Find the general solutions of the following equations:
Question: Find the general solutions of the following equations: (i) $\sin 2 x=\frac{\sqrt{3}}{2}$ (ii) $\cos 3 x=\frac{1}{2}$ (iii) $\sin 9 x=\sin \mathrm{x}$ (iv) $\sin 2 x=\cos 3 x$ (v) $\tan x+\cot 2 x=0$ (vi) $\tan 3 x=\cot x$ (vii) $\tan 2 x \tan x=1$ (viii) $\tan m x+\cot n x=0$ (ix) $\tan p x=\cot q x$ (x) $\sin 2 x+\cos x=0$ (xi) $\sin x=\tan x$ (xii) $\sin 3 x+\cos 2 x=0$ Solution: We have: (i) $\sin 2 x=\frac{\sqrt{3}}{2}$ $\Rightarrow \sin 2 x=\sin \frac{\pi}{3}$ $\Rightarrow 2 x=n \...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$ Solution: In the given question, we need to prove $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$ Here, we will first solve the LHS. Now, using $\cot \theta=\frac{\cos \theta}{\sin \theta}$, we get $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\left(\frac{\cos A}{\sin A}-\cos A\right)}{\left(\frac{\cos...
Read More →Find the general solutions of the following equations:
Question: Find the general solutions of the following equations: (i) $\sin 2 x=\frac{\sqrt{3}}{2}$ (ii) $\cos 3 x=\frac{1}{2}$ (iii) $\sin 9 x=\sin \mathrm{x}$ (iv) $\sin 2 x=\cos 3 x$ (v) $\tan x+\cot 2 x=0$ (vi) $\tan 3 x=\cot x$ (vii) $\tan 2 x \tan x=1$ (viii) $\tan m x+\cot n x=0$ (ix) $\tan p x=\cot q x$ (x) $\sin 2 x+\cos x=0$ (xi) $\sin x=\tan x$ (xii) $\sin 3 x+\cos 2 x=0$ Solution: We have: (i) $\sin 2 x=\frac{\sqrt{3}}{2}$ $\Rightarrow \sin 2 x=\sin \frac{\pi}{3}$ $\Rightarrow 2 x=n \...
Read More →State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
Question: State which of the following arenotthe probability distributions of a random variable. Give reasons for your answer. (i) (ii) (iii) (iv) Solution: It is known that the sum of all the probabilities in a probability distribution is one. (i)Sum of the probabilities = 0.4 + 0.4 + 0.2 = 1 Therefore, the given table is a probability distribution of random variables. (ii)It can be seen that for X = 3, P (X) = 0.1 It is known that probability of any observation is not negative. Therefore, the ...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\tan ^{2} A}{1+\tan ^{2} A}+\frac{\cot ^{2} A}{1+\cot ^{2} A}=1$ Solution: In the given question, we need to prove $\frac{\tan ^{2} A}{1+\tan ^{2} A}+\frac{\cot ^{2} A}{1+\cot ^{2} A}=1$. Here, we will first solve the LHS. Now, using $\tan \theta=\frac{\sin \theta}{\cos \theta}$ and $\cot \theta=\frac{\cos \theta}{\sin \theta}$, we get $\frac{\tan ^{2} A}{1+\tan ^{2} A}+\frac{\cot ^{2} A}{1+\cot ^{2} A}=\frac{\left(\frac{\sin ^{2} A}...
Read More →find the value of
Question: If $x=\sqrt{13}+2 \sqrt{3}$, find the value of $x-\frac{1}{x}$. Solution: $x=\sqrt{13}+2 \sqrt{3} \quad \ldots .(1)$ $\Rightarrow \frac{1}{x}=\frac{1}{\sqrt{13}+2 \sqrt{3}}$ $\Rightarrow \frac{1}{x}=\frac{1}{\sqrt{13}+2 \sqrt{3}} \times \frac{\sqrt{13}-2 \sqrt{3}}{\sqrt{13}-2 \sqrt{3}}$ $\Rightarrow \frac{1}{x}=\frac{\sqrt{13}-2 \sqrt{3}}{(\sqrt{13})^{2}-(2 \sqrt{3})^{2}}$ $\Rightarrow \frac{1}{x}=\frac{\sqrt{13}-2 \sqrt{3}}{13-12}$ $\Rightarrow \frac{1}{x}=\sqrt{13}-2 \sqrt{3}$ .........
