Find the shortest distance between lines
Question: Find the shortest distance between lines $\vec{r}=6 \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k})$ and $\vec{r}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k})$. Solution: The given lines are $\vec{r}=6 \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k})$ $\vec{r}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k})$ It is known that the shortest distance between two lines, $\vec{r}=\vec{a}_{1}+\lambda \vec{b}_{1}$ and $\vec{r}=\vec{a}_{2}+\lambda...
Read More →If A=2 sin
Question: If $A=2 \sin ^{2} x-\cos 2 x$, then $A$ lies in the interval (a) $[-1,3]$ (b) $[1,2]$ (c) $[-2,4]$ (d) none of these Solution: (a) $[-1,3]$ $A=2 \sin ^{2} x-\cos 2 x$ $=2 \sin ^{2} x-\left(1-2 \sin ^{2} x\right)$ $=4 \sin ^{2} x-1$ $\because 0 \leq \sin ^{2} x \leq 1$ $\Rightarrow 4 \times 0 \leq 4 \times \sin ^{2} x \leq 4 \times 1$ $\Rightarrow 0 \leq 4 \sin ^{2} x \leq 4$ $\Rightarrow 0-1 \leq 4 \sin ^{2} x-1 \leq 4-1$ $\Rightarrow-1 \leq 4 \sin ^{2} x-1 \leq 3$ $\Rightarrow-1 \leq ...
Read More →If A=2 sin
Question: If $A=2 \sin ^{2} x-\cos 2 x$, then $A$ lies in the interval (a) $[-1,3]$ (b) $[1,2]$ (c) $[-2,4]$ (d) none of these Solution: (a) $[-1,3]$ $A=2 \sin ^{2} x-\cos 2 x$ $=2 \sin ^{2} x-\left(1-2 \sin ^{2} x\right)$ $=4 \sin ^{2} x-1$ $\because 0 \leq \sin ^{2} x \leq 1$ $\Rightarrow 4 \times 0 \leq 4 \times \sin ^{2} x \leq 4 \times 1$ $\Rightarrow 0 \leq 4 \sin ^{2} x \leq 4$ $\Rightarrow 0-1 \leq 4 \sin ^{2} x-1 \leq 4-1$ $\Rightarrow-1 \leq 4 \sin ^{2} x-1 \leq 3$ $\Rightarrow-1 \leq ...
Read More →The percentage of marks obtained by a student in the monthly unit tests are given below:
Question: The percentage of marks obtained by a student in the monthly unit tests are given below: Find the probability that the student gets 1. More than 70% marks 2. Less than 70% marks 3. A distinction Solution: 1:Let E be the event of getting more than 70% marks No of times E happens = 3 Probability (Getting more than 70\%) $=\frac{\text { Number of times student got more than 70 }}{\text { Total no of exams taken }}$ = 3/5= 0.6 2.Let F be the event of getting less than 70% marks No of times...
Read More →The value of 2 cosx
Question: The value of $2 \cos x-\cos 3 x-\cos 5 x-16 \cos ^{3} x \sin ^{2} x$ is (a) 2 (b) 1 (c) 0 (d) 1 Solution: (c) 0 We have, $2 \cos x-\cos 3 x-\cos 5 x-16 \cos ^{3} x \sin ^{2} x$ $=2 \cos x-\cos 3 x-\cos 5 x-16\left[\frac{\cos 3 x+3 \cos x}{4} \times \frac{(1-\cos 2 x)}{2}\right]$ $=2 \cos x-\cos 3 x-\cos 5 x-2[(\cos 3 x+3 \cos x)(1-\cos 2 x)]$ $=2 \cos x-\cos 3 x-\cos 5 x-2[\cos 3 x-\cos 3 x \cos 2 x+3 \cos x-3 \cos x \cos 2 x]$ $=2 \cos x-\cos 3 x-\cos 5 x-2[\cos 3 x+3 \cos x]+2 \cos 3...
Read More →The value of 2 cosx
Question: The value of $2 \cos x-\cos 3 x-\cos 5 x-16 \cos ^{3} x \sin ^{2} x$ is (a) 2 (b) 1 (c) 0 (d) 1 Solution: (c) 0 We have, $2 \cos x-\cos 3 x-\cos 5 x-16 \cos ^{3} x \sin ^{2} x$ $=2 \cos x-\cos 3 x-\cos 5 x-16\left[\frac{\cos 3 x+3 \cos x}{4} \times \frac{(1-\cos 2 x)}{2}\right]$ $=2 \cos x-\cos 3 x-\cos 5 x-2[(\cos 3 x+3 \cos x)(1-\cos 2 x)]$ $=2 \cos x-\cos 3 x-\cos 5 x-2[\cos 3 x-\cos 3 x \cos 2 x+3 \cos x-3 \cos x \cos 2 x]$ $=2 \cos x-\cos 3 x-\cos 5 x-2[\cos 3 x+3 \cos x]+2 \cos 3...
