Trigonometry · NDA previous year questions

What is $\frac{1+{\tan }^2\theta }{1+{\cot }^2\theta }-{\left(\frac{1-\tan \theta }{1-\cot \theta }\right)}^2$ equal to?

1. A.0
2. B.1
3. C.$2\tan \theta$
4. D.$2\cot \theta$

What is the interior angle of a regular octagon of side length 2 cm?

1. A.$\frac{\pi }{2}$
2. B.$\frac{3\pi }{4}$
3. C.$\frac{3\pi }{5}$
4. D.$\frac{3\pi }{8}$

If 7 sinθ + 24 cosθ = 25, then what is the value of (sin θ + cos θ)?

1. A.1
2. B.$\frac{26}{25}$
3. C.$\frac{6}{5}$
4. D.$\frac{31}{25}$

A ladder 6 m long reaches a point 6 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is 75°. What is the height of the Flagstaff?

1. A.11 m
2. B.9 m
3. C.(6 + √3) m
4. D.(6 + 3√3) m

The shadow of a tower is found to be x meter longer, when the angle of elevation of the sun changes from 60° to 45°. If the height of the tower is 5(3 + √3) m, then what is x equal to?

1. A.8 m
2. B.10 m
3. C.12 m
4. D.15 m

If 3cosθ = 4sinθ, then what is the value of tan (45° + θ)?

1. A.10
2. B.7
3. C.$\frac{7}{2}$
4. D.$\frac{7}{4}$

The smallest positive integer n for which where i = √-1, is

1. A.2
2. B.4
3. C.6
4. D.8

The value of x, satisfying the equation ${\log }_{\cos x}\sin x=1$, where $0<x<\frac{\pi }{2}$, is

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

What is cot 2x cot 4x - cot 4x cot 6x - cot 6x cot 2x equal to

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

Consider the following statements in respect of the function $f(x)=\sin x$: 1. $f(x)$ increases in the interval $(0,\pi )$. 2. $f(x)$ decreases in the interval $\left(\frac{5\pi }{2},3\pi \right)$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is $\int \frac{\mathrm{d}x}{\sec x+\tan x}$ equal to?

1. A.$\ln (\sec x)+\ln |\sec x+\tan x|+c$
2. B.$\ln (\sec x)-\ln |\sec x+\tan x|+c$
3. C.$\sec x\tan x-\ln |\sec x-\tan x|+c$
4. D.$\ln |\sec x+\tan x|-\ln |\sec x|+c$

What is $\int \frac{\mathrm{d}x}{{\sec }^2({\tan }^{-1}x)}$ equal to?

1. A.${\sin }^{-1}x+c$
2. B.${\tan }^{-1}x+c$
3. C.${\sec }^{-1}x+c$
4. D.${\cos }^{-1}x+c$

Consider the following statements: 1. If ABC is a right-angled triangle, right-angled at A and if $\sin B=\frac{1}{3}$, then $\mathrm{cosec}C=3$. 2. If $b\cos B=c\cos C$ and if the triangle ABC is not right-angled, then ABC must be isosceles. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Consider the following statements: 1. If in a triangle ABC, $A=2B$ and $b=c$, then it must be an obtuse-angled triangle. 2. There exists no triangle ABC with $A=40^{\circ }$, $B=65^{\circ }$ and $\frac{a}{c}=\sin 40^{\circ }\mathrm{cosec}15^{\circ }$. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $a{\sin }^{2}x+b{\cos }^{2}x=c$; $b{\sin }^{2}y+a{\cos }^{2}y=d$ and $p\tan x=q\tan y$. What is ${\tan }^{2}x$ equal to?

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

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $a{\sin }^{2}x+b{\cos }^{2}x=c$; $b{\sin }^{2}y+a{\cos }^{2}y=d$ and $p\tan x=q\tan y$. What is ${\tan }^{2}x$ equal to? What is $\frac{d-a}{b-d}$ equal to?

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

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $a{\sin }^{2}x+b{\cos }^{2}x=c$; $b{\sin }^{2}y+a{\cos }^{2}y=d$ and $p\tan x=q\tan y$. What is ${\tan }^{2}x$ equal to? What is $\frac{p^2}{q^2}$ equal to?

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

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $t_n={\sin }^n\theta +{\cos }^n\theta$. What is $\frac{t_3-t_5}{t_5-t_7}$ equal to?

1. A.$\frac{t_1}{t_3}$
2. B.$\frac{t_3}{t_5}$
3. C.$\frac{t_5}{t_7}$
4. D.$\frac{t_1}{t_7}$

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $t_n={\sin }^n\theta +{\cos }^n\theta$. What is $t_1^2-t_2$ equal to?

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

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $t_n={\sin }^n\theta +{\cos }^n\theta$. What is the value of $t_{10}$ where $\theta =45^{\circ }$ ?

1. A.1
2. B.$\frac{1}{4}$
3. C.$\frac{1}{16}$
4. D.$\frac{1}{32}$

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $\alpha =\beta =15^{\circ }$. What is the value of $\sin \alpha +\cos \beta$ ?

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

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $\alpha =\beta =15^{\circ }$. What is the value of $\sin 7\alpha -\cos 7\beta$ ?

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

Directions for the following three (03) items: Read the following information and answer the three items that follow: Let $\alpha =\beta =15^{\circ }$. What is the value of $\sin \alpha +\cos \beta$ ? What is $\sin (\alpha +1^{\circ })+\cos (\beta +1^{\circ })$ equal to?

1. A.$\sqrt{3}\cos 1^{\circ }+\sin 1^{\circ }$
2. B.$\sqrt{3}\cos 1^{\circ }-\frac{1}{2}\sin 1^{\circ }$
3. C.$\frac{1}{\sqrt{2}}(\sqrt{3}\cos 1^{\circ }-\sin 1^{\circ })$
4. D.$\frac{1}{2}(\sqrt{3}\cos 1^{\circ }+\sin 1^{\circ })$

If $\sin x+\sin y=\cos y-\cos x$, where $0<y<x<\frac{\pi }{2}$, then what is $\tan \left(\frac{x-y}{2}\right)$ equal to?

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

The angle of elevation of a tower of height $h$ from a point A due South of it is $x$ and from a point B due East of A is $y$. If AB = $z$, then which one of the following is correct?

1. A.$h^2({\cot }^{2}y-{\cot }^{2}x)=z^2$
2. B.$z^2({\cot }^{2}y-{\cot }^{2}x)=h^2$
3. C.$h^2({\tan }^{2}y-{\tan }^{2}x)=z^2$
4. D.$z^2({\tan }^{2}y-{\tan }^{2}x)=h^2$