Number of solutions of the equation tan x + sec x = 2 cos x lying in the interval
Question: Number of solutions of the equation tanx+ secx= 2 cosxlying in the interval [0, 2] is (a) 0 (b) 1 (c) 2 (d) 3 Solution: tanx+ secx= 2 cosxin[0, 2] i.e $\frac{\sin x}{\cos x}+\frac{1}{\cos x}=2 \cos x$ where $x \notin(2 x+1) \frac{\pi}{2}$ ( $\because \cos x$ is not defined) i.e sinx+ 1 = 2 cos2x ⇒ sinx+ 1 = 2(1 sin2x) (∵ sin2x+ cos2x= 1) i.e 2 sin2x+ sinx 1 = 0 i.e 2 sin2x+ 2 sinx sinx 1 = 0 i.e 2 sinx(sinx+ 1)1 (sinx+ 1) = 0 i.e (2 sinx 1) (sinx+ 1) = 0 i.e $\sin x=\frac{1}{2}$ or $\s...
Read More →Number of solutions of the equation tan x + sec x = 2 cos x lying in the interval
Question: Number of solutions of the equation tanx+ secx= 2 cosxlying in the interval [0, 2] is (a) 0 (b) 1 (c) 2 (d) 3 Solution: tanx+ secx= 2 cosxin[0, 2] i.e $\frac{\sin x}{\cos x}+\frac{1}{\cos x}=2 \cos x$ where $x \notin(2 x+1) \frac{\pi}{2}$ ( $\because \cos x$ is not defined) i.e sinx+ 1 = 2 cos2x ⇒ sinx+ 1 = 2(1 sin2x) (∵ sin2x+ cos2x= 1) i.e 2 sin2x+ sinx 1 = 0 i.e 2 sin2x+ 2 sinx sinx 1 = 0 i.e 2 sinx(sinx+ 1)1 (sinx+ 1) = 0 i.e (2 sinx 1) (sinx+ 1) = 0 i.e $\sin x=\frac{1}{2}$ or $\s...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $a \cos ^{3} \theta+3 a \cos \theta \sin ^{2} \theta=m, a \sin ^{3} \theta+3 a \cos ^{2} \theta \sin \theta=n$, prove that $(m+n)^{2 / 3}+(m-n)^{2 / 3}=2 a^{2 / 3}$ Solution: Given that, $a \cos ^{3} \theta+3 a \cos \theta \sin ^{2} \theta=m$ $a \sin ^{3} \theta+3 a \cos ^{2} \theta \sin \theta=n$ We have to prove $(m+n)^{\frac{2}{3}}+(m-n)^{\frac{2}{3}}=2 a^{\frac{2}{3}}$ Adding both the equations, we get $m+n=a \cos ^{3} \theta+3 a \co...
Read More →A bag consists of 10 balls each marked with one of the digits 0 to 9.
Question: A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0? Solution: Let X denote the number of balls marked with the digit 0 among the 4 balls drawn. Since the balls are drawn with replacement, the trials are Bernoulli trials. $X$ has a binomial distribution with $n=4$ and $p=\frac{1}{10}$ $\therefore q=1-p=1-\frac{1}{10}=\frac{9}{10}$ $\therefor...
Read More →The number of values of x in the interval [0, 5 π]
Question: The number of values of $x$ in the interval $[0,5 \pi]$ satisfying the equation $3 \sin ^{2} x-7 \sin x+2=0$ is (a) 0 (b) 5 (c) 6 (d) 10 Solution: (c) 6 Given: $3 \sin ^{2} x-7 \sin x+2=0$ $\Rightarrow 3 \sin ^{2} x-6 \sin x-\sin x+2=0$ $\Rightarrow 3 \sin x(\sin x-2)-1(\sin x-2)=0$ $\Rightarrow(3 \sin x-1)(\sin x-2)=0$ $\Rightarrow 3 \sin x-1=0$ or $\sin x-2=0$ Now, $\sin x=2$ is not possible, as the value of $\sin x$ lies between $-1$ and 1 . $\Rightarrow \sin x=\frac{1}{a}$ Also, $\...
