Evaluate each of the following

Question: Evaluate each of the following $\frac{\tan ^{2} 60^{\circ}+4 \cos ^{2} 45^{\circ}+3 \sec ^{2} 30^{\circ}+5 \cos ^{2} 90^{\circ}}{\operatorname{cosec} 30^{\circ}+\sec 60^{\circ}-\cot ^{2} 30^{\circ}}$ Solution: We have, $\frac{\tan ^{2} 60^{\circ}+4 \cos ^{2} 45^{\circ}+3 \sec ^{2} 30^{\circ}+5 \cos ^{2} 90^{\circ}}{\operatorname{cosec} 30^{\circ}+\sec 60^{\circ}-\cot ^{2} 30^{\circ}}$.....(1) Now, $\tan 60^{\circ}=\cot 30^{\circ}=\sqrt{3}, \cos 45^{\circ}=\frac{1}{\sqrt{2}}, \sec 30^{\...

If tan

Question: If $\tan \frac{x}{2}=\frac{\sqrt{1-e}}{1+e} \tan \frac{\alpha}{2}$, then $\cos \alpha=$ (a) $1-e \cos (\cos x+e)$ (b) $\frac{1+e \cos x}{\cos x-e}$ (c) $\frac{1-e \cos x}{\cos x-e}$ (d) $\frac{\cos x-e}{1-e \cos x}$ Solution: (d) $\frac{\cos x-e}{1-e \cos x}$ Given : $\tan \frac{x}{2}=\sqrt{\frac{1-e}{1+e}} \tan \frac{\alpha}{2}$ $\Rightarrow \frac{\tan \frac{x}{2}}{\tan \frac{\alpha}{2}}=\sqrt{\frac{1-\mathrm{e}}{1+\mathrm{e}}}$ Squaring both sides, we get, $\frac{\tan ^{2} \frac{x}{2...

Evaluate each of the following

Question: Evaluate each of the following $4\left(\sin ^{4} 30^{\circ}+\cos ^{2} 60^{\circ}\right)-3\left(\cos ^{2} 45^{\circ}-\sin ^{2} 90^{\circ}\right)-\sin ^{2} 60^{\circ}$ Solution: We have, $4\left(\sin ^{4} 30^{\circ}+\cos ^{2} 60^{\circ}\right)-3\left(\cos ^{2} 45^{-}-\sin ^{2} 90^{\circ}\right)-\sin ^{2} 60^{\circ} \ldots \ldots$ (1) Now, $\sin 30^{\circ}=\cos 60^{\circ}=\frac{1}{2}, \sin 90^{\circ}=1, \cos 45^{\circ}=\frac{1}{\sqrt{2}}, \sin 60^{\circ}=\frac{\sqrt{3}}{2}$ So by substitu...

Find the equation of the plane passing through the point

Question: Find the equation of the plane passing through the point $(-1,3,2)$ and perpendicular to each of the planes $x+2 y+3 z=5$ and $3 x+3 y+z=0$. Solution: The equation of the plane passing through the point (1, 3, 2) is a(x+ 1) +b(y 3) +c(z 2) = 0 (1) where,a,b,care the direction ratios of normal to the plane. It is known that two planes, $a_{1} x+b_{1} y+c_{1} z+d_{1}=0$ and $a_{2} x+b_{2} y+c_{2} z+d_{2}=0$, are perpendicular, if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$ Plane (1) is perpe...

A company selected 2400 families at random and surveys them to determine a relationship between income

Question: A company selected 2400 families at random and surveys them to determine a relationship between income level and the number of vehicles in a home. The information gathered is listed below: Vechicles Per Family: If a family is chosen at random find the probability that the family is: 1. Earning Rs 10000 13000 per month and owning exactly 2 vehicles. 2. Earning Rs 16000 or more per month and owning exactly 1 vehicle. 3. Earning less than Rs 7000 per month and does not own any vehicle. 4....

If α and β are acute angles satisfying cos 2 α =

Question: If $\alpha$ and $\beta$ are acute angles satisfying $\cos 2 \alpha=\frac{3 \cos 2 \beta-1}{3-\cos 2 \beta}$, then $\tan \alpha=$ (a) $\sqrt{2} \tan \beta$ (b) $\frac{1}{\sqrt{2}} \tan \beta$ (c) $\sqrt{2} \cot \beta$ (d) $\frac{1}{\sqrt{2}} \cot \beta$ Solution: (a) $\sqrt{2} \tan \beta$ Given: $\cos 2 \alpha=\frac{3 \cos 2 \beta-1}{3-\cos 2 \beta}$ $\Rightarrow \frac{\cos 2 \alpha-1}{\cos 2 \alpha+1}=\frac{(3 \cos 2 \beta-1)-(3-\cos 2 \beta)}{(3 \cos 2 \beta-1)+(3-\cos 2 \beta)} \quad...

