Trigonometry · NDA previous year questions

If $\tan \theta =-\frac{5}{12}$, then what can be the value of $\sin \theta$?

1. A.$\frac{5}{13}$ but cannot be $-\frac{5}{13}$
2. B.$-\frac{5}{13}$ but cannot be $\frac{5}{13}$
3. C.$\frac{5}{13}$ or $-\frac{5}{13}$
4. D.None of the above

What is the value of ${\cos }^4\frac{7\pi }{8}+{\cos }^4\frac{5\pi }{8}$?

1. A.$\frac{3}{2}$
2. B.$\frac{3}{4}$
3. C.$\frac{3}{8}$
4. D.$\frac{3}{16}$

What is ${\sin }^2\left(\frac{\pi }{4}+\theta \right)-{\sin }^2\left(\frac{\pi }{4}-\theta \right)$ equal to?

1. A.$\sin 2\theta$
2. B.$\cos 2\theta$
3. C.$2\sin \theta$
4. D.$2\cos \theta$

The shadow of a tower becomes x metre longer, when the angle of elevation of sun changes from $60^{\circ }$ to $\theta$. If the height of the tower is $\sqrt{3}x$ metre, then which one of the following is correct ?

1. A.$0<\theta <30^{\circ }$
2. B.$30^{\circ }<\theta <45^{\circ }$
3. C.$45^{\circ }<\theta <60^{\circ }$
4. D.$60^{\circ }<\theta <90^{\circ }$

If ${\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^{-1}\left(\frac{x}{3}\right)=\frac{\pi }{4}$, where $0<x<6$, then what is x equal to?

1. A.1
2. B.2
3. C.3
4. D.5

If $3{\sin }^{-1}x+{\cos }^{-1}x=\pi$, then what is x equal to?

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

If tan α + tan β = 1 - tan α.tan β, where tan α tan β ≠ 1, then which of the following is one of the values of (α + β) ?

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

If (1 + tan θ) (1 + tan 9θ) = 2, then what is the value of tan(10θ)?

1. A.0
2. B.1
3. C.2
4. D.Infinite

What is the value of sin 0° + sin 10° + sin 20° + sin 30° + ⋯ + sin 360°?

1. A.-1
2. B.0
3. C.1
4. D.2

Let a, b, c be the lengths of sides BC, CA, AB respectively of a triangle ABC. If p is the perimeter and q is the area of the triangle, then what is $p(p-2a)\tan \left(\frac{A}{2}\right)$ equal to ?

1. A.q
2. B.2q
3. C.3q
4. D.4q

If $\sin (A+B)=1$ and $2\sin (A-B)=1$, where $0<A,B<\frac{\pi }{2}$, then what is $\tan A:\tan B$ equal to?

1. A.$1:2$
2. B.$2:1$
3. C.$1:3$
4. D.$3:1$

Let ${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{3\pi }{2}$ for $0\le x,y,z\le 1$. What is the value of $x^{1000}+y^{1001}+z^{1002}$?

1. A.0
2. B.1
3. C.3
4. D.6

Let $\sin x+\sin y=\cos x+\cos y$ for all $x,y\in \mathbb{R}$. What is $\tan \left(\frac{x}{2}+\frac{y}{2}\right)$ equal to?

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

Let ABC be a triangle. If cos2A + cos2B + cos2C = -1 then which one of the following is correct?

1. A.sin A sin B sin C = 0
2. B.sin A sin B cos C = 0
3. C.cos A sin B sin C = 0
4. D.cos A cos B cos C = 0

What is the equation of the altitude through B on AC?

1. A.x - 3y + 1 = 0
2. B.x - 3y + 4 = 0
3. C.2x - y + 4 = 0
4. D.2x - y - 5 = 0

If $y=\cos x\cdot \cos 4x\cdot \cos 8x$, then what is $\frac{1}{y}\frac{\mathrm{d}y}{\mathrm{d}x}$ at $x=\frac{\pi }{4}$ equal to?

1. A.-1
2. B.0
3. C.1
4. D.3

If $\tan A=\frac{1}{7}$, then what is $\cos 2A$ equal to?

1. A.$\frac{24}{25}$
2. B.$\frac{18}{25}$
3. C.$\frac{12}{25}$
4. D.$\frac{6}{25}$

The sides of a triangle are $m$, $n$ and $\sqrt{m^2+n^2+mn}$. What is the sum of the acute angles of the triangle?

1. A.$45^{\circ }$
2. B.$60^{\circ }$
3. C.$75^{\circ }$
4. D.$90^{\circ }$

What is the area of the triangle ABC with sides a = 10cm  and c = 4cm angle B = 30°?

1. A.$16\mathrm{c}{\mathrm{m}}^2$
2. B.$12\mathrm{c}{\mathrm{m}}^2$
3. C.$10\mathrm{c}{\mathrm{m}}^2$
4. D.$8\mathrm{c}{\mathrm{m}}^2$

If $\tan x=-\frac{3}{4}$ and $x$ is in the second quadrant, then what is the value of $\sin x\cdot \cos x$?

1. A.$\frac{6}{25}$
2. B.$\frac{12}{25}$
3. C.$-\frac{6}{25}$
4. D.$-\frac{12}{25}$

What is the value of the following? $\mathrm{cosec}\left(\frac{7\pi }{6}\right)\sec \left(\frac{5\pi }{3}\right)$

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

What is the value of the following? tan 31° tan 33° tan 35° _ _ _ _ _ tan 57° tan 59°

1. A.-1
2. B.0
3. C.1
4. D.2

If then what is f(-1) + f(0) + f(1) equal to?

1. A.0
2. B.1
3. C.100
4. D.-100

What is the value of the following? (sin 24° + cos 66°)(sin 24° - cos 66°)

1. A.-1
2. B.0
3. C.1
4. D.2

What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?

1. A.1
2. B.2
3. C.3
4. D.4