Suppose that two cards are drawn at random from a deck of cards.
Question: Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is (A) $\frac{37}{221}$ (B) $\frac{5}{13}$ (C) $\frac{1}{13}$ (D) $\frac{2}{13}$ Solution: Let X denote the number of aces obtained. Therefore, X can take any of the values of 0, 1, or 2. In a deck of 52 cards, 4 cards are aces. Therefore, there are 48 non-ace cards. $\therefore \mathrm{P}(\mathrm{X}=0)=\mathrm{P}(0$ ace and 2 non-ace cards $)=\frac{{ }^{4} \mat...
Read More →The smallest value of x satisfying the equation
Question: The smallest value of $x$ satisfying the equation $\sqrt{3}(\cot x+\tan x)=4$ is (a) $2 \pi / 3$ (b) $\pi / 3$ (c) $\pi / 6$ (d) $\pi / 12$ Solution: (c) $\pi / 6$ Given: $\sqrt{3}(\cot x+\tan x)=4$ $\Rightarrow \sqrt{3}\left(\frac{\cos x}{\sin x}+\frac{\sin x}{\cos x}\right)=4$ $\Rightarrow \sqrt{3}\left(\cos ^{2} x+\sin ^{2} x\right)=4 \sin x \cos x$ $\Rightarrow \sqrt{3}=2 \sin 2 x \quad[\sin 2 x=2 \sin x \cos x]$ $\Rightarrow \sin 2 x=\frac{\sqrt{3}}{2}$ $\Rightarrow \sin 2 x=\sin ...
Read More →find the value of
Question: If $x=\frac{1}{2-\sqrt{3}}$, find the value of $x^{3}-2 x^{2}-7 x+5$ Solution: $x=\frac{1}{2-\sqrt{3}}$ $\Rightarrow x=\frac{1}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2+\sqrt{3}}$ $\Rightarrow x=\frac{2+\sqrt{3}}{(2)^{2}-(\sqrt{3})^{2}}$ $\Rightarrow x=\frac{2+\sqrt{3}}{4-3}$ $\Rightarrow x=2+\sqrt{3} \quad \ldots(1)$ Now, $x^{2}=(2+\sqrt{3})^{2}$ $\Rightarrow x^{2}=(2)^{2}+(\sqrt{3})^{2}+2(2)(\sqrt{3})$ $\Rightarrow x^{2}=4+3+4 \sqrt{3}$ $\Rightarrow x^{2}=7+4 \sqrt{3} \quad \ldots(2)$ ...
Read More →The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is
Question: The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is (A) 1 (B) 2 (C) 5 (D) $\frac{8}{3}$ Solution: Let X be the random variable representing a number on the die. The total number of observations is six. $\therefore P(X=1)=\frac{3}{6}=\frac{1}{2}$ $\mathrm{P}(\mathrm{X}=2)=\frac{2}{6}=\frac{1}{3}$ $\mathrm{P}(\mathrm{X}=5)=\frac{1}{6}$ Therefore, the probability distribution is as follows. Mean $=\mathrm{E}(\mathrm{X})=\...
Read More →Solve the following equations:
Question: Solve the following equations: $2^{\sin ^{2} x}+2^{\cos ^{2} x}=2 \sqrt{2}$ Solution: $2^{\sin ^{2} x}+2^{\cos ^{2} x}=2 \sqrt{2}$ $\Rightarrow 2^{\sin ^{2} x}+2^{1-\sin ^{2} x}=2 \sqrt{2}$ $\Rightarrow 2^{\sin ^{2} x}+\frac{2}{2^{\sin ^{2} x}}=2 \sqrt{2}$ Let $2^{\sin ^{2} x}=y$ $\Rightarrow y+\frac{2}{y}=2 \sqrt{2}$ $\Rightarrow y^{2}+2=2 \sqrt{2} y$ $\Rightarrow y^{2}-2 \sqrt{2} y+2=0$ $\Rightarrow y^{2}-\sqrt{2} y-\sqrt{2} y+2=0$ $\Rightarrow y(y-\sqrt{2})-\sqrt{2}(y-\sqrt{2})=0$ $...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\left(\frac{1}{\sec ^{2} \theta-\cos ^{2} \theta}+\frac{1}{\operatorname{cosec}^{2} \theta-\sin ^{2} \theta}\right) \sin ^{2} \theta \cos ^{2} \theta=\frac{1-\sin ^{2} \theta \cos ^{2} \theta}{2+\sin ^{2} \theta \cos ^{2} \theta}$ Solution: In the given question, we need to prove $\left(\frac{1}{\sec ^{2} \theta-\cos ^{2} \theta}+\frac{1}{\operatorname{cosec}^{2} \theta-\sin ^{2} \theta}\right) \sin ^{2} \theta \cos ^{2} \theta=\left(\frac...