Question: $2\left(1-2 \sin ^{2} 7 x\right) \sin 3 x$ is equal to (a) $\sin 17 x-\sin 11 x$ (b) $\sin 11 x-\sin 17 x$ (c) $\cos 17 x-\cos 11 x$ (d) $\cos 17 x+\cos 11 x$ Solution: (a) $\sin 17 x-\sin 11 x$ We have, $2\left(1-2 \sin ^{2} 7 x\right) \sin 3 x=2(\cos 14 x) \sin 3 x$ $\left[\because \cos 2 x=1-2 \sin ^{2} x\right]$ $=2 \sin 3 x \cos 14 x$ $=\sin 17 x-\sin 11 x$ $[\because 2 \sin \mathrm{A} \cos \mathrm{B}=\sin (\mathrm{A}+\mathrm{B})-\sin (\mathrm{A}-\mathrm{B})]$ $\therefore 2\left(1...

Find the coordinates of the point where the line through

Question: Find the coordinates of the point where the line through (3, 4, 5) and (2, 3, 1) crosses the plane 2x+y+z= 7). Solution: It is known that the equation of the line through the points, $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$, is $\frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}}$ Since the line passes through the points, (3, 4, 5) and (2, 3, 1), its equation is given by, $\frac{x-3}{2-3}=\frac{y+4}{-3+4}=\frac{z+5}{1+5}$...

Evaluate each of the following

Question: Evaluate each of the following $\frac{4}{\cot ^{2} 30^{\circ}}+\frac{1}{\sin ^{2} 60^{\circ}}-\cos ^{2} 45^{\circ}$ Solution: We have, $\frac{4}{\cot ^{2} 30^{\circ}}+\frac{1}{\sin ^{2} 60^{\circ}}-\cos ^{2} 45^{\circ}$.....(1) Now, $\cot 30^{\circ}=\sqrt{3}, \cos 45^{\circ}=\frac{1}{\sqrt{2}}, \sin 60^{\circ}=\frac{\sqrt{3}}{2}$ So by substituting above values in equation (1) We get, $\frac{4}{\cot ^{2} 30^{\circ}}+\frac{1}{\sin ^{2} 60^{\circ}}-\cos ^{2} 45^{\circ}$ $=\frac{4}{(\sqrt...

The value of

Question: The value of $\frac{2\left(\sin 2 x+2 \cos ^{2} x-1\right)}{\cos x-\sin x-\cos 3 x+\sin 3 x}$ is (a) cosx (b) secx (c) cosecx (d) sinx Solution: (c) cosecx We have, $\frac{2\left(\sin 2 x+2 \cos ^{2} x-1\right)}{\cos x-\sin x-\cos 3 x+\sin 3 x}$ $=\frac{2(\sin 2 x+\cos 2 x)}{\cos x-\sin x-4 \cos ^{3} x+3 \cos x+3 \sin x-4 \sin ^{3} x}$ $=\frac{2(\sin 2 x+\cos 2 x)}{4 \cos x-4 \cos ^{3} x+2 \sin x-4 \sin ^{3} x}$ $=\frac{2(\sin 2 x+\cos 2 x)}{4 \cos x\left(1-\cos ^{2} x\right)+2 \sin x\...

Given below is the frequency distribution table regarding the concentration of SO2 in the air in parts per million of a certain city for 30 days.

Question: Given below is the frequency distribution table regarding the concentration ofSO2in the air in parts per million of a certain city for 30 days. Find the probability of the concentration ofSO2 in the interval 0.12-0.16 on any of these days. Solution: Total no of days: 30 Probability of concentration ofSO2in interval 0.12 - 0.16 $=\frac{\text { Favorable out come }}{\text { Total out come }}$ =2/30 =1/15 = 0.06...

The following table show the birth month of 40 students in class IX:

Question: The following table show the birth month of 40 students in class IX: Find the probability that a student is born in October Solution: 1. Probability that a student is born in the month of October $=\frac{\text { No of students bornin October }}{\text { Total no of students }}$ =6/40 =3/20 = 0.15...

