Trigonometry · NDA previous year questions

Consider the following for the next three (03) items that follow : Let $I_1={\int }_0^{\pi }\frac{x}{1+{\cos }^{2}x}\mathrm{dx}$ and $I_2={\int }_0^{\pi }\frac{1}{1+{\sin }^{2}x}\mathrm{dx}$ What is the value of $\frac{I_1+I_2}{I_1-I_2}$ ?

1. A.1
2. B.$\pi$
3. C.${\pi }^2$
4. D.$\frac{\pi +2}{\pi -2}$

1. A.$\pi$
2. B.${\pi }^2$
3. C.${\pi }^3$
4. D.${\pi }^4$

1. A.$\frac{\pi }{\sqrt{2}}$
2. B.$\frac{\pi }{2\sqrt{2}}$
3. C.$\frac{3\pi }{2\sqrt{2}}$
4. D.$\frac{\pi }{4\sqrt{2}}$

Consider the following for the next two (02) items that follow : Let $l={\int }_a^b\frac{|x|}{x}\mathrm{dx}$, $a<b$ What is l equal to when $a<0<b$ ?

1. A.$a+b$
2. B.$a-b$
3. C.$b-a$
4. D.$\frac{(a+b)}{2}$

Consider the following for the next two (02) items that follow : Let $l={\int }_a^b\frac{|x|}{x}\mathrm{dx}$, $a<b$ What is l equal to when $a<b<0$ ?

1. A.$a+b$
2. B.$a-b$
3. C.$b-a$
4. D.$\frac{(a+b)}{2}$

What is cos A + cos B + cos C equal to ?

1. A.1
2. B.$\frac{41}{25}$
3. C.$\frac{37}{25}$
4. D.$\frac{33}{25}$

At what height is the top of the tower above the ground level ?

1. A.$\frac{x-y}{2\sqrt{3}}$
2. B.$\frac{x-y}{4\sqrt{3}}$
3. C.$\frac{x-y}{4}$
4. D.$\frac{x-y}{2}$

If θ is the inclination of the tower to the horizontal, then what is cot θ equal to ?

1. A.$2+\frac{\sqrt{3}(x-y)}{x+y}$
2. B.$2-\frac{\sqrt{3}(x-y)}{x+y}$
3. C.$2+\frac{\sqrt{3}(x+y)}{x-y}$
4. D.$2-\frac{\sqrt{3}(x+y)}{x-y}$

What is the length of the tower ?

1. A.$\frac{x-y}{2\sqrt{3}}\sqrt{1+{\left\{2+\frac{\sqrt{3}(x+y)}{x-y}\right\}}^2}$
2. B.$\frac{x-y}{2\sqrt{3}}\sqrt{1+{\left\{2-\frac{\sqrt{3}(x+y)}{x-y}\right\}}^2}$
3. C.$\frac{x-y}{4\sqrt{3}}\sqrt{1+{\left\{2+\frac{\sqrt{3}(x+y)}{x-y}\right\}}^2}$
4. D.$\frac{x-y}{4\sqrt{3}}\sqrt{1+{\left\{2-\frac{\sqrt{3}(x+y)}{x-y}\right\}}^2}$

What is the value of $\mathrm{cosec}(-\frac{73\pi }{3})$ ?

1. A.$\frac{2}{\sqrt{3}}$
2. B.$-\frac{2}{\sqrt{3}}$
3. C.2
4. D.-2

What is the value of $\cos \left(\frac{5\pi }{17}\right)+\cos \left(\frac{7\pi }{17}\right)+2\cos \left(\frac{11\pi }{17}\right)\cos \left(\frac{\pi }{17}\right)$ ?

1. A.0
2. B.1
3. C.$4\cos \left(\frac{6\pi }{17}\right)\cos \left(\frac{\pi }{17}\right)$
4. D.$4\cos \left(\frac{11\pi }{17}\right)\cos \left(\frac{\pi }{17}\right)$

What is the value of $\tan \left(\frac{3\pi }{8}\right)$ ?