Read More →The number of values of x in the interval [0, 5 π]
Question: The number of values of $x$ in the interval $[0,5 \pi]$ satisfying the equation $3 \sin ^{2} x-7 \sin x+2=0$ is (a) 0 (b) 5 (c) 6 (d) 10 Solution: (c) 6 Given: $3 \sin ^{2} x-7 \sin x+2=0$ $\Rightarrow 3 \sin ^{2} x-6 \sin x-\sin x+2=0$ $\Rightarrow 3 \sin x(\sin x-2)-1(\sin x-2)=0$ $\Rightarrow(3 \sin x-1)(\sin x-2)=0$ $\Rightarrow 3 \sin x-1=0$ or $\sin x-2=0$ Now, $\sin x=2$ is not possible, as the value of $\sin x$ lies between $-1$ and 1 . $\Rightarrow \sin x=\frac{1}{a}$ Also, $\...
Read More →Simplify the product
Question: Simplify the product $\sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32}$. Solution: $\sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32}=(2)^{\frac{1}{3}} \cdot(2)^{\frac{1}{4}} \cdot(32)^{\frac{1}{12}}$ $=(2)^{\frac{1}{3}} \cdot(2)^{\frac{1}{4}} \cdot\left(2^{5}\right)^{\frac{1}{12}}$ $=(2)^{\frac{1}{3}} \cdot(2)^{\frac{1}{4}} \cdot(2)^{\frac{5}{12}}$ $=(2)^{\frac{1}{3}+\frac{1}{4}+\frac{5}{12}}$ $=(2)^{\frac{4+3+5}{12}}$ $=(2)^{\frac{12}{12}}$ $=(2)^{1}$ $=2$...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $\operatorname{cosec} \theta-\sin \theta=a^{3}, \sec \theta-\cos \theta=b^{3}$, prove that $a^{2} b^{2}\left(a^{2}+b^{2}\right)=1$ Solution: Given that, $\operatorname{cosec} \theta-\sin \theta=a^{3}$.......(1) $\sec \theta-\cos \theta=b^{3}$........(2) We have to prove $a^{2} b^{2}\left(a^{2}+b^{2}\right)=1$ We know that $\sin ^{2} \theta+\cos ^{2} \theta=1$ Now from the first equation, we have $\operatorname{cosec} \theta-\sin \theta=a...
Read More →Simplify
Question: Simplify $\sqrt[4]{\sqrt[3]{x^{2}}}$ and express the result in the exponential form of $x$. Solution: $\sqrt[4]{\sqrt[3]{x^{2}}}=\left[\left(x^{2}\right)^{\frac{1}{3}}\right]^{\frac{1}{4}}$ $=\left[x^{\frac{2}{3}}\right]^{\frac{1}{4}}$ $=x^{\frac{2}{12}}$ $=x^{\frac{1}{6}}$ Hence, the result in the exponential form is $x^{\frac{1}{6}}$....
Read More →If cos x=
Question: If $\cos x=-\frac{1}{2}$ and $0x2 \pi$, then the solutions are (a) $x=\frac{\pi}{3}, \frac{4 \pi}{3}$ (b) $x=\frac{2 \pi}{3}, \frac{4 \pi}{3}$ (c) $x=\frac{2 \pi}{3}, \frac{7 \pi}{6}$ (d) $\theta=\frac{2 \pi}{3}, \frac{5 \pi}{3}$ Solution: (b) $x=\frac{2 \pi}{3}, \frac{4 \pi}{3}$ Given equation: $\cos x=-\frac{1}{2}$ $\Rightarrow \cos x=\cos \frac{2 \pi}{3}$ $\Rightarrow x=\frac{2 \pi}{2}$ Or, $\cos x=\cos \frac{4 \pi}{3}$ $\Rightarrow x=\frac{4 \pi}{3}$ So, both $\frac{2 \pi}{3}$ and ...
