The value of tan x tan

Question: The value of $\tan x \tan \left(\frac{\pi}{3}-x\right) \tan \left(\frac{\pi}{3}+x\right)$ is (a) cot 3x (b) 2cot 3x (c) tan 3x (d) 3 tan 3x Solution: (c) tan 3x $\frac{\pi}{3}=60^{\circ}$ $\tan x \tan \left(60^{\circ}-x\right) \tan \left(60^{\circ}+x\right)=\tan x \times \frac{\tan 60^{\circ}-\tan x}{1+\tan 60^{\circ} \tan x} \times \frac{\tan 60^{\circ}+\tan x}{1-\tan 60^{\circ} \tan x}$ $=\tan x \times \frac{\sqrt{3}-\tan x}{1+\sqrt{3} \tan x} \times \frac{\sqrt{3}+\tan x}{1-\sqrt{3}...

Find the vector equation of the line passing through $(1,2,3)$ and parallel to the planes

Question: Find the vector equation of the line passing through $(1,2,3)$ and parallel to the planes $\vec{r}=(\hat{i}-\hat{j}+2 \hat{k})=5$ and $\vec{r} \cdot(3 \hat{i}+\hat{j}+\hat{k})=6$. Solution: Let the required line be parallel to vectorgiven by, $\vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}$ The position vector of the point $(1,2,3)$ is $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ The equation of line passing through $(1,2,3)$ and parallel to $\vec{b}$ is given by, $\vec{r}=\vec{a}+\lambda ...

If A = B = 60°, verify that

Question: If $A=B=60^{\circ}$, verify that (i) $\cos (A-B)=\cos A \cos B+\sin A \sin B$ (ii) $\sin (A-B)=\sin A \cos B-\cos A \sin B$ (iii) $\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}$ Solution: (i) Given: $A=B=60^{\circ} \ldots \ldots(1)$ To verify: $\cos (A-B)=\cos A \cos B+\sin A \sin B$....(2) Now consider left hand side of the expression to be verified in equation (2) Therefore, $\cos (A-B)=\cos (60-60)$ $=\cos 0$ $=1$ Now consider right hand side of the expression to be verified in e...

If θ = 30°, verify that

Question: If $\theta=30^{\circ}$, verify that (i) $\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}$ (ii) $\sin 2 \theta=\frac{2 \tan \theta}{1+\tan ^{2} \theta}$ (iii) $\cos 2 \theta=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}$ (iv) $\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta$ Solution: (i) Given:' $\theta=30^{\circ}$....(1) To verify: $\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}$....(2) Now consider LHS of the expression to be verified in equation (2) Therefore, LHS $=\t...

Find the distance of the point $(-1,-5,-10)$ from the point of intersection of the line

Question: Find the distance of the point $(-1,-5,-10)$ from the point of intersection of the line $\vec{r}=2 \hat{i}-\hat{j}+2 \hat{k}+\lambda(3 \hat{i}+4 \hat{j}+2 \hat{k})$ and the plane $\vec{r} \cdot(\hat{i}-\hat{j}+\hat{k})=5$. Solution: The equation of the given line is $\vec{r} \cdot=2 \hat{i}-\hat{j}+2 \hat{k}+\lambda(3 \hat{i}+4 \hat{j}+2 \hat{k})$ $\ldots(1)$ The equation of the given plane is $\vec{r} \cdot(\hat{i}-\hat{j}+\hat{k})=5$ $\ldots(2)$ Substituting the value of $\vec{r}$ fr...

Find the value of x in each of the following :

Question: Find the value ofxin each of the following : $\cos 2 x=\cos 60^{\circ} \cos 30^{\circ}+\sin 60^{\circ} \sin 30^{\circ}$ Solution: We have, $\cos 2 x=\cos 60^{\circ} \cos 30^{\circ}+\sin 60^{\circ} \sin 30^{\circ}$ Now we know that $\sin 60^{\circ}=\cos 30^{\circ}=\frac{\sqrt{3}}{2}$ and $\sin 30^{\circ}=\cos 60^{\circ}=\frac{1}{2}$ Now by substituting above values in equation (1), we get, $\cos 2 x=\cos 60^{\circ} \cos 30^{\circ}+\sin 60^{\circ} \sin 30^{\circ}$ $\cos 2 x=\frac{1}{2} \...