Read More →If A and B are two events such that
Question: If $A$ and $B$ are two events such that $A \subset B$ and $P(B) \neq 0$, then which of the following is correct? A. $P(A \mid B)=\frac{P(B)}{P(A)}$ B. $\mathrm{P}(\mathrm{A} \mid \mathrm{B})\mathrm{P}(\mathrm{A})$ C. $P(A \mid B) \geq P(A)$ D. None of these Solution: If $A \subset B$, then $A \cap B=A$ $\Rightarrow P(A \cap B)=P(A)$ Also, $P(A)P(B)$ Consider $P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{P(A)}{P(B)} \neq \frac{P(B)}{P(A)} \ldots$(1) Consider $\mathrm{P}(\mathrm{A} \mid \m...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\left(1+\tan ^{2} A\right)+\left(1+\frac{1}{\tan ^{2} A}\right)=\frac{1}{\sin ^{2} A-\sin ^{4} A}$ Solution: We need to prove $\left(1+\tan ^{2} A\right)+\left(1+\frac{1}{\tan ^{2} A}\right)=\frac{1}{\sin ^{2} A-\sin ^{4} A}$. Using the property $1+\tan ^{2} \theta=\sec ^{2} \theta$, we get $\left(1+\tan ^{2} A\right)+\left(1+\frac{1}{\tan ^{2} A}\right)=\sec ^{2} A+\left(\frac{\tan ^{2} A+1}{\tan ^{2} A}\right)$ $=\sec ^{2} A+\left(\frac{...
Read More →find the value
Question: If $a=3-2 \sqrt{2}$, find the value of $a^{2}-\frac{1}{a^{2}}$ Solution: $a=3-2 \sqrt{2}$ $\Rightarrow a^{2}=(3-2 \sqrt{2})^{2}$ $\Rightarrow a^{2}=9+8-12 \sqrt{2}$ $\Rightarrow a^{2}=17-12 \sqrt{2}$ ..........(1) $\therefore \frac{1}{a^{2}}=\frac{1}{17-12 \sqrt{2}}$ $\Rightarrow \frac{1}{a^{2}}=\frac{1}{17-12 \sqrt{2}} \times \frac{17+12 \sqrt{2}}{17+12 \sqrt{2}}$ $\Rightarrow \frac{1}{a^{2}}=\frac{17+12 \sqrt{2}}{17^{2}-(12 \sqrt{2})^{2}}$ $\Rightarrow \frac{1}{a^{2}}=\frac{17+12 \sq...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\operatorname{cosec} A}{\operatorname{cosec} A-1}+\frac{\operatorname{cosec} A}{\operatorname{cosec} A+1}=2 \sec ^{2} A$ Solution: We need to prove $\frac{\operatorname{cosec} A}{\operatorname{cosec} A-1}+\frac{\operatorname{cosec} A}{\operatorname{cosec} A+1}=2 \sec ^{2} A$ Using the identity $a^{2}-b^{2}=(a+b)(a-b)$, we get $\frac{\operatorname{cosec} A}{\operatorname{cosec} A-1}+\frac{\operatorname{cosec} A}{\operatorname{cosec} A...
Read More →Probability that A speaks truth is
Question: Probability that A speaks truth is $\frac{4}{5}$. A coin is tossed. A reports that a head appears. The probability that actually there was head is A. $\frac{4}{5}$ B. $\frac{1}{2}$ C. $\frac{1}{5}$ D. $\frac{2}{5}$ Solution: Let $E_{1}$ and $E_{2}$ be the events such that $E_{1}$ : A speaks truth $\mathrm{E}_{2}$ : A speaks false Let X be the event that a head appears. $P\left(E_{1}\right)=\frac{4}{5}$ Therefore, $P\left(E_{2}\right)=1-P\left(E_{1}\right)=1-\frac{4}{5}=\frac{1}{5}$ If ...
