 \cos x-1)=0$ $\Rightarrow 2 \cos x+3=0$ or, $2...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\sin ^{2} A \cos ^{2} B-\cos ^{2} A \sin ^{2} B=\sin ^{2} A-\sin ^{2} B$ Solution: We know that, $\sin ^{2} A+\cos ^{2} A=1$ So have, $\sin ^{2} A \cos ^{2} B-\cos ^{2} A \sin ^{2} B=\sin ^{2} A\left(1-\sin ^{2} B\right)-\left(1-\sin ^{2} A\right) \sin ^{2} B$ $=\sin ^{2} A-\sin ^{2} A \sin ^{2} B-\sin ^{2} B+\sin ^{2} A \sin ^{2} B$ $=\sin ^{2} A-\sin ^{2} B$ Hence proved....
Read More →Simplify
Question: Simplify (i) $\left(\frac{81}{49}\right)^{-\frac{3}{2}}$ (ii) $(14641)^{0.25}$ (iii) $\left(\frac{32}{243}\right)^{-\frac{4}{5}}$ (iv) $\left(\frac{7776}{243}\right)^{-\frac{3}{5}}$ Solution: (i) $\left(\frac{81}{49}\right)^{-\frac{3}{2}}$ $\left(\frac{81}{49}\right)^{-\frac{3}{2}}=\left[\left(\frac{9}{7}\right)^{2}\right]^{-\frac{3}{2}}$ $=\left(\frac{9}{7}\right)^{-3}$ $=\left(\frac{7}{9}\right)^{3}$ $=\frac{343}{729}$ (ii) $(14641)^{0.25}$ $(14641)^{0.25}=(14641)^{\frac{25}{100}}$ $...
Read More →A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
Question: A die is thrown 6 times. If getting an odd number is a success, what is the probability of (i) 5 successes? (ii) at least 5 successes? (iii) at most 5 successes? Solution: The repeated tosses of a die are Bernoulli trials. Let X denote the number of successes of getting odd numbers in an experiment of 6 trials. Probability of getting an odd number in a single throw of a die is, $p=\frac{3}{6}=\frac{1}{2}$ $\therefore q=1-p=\frac{1}{2}$ X has a binomial distribution. Therefore, $\mathrm...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $(1+\cot A+\tan A)(\sin A-\cos A)=\frac{\sec A}{\operatorname{cosec}^{2} A}-\frac{\operatorname{cosec} A}{\sec ^{2} A}=\sin A \tan A$ $-\cot A \cos A$ Solution: We have prove that $(1+\cot A+\tan A)(\sin A-\cos A)=\frac{\sec A}{\operatorname{cosec}^{2} A}-\frac{\operatorname{cosec} A}{\sec ^{2} A}=\sin A \tan A-\cot A \cos A$ We know that, $\sin ^{2} A+\cos ^{2} A=1$ So, $(1+\cot A+\tan A)(\sin A-\cos A)$ $=\left(1+\frac{\cos A}{\sin A}+\fr...
Read More →In (0, π), the number of solutions of the equation tan
Question: In $(0, \pi)$, the number of solutions of the equation $\tan x+\tan 2 x+\tan 3 x=\tan x \tan 2 x \tan 3 x$ is (a) 7 (b) 5 (c) 4 (d) 2. Solution: (d) 2. Given equation: $\tan x+\tan 2 x+\tan 3 x=\tan x \tan 2 x \tan 3 x$ $\Rightarrow \tan x+\tan 2 x=-\tan 3 x+\tan x \tan 2 x \tan 3 x$ $\Rightarrow \tan x+\tan 2 x=-\tan 3 x(1-\tan x \tan 2 x)$ $\Rightarrow \frac{\tan x+\tan 2 x}{1-\tan x \tan 2 x}=-\tan 3 x$ $\Rightarrow \tan (x+2 x)=-\tan 3 x$ $\Rightarrow \tan 3 x=-\tan 3 x$ $\Rightarr...