Read More →Find the equation of the plane passing through
Question: Find the equation of the plane passing through $(a, b, c)$ and parallel to the plane $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=2$ Solution: Any plane parallel to the plane, $\vec{r}_{1} \cdot(\hat{i}+\hat{j}+\hat{k})=2$, is of the form $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=\lambda$ ...(1) The plane passes through the point $(a, b, c)$. Therefore, the position vector $\vec{r}$ of this point is $\vec{r}=a \hat{i}+b \hat{j}+c \hat{k}$ Therefore, equation (1) becomes $(a \hat{i}+b \hat{j}+...
Read More →In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays. Find the probability that:
Question: In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays. Find the probability that: 1. He hit a boundary 2. He did not hit a boundary. Solution: Number of times the batsman hits a boundary = 6 Total number of balls played = 30 Number of times the batsman did not hit a boundary = 30 - 6 = 24 1. Probability that the batsman hits a boundary $=\frac{\text { Number of times hehita boundary }}{\text { Total no of balls }}$ = 6/30 = 15 2. Probability that the batsman do...
Read More →If 5 sin α=3 sin
Question: If $5 \sin \alpha=3 \sin (\alpha+2 \beta) \neq 0$, then tan $(\alpha+\beta)$ is equal to (a) $2 \tan \beta$ (b) $3 \tan \beta$ (c) $4 \tan \beta$ (d) $6 \tan \beta$ Solution: (c) $4 \tan \beta$ We have, $5 \sin \alpha=3 \sin (\alpha+2 \beta)$ $\Rightarrow \frac{5}{3}=\frac{\sin (\alpha+2 \beta)}{\sin \alpha}$ $\Rightarrow \frac{5-3}{5+3}=\frac{\sin (\alpha+2 \beta)-\sin \alpha}{\sin (\alpha+2 \beta)+\sin \alpha} \quad$ (Using componendo and dividendo) $\Rightarrow \frac{2}{8}=\frac{\si...
Read More →Evaluate each of the following
Question: Evaluate each of the following $\cot ^{2} 30^{\circ}-2 \cos ^{2} 60^{\circ}-\frac{3}{4} \sec ^{2} 45^{\circ}-4 \sec ^{2} 30^{\circ}$ Solution: We have, $\cot ^{2} 30^{\circ}-2 \cos ^{2} 60^{\circ}-\frac{3}{4} \sec ^{2} 45^{\circ}-4 \sec ^{2} 30^{\circ}$ (1) Now, $\cot 30^{\circ}=\sqrt{3}, \cos 60^{\circ}=\frac{1}{2}, \sec 45^{\circ}=\sqrt{2}, \sec 30^{\circ}=\frac{2}{\sqrt{3}}$ So by substituting above values in equation (1) We get, $\cot ^{2} 30^{\circ}-2 \cos ^{2} 60^{\circ}-\frac{3}...
Read More →1500 families with 2 children were selected randomly, and the following data were recorded:
Question: 1500 families with 2 children were selected randomly, and the following data were recorded: If a family is chosen at random, compute the probability that it has: 1. No girl 2. 1 girl 3. 2 girls 4. At most one girl 5. More girls than boys Solution: 1. Probability of having no girl in a family $=\frac{\text { No of families having no girl }}{\text { Total no of families }}$ $=\frac{211}{1500}=0.1406$ 2. Probability of having 1 girl in a family $=\frac{\text { No of families having } 1 \t...
Read More →Find the vector equation of the line passing through
Question: Find the vector equation of the line passing through $(1,2,3)$ and perpendicular to the plane $\vec{r} \cdot(\hat{i}+2 \hat{j}-5 \hat{k})+9=0$ Solution: The position vector of the point $(1,2,3)$ is $\vec{r}_{1}=\hat{i}+2 \hat{j}+3 \hat{k}$ The direction ratios of the normal to the plane, $\vec{r} \cdot(\hat{i}+2 \hat{j}-5 \hat{k})+9=0$, are 1,2, and $-5$ and the normal vector is $\vec{N}=\hat{i}+2 \hat{j}-5 \hat{k}$ The equation of a line passing through a point and perpendicular to t...
