A and B are two events such that
Question: A and B are two events such that P (A) 0. Find P (B|A), if (i) $A$ is a subset of $B$ (ii) $A \cap B=\Phi$ Solution: It is given that, $P(A) \neq 0$ (i) A is a subset of B. $\Rightarrow \mathrm{A} \cap \mathrm{B}=\mathrm{A}$ $\therefore \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{B} \cap \mathrm{A})=\mathrm{P}(\mathrm{A})$ $\therefore \mathrm{P}(\mathrm{B} \mid \mathrm{A})=\frac{\mathrm{P}(\mathrm{B} \cap \mathrm{A})}{\mathrm{P}(\mathrm{A})}=\frac{\mathrm{P}(\mathrm{A})}{...
Read More →Write the following in ascending order of magnitude.
Question: Write the following in ascending order of magnitude. $\sqrt[6]{6}, \sqrt[3]{7}, \sqrt[4]{8}$ Solution: $\sqrt[6]{6}, \sqrt[3]{7}, \sqrt[4]{8}$ $\sqrt[6]{6}=(6)^{\frac{1}{6}}=(6)^{\frac{2}{12}}=\left(6^{2}\right)^{\frac{1}{12}}=(36)^{\frac{1}{12}} \quad \ldots(1)$ $\sqrt[3]{7}=(7)^{\frac{1}{3}}=(7)^{\frac{4}{12}}=\left(7^{4}\right)^{\frac{1}{12}}=(2401)^{\frac{1}{12}} \ldots(2)$ $\sqrt[4]{8}=(8)^{\frac{1}{4}}=(8)^{\frac{3}{12}}=\left(8^{3}\right)^{\frac{1}{12}}=(512)^{\frac{1}{12}} \ldo...
Read More →Write the number of values of x in [0, 2π]
Question: Write the number of values of $x$ in $[0,2 \pi]$ that satisfy the equation $\sin x-\cos x=\frac{1}{4}$. Solution: Given equation: $\sin ^{2} x-\cos x=\frac{1}{4}$ Now, $\left(1-\cos ^{2} x\right)-\cos x=\frac{1}{4}$ $\Rightarrow 4-4 \cos ^{2} x-4 \cos x=1$ $\Rightarrow 4 \cos ^{2} x+4 \cos x-3=0$ $\Rightarrow 4 \cos ^{2} x+6 \cos x-2 \cos x-3=0$ $\Rightarrow 2 \cos x(2 \cos x+3)-1(2 \cos x+3)=0$ $\Rightarrow(2 \cos x+3)(2 \cos x-1)=0$ Here, $2 \cos x+3=0 \Rightarrow \cos x=-\frac{3}{2}...
Read More →The probability that a student is not a swimmer
Question: The probability that a student is not a swimmer is $\frac{1}{5}$. Then the probability that out of five students, four are swimmers is (A) ${ }^{5} \mathrm{C}_{4}\left(\frac{4}{5}\right)^{4} \frac{1}{5}$ (B) $\left(\frac{4}{5}\right)^{4} \frac{1}{5}$ (C) ${ }^{5} \mathrm{C}_{1} \frac{1}{5}\left(\frac{4}{5}\right)^{4}$ (D) None of these Solution: The repeated selection of students who are swimmers are Bernoulli trials. Let X denote the number of students, out of 5 students, who are swim...
Read More →prove that
Question: If $\frac{9^{n} \times 3^{2} \times\left(3^{\frac{-n}{2}}\right)^{-2}-(27)^{n}}{3^{3 m} \times 2^{3}}=\frac{1}{27}$, prove that $m-n=1$ Solution: $\frac{9^{n} \times 3^{2} \times\left(3^{\frac{-n}{2}}\right)^{-2}-(27)^{n}}{3^{3 m} \times 2^{3}}=\frac{1}{27}$ $\Rightarrow \frac{\left(3^{2}\right)^{n} \times 3^{2} \times\left(3^{-n}\right)^{-1}-\left(3^{3}\right)^{n}}{3^{3 m} \times 2^{3}}=\frac{1}{3^{3}}$ $\Rightarrow \frac{3^{2 n} \times 3^{2} \times 3^{n}-3^{3 n}}{3^{3 m} \times 2^{3}...