Read More →find the value
Question: If $x=\frac{5-\sqrt{21}}{2}$, find the value of $x+\frac{1}{x}$ Solution: $x=\frac{5-\sqrt{21}}{2} \quad \ldots \ldots(1)$ $\Rightarrow \frac{1}{x}=\frac{1}{\frac{5-\sqrt{21}}{2}}$ $\Rightarrow \frac{1}{x}=\frac{2}{5-\sqrt{21}}$ $\Rightarrow \frac{1}{x}=\frac{2}{5-\sqrt{21}} \times \frac{5+\sqrt{21}}{5+\sqrt{21}}$ $\Rightarrow \frac{1}{x}=\frac{2(5+\sqrt{21})}{5^{2}-(\sqrt{21})^{2}}$ $\Rightarrow \frac{1}{x}=\frac{2(5+\sqrt{21})}{25-21}$ $\Rightarrow \frac{1}{x}=\frac{2(5+\sqrt{21})}{4...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\sin A+\cos A$ Solution: We need to prove $\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\sin A+\cos A$ Solving the L.H.S, we get $\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\frac{\cos A}{1-\frac{\sin A}{\cos A}}+\frac{\sin A}{1-\frac{\cos A}{\sin A}}$ $=\frac{\frac{\cos A}{\cos A-\sin A}}{\frac{\cos A}{\sin A}}+\frac{\sin A}{\frac{\sin A-\cos A}{\sin }}$ $=\frac{\cos ^{2} A}{\cos A-\s...
Read More →Find the general solutions of the following equations:
Question: Find the general solutions of the following equations: (i) $\sin x=\frac{1}{2}$ (ii) $\cos x=-\frac{\sqrt{3}}{2}$ (iii) $\operatorname{cosec} x=-\sqrt{2}$ (iv) $\sec x=\sqrt{2}$ (v) $\tan x=-\frac{1}{\sqrt{3}}$ (vi) $\sqrt{3} \sec x=2$ Solution: We have: (i) $\sin x=\frac{1}{2}$ The value of $x$ satisfying $\sin x=\frac{1}{2}$ is $\frac{\pi}{6}$. $\therefore \sin x=\frac{1}{2}$ $\Rightarrow \sin x=\sin \frac{\pi}{6}$ $\Rightarrow x=n \pi+(-1)^{n} \frac{\pi}{6}, n \in Z$ (ii) $\cos x=-\...
Read More →Find the general solutions of the following equations:
Question: Find the general solutions of the following equations: (i) $\sin x=\frac{1}{2}$ (ii) $\cos x=-\frac{\sqrt{3}}{2}$ (iii) $\operatorname{cosec} x=-\sqrt{2}$ (iv) $\sec x=\sqrt{2}$ (v) $\tan x=-\frac{1}{\sqrt{3}}$ (vi) $\sqrt{3} \sec x=2$ Solution: We have: (i) $\sin x=\frac{1}{2}$ The value of $x$ satisfying $\sin x=\frac{1}{2}$ is $\frac{\pi}{6}$. $\therefore \sin x=\frac{1}{2}$ $\Rightarrow \sin x=\sin \frac{\pi}{6}$ $\Rightarrow x=n \pi+(-1)^{n} \frac{\pi}{6}, n \in Z$ (ii) $\cos x=-\...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A$ Solution: We need to prove $\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A$ Solving the L.H.S, we get $\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=\frac{\sec A+1+\sec A-1}{(\sec A-1)(\sec A+1)}$ $=\frac{2 \sec A}{\sec ^{2} A-1}$ Further using the property $1+\tan ^{2} \theta=\sec ^{2} \theta$, we get So, $\frac{2 \sec A}{\sec ^{2} A-1}=\frac{2 \sec A}{\...
Read More →A card from a pack of 52 cards is lost.
Question: A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond. Solution: Let E1and E2be the respective events of choosing a diamond card and a card which is not diamond. Let A denote the lost card. Out of 52 cards, 13 cards are diamond and 39 cards are not diamond. $\therefore P\left(E_{1}\right)=\frac{13}{52}=\frac{1}{4}$ $\mathrm{P}\left(\mathrm{E}_{2}\right)=\...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\frac{1-\cos A}{1+\cos A}=(\cot A-\operatorname{cosec} A)^{2}$ Solution: We need to prove $\frac{1-\cos A}{1+\cos A}=(\cot A-\operatorname{cosec} A)^{2}$ Now, rationalising the L.H.S, we get $\frac{1-\cos A}{1+\cos A}=\left(\frac{1-\cos A}{1+\cos A}\right)\left(\frac{1-\cos A}{1-\cos A}\right)$ $=\frac{(1-\cos A)^{2}}{1-\cos ^{2} A}$ $\left(\right.$ Using $\left.a^{2}-b^{2}=(a+b)(a-b)\right)$ $=\frac{1+\cos ^{2} A-2 \cos A}{\sin ^{2} A}$ (...