Read More →sin
Question: $\sin ^{2}\left(\frac{\pi}{18}\right)+\sin ^{2}\left(\frac{\pi}{9}\right)+\sin ^{2}\left(\frac{7 \pi}{18}\right)+\sin ^{2}\left(\frac{4 \pi}{9}\right)=$ (a) 1 (b) 2 (c) 4 (d) none of these. Solution: (b) 2 We have, $\sin ^{2}\left(\frac{\pi}{18}\right)+\sin ^{2}\left(\frac{\pi}{9}\right)+\sin ^{2}\left(\frac{7 \pi}{18}\right)+\sin ^{2}\left(\frac{4 \pi}{9}\right)$ $=\frac{1}{2}\left[1-\cos \left(\frac{\pi}{9}\right)+1-\cos \left(\frac{2 \pi}{9}\right)+1-\cos \frac{7 \pi}{9}+1-\cos \fra...
Read More →sin
Question: $\sin ^{2}\left(\frac{\pi}{18}\right)+\sin ^{2}\left(\frac{\pi}{9}\right)+\sin ^{2}\left(\frac{7 \pi}{18}\right)+\sin ^{2}\left(\frac{4 \pi}{9}\right)=$ (a) 1 (b) 2 (c) 4 (d) none of these. Solution: (b) 2 We have, $\sin ^{2}\left(\frac{\pi}{18}\right)+\sin ^{2}\left(\frac{\pi}{9}\right)+\sin ^{2}\left(\frac{7 \pi}{18}\right)+\sin ^{2}\left(\frac{4 \pi}{9}\right)$ $=\frac{1}{2}\left[1-\cos \left(\frac{\pi}{9}\right)+1-\cos \left(\frac{2 \pi}{9}\right)+1-\cos \frac{7 \pi}{9}+1-\cos \fra...
Read More →If the lines
Question: If the lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular, find the value of $k$. Solution: The direction of ratios of the lines, $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-1}{1}=\frac{z-6}{-5}$, are $-3,2 k, 2$ and $3 k, 1,-5$ respectively. It is known that two lines with direction ratios, $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$, are perpendicular, if $a_{1} a_{2}+b_{1} b_{2...
Read More →Evaluate each of the following
Question: Evaluate each of the following $\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}$ Solution: We have, $\operatorname{cosec}^{3} 30^{\circ} \cos 60^{\circ} \tan ^{3} 45^{\circ} \sin ^{2} 90^{\circ} \sec ^{2} 45^{\circ} \cot 30^{\circ}$....(1) Now, $\operatorname{cosec} 30^{\circ}=2, \cos 60^{\circ}=\frac{1}{2}, \sec 45^{\circ}=\sqrt{2}, \tan 45^{\circ}=1, \sin 90^{\circ}=1, \cot 30^{\circ}=\sqrt{3}$ So by s...
Read More →If the coordinates of the points A, B, C, D be
Question: If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (4, 3, 6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD. Solution: The coordinates of A, B, C, and D are (1, 2, 3), (4, 5, 7), (4, 3, 6), and (2, 9, 2) respectively. The direction ratios of AB are (4 1) = 3, (5 2) = 3, and (7 3) = 4 The direction ratios of CD are (2 ( 4)) = 6, (9 3) = 6, and (2 (6)) = 8 It can be seen that, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}=\fra...
Read More →The value of tan x sin
Question: The value of $\tan x \sin \left(\frac{\pi}{2}+x\right) \cos \left(\frac{\pi}{2}-x\right)$ (a) 1 (b) $-1$ (c) $\frac{1}{2} \sin 2 x$ (d) none of these. Solution: (d) none of these. We have: $\tan \theta \sin \left(\frac{\pi}{2}+x\right) \cos \left(\frac{\pi}{2}-x\right)$ $=\frac{\sin x}{\cos x} \cos x \sin x$ $=\sin ^{2} x$...
Read More →Evaluate each of the following
Question: Evaluate each of the following $\left(\operatorname{cosec}^{2} 45^{\circ} \sec ^{2} 30^{\circ}\right)\left(\sin ^{2} 30^{\circ}+4 \cot ^{2} 45^{\circ}-\sec ^{2} 60^{\circ}\right)$ Solution: We have, $\left(\operatorname{cosec}^{2} 45^{\circ} \sec ^{2} 30^{\circ}\right)\left(\sin ^{2} 30^{\circ}+4 \cot ^{2} 45^{\circ}-\sec ^{2} 60^{-}\right)$...(1) Now, $\sin 30^{\circ}=\frac{1}{2}, \operatorname{cosec} 45^{\circ}=\sqrt{2}, \sec 30^{\circ}=\frac{2}{\sqrt{3}}, \sec 60^{\circ}=2, \cot 45^...