Read More →In (0, π), the number of solutions of the equation tan
Question: In $(0, \pi)$, the number of solutions of the equation $\tan x+\tan 2 x+\tan 3 x=\tan x \tan 2 x \tan 3 x$ is (a) 7 (b) 5 (c) 4 (d) 2. Solution: (d) 2. Given equation: $\tan x+\tan 2 x+\tan 3 x=\tan x \tan 2 x \tan 3 x$ $\Rightarrow \tan x+\tan 2 x=-\tan 3 x+\tan x \tan 2 x \tan 3 x$ $\Rightarrow \tan x+\tan 2 x=-\tan 3 x(1-\tan x \tan 2 x)$ $\Rightarrow \frac{\tan x+\tan 2 x}{1-\tan x \tan 2 x}=-\tan 3 x$ $\Rightarrow \tan (x+2 x)=-\tan 3 x$ $\Rightarrow \tan 3 x=-\tan 3 x$ $\Rightarr...
Read More →find the values of
Question: If $a=2, b=3$, find the values of (i) $\left(a^{b}+b^{a}\right)^{-1}$ (ii) $\left(a^{a}+b^{b}\right)^{-1}$ Solution: (i) $\left(a^{b}+b^{a}\right)^{-1}$ $\left(a^{b}+b^{a}\right)^{-1}=\left(2^{3}+3^{2}\right)^{-1}$ $=(8+9)^{-1}$ $=(17)^{-1}$ $=\frac{1}{17}$ (ii) $\left(a^{a}+b^{b}\right)^{-1}$ $\left(a^{a}+b^{b}\right)^{-1}=\left(2^{2}+3^{3}\right)^{-1}$ $=(4+27)^{-1}$ $=(31)^{-1}$ $=\frac{1}{31}$...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\cot ^{2} A(\sec A-1)}{1+\sin A}=\sec ^{2} A\left(\frac{1-\sin A}{1+\sec A}\right)$ Solution: We have to prove $\frac{\cot ^{2} A(\sec A-1)}{1+\sin A}=\sec ^{2} A\left(\frac{1-\sin A}{1+\sec A}\right)$. We know that, $\sin ^{2} A+\cos ^{2} A=1$ So, $\frac{\cot ^{2} A(\sec A-1)}{1+\sin A}=\sec ^{2} A\left(\frac{1-\sin A}{1+\sec A}\right)$ $=\frac{\frac{\cos ^{2} A}{\sin ^{2} A}\left(\frac{1}{\cos A}-1\right)}{1+\sin A}$ $=\frac{\frac{...
Read More →A value of x satisfying cos x
Question: A value of $x$ satisfying $\cos x+\sqrt{3} \sin x=2$ is (a) $\frac{5 \pi}{3}$ (b) $\frac{4 \pi}{3}$ (c) $\frac{2 \pi}{3}$ (d) $\frac{\pi}{3}$ Solution: (d) $\frac{\pi}{3}$ Given equation: $\cos x+\sqrt{3} \sin x=2 \quad \ldots$ (i) Thus, the equation is of the form $a \cos x+b \sin x=c$, where $a=1, b=\sqrt{3}$ and $c=3$. Let: $a=r \cos \alpha$ and $b=r \sin \alpha$ $1=r \cos \alpha$ and $\sqrt{3}=r \sin \alpha$ $\Rightarrow r=\sqrt{a^{2}+b^{2}}=\sqrt{(\sqrt{3})^{2}+1^{2}}=2$ and $\tan...
Read More →Evaluate
Question: Evaluate (i) $(125)^{\frac{1}{3}}$ (ii) $(64)^{\frac{1}{6}}$ (iii) $(25)^{\frac{3}{2}}$ (iv) $(81)^{\frac{3}{4}}$ (v) $(64)^{-\frac{1}{2}}$ (vi) $(8)^{-\frac{1}{3}}$ Solution: $(\mathrm{i})(125)^{\frac{1}{3}}=\left(5^{3}\right)^{\frac{1}{3}}=5^{3 \times \frac{1}{3}}=5$ $(\mathrm{ii})(64)^{\frac{1}{6}}=\left(2^{6}\right)^{\frac{1}{6}}=2^{6 \times \frac{1}{6}}=2$ $($ iii $)(25)^{\frac{3}{2}}=5^{2 \times \frac{3}{2}}=5^{3}=125 \quad\left[\left(a^{m}\right)^{n}=a^{m n}\right]$ $($ iv $)(81...