Read More →Write the solution set of the equation
Question: Write the solution set of the equation $(2 \cos x+1)(4 \cos x+5)=0$ in the interval $[0,2 \pi]$. Solution: Given: $(2 \cos x+1)(4 \cos x+5)=0$ Now, $2 \cos x+1=0$ or $4 \cos x+5=0$ $\Rightarrow \cos x=-\frac{1}{2}$ or $\cos x=-\frac{5}{4}$ $\cos x=-\frac{5}{4}$ is not possible. Thus, we have: $\cos x=-\frac{1}{2}$ $\Rightarrow \cos x=\cos \frac{2 \pi}{3}$ $\Rightarrow x=2 n \pi \pm \frac{2 \pi}{3}$ By puttingn= 0 andn= 1 in the above equation, we get: $x=\frac{2 \pi}{3}$ or $x=\frac{4 ...
Read More →If cos θ=45, find all other trigonometric ratios of angle θ.
Question: If $\cos \theta=\frac{4}{5}$, find all other trigonometric ratios of angle $\theta$. Solution: Given: $\cos \theta=\frac{4}{5}$ Now, we have to find all the other trigonometric ratios. We have the following right angle triangle. From the above figure, Perpendicular $=\sqrt{\text { Hypotenuse }^{2}-\text { Base }^{2}}$ $\Rightarrow A B=\sqrt{A C^{2}-B C^{2}}$ $\Rightarrow A B=\sqrt{5^{2}-4^{2}}$ $\Rightarrow A B=3$ Therefore, $\sin \theta=\frac{A B}{A C}=\frac{3}{5}$ $\operatorname{cose...
Read More →Write the number of points of intersection of the curves 2y
Question: Write the number of points of intersection of the curves $2 y=-1$ and $y=\operatorname{cosec} x$. Solution: Given; $2 y=-1$ and $y=\operatorname{cosec} x$ Now, $2 y=-1 \Rightarrow y=-\frac{1}{2}$ Also, $\operatorname{cosec} x=y$ $\Rightarrow \operatorname{cosec} x=-\frac{1}{2}$ $\Rightarrow \frac{1}{\sin x}=-\frac{1}{2}$ $\Rightarrow \sin x=-2$ The value of sine function lies between $-1$ and 1 . Therefore, the two curves will not intersect at any point. Hence, the number of points of ...
Read More →In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
Question: In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is (A) $10^{-1}$ (B) $\left(\frac{1}{2}\right)^{5}$ (C) $\left(\frac{9}{10}\right)^{5}$ (D) $\frac{9}{10}$ Solution: The repeated selections of defective bulbs from a box are Bernoulli trials. Let X denote the number of defective bulbs out of a sample of 5 bulbs. Probability of getting a defective bulb, $p=\frac{10}{100}=\frac{1}{10}$ $\therefore q=1-p=1-\frac{1}{10}=\fra...
Read More →If x is a positive real number and exponents are rational numbers, simplify
Question: Ifxis a positive real number and exponents are rational numbers, simplify $\left(\frac{x^{b}}{x^{c}}\right)^{b+c-a} \cdot\left(\frac{x^{c}}{x^{a}}\right)^{c+a-b} \cdot\left(\frac{x^{a}}{x^{b}}\right)^{a+b-c}$ Solution: $\left(\frac{x^{b}}{x^{c}}\right)^{b+c-a} \cdot\left(\frac{x^{c}}{x^{a}}\right)^{c+a-b} \cdot\left(\frac{x^{a}}{x^{b}}\right)^{a+b-c}$ $=\left(x^{b-c}\right)^{b+c-a} \cdot\left(x^{c-a}\right)^{c+a-b} \cdot\left(x^{a-b}\right)^{a+b-c}$ $=\left[\left(x^{b-c}\right)^{b} \cd...