Read More →If cos x=
Question: If $\cos x=-\frac{1}{2}$ and $0x2 \pi$, then the solutions are (a) $x=\frac{\pi}{3}, \frac{4 \pi}{3}$ (b) $x=\frac{2 \pi}{3}, \frac{4 \pi}{3}$ (c) $x=\frac{2 \pi}{3}, \frac{7 \pi}{6}$ (d) $\theta=\frac{2 \pi}{3}, \frac{5 \pi}{3}$ Solution: (b) $x=\frac{2 \pi}{3}, \frac{4 \pi}{3}$ Given equation: $\cos x=-\frac{1}{2}$ $\Rightarrow \cos x=\cos \frac{2 \pi}{3}$ $\Rightarrow x=\frac{2 \pi}{2}$ Or, $\cos x=\cos \frac{4 \pi}{3}$ $\Rightarrow x=\frac{4 \pi}{3}$ So, both $\frac{2 \pi}{3}$ and ...
Read More →Prove that
Question: Prove that (i) $\left[8^{-\frac{2}{3}} \times 2^{\frac{1}{2}} \times 25^{-\frac{5}{4}}\right] \div\left[32^{-\frac{2}{5}} \times 125^{-\frac{5}{6}}\right]=\sqrt{2}$ (ii) $\left(\frac{64}{125}\right)^{-\frac{2}{3}}$ $+\frac{1}{\left(\frac{256}{625}\right)^{\frac{1}{4}}}+\frac{\sqrt{25}}{\sqrt[3]{64}}=\frac{65}{16}$ (iii) $\left[7\left\{(81)^{\frac{1}{4}}+(256)^{\frac{1}{4}}\right\}^{\frac{1}{4}}\right]^{4}=16807$ Solution: (i) $\left[8^{-\frac{2}{3}} \times 2^{\frac{1}{2}} \times 25^{-\...
Read More →The solution of the equation cos
Question: The solution of the equation $\cos ^{2} x+\sin x+1=0$ lies in the interval (a) $(-\pi / 4, \pi / 4)$ (b) $(\pi / 4,3 \pi / 4)$ (c) $(3 \pi / 4,5 \pi / 4)$ (d) $(5 \pi / 4,7 \pi / 4)$ Solution: Given equation: $\cos ^{2} x+\sin x+1=0$ $\Rightarrow\left(1-\sin ^{2} x\right)+\sin x+1=0$ $\Rightarrow 2-\sin ^{2} x+\sin x=0$ $\Rightarrow \sin ^{2} x-\sin x-2=0$ $\Rightarrow \sin ^{2} x-2 \sin x+\sin x-2=0$ $\Rightarrow \sin x(\sin x-2)+1(\sin x-2)=0$ $\Rightarrow(\sin x-2)(\sin x+1)=0$ $\Ri...
Read More →The solution of the equation cos
Question: The solution of the equation $\cos ^{2} x+\sin x+1=0$ lies in the interval (a) $(-\pi / 4, \pi / 4)$ (b) $(\pi / 4,3 \pi / 4)$ (c) $(3 \pi / 4,5 \pi / 4)$ (d) $(5 \pi / 4,7 \pi / 4)$ Solution: Given equation: $\cos ^{2} x+\sin x+1=0$ $\Rightarrow\left(1-\sin ^{2} x\right)+\sin x+1=0$ $\Rightarrow 2-\sin ^{2} x+\sin x=0$ $\Rightarrow \sin ^{2} x-\sin x-2=0$ $\Rightarrow \sin ^{2} x-2 \sin x+\sin x-2=0$ $\Rightarrow \sin x(\sin x-2)+1(\sin x-2)=0$ $\Rightarrow(\sin x-2)(\sin x+1)=0$ $\Ri...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$ and $\frac{x}{a} \sin \theta-\frac{y}{b} \cos \theta=1$, prove that $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=2$ Solution: Given that, $\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$ ........(1) $\frac{x}{a} \sin \theta-\frac{y}{b} \cos \theta=1$.........(2) We have to prove $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=2$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$ Squaring and t...