Read More →A value of x satisfying cos x
Question: A value of $x$ satisfying $\cos x+\sqrt{3} \sin x=2$ is (a) $\frac{5 \pi}{3}$ (b) $\frac{4 \pi}{3}$ (c) $\frac{2 \pi}{3}$ (d) $\frac{\pi}{3}$ Solution: (d) $\frac{\pi}{3}$ Given equation: $\cos x+\sqrt{3} \sin x=2 \quad \ldots$ (i) Thus, the equation is of the form $a \cos x+b \sin x=c$, where $a=1, b=\sqrt{3}$ and $c=3$. Let: $a=r \cos \alpha$ and $b=r \sin \alpha$ $1=r \cos \alpha$ and $\sqrt{3}=r \sin \alpha$ $\Rightarrow r=\sqrt{a^{2}+b^{2}}=\sqrt{(\sqrt{3})^{2}+1^{2}}=2$ and $\tan...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\sec ^{4} A\left(1-\sin ^{4} A\right)-2 \tan ^{2} A=1$ Solution: We have to prove $\sec ^{4} A\left(1-\sin ^{4} A\right)-2 \tan ^{2} A=1$ We know that, $\sin ^{2} A+\cos ^{2} A=1$ So, $\sec ^{4} A\left(1-\sin ^{4} A\right)-2 \tan ^{2} A=\frac{1}{\cos ^{4} A}\left(1-\sin ^{4} A\right)-2 \frac{\sin ^{2} A}{\cos ^{2} A}$ $=\left(\frac{1}{\cos ^{4} A}-\frac{\sin ^{4} A}{\cos ^{4} A}\right)-2 \frac{\sin ^{2} A}{\cos ^{2} A}$ $=\left(\frac{1-\si...
Read More →If cot x −tan x=sec x,
Question: If $\cot x-\tan x=\sec x$, then, $x$ is equal to (a) $2 n \pi+\frac{3 \pi}{2}, n \in Z$ (b) $n \pi+(-1)^{n} \frac{\pi}{6}, n \in Z$ (c) $n \pi+\frac{\pi}{2}, n \in Z$ (d) none of these. Solution: (b) $n \pi+(-1)^{n} \frac{\pi}{6}, n \in Z$ Given equation: $\cot x-\tan x=\sec x$ $\Rightarrow \frac{\cos x}{\sin x}-\frac{\sin x}{\cos x}=\frac{1}{\cos x}$ $\Rightarrow \frac{\cos ^{2} x-\sin ^{2} x}{\sin x \cos x}=\frac{1}{\cos x}$ $\Rightarrow \cos ^{2} x-\sin ^{2} x=\sin x$ $\Rightarrow\l...
Read More →If cot x −tan x=sec x,
Question: If $\cot x-\tan x=\sec x$, then, $x$ is equal to (a) $2 n \pi+\frac{3 \pi}{2}, n \in Z$ (b) $n \pi+(-1)^{n} \frac{\pi}{6}, n \in Z$ (c) $n \pi+\frac{\pi}{2}, n \in Z$ (d) none of these. Solution: (b) $n \pi+(-1)^{n} \frac{\pi}{6}, n \in Z$ Given equation: $\cot x-\tan x=\sec x$ $\Rightarrow \frac{\cos x}{\sin x}-\frac{\sin x}{\cos x}=\frac{1}{\cos x}$ $\Rightarrow \frac{\cos ^{2} x-\sin ^{2} x}{\sin x \cos x}=\frac{1}{\cos x}$ $\Rightarrow \cos ^{2} x-\sin ^{2} x=\sin x$ $\Rightarrow\l...
Read More →Simplify:
Question: Simplify: (i) $\left(3^{4}\right)^{1 / 4}$ (ii) $\left(3^{1 / 3}\right)^{4}$ (iii) $\left(\frac{1}{3^{4}}\right)$ Solution: $(\mathrm{i})\left(3^{4}\right)^{\frac{1}{4}}=3^{4 \times \frac{1}{4}}=3$ $\left[\left((a)^{m}\right)^{n}=a^{m n}\right]$ $(\mathrm{ii})\left(3^{\frac{1}{3}}\right)^{4}=3^{\frac{1}{3} \times 4}=3^{\frac{4}{3}}$ $($ iii $)\left(\frac{1}{3^{4}}\right)^{\frac{1}{2}}=\left(\frac{1}{3}\right)^{4 \times \frac{1}{2}}=\left(\frac{1}{3}\right)^{2}=\frac{1}{9}=3^{-2}$...
Read More →