Read More →Write the values of x in [0, π] for which sin 2x,
Question: Write the values of $x$ in $[0, \pi]$ for which $\sin 2 x, \frac{1}{2}$ and $\cos 2 x$ are in A.P. Solution: (i) $\sin 2 x, \frac{1}{2}$ and $\cos 2 x$ are in AP. $\therefore \sin 2 x+\cos 2 x=2 \times \frac{1}{2}$ $\Rightarrow \sin 2 x+\cos 2 x=1 \quad \ldots$ (1) This equation is of the form $a \sin \theta+b \cos \theta=c$, where $a=1, b=1$ and $c=1$. Now, Let: $a=r \sin \alpha$ and $b=r \cos \alpha$ Thus, we have: $r=\sqrt{a^{2}+b^{2}}=\sqrt{1^{2}+1^{2}}=\sqrt{2}$ and $\tan \alpha=1...
Read More →It is known that 10% of certain articles manufactured are defective.
Question: It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective? Solution: The repeated selections of articles in a random sample space are Bernoulli trails. Let X denote the number of times of selecting defective articles in a random sample space of 12 articles. Clearly, $X$ has a binomial distribution with $n=12$ and $p=10 \%=\frac{10}{100}=\frac{1}{10}$ $\therefore q=1-p=1-\frac{1}{10}=\frac{9}...
Read More →If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, show that x2a2+y2b2−z2c2=1.
Question: If $x=a \sec \theta \cos \phi, y=b \sec \theta \sin \phi$ and $z=c \tan \theta$, show that $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1$. Solution: Given: $x=a \sec \theta \cos \phi$ $\Rightarrow \frac{x}{a}=\sec \theta \cos \phi$ .......(1) $y=b \sec \theta \sin \phi$ $\Rightarrow \frac{y}{b}=\sec \theta \sin \phi$ .......(2) $\Rightarrow \frac{z}{c}=\tan \theta$ ........(3) We have to prove that $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1$. Squari...
Read More →Prove that
Question: Prove that (i) $\sqrt{x^{-1} y} \cdot \sqrt{y^{-1} z} \cdot \sqrt{z^{-1} x}=1$. (ii) $\left(x^{\frac{1}{a-b}}\right)^{\frac{1}{a-c}} \cdot\left(x^{\frac{1}{b-c}}\right)^{\frac{1}{b-a}} \cdot\left(x^{\frac{1}{c-a}}\right)^{\frac{1}{c-b}}=1$ (iii) $\frac{x^{a(b-c)}}{x^{b(a-c)}} \div\left(\frac{x^{b}}{x^{a}}\right)^{c}=1$ (iv) $\frac{\left(x^{a+b}\right)^{2}\left(x^{b+c}\right)^{2}\left(x^{c+a}\right)^{2}}{\left(x^{a} x^{b} x^{c}\right)^{4}}=1$ Solution: (i) $\sqrt{x^{-1} y} \cdot \sqrt{y...
Read More →Write the number of points of intersection of the curves
Question: Write the number of points of intersection of the curves $2 y=1$ and $y=\cos x, 0 \leq x \leq 2 \pi$. Solution: Given curves: $2 y=1$ and $y=\cos x$ Now, $2 y=1 \Rightarrow y=\frac{1}{2}$ Also, $\cos x=y$ $\Rightarrow \cos x=\frac{1}{2}$ $\Rightarrow \cos x=\cos \left(\frac{\pi}{3}\right)$ and $\cos x=\cos \left(\frac{4 \pi}{3}\right)$ $\Rightarrow x=2 n \pi \pm \frac{\pi}{3} \quad$ or $\quad x=2 n \pi \pm \frac{4 \pi}{3}$ By putting $n=0$, we get: $x=\frac{\pi}{3}$ and $x=\frac{2 \pi}...