1. A.$\sqrt{2}-1$
2. B.$\sqrt{2}+1$
3. C.$1-\sqrt{2}$
4. D.$-(\sqrt{2}+1)$

What is ${\tan }^{-1}\cot ({\mathrm{cosec}}^{-1}2)$ equal to ?

1. A.$\frac{\pi }{8}$
2. B.$\frac{\pi }{6}$
3. C.$\frac{\pi }{4}$
4. D.$\frac{\pi }{3}$

In a triangle ABC, a = 4, b = 3, c = 2. What is cos 3C equal to ?

1. A.$\frac{7}{128}$
2. B.$\frac{11}{128}$
3. C.$\frac{7}{64}$
4. D.$\frac{11}{64}$

What is cos 36° − cos 72° equal to ?

1. A.$\frac{\sqrt{5}}{2}$
2. B.$-\frac{\sqrt{5}}{2}$
3. C.$\frac{1}{2}$
4. D.$-\frac{1}{2}$

If $\sec x=\frac{25}{24}$ and x lies in the fourth quadrant, then what is the value of $\tan x+\sin x$ ?

1. A.$-\frac{625}{168}$
2. B.$-\frac{343}{600}$
3. C.$\frac{625}{168}$
4. D.$\frac{343}{600}$

What is the value of ${\tan }^{2}165^{\circ }+{\cot }^{2}165^{\circ }$ ?

1. A.7
2. B.14
3. C.$4\sqrt{3}$
4. D.$8\sqrt{3}$

What is the value of $\sin \left(2n\pi +\frac{5\pi }{6}\right)\sin \left(2n\pi -\frac{5\pi }{6}\right)$ where $n\in Z$ ?

1. A.$-\frac{1}{4}$
2. B.$-\frac{3}{4}$
3. C.$\frac{1}{4}$
4. D.$\frac{3}{4}$

If 1 + 2(sin x + cos x)(sin x − cos x) = 0 where 0 < x < 360°, then how many values does x take ?

1. A.Only one value
2. B.Only two values
3. C.Only three values
4. D.Four values

What is ${\lim }_{h\to 0}\frac{{\sin }^2(x+h)-{\sin }^{2}x}{h}$ equal to ?

1. A.${\sin }^{2}x$
2. B.${\cos }^{2}x$
3. C.$\sin 2x$
4. D.$\cos 2x$

What is ${\int }_{-\pi /2}^{\pi /2}(e^{\cos x}\sin x+e^{\sin x}\cos x)\mathrm{dx}$ equal to ?

1. A.$\frac{e^2-1}{e}$
2. B.$\frac{e^2+1}{e}$
3. C.$\frac{1-e^2}{e}$
4. D.0

In a triangle ABC, sin A - cos B - cos C = 0. What is angle B equal to?

1. A.$\frac{\pi }{6}$
2. B.$\frac{\pi }{4}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{2}$

If $\alpha +\beta =\frac{\pi }{4}$ and $2\tan \alpha =1$, then what is $\tan 2\beta$ equal to?

1. A.$\frac{1}{3}$
2. B.$\frac{2}{3}$
3. C.$\frac{3}{4}$
4. D.$\frac{3}{5}$

If $\tan (45^{\circ }+\theta )=1+\sin 2\theta$, where $-\frac{\pi }{4}<\theta <\frac{\pi }{4}$, then what is the value of $\cos 2\theta$?

1. A.0
2. B.$\frac{1}{2}$
3. C.1
4. D.2

Let $\sin 2\theta =\cos 3\theta$, where $\theta$ is acute angle. What is the value of $1+4\sin \theta$? (given that $\sin 18^{\circ }=\frac{\sqrt{5}-1}{4}$)

1. A.$\sqrt{3}$
2. B.2
3. C.$\sqrt{5}$
4. D.3