Find the equation of the plane which contains the line of intersection of the planes

Question: Find the equation of the plane which contains the line of intersection of the planes $\vec{r} \cdot(\hat{i}+2 \hat{j}+3 \hat{k})-4=0, \vec{r} \cdot(2 \hat{i}+\hat{j}-\hat{k})+5=0$ and which is perpendicular to the plane $\vec{r} \cdot(5 \hat{i}+3 \hat{j}-6 \hat{k})+8=0$. Solution: The equations of the given planes are $\vec{r} \cdot(\hat{i}+2 \hat{j}+3 \hat{k})-4=0$ $\vec{r} \cdot(2 \hat{i}+\hat{j}-\hat{k})+5=0$ The equation of the plane passing through the line intersection of the pla...

Find the value of x in each of the following :

Question: Find the value ofxin each of the following : $\sqrt{3} \tan 2 x=\cos 60^{\circ}+\sin 45^{\circ} \cos 45^{\circ}$ Solution: We have, $\sqrt{3} \tan 2 x=\cos 60^{\circ}+\sin 45^{\circ} \cos 45^{\circ} \ldots \ldots$ (1) Now we know that $\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ and $\cos 60^{\circ}=\frac{1}{2}$ Now by substituting above values in equation (1), we get, $\sqrt{3} \tan 2 x=\cos 60^{\circ}+\sin 45^{\circ} \cos 45^{\circ}$ $\sqrt{3} \tan 2 x=\frac{1}{2}+\frac{1}{\s...

Question: Find the equation of the plane which contains the line of intersection of the planes $\vec{r} \cdot(\hat{i}+2 \hat{j}+3 \hat{k})-4=0, \vec{r} \cdot(2 \hat{i}+\hat{j}-\hat{k})+5=0$ and which is perpendicular to the plane Solution: The equations of the given planes are $\vec{r} \cdot(\hat{i}+2 \hat{j}+3 \hat{k})-4=0$ $\vec{r} \cdot(2 \hat{i}+\hat{j}-\hat{k})+5=0$ The equation of the plane passing through the line intersection of the plane given in equation (1) and equation (2) is $[\vec{...

Find the value of x in each of the following :

Question: Find the value ofxin each of the following : $\tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ}$ Solution: We have, $\tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ} \ldots \ldots$ (1) Now we know that $\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ and $\sin 30^{\circ}=\frac{1}{2}$ Now by substituting above values in equation (1), we get, $\tan x=\sin 45^{\circ} \cos 45^{\circ}+\sin 30^{\circ}$ $\tan x=\frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}+\frac{1}{2}$ $=\frac{1...

Find the value of x in each of the following :

Question: Find the value ofxin each of the following : $\sqrt{3} \sin x=\cos x$ Solution: We have, $\sqrt{3} \sin x=\cos x$ Now by cross multiplying we get, $\sqrt{3} \sin x=\cos x$ $\Rightarrow \frac{\sin x}{\cos x}=\frac{1}{\sqrt{3}}$.....(1) Now we know that $\frac{\sin x}{\cos x}=\tan x$....(2) Therefore from equation (1) and (2) We get, $\tan x=\frac{1}{\sqrt{3}}$.....(3) Since, $\tan 30^{\circ}=\frac{1}{\sqrt{3}}$....(4) Therefore, by comparing equation (3) and (4) we get, $x=30^{\circ}$ T...

Find the value of x in each of the following :

Question: Find the value ofxin each of the following : $2 \sin \frac{x}{2}=1$ Solution: We have, $2 \sin \frac{x}{2}=1$ $\Rightarrow \sin \frac{x}{2}=\frac{1}{2}$ Since, $\sin 30^{\circ}=\frac{1}{2}$ Therefore, $\sin \frac{x}{2}=\frac{1}{2}$ $\Rightarrow \frac{x}{2}=30^{\circ}$ $\Rightarrow x=2 \times 30^{\circ}$ $\Rightarrow x=60^{\circ}$ Therefore, $x=60^{\circ}$...

Find the value of x in each of the following :

Question: Find the value ofxin each of the following : $2 \sin 3 x=\sqrt{3}$ Solution: We have, $2 \sin 3 x=\sqrt{3}$ $\Rightarrow \sin 3 x=\frac{\sqrt{3}}{2}$ Since, $\sin 60^{\circ}=\frac{\sqrt{3}}{2}$ Therefore, $\sin 3 x=\frac{\sqrt{3}}{2}$ $\Rightarrow 3 x=60^{\circ}$ $\Rightarrow x=\frac{60^{\circ}}{3}$ $\Rightarrow x=20^{\circ}$ Therefore, $x=20^{\circ}$...