Read More →If sin θ + cos θ = x, prove that sin6θ+cos6θ=4−3 (x2−1)24.
Question: If $\sin \theta+\cos \theta=x$, prove that $\sin ^{6} \theta+\cos ^{6} \theta=\frac{4-3\left(x^{2}-1\right)^{2}}{4}$. Solution: Given: $\sin \theta+\cos \theta=x$ Squaring the given equation, we have $(\sin \theta+\cos \theta)^{2}=x^{2}$ $\Rightarrow \sin ^{2} \theta+2 \sin \theta \cos \theta+\cos ^{2} \theta=x^{2}$ $\Rightarrow\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+2 \sin \theta \cos \theta=x^{2}$ $\Rightarrow \quad 1+2 \sin \theta \cos \theta=x^{2}$ $\Rightarrow \quad 2 \sin ...
Read More →Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Question: Find the probability of throwing at most 2 sixes in 6 throws of a single die. Solution: The repeated tossing of the die are Bernoulli trials. Let X represent the number of times of getting sixes in 6 throws of the die. Probability of getting six in a single throw of die, $p=\frac{1}{6}$ $\therefore q=1-p=1-\frac{1}{6}=\frac{5}{6}$ Clearly, X has a binomial distribution withn= 6 $\therefore \mathrm{P}(\mathrm{X}=x)={ }^{1} \mathrm{C}_{x} q^{n-x} p^{x}={ }^{6} \mathrm{C}_{x}\left(\frac{5...
Read More →If cos x = k has exactly one solution in [0, 2π],
Question: If cosx=khas exactly one solution in [0, 2], then write the values(s) ofk. Solution: Given: $\cos x=k$ If $k=0$, then $\cos x=0$ $\Rightarrow \cos x=\cos \frac{\pi}{2}$ $\Rightarrow x=(2 n+1) \frac{\pi}{2}, n \in Z$ Now, $x=\frac{3 \pi}{2}, \frac{5 \pi}{2}, \frac{7 \pi}{2}, \ldots$ for $n=1,2,3, \ldots$ If $k=1$, then $\cos x=1$ $\Rightarrow \cos x=\cos 0$ $\Rightarrow x=2 m \pi, \mathrm{m} \in \mathrm{Z}$ Now, $x=2 \pi, 4 \pi, 6 \pi, 8 \pi, \ldots$ for $m=1,2,3,4, \ldots$ If $k=-1$, t...
Read More →Given that:
Question: Given that: $(1+\cos \alpha)(1+\cos \beta)(1+\cos \gamma)=(1-\cos \alpha)(1-\cos \alpha)(1-\cos \beta)(1-\cos \gamma)$ Show that one of the values of each member of this equality is $\sin \alpha \sin \beta \sin \gamma$ Solution: Given: $(1+\cos \alpha)(1+\cos \beta)(1+\cos \gamma)=(1-\cos \alpha)(1-\cos \beta)(1-\cos \gamma)$ Let us assume that $(1+\cos \alpha)(1+\cos \beta)(1+\cos \gamma)=(1-\cos \alpha)(1-\cos \beta)(1-\cos \gamma)=L$ We know that, $\sin ^{2} \theta+\cos ^{2} \theta=...
Read More →Write the set of values of a for which the equation
Question: Write the set of values of a for which the equation $\sqrt{3} \sin x-\cos x=a$ has no solution. Solution: Given; $\sqrt{3} \sin x-\cos x=a$ $\Rightarrow \frac{\sqrt{3} \sin x-\cos x}{2}=\frac{a}{2}$ $\Rightarrow \frac{\sqrt{3}}{2} \sin x-\frac{1}{2} \cos x=\frac{a}{2}$ $\Rightarrow \cos 30^{\circ} \sin x-\sin 30^{\circ} \cos x=\frac{a}{2}$ $\Rightarrow \sin \left(x-30^{\circ}\right)=\frac{a}{2}$ $\Rightarrow x-30^{\circ}=\sin ^{-1}\left(\frac{a}{2}\right)$ $\Rightarrow x=\sin ^{-1}\lef...