Read More →Solve the following equations:
Question: Solve the following equations: $3 \sin ^{2} x-5 \sin x \cos x+8 \cos ^{2} x=2$ Solution: $3 \sin ^{2} x-5 \sin x \cos x+8 \cos ^{2} x=2$ $\Rightarrow 3 \sin ^{2} x-5 \sin x \cos x+3 \cos ^{2} x+5 \cos ^{2} x-2=0$ $\Rightarrow 3\left(\sin ^{2} x+\cos ^{2} x\right)-5 \sin x \cos x+5 \cos ^{2} x-2=0$ $\Rightarrow 3-5 \sin x \cos x+5 \cos ^{2} x-2=0$ $\Rightarrow 5 \cos ^{2} x-5 \sin x \cos x+1=0$ $\Rightarrow 5\left(1-\sin ^{2} x\right)-5 \sin x \cos x+1=0$ $\Rightarrow 5-5 \sin ^{2} x-5 ...
Read More →find the value of
Question: Given, $\sqrt{2}=1.414$ and $\sqrt{6}=2.449$, find the value of $\frac{1}{\sqrt{3}-\sqrt{2}-1}$ correct to 3 places of decimal. Solution: $\frac{1}{\sqrt{3}-\sqrt{2}-1}$ $=\frac{1}{\sqrt{3}-(\sqrt{2}+1)} \times \frac{\sqrt{3}+(\sqrt{2}+1)}{\sqrt{3}+(\sqrt{2}+1)}$ $=\frac{\sqrt{3}+(\sqrt{2}+1)}{(\sqrt{3})^{2}-(\sqrt{2}+1)^{2}}$ $=\frac{\sqrt{3}+(\sqrt{2}+1)}{(\sqrt{3})^{2}-(\sqrt{2})^{2}-2(\sqrt{2})(1)-(1)^{2}}$ $=\frac{\sqrt{3}+(\sqrt{2}+1)}{3-2-2 \sqrt{2}-1}$ $=\frac{\sqrt{3}+(\sqrt{2...
Read More →Solve the following equations:
Question: Solve the following equations: (i) $\cot x+\tan x=2$ [NCERT EXEMPLAR] (ii) $2 \sin ^{2} x=3 \cos x, 0 \leq x \leq 2 \pi$ [NCERT EXEMPLAR] (iii) $\sec x \cos 5 x+1=0,0x\frac{\pi}{2}$ [NCERT EXEMPLAR] (iv) $5 \cos ^{2} x+7 \sin ^{2} x-6=0$ [NCERT EXEMPLAR] (v) $\sin x-3 \sin 2 x+\sin 3 x=\cos x-3 \cos 2 x+\cos 3 x$ [NCERT EXEMPLAR] (vi) $4 \sin x \cos x+2 \sin x+2 \cos x+1=0$ [NCERT EXEMPLAR] (vii) $\cos x+\sin x=\cos 2 x+\sin 2 x$ [NCERT EXEMPLAR] (viii) $\sin x \tan x-1=\tan x-\sin x$ ...
Read More →In a meeting, 70% of the members favour and 30% oppose a certain proposal.