Given below is the frequency distribution of wages (in Rs) of 30 workers in certain factory:

Question: Given below is the frequency distribution of wages (in Rs) of 30 workers in certain factory: A worker is selected at random. Find the probability that his wages are: 1. Less than Rs.150 2. At least Rs.210 3. More than or equal to 150 but less than 210 Solution: Total number of workers = 30 1.Probability that the worker wages are less than Rs.150 $=\frac{\text { No of workers having wages below Rs. } 150}{\text { Total no of workers }}$ $=\frac{3+4}{30}=\frac{7}{30}$ 2. Probability that...

Evaluate each of the following

Question: Evaluate each of the following $\frac{\tan 45^{\circ}}{\operatorname{cosec} 30^{\circ}}+\frac{\sec 60^{\circ}}{\cot 45^{\circ}}-\frac{5 \sin 90^{\circ}}{2 \cos 0^{\circ}}$ Solution: We have, $\frac{\tan 45^{\circ}}{\operatorname{cosec} 30^{\circ}}+\frac{\sec 60^{\circ}}{\cot 45^{\circ}}-\frac{5 \sin 90^{\circ}}{2 \cos 0^{\circ}}$....(1) Now, $\sin 90^{\circ}=\cos 0^{\circ}=1, \tan 45^{\circ}=\cot 45^{\circ}=1, \operatorname{cosec} 30^{\circ}=\sec 60^{\circ}=2$ We get, $\frac{\tan 45^{\...

If O be the origin and the coordinates of P be

Question: If O be the origin and the coordinates of P be (1, 2, 3), then find the equation of the plane passing through P and perpendicular to OP. Solution: The coordinates of the points, O and P, are (0, 0, 0) and (1, 2, 3) respectively. Therefore, the direction ratios of OP are (1 0) = 1, (2 0) = 2, and (3 0) = 3 It is known that the equation of the plane passing through the point (x1,y1z1) is $a\left(x-x_{1}\right)+b\left(y-y_{1}\right)+c\left(z-z_{1}\right)=0$ where, $\mathrm{a}, b$, and $c$...

The following table gives the life time of 400 neon lamps:

Question: The following table gives the life time of 400 neon lamps: A bulb is selected at random. Find the probability that the lifetime of a selected bulb: 1. Less than 400 hrs 2. between 300 - 800 hours 3. At least 700 hours Solution: Total number of bulbs = 400 1.Probability that the life of the selected bulb is less than 400 hrs $=\frac{\text { No of bulbs having life less than } 400 \mathrm{hrs}}{\text { Total no of bulbs }}$ =14/400 =7/200 2. Probability that the life of the selected bulb...

Find the equation of the plane passing through the line of intersection of the planes

Question: Find the equation of the plane passing through the line of intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $\vec{r} \cdot(2 \hat{i}+3 \hat{j}-\hat{k})+4=0$ and parallel to $x$-axis. Solution: The given planes are $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ $\Rightarrow \vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})-1=0$ $\vec{r} \cdot(2 \hat{i}+3 \hat{j}-\hat{k})+4=0$ The equation of any plane passing through the line of intersection of these planes is $[\vec{r} \cdot(...

Evaluate each of the following

Question: Evaluate each of the following $\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}$ Solution: We have, $\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}$....(1) Now, $\sin 45^{\circ}=\frac{1}{\sqrt{2}}, \sin 30^{\circ}=\frac{1}{2}, \sin 90^{\circ}=1, \tan 45^{\circ}=\co...

The value of cos

Question: The value of $\cos \left(36^{\circ}-A\right) \cos \left(36^{\circ}+A\right)+\cos \left(54^{\circ}-A\right) \cos \left(54^{\circ}+A\right)$ is (a) cos 2A (b) sin 2A (c) cosA (d) 0 Solution: (a) cos 2A $\cos \left(36^{\circ}-\mathrm{A}\right) \cos \left(36^{\circ}+\mathrm{A}\right)+\cos \left(54^{\circ}-\mathrm{A}\right) \cos \left(54^{\circ}+\mathrm{A}\right)$ $=\cos \left[90^{\circ}-\left(54^{\circ}+\mathrm{A}\right)\right] \cos \left[90^{\circ}-\left(54^{\circ}-\mathrm{A}\right)\right...