The value of 2 sin

Question: The value of $2 \sin ^{2} B+4 \cos (A+B) \sin A \sin B+\cos 2(A+B)$ is (a) 0 (b) cos 3A (c) cos 2A (d) none of these Solution: (c) cos 2A We have, $2 \sin ^{2} B+4 \cos (A+B) \sin A \sin B+\cos 2(A+B)$ $=1-\cos 2 B+\cos 2(A+B)+4 \cos (A+B) \sin A \sin B$ $=1+(\cos 2(A+B)-\cos 2 B)+4 \cos (A+B) \sin A \sin B$ $=1-2 \sin A \sin (A+2 B)+4 \cos (A+B) \sin A \sin B$ $\left[\because \cos C-\cos D=-2 \sin \frac{C+D}{2} \sin \frac{C-D}{2}\right]$ $=1-2 \sin A[\sin (A+2 B)-2 \sin B \cos (A+B)]$...

Evaluate each of the following

Question: Evaluate each of the following $\frac{\sin 30^{\circ}-\sin 90^{\circ}+2 \cos 0^{\circ}}{\tan 30^{\circ} \tan 60^{\circ}}$ Solution: We have, $\frac{\sin 30^{\circ}-\sin 90^{\circ}+2 \cos 0^{\circ}}{\tan 30^{\circ} \tan 60^{\circ}}$....(1) Now, $\sin 30^{\circ}=\frac{1}{2}, \sin 90^{\circ}=\cos 0^{\circ}=1, \tan 30^{\circ}=\frac{1}{\sqrt{3}}, \tan 60^{\circ}=\sqrt{3}$ So by substituting above values in equation (1) We get, $\frac{\sin 30^{\circ}-\sin 90^{\circ}+2 \cos 0^{\circ}}{\tan 30...

Eleven bags of wheat flour, each marked 5kg, actually contained the following weights of flour (in Kg)

Question: Eleven bags of wheat flour, each marked 5kg, actually contained the following weights of flour (in Kg) Find the probability that any of these bags chosen at random contains more than 5 kg of flour. Solution: Number of bags weighing more than 5 kgs = 7 Total no of bags = 11 Probability of having more than 10 kgs of rice $=\frac{\text { No of bags weighing more than } 5 \mathrm{~kg}}{\text { Total no of bags }}$ = 7/11 = 0.63...

Question: $\frac{\sin 3 x}{1+2 \cos 2 x}$ is equal to (a) $\cos x$ (b) $\sin x$ (c) $-\cos x$ (d) $\sin x$ Solution: (b) $\sin x$ We have, $\frac{\sin 3 x}{1+2 \cos 2 x}=\frac{3 \sin x-4 \sin ^{3} x}{1+2\left(1-2 \sin ^{2} x\right)}$ $=\frac{3 \sin x-4 \sin ^{3} x}{1+2-4 \sin ^{2} x}$ $=\frac{\sin x\left(3-4 \sin ^{2} x\right)}{\left(3-4 \sin ^{2} x\right)}$ $=\sin x$...

Find the coordinates of the point where the line through

Question: Find the coordinates of the point where the line through $(5,1,6)$ and $(3,4,1)$ crosses the $Z X$ - plane. Solution: It is known that the equation of the line passing through the points, $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$, is $\frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}}$ The line passing through the points, (5, 1, 6) and (3, 4, 1), is given by, $\frac{x-5}{3-5}=\frac{y-1}{4-1}=\frac{z-6}{1-6}$ $\Rightarrow ...

The value of cos

Question: The value of $\cos ^{2}\left(\frac{\pi}{6}+x\right)-\sin ^{2}\left(\frac{\pi}{6}-x\right)$ is (a) $\frac{1}{2} \cos 2 x$ (b) 0 (c) $-\frac{1}{2} \cos 2 x$ (d) $\frac{1}{2}$ Solution: (a) $\frac{1}{2} \cos 2 x$ We have, $\cos ^{2}\left(\frac{\pi}{6}+x\right)-\sin ^{2}\left(\frac{\pi}{6}-x\right)$ $=\cos ^{2}\left(\frac{\pi}{6}+x\right)-\cos ^{2}\left[\frac{\pi}{2}-\left(\frac{\pi}{6}-x\right)\right]$ $=\cos ^{2}\left(\frac{\pi}{6}+x\right)-\cos ^{2}\left(\frac{\pi}{3}+x\right)$ $=\left[...