Question: In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour. Find E(X) and Var(X). Solution: It is given that $P(X=0)=30 \%=\frac{30}{100}=0.3$ $\mathrm{P}(\mathrm{X}=1)=70 \%=\frac{70}{100}=0.7$ Therefore, the probability distribution is as follows. Then, $E(X)=\sum X_{i} P\left(X_{i}\right)$ $=0 \times 0.3+1 \times 0.7$ $=0.7$ $\mathrm{E}\left(\mathrm{X}^{2}\right)=\sum \mathrm...
Read More →Solve the following equations:
Question: Solve the following equations: (i) $\cot x+\tan x=2$ [NCERT EXEMPLAR] (ii) $2 \sin ^{2} x=3 \cos x, 0 \leq x \leq 2 \pi$ [NCERT EXEMPLAR] (iii) $\sec x \cos 5 x+1=0,0x\frac{\pi}{2}$ [NCERT EXEMPLAR] (iv) $5 \cos ^{2} x+7 \sin ^{2} x-6=0$ [NCERT EXEMPLAR] (v) $\sin x-3 \sin 2 x+\sin 3 x=\cos x-3 \cos 2 x+\cos 3 x$[NCERT EXEMPLAR] (vi) $4 \sin x \cos x+2 \sin x+2 \cos x+1=0$ [NCERT EXEMPLAR] (vii) $\cos x+\sin x=\cos 2 x+\sin 2 x$ [NCERT EXEMPLAR] (viii) $\sin x \tan x-1=\tan x-\sin x$ [...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\left(\tan \theta+\frac{1}{\cos \theta}\right)^{2}+\left(\tan \theta-\frac{1}{\cos \theta}\right)^{2}=2\left(\frac{1+\sin ^{2} \theta}{1-\sin ^{2} \theta}\right)$ Solution: In the given question, we need to prove $\left(\tan \theta+\frac{1}{\cos \theta}\right)^{2}+\left(\tan \theta-\frac{1}{\cos \theta}\right)^{2}=2\left(\frac{1+\sin ^{2} \theta}{1-\sin ^{2} \theta}\right)$ Now, using the identity $(a+b)^{2}=a^{2}+b^{2}+2 a b$ in L.H.S, we...
Read More →A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years
Question: A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X. Solution: There are 15 students in the class. Each student has the same chance to be chosen. Therefore, the probability of each stud...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. If $\mathrm{T}_{n}=\sin ^{n} \theta+\cos ^{n} \theta$, prove that $\frac{\mathrm{T}_{3}-\mathrm{T}_{5}}{\mathrm{~T}_{1}}=\frac{\mathrm{T}_{5}-\mathrm{T}_{7}}{\mathrm{~T}_{3}}$ Solution: In the given question, we are given $T_{n}=\sin ^{n} \theta+\cos ^{n} \theta$ We need to prove $\frac{T_{3}-T_{5}}{T_{1}}=\frac{T_{5}-T_{7}}{T_{3}}$ Here L.H.S is $\frac{T_{3}-T_{5}}{T_{1}}=\frac{\left(\sin ^{3} \theta+\cos ^{3} \theta\right)-\left(\sin ^{5}...
Read More →Rationalise the denominator of each of the following.