Read More →Find the equation of a line parallel to x-axis and passing through the origin.
Question: Find the equation of a line parallel tox-axis and passing through the origin. Solution: The line parallel tox-axis and passing through the origin isx-axis itself. Let A be a point onx-axis. Therefore, the coordinates of A are given by (a, 0, 0), wherea R. Direction ratios of OA are (a 0) =a, 0, 0 The equation of OA is given by, $\frac{x-0}{a}=\frac{y-0}{0}=\frac{z-0}{0}$ $\Rightarrow \frac{x}{1}=\frac{y}{0}=\frac{z}{0}=a$ Thus, the equation of line parallel tox-axis and passing through...
Read More →Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes:
Question: Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes: If the three coins are tossed simultaneously again, compute the probability of: 1. heads coming up 2. heads coming up 3. At least one Head coming up 4. Getting more Tails than Heads 5. Getting more heads than tails Solution: 1. Probability of 2 Heads coming up $=\frac{\text { Favorable out come }}{\text { Total out come }}$ $=\frac{36}{100}=0.36$ 2. Probability of 3 Heads coming up $=\...
Read More →The value of
Question: The value of $\left(\cot \frac{x}{2}-\tan \frac{x}{2}\right)^{2}(1-2 \tan x \cot 2 x)$ is (a) 1 (b) 2 (c) 3 (d) 4 Solution: (d) 4 We have: $\left(\cot \frac{x}{2}-\tan \frac{x}{2}\right)^{2}(1-2 \tan x \cot 2 x)$ $\left(\cot ^{2} \frac{x}{2}-2 \cot \frac{x}{2} \tan \frac{x}{2}+\tan ^{2} \frac{x}{2}\right)\left\{1-2 \tan x\left(\frac{\cot ^{2} x-1}{2 \cot x}\right)\right\}$ $\left(\cot ^{2} \frac{x}{2}-2+\tan ^{2} \frac{x}{2}\right)\left\{1-\tan x\left(\frac{\cot ^{2} x-1}{\cot x}\right...
Read More →The value of
Question: The value of $\left(\cot \frac{x}{2}-\tan \frac{x}{2}\right)^{2}(1-2 \tan x \cot 2 x)$ is (a) 1 (b) 2 (c) 3 (d) 4 Solution: (d) 4 We have: $\left(\cot \frac{x}{2}-\tan \frac{x}{2}\right)^{2}(1-2 \tan x \cot 2 x)$ $\left(\cot ^{2} \frac{x}{2}-2 \cot \frac{x}{2} \tan \frac{x}{2}+\tan ^{2} \frac{x}{2}\right)\left\{1-2 \tan x\left(\frac{\cot ^{2} x-1}{2 \cot x}\right)\right\}$ $\left(\cot ^{2} \frac{x}{2}-2+\tan ^{2} \frac{x}{2}\right)\left\{1-\tan x\left(\frac{\cot ^{2} x-1}{\cot x}\right...
Read More →Find the angle between the lines whose direction ratios are a, b, c and b − c,
Question: Find the angle between the lines whose direction ratios area,b,candbc,ca,ab. Solution: The angleQbetween the lines with direction cosines,a,b,candbc,ca,ab, is given by, $\cos Q=\left|\frac{a(b-c)+b(c-a)+c(a-b)}{\sqrt{a^{2}+b^{2}+c^{2}}+\sqrt{(b-c)^{2}+(c-a)^{2}+(a-b)^{2}}}\right|$ $\Rightarrow \cos Q=0$ $\Rightarrow Q=\cos ^{-1} 0$ $\Rightarrow Q=90^{\circ}$ Thus, the angle between the lines is 90....
Read More →Evaluate each of the following
Question: Evaluate each of the following $4\left(\sin ^{4} 60^{\circ}+\cos ^{4} 30^{\circ}\right)-3\left(\tan ^{2} 60^{\circ}-\tan ^{2} 45^{\circ}\right)+5 \cos ^{2} 45^{\circ}$ Solution: We have, $4\left(\sin ^{4} 60^{\circ}+\cos ^{4} 30^{\circ}\right)-3\left(\tan ^{2} 60^{\circ}-\tan ^{2} 45^{\circ}\right)+5 \cos ^{2} 45^{\circ}$ (1) Now, $\sin 60^{\circ}=\cos 30^{\circ}=\frac{\sqrt{3}}{2}, \cos 45^{\circ}=\frac{1}{\sqrt{2}}, \tan 60^{\circ}=\sqrt{3}, \tan 45^{\circ}=1$ So by substituting abov...
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