The Blood group table of 30 students of class IX is recorded as follows:

Question: The Blood group table of 30 students of class IX is recorded as follows: A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O A student is selected at random from the class from blood donation. Find the probability that the blood group of the student chosen is: 1: A 2: B 3: AB 4: O Solution: 1. Probability of a student having blood group A $=\frac{\text { Favorable out come }}{\text { Total out come }}$ =9/30 = 0.3 2. Probability of a student havin...

Evaluate each of the following

Question: Evaluate each of the following $\left(\cos 0^{\circ}+\sin 45^{\circ}+\sin 30^{\circ}\right)\left(\sin 90^{\circ}-\cos 45^{\circ}+\cos 60^{\circ}\right)$ Solution: We have, $\left(\cos 0^{\circ}+\sin 45^{\circ}+\sin 30^{\circ}\right)\left(\sin 90^{\circ}-\cos 45^{\circ}+\cos 60^{\circ}\right) \ldots \ldots$ (1) Now, $\sin 90^{\circ}=\cos 0^{\circ}=1, \sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}, \sin 30^{\circ}=\cos 60^{\circ}=\frac{1}{2}$ So by substituting above values in equati...

If tan

Question: If $\tan (\pi / 4+x)+\tan (\pi / 4-x)=\lambda \sec 2 x$, then (a) 3 (b) 4 (c) 1 (d) 2 Solution: (d) 2 Given: $\tan \left(\frac{\pi}{4}+x\right)+\tan \left(\frac{\pi}{4}-x\right)=\lambda \sec 2 x$ $\Rightarrow \frac{\tan \frac{\pi}{4}+\tan x}{1-\tan \frac{\pi}{4} \times \tan x}+\frac{\tan \frac{\pi}{4}-\tan x}{1+\tan \frac{\pi}{4} \times \tan x}=\lambda \sec 2 x$ $\Rightarrow \frac{1+\tan x}{1-\tan x}+\frac{1-\tan x}{1+\tan x}=\lambda$ sec $2 x$ $\Rightarrow \frac{(1+\tan x)^{2}+(1-\tan...

Find the coordinates of the point where the line through

Question: Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane Solution: It is known that the equation of the line passing through the points, $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$, is $\frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}}$ The line passing through the points, (5, 1, 6) and (3, 4, 1), is given by, $\frac{x-5}{3-5}=\frac{y-1}{4-1}=\frac{z-6}{1-6}$ $\Rightarrow \frac{...

If tan

Question: If $\tan (\pi / 4+x)+\tan (\pi / 4-x)=\lambda \sec 2 x$, then (a) 3 (b) 4 (c) 1 (d) 2 Solution: (d) 2 Given: $\tan \left(\frac{\pi}{4}+x\right)+\tan \left(\frac{\pi}{4}-x\right)=\lambda \sec 2 x$ $\Rightarrow \frac{\tan \frac{\pi}{4}+\tan x}{1-\tan \frac{\pi}{4} \times \tan x}+\frac{\tan \frac{\pi}{4}-\tan x}{1+\tan \frac{\pi}{4} \times \tan x}=\lambda \sec 2 x$ $\Rightarrow \frac{1+\tan x}{1-\tan x}+\frac{1-\tan x}{1+\tan x}=\lambda$ sec $2 x$ $\Rightarrow \frac{(1+\tan x)^{2}+(1-\tan...

The value of

Question: The value of $\frac{\cos 3 x}{2 \cos 2 x-1}$ is equal to (a) cosx (b) sinx (c) tanx (d) none of these Solution: (a) cosx We have, $\therefore \frac{\cos 3 x}{2 \cos 2 x-1}=\frac{4 \cos ^{3} x-3 \cos x}{2\left(2 \cos ^{2} x-1\right)-1} \quad\left[\because \cos 3 x=4 \cos ^{3} x-3 \cos x\right]$ $=\frac{4 \cos ^{3} x-3 \cos x}{4 \cos ^{2} x-2-1}$ $=\frac{4 \cos ^{3} x-3 \cos x}{4 \cos ^{2} x-3}$ $=\cos x\left(\frac{4 \cos ^{2} x-3}{4 \cos ^{2} x-3}\right)$ $=\cos x$...

To know the opinion of the students about Mathematics, a survey of 200 students were conducted.

Question: To know the opinion of the students about Mathematics, a survey of 200 students were conducted. The data was recorded in the following table Find the probability that student chosen at random: 1. Likes Mathematics 2. Does not like it. Solution: 1. Probability that a student likes mathematics $=\frac{\text { Favorable out come }}{\text { Total out come }}$ = 135/200 = 0.675 2. Probability that a student does not like mathematics $=\frac{\text { Favorable out come }}{\text { Total out co...