Read More →The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. What is the probability that out of 5 such bulbs
Question: The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. What is the probability that out of 5 such bulbs (i) none (ii) not more than one (iii) more than one (iv) at least one will fuse after 150 days of use. Solution: Let X represent the number of bulbs that will fuse after 150 days of use in an experiment of 5 trials. The trials are Bernoulli trials. It is given that,p= 0.05 $\therefore q=1-p=1-0.05=0.95$ X has a binomial distribution withn= 5 andp= ...
Read More →General solution of tan 5 x=cot 2 x is
Question: General solution of $\tan 5 x=\cot 2 x$ is (a) $\frac{n \pi}{7}+\frac{\pi}{2}, n \in Z$ (b) $x=\frac{n \pi}{7}+\frac{\pi}{3}, n \in Z$ (c) $x=\frac{n \pi}{7}+\frac{\pi}{14}, n \in Z$ (d) $x=\frac{n \pi}{7}-\frac{\pi}{14}, n \in Z$ Solution: (c) $x=\frac{n \pi}{7}+\frac{\pi}{14}, n \in Z$ Given; $\tan 5 x=\cot 2 x$ $\Rightarrow \tan 5 x=\tan \left(\frac{\pi}{2}-2 x\right)$ $\Rightarrow 5 x=n \pi+\frac{\pi}{2}-2 x$ $\Rightarrow 7 x=\mathrm{n} \pi+\frac{\pi}{2}$ $\Rightarrow x=\frac{\math...
Read More →If
Question: If $\sqrt{3} \cos x+\sin x=\sqrt{2}$, then general value of $x$ is (a) $n \pi+(-1)^{n} \frac{\pi}{4}, n \in Z$ (b) $(-1)^{n} \frac{\pi}{4}-\frac{\pi}{3}, n \in Z$ (c) $n \pi+\frac{\pi}{4}-\frac{\pi}{3}, n \in Z$ (d) $n \pi+(-1)^{n} \frac{\pi}{4}-\frac{\pi}{3}, n \in Z$ Solution: (d) $n \pi+(-1)^{n} \frac{\pi}{4}-\frac{\pi}{3}, n \in Z$ Given equation: $\sqrt{3} \cos x+\sin x=\sqrt{2} \ldots$.(i) This is of the form $a \cos x+b \sin x=c$, where $a=\sqrt{3}, b=1$ and $c=\sqrt{2}$. Let: $...
Read More →If
Question: If $\sqrt{3} \cos x+\sin x=\sqrt{2}$, then general value of $x$ is (a) $n \pi+(-1)^{n} \frac{\pi}{4}, n \in Z$ (b) $(-1)^{n} \frac{\pi}{4}-\frac{\pi}{3}, n \in Z$ (c) $n \pi+\frac{\pi}{4}-\frac{\pi}{3}, n \in Z$ (d) $n \pi+(-1)^{n} \frac{\pi}{4}-\frac{\pi}{3}, n \in Z$ Solution: (d) $n \pi+(-1)^{n} \frac{\pi}{4}-\frac{\pi}{3}, n \in Z$ Given equation: $\sqrt{3} \cos x+\sin x=\sqrt{2} \ldots$.(i) This is of the form $a \cos x+b \sin x=c$, where $a=\sqrt{3}, b=1$ and $c=\sqrt{2}$. Let: $...
Read More →Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that
Question: Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that (i) all the five cards are spades? (ii) only 3 cards are spades? (iii) none is a spade? Solution: Let X represent the number of spade cards among the five cards drawn. Since the drawing of card is with replacement, the trials are Bernoulli trials. In a well shuffled deck of 52 cards, there are 13 spade cards. $\Rightarrow p=\frac{13}{52}=\frac{1}{4}$ $\therefore q=1-\f...