Question: Rationalise the denominator of each of the following. (i) $\frac{1}{\sqrt{7}+\sqrt{6}-\sqrt{13}}$ (ii) $\frac{3}{\sqrt{3}+\sqrt{5}-\sqrt{2}}$ (iii) $\frac{4}{2+\sqrt{3}+\sqrt{7}}$ Solution: (i) $\frac{1}{\sqrt{7}+\sqrt{6}-\sqrt{13}}$ $=\frac{1}{(\sqrt{7}+\sqrt{6})-\sqrt{13}} \times \frac{(\sqrt{7}+\sqrt{6})+\sqrt{13}}{(\sqrt{7}+\sqrt{6})+\sqrt{13}}$ $=\frac{(\sqrt{7}+\sqrt{6})+\sqrt{13}}{(\sqrt{7}+\sqrt{6})^{2}-(\sqrt{13})^{2}}$ $=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{(\sqrt{7})^{2}+(\sqr...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\tan ^{3} \theta}{1+\tan ^{2} \theta}+\frac{\cot ^{3} \theta}{1+\cot ^{2} \theta}=\sec \theta \operatorname{cosec} \theta-2 \sin \theta \cos \theta$ Solution: In the given question, we need to prove $\frac{\tan ^{3} \theta}{1+\tan ^{2} \theta}+\frac{\cot ^{3} \theta}{1+\cot ^{2} \theta}=\sec \theta \operatorname{cosec} \theta-2 \sin \theta \cos \theta$ Using the property $1+\tan ^{2} \theta=\sec ^{2} \theta$ and $1+\cot ^{2} \theta=\...
Read More →Let X denotes the sum of the numbers obtained when two fair dice are rolled.
Question: Let X denotes the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X. Solution: When two fair dice are rolled, 6 6 = 36 observations are obtained. $P(X=2)=P(1,1)=\frac{1}{36}$ $P(X=3)=P(1,2)+P(2,1)=\frac{2}{36}=\frac{1}{18}$ $P(X=4)=P(1,3)+P(2,2)+P(3,1)=\frac{3}{36}=\frac{1}{12}$ $P(X=5)=P(1,4)+P(2,3)+P(3,2)+P(4,1)=\frac{4}{36}=\frac{1}{9}$ $P(X=6)=P(1,5)+P(2,4)+P(3,3)+P(4,2)+P(5,1)=\frac{5}{36}$ $P(X=7)=P(1,6)+P(2,5)+P(3,4)+P(4,3)+...
Read More →Let X denotes the sum of the numbers obtained when two fair dice are rolled.
Question: Let X denotes the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X. Solution: When two fair dice are rolled, 6 6 = 36 observations are obtained. $P(X=2)=P(1,1)=\frac{1}{36}$ $P(X=3)=P(1,2)+P(2,1)=\frac{2}{36}=\frac{1}{18}$ $P(X=4)=P(1,3)+P(2,2)+P(3,1)=\frac{3}{36}=\frac{1}{12}$ $P(X=5)=P(1,4)+P(2,3)+P(3,2)+P(4,1)=\frac{4}{36}=\frac{1}{9}$ $P(X=6)=P(1,5)+P(2,4)+P(3,3)+P(4,2)+P(5,1)=\frac{5}{36}$ $P(X=7)=P(1,6)+P(2,5)+P(3,4)+P(4,3)+...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{1+\cos \theta-\sin ^{2} \theta}{\sin \theta(1+\cos \theta)}=\cot \theta$ Solution: In the given question, we need to prove $\frac{1+\cos \theta-\sin ^{2} \theta}{\sin \theta(1+\cos \theta)}=\cot \theta$. Using the property $\sin ^{2} \theta+\cos ^{2} \theta=1$, we get So, $\frac{1+\cos \theta-\sin ^{2} \theta}{\sin \theta(1+\cos \theta)}$ $=\frac{1+\cos \theta-\left(1-\cos ^{2} \theta\right)}{\sin \theta(1+\cos \theta)}$ $=\frac{\cos...
Read More →Two numbers are selected at random (without replacement) from the first six positive integers.
Question: Two numbers are selected at random (without replacement) from the first six positive integers. Let X denotes the larger of the two numbers obtained. Find E(X). Solution: The two positive integers can be selected from the first six positive integers without replacement in 6 5 = 30 ways X represents the larger of the two numbers obtained. Therefore, X can take the value of 2, 3, 4, 5, or 6. For X = 2, the possible observations are (1, 2) and (2, 1). $\therefore \mathrm{P}(\mathrm{X}=2)=\...