Read More →find the value of
Question: If $x=9-4 \sqrt{5}$, find the value of $x^{2}+\frac{1}{x^{2}}$. Solution: $x=9-4 \sqrt{5}$ .......(1) $\Rightarrow \frac{1}{x}=\frac{1}{9-4 \sqrt{5}}$ $\Rightarrow \frac{1}{x}=\frac{1}{9-4 \sqrt{5}} \times \frac{9+4 \sqrt{5}}{9+4 \sqrt{5}}$ $\Rightarrow \frac{1}{x}=\frac{9+4 \sqrt{5}}{9^{2}-(4 \sqrt{5})^{2}}$ $\Rightarrow \frac{1}{x}=\frac{9+4 \sqrt{5}}{81-80}$ $\Rightarrow \frac{1}{x}=9+4 \sqrt{5} \quad \ldots(2)$ Adding (1) and (2), we get $x+\frac{1}{x}=9-4 \sqrt{5}+9+4 \sqrt{5}$ $\...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $(\sec A-\tan A)^{2}=\frac{1-\sin A}{1+\sin A}$ Solution: We need to prove $(\sec A-\tan A)^{2}=\frac{1-\sin A}{1+\sin A}$ Here, we will first solve the L.H.S. Now, using $\sec \theta=\frac{1}{\cos \theta}$ and $\tan \theta=\frac{\sin \theta}{\cos \theta}$, we get $(\sec A-\tan A)^{2}=\left(\frac{1}{\cos A}-\frac{\sin A}{\cos A}\right)^{2}$ $=\left(\frac{1-\sin A}{\cos A}\right)^{2}$ $=\frac{(1-\sin A)^{2}}{(\cos A)^{2}}$ Further using the ...
Read More →A manufacturer has three machine operators A, B and C.
Question: A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that was produced by A? Solution: Let E1, E2, and E3be the respective events of the time consumed by machines A, B, and C for th...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. $\sqrt{\frac{1-\cos A}{1+\cos A}}+\sqrt{\frac{1+\cos A}{1-\cos A}}=2 \operatorname{cosec} A$ Solution: We need to prove $\sqrt{\frac{1-\cos A}{1+\cos A}}=\operatorname{cosec} A-\cot A$ Here, rationaliaing the L.H.S, we get $\sqrt{\frac{1-\cos A}{1+\cos A}}=\sqrt{\frac{1-\cos A}{1+\cos A}} \times \sqrt{\frac{1-\cos A}{1-\cos A}}$ $=\sqrt{\frac{(1-\cos A)^{2}}{1-\cos ^{2} A}}$ Further using the property, $\sin ^{2} \theta+\cos ^{2} \theta=1$,...
Read More →Suppose a girl throws a die. If she gets a 5 or 6,
Question: Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die? Solution: Let E1be the event that the outcome on the die is 5 or 6 and E2be the event that the outcome on the die is 1, 2, 3, or 4. $\therefore \mathrm{P}\left(\mathrm{E}_{1}\right)=\...
Read More →Solve this
Question: If $x=2-\sqrt{3}$, find value of $\left(x-\frac{1}{x}\right)^{3} .$ Solution: $x=2-\sqrt{3} \quad \ldots .(1)$ $\Rightarrow \frac{1}{x}=\frac{1}{2-\sqrt{3}}$ $\Rightarrow \frac{1}{x}=\frac{1}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2+\sqrt{3}}$ $\Rightarrow \frac{1}{x}=\frac{2+\sqrt{3}}{2^{2}-(\sqrt{3})^{2}}$ $\Rightarrow \frac{1}{x}=\frac{2+\sqrt{3}}{4-3}$ $\Rightarrow \frac{1}{x}=2+\sqrt{3} \quad \ldots(2)$ Subtracting (2) from (1), we get $x-\frac{1}{x}=(2-\sqrt{3})-(2+\sqrt{3})$ $\Rig...
Read More →Prove the following trigonometric identities.
Question: Prove the following trigonometric identities. (i) $\sqrt{\frac{1+\sin A}{1-\sin A}}=\sec A+\tan A$ (ii) $\sqrt{\frac{1-\cos A}{1+\cos A}}=\operatorname{cosec} A-\cot A$ Solution: (i) We need to prove $\sqrt{\frac{1+\sin A}{1-\sin A}}=\sec A+\tan A$ Here, rationalising the L.H.S, we get $\sqrt{\frac{1+\sin A}{1-\sin A}}=\sqrt{\frac{1+\sin A}{1-\sin A}} \times \sqrt{\frac{1+\sin A}{1+\sin A}}$ $=\sqrt{\frac{(1+\sin A)^{2}}{1-\sin ^{2} A}}$ Further using the property, $\sin ^{2} \theta+\c...
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