Read More →Evaluate
Question: Evaluate (i) $\left(1^{3}+2^{3}+3^{3}\right)^{\frac{1}{2}}$ (ii) $\left[5\left(8^{\frac{1}{3}}+27^{\frac{1}{3}}\right)^{3}\right]^{\frac{1}{4}}$ (iii) $\frac{2^{0}+7^{0}}{5^{0}}$ (iv) $\left[(16)^{\frac{1}{2}}\right]^{\frac{1}{2}}$ Solution: (i) $\left(1^{3}+2^{3}+3^{3}\right)^{\frac{1}{2}}$ $\left(1^{3}+2^{3}+3^{3}\right)^{\frac{1}{2}}=(1+8+27)^{\frac{1}{2}}$ $=(36)^{\frac{1}{2}}$ $=\left[(6)^{2}\right]^{\frac{1}{2}}$ $=6$ (ii) $\left[5\left(8^{\frac{1}{3}}+27^{\frac{1}{3}}\right)^{3}...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $x=a \sec \theta+b \tan \theta$ and $y=a \tan \theta+b \sec \theta$, prove that $x^{2}-y^{2}=a^{2}-b^{2}$ Solution: Given that, $x=a \sec \theta+b \tan \theta$, $y=a \tan \theta+b \sec \theta$ We have to prove $x^{2}-y^{2}=a^{2}-b^{2}$ We know that, $\sec ^{2} \theta-\tan ^{2} \theta=1$ So, $x^{2}-y^{2}$ $=(a \sec \theta+b \tan \theta)^{2}-(a \tan \theta+b \sec \theta)^{2}$ $=\left(a^{2} \sec ^{2} \theta+2 a b \sec \theta \tan \theta+b^{...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\tan ^{2} A \sec ^{2} B-\sec ^{2} A \tan ^{2} B=\tan ^{2} A-\tan ^{2} B$ Solution: We have to prove $\tan ^{2} A \sec ^{2} B-\sec ^{2} A \tan ^{2} B=\tan ^{2} A-\tan ^{2} B$ We know that, $\sec ^{2} A-\tan ^{2} A=1$ So, $\tan ^{2} A \sec ^{2} B-\sec ^{2} A \tan ^{2} B=\tan ^{2} A\left(1+\tan ^{2} B\right)-\left(1+\tan ^{2} A\right) \tan ^{2} B$ $=\tan ^{2} A+\tan ^{2} A \tan ^{2} B-\tan ^{2} B-\tan ^{2} A \tan ^{2} B$ $=\tan ^{2} A-\tan ^{...
Read More →The equation 3 cos x+4 sin x=6 has
Question: The equation $3 \cos x+4 \sin x=6$ has .... solution. (a) finite (b) infinite (c) one (d) no Solution: (d) no Given equation: $3 \cos x+4 \sin x=6 \quad \ldots(\mathrm{i})$ Thus, the equation is of the form $a \cos x+b \sin x=c$, where $a=3, b=4$ and $c=6$. Let: $a=3=r \cos \alpha$ and $b=4=r \sin \alpha$ Now, $\tan \alpha=\frac{b}{a}=\frac{4}{3}$ $\Rightarrow \alpha=\tan ^{-1}\left(\frac{4}{3}\right)$ Aso, $r=\sqrt{a^{2}+b^{2}}=\sqrt{9+16}=\sqrt{25}=5$ On putting $a=3=r \cos \alpha$ a...
Read More →The equation 3 cos x+4 sin x=6 has
Question: The equation $3 \cos x+4 \sin x=6$ has .... solution. (a) finite (b) infinite (c) one (d) no Solution: (d) no Given equation: $3 \cos x+4 \sin x=6 \quad \ldots(\mathrm{i})$ Thus, the equation is of the form $a \cos x+b \sin x=c$, where $a=3, b=4$ and $c=6$. Let: $a=3=r \cos \alpha$ and $b=4=r \sin \alpha$ Now, $\tan \alpha=\frac{b}{a}=\frac{4}{3}$ $\Rightarrow \alpha=\tan ^{-1}\left(\frac{4}{3}\right)$ Aso, $r=\sqrt{a^{2}+b^{2}}=\sqrt{9+16}=\sqrt{25}=5$ On putting $a=3=r \cos \alpha$ a...
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