The value of cos

Question: The value of $\cos ^{4} x+\sin ^{4} x-6 \cos ^{2} x \sin ^{2} x$ is (a) cos 2x (b) sin 2x (c) cos 4x (d) none of these Solution: (c) cos 4x $\cos ^{4} x+\sin ^{4} x-6 \cos ^{2} x \sin ^{2} x=\cos ^{4} x+\sin ^{4} x-2 \cos ^{2} x \sin ^{2} x-4 \cos ^{2} x \sin ^{2} x$ $=\left(\cos ^{2} x-\sin ^{2} x\right)^{2}-(2 \sin x \cos x)^{2}$ $=\cos ^{2} 2 x-\sin ^{2} 2 x$ $=\cos 4 x$...

If tan x=t

Question: If $\tan x=t$ then $\tan 2 x+\sec 2 x=$ (a) $\frac{1+t}{1-t}$ (b) $\frac{1-t}{1+t}$ (c) $\frac{2 t}{1-t}$ (d) $\frac{2 t}{1+t}$ Solution: (a) $\frac{1+t}{1-t}$ $\tan 2 x+\sec 2 x=\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1+\tan ^{2} x}{1-\tan ^{2} x}$ $=\frac{2 \tan x+1+\tan ^{2} x}{1-\tan ^{2} x}$ $=\frac{(1+\tan x)^{2}}{1-\tan ^{2} x}$ $=\frac{(1+\tan x)(1+\tan x)}{(1+\tan x)(1-\tan x)}$ $=\frac{1+\tan x}{1-\tan x}$ $=\frac{1+t}{1-t} \quad[\tan x=t$ (given) $]$...

If

Question: If $\left(2^{n}+1\right) x=\pi$, then $2^{n} \cos x \cos 2 x \cos 2^{2} x \ldots \cos 2^{n-1} x=1$ (a) -1 (b) 1 (c) 1/2 (d) None of these Solution: (b) 1 $\left(2^{n}+1\right) x=\pi \quad$ (Given) $\Rightarrow 2^{n} x+x=\pi$ $\Rightarrow 2^{n} x=\pi-x$ $\Rightarrow \sin 2^{n} x=\sin (\pi-x)$ $\Rightarrow \sin 2^{n} x=\sin x$ $2^{n} \cos x \cos 2 x \cos 2^{2} x \ldots \cos 2^{n-1} x=2^{\mathrm{n}} \times \frac{\sin 2^{n} x}{2^{n} \sin x}$ $=\frac{\sin 2^{n} x}{\sin x}$ $=\frac{\sin x}{\...

If the points

Question: If the points $(1,1, p)$ and $(-3,0,1)$ be equidistant from the plane $\vec{r} \cdot(3 \hat{i}+4 \hat{j}-12 \hat{k})+13=0$, then find the value of $p$. Solution: The position vector through the point $(1,1, p)$ is $\vec{a}_{1}=\hat{i}+\hat{j}+p \hat{k}$ Similarly, the position vector through the point $(-3,0,1)$ is $\vec{a}_{2}=-4 \hat{i}+\hat{k}$ The equation of the given plane is $\vec{r} \cdot(3 \hat{i}+4 \hat{j}-12 \hat{k})+13=0$ It is known that the perpendicular distance between ...

If

Question: If $\left(2^{n}+1\right) x=\pi$, then $2^{n} \cos x \cos 2 x \cos 2^{2} x \ldots \cos 2^{n-1} x=1$ (a) -1 (b) 1 (c) 1/2 (d) None of these Solution: (b) 1 $\left(2^{n}+1\right) x=\pi \quad$ (Given) $\Rightarrow 2^{n} x+x=\pi$ $\Rightarrow 2^{n} x=\pi-x$ $\Rightarrow \sin 2^{n} x=\sin (\pi-x)$ $\Rightarrow \sin 2^{n} x=\sin x$ $2^{n} \cos x \cos 2 x \cos 2^{2} x \ldots \cos 2^{n-1} x=2^{\mathrm{n}} \times \frac{\sin 2^{n} x}{2^{n} \sin x}$ $=\frac{\sin 2^{n} x}{\sin x}$ $=\frac{\sin x}{\...