Read More →Solve the following equations:
Question: Solve the following equations: (i) $\sin \mathrm{x}+\cos \mathrm{x}=\sqrt{2}$ (ii) $\sqrt{3} \cos x+\sin x=1$ (iii) $\sin x+\cos x=1$ (iv) $\operatorname{cosec} x=1+\cot x$ Solution: (i) Given: $\sin x+\cos x=\sqrt{2} \ldots$.(i) The equation is of the form $a \sin x+b \cos x=c$, where $a=1, b=1$ and $c=\sqrt{2}$. Let: $a=r \sin \alpha$ and $b=r \cos \alpha$ Now, $r=\sqrt{a^{2}+b^{2}}=\sqrt{1^{2}+1^{2}}=\sqrt{2}$ and $\tan \alpha=1 \Rightarrow \alpha=\frac{\pi}{4}$ On putting $a=1=r \s...
Read More →Solve the following equations:
Question: Solve the following equations: (i) $\sin \mathrm{x}+\cos \mathrm{x}=\sqrt{2}$ (ii) $\sqrt{3} \cos x+\sin x=1$ (iii) $\sin x+\cos x=1$ (iv) $\operatorname{cosec} x=1+\cot x$ Solution: (i) Given: $\sin x+\cos x=\sqrt{2} \ldots$.(i) The equation is of the form $a \sin x+b \cos x=c$, where $a=1, b=1$ and $c=\sqrt{2}$. Let: $a=r \sin \alpha$ and $b=r \cos \alpha$ Now, $r=\sqrt{a^{2}+b^{2}}=\sqrt{1^{2}+1^{2}}=\sqrt{2}$ and $\tan \alpha=1 \Rightarrow \alpha=\frac{\pi}{4}$ On putting $a=1=r \s...
Read More →Solve the following equations:
Question: Solve the following equations: (i) $\sin \mathrm{x}+\cos \mathrm{x}=\sqrt{2}$ (ii) $\sqrt{3} \cos x+\sin x=1$ (iii) $\sin x+\cos x=1$ (iv) $\operatorname{cosec} x=1+\cot x$ Solution: (i) Given: $\sin x+\cos x=\sqrt{2} \ldots$.(i) The equation is of the form $a \sin x+b \cos x=c$, where $a=1, b=1$ and $c=\sqrt{2}$. Let: $a=r \sin \alpha$ and $b=r \cos \alpha$ Now, $r=\sqrt{a^{2}+b^{2}}=\sqrt{1^{2}+1^{2}}=\sqrt{2}$ and $\tan \alpha=1 \Rightarrow \alpha=\frac{\pi}{4}$ On putting $a=1=r \s...
Read More →Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Question: Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X. Solution: Here, X represents the number of sixes obtained when two dice are thrown simultaneously. Therefore, X can take the value of 0, 1, or 2. $\therefore \mathrm{P}(\mathrm{X}=0)=\mathrm{P}$ (not getting six on any of the dice) $=\frac{25}{36}$ P (X = 1) = P (six on first die and no six on second die) + P (no six on first die and six on second die) $=2\left(\frac{1}{6} \times \frac{5}{6...
Read More →Solve this
Question: If $p=\frac{3-\sqrt{5}}{3+\sqrt{5}}$ and $q=\frac{3+\sqrt{5}}{3-\sqrt{5}}$, find the value of $p^{2}+q^{2} .$ Solution: According to question, $p=\frac{3-\sqrt{5}}{3+\sqrt{5}}$ and $q=\frac{3+\sqrt{5}}{3-\sqrt{5}}$ $p=\frac{3-\sqrt{5}}{3+\sqrt{5}}$ $=\frac{3-\sqrt{5}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}}$ $=\frac{(3-\sqrt{5})^{2}}{(3)^{2}-(\sqrt{5})^{2}}$ $=\frac{(3)^{2}+(\sqrt{5})^{2}-2(3)(\sqrt{5})}{9-5}$ $=\frac{9+5-6 \sqrt{5}}{4}$ $=\frac{14-6 \sqrt{5}}{4} \quad \cdots(...
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