Read More →Write the general solutions of tan
Question: Write the general solutions of tan22x= 1. Solution: Given: $\tan ^{2} 2 x=1$ $\Rightarrow \tan 2 x=\tan \frac{\pi}{4}$ $\Rightarrow 2 x=n \pi+\frac{\pi}{4}$ $\Rightarrow x=\frac{n \pi}{2}+\frac{\pi}{8}, n \in Z$ Hence, the general solution of the equation is $\frac{n \pi}{2}+\frac{\pi}{8}, n \in Z$....
Read More →Find the probability of getting 5 exactly twice in 7 throws of a die.
Question: Find the probability of getting 5 exactly twice in 7 throws of a die. Solution: The repeated tossing of a die are Bernoulli trials. Let X represent the number of times of getting 5 in 7 throws of the die. Probability of getting 5 in a single throw of the die, $p=\frac{1}{6}$ $\therefore q=1-p=1-\frac{1}{6}=\frac{5}{6}$ Clearly, $\mathrm{X}$ has the probability distribution with $n=7$ and $p=\frac{1}{6}$ $\therefore \mathrm{P}(\mathrm{X}=x)={ }^{9} \mathrm{C}_{x} q^{n-x} p^{x}={ }^{7} \...
Read More →If cos θ + cos2 θ = 1, prove that sin12 θ + 3 sin10 θ + 3 sin8 θ + sin6 θ + 2 sin4 θ + 2 sin2 θ − 2 = 1
Question: If $\cos \theta+\cos ^{2} \theta=1$, prove that $\sin ^{12} \theta+3 \sin ^{10} \theta+3 \sin ^{8} \theta+\sin ^{6} \theta+2 \sin ^{4} \theta+2 \sin ^{2} \theta-2=1$ Solution: Given: $\cos \theta+\cos ^{2} \theta=1$ We have to prove $\sin ^{12} \theta+3 \sin ^{10} \theta+3 \sin ^{8} \theta+\sin ^{6} \theta+2 \sin ^{4} \theta+2 \sin ^{2} \theta-2=1$ From the given equation, we have $\cos \theta+\cos ^{2} \theta=1$ $\Rightarrow \quad \cos \theta=1-\cos ^{2} \theta$ $\Rightarrow \quad \co...
Read More →Write the number of solutions of the equation
Question: Write the number of solutions of the equation $4 \sin x-3 \cos x=7$. Solution: We have: $4 \sin x-3 \cos x=7 \ldots$..(i) The equation is of the form $a \sin x+b \cos x=c$, where $a=4, b=-3$ and $c=7$.Now, Let: $a=r \sin \alpha$ and $b=r \cos \alpha$ Thus, we have: $r=\sqrt{a^{2}+b^{2}}=\sqrt{4^{2}+3^{2}}=5$ and $\tan \alpha=\frac{-4}{3} \Rightarrow \alpha=\tan ^{-1}\left(-\frac{4}{3}\right)$ By putting $a=4=r \sin \alpha$ and $b=-3=r \cos \alpha$ in equation (i), we get: $r \sin \alph...
Read More →Write the number of solutions of the equation
Question: Write the number of solutions of the equation $4 \sin x-3 \cos x=7$. Solution: We have: $4 \sin x-3 \cos x=7 \ldots$..(i) The equation is of the form $a \sin x+b \cos x=c$, where $a=4, b=-3$ and $c=7$.Now, Let: $a=r \sin \alpha$ and $b=r \cos \alpha$ Thus, we have: $r=\sqrt{a^{2}+b^{2}}=\sqrt{4^{2}+3^{2}}=5$ and $\tan \alpha=\frac{-4}{3} \Rightarrow \alpha=\tan ^{-1}\left(-\frac{4}{3}\right)$ By putting $a=4=r \sin \alpha$ and $b=-3=r \cos \alpha$ in equation (i), we get: $r \sin \alph...
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