Trigonometry · NDA previous year questions

If the angles of a triangle ABC are in the ratio 1 : 2 : 3, then the corresponding sides are in the ratio

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

Consider the following statements: 1. For an equation of a line, $x\cos \theta +y\sin \theta =p$, in normal form, the length of the perpendicular from the point $(\alpha ,\beta )$ to the line is $|\alpha \cos \theta +\beta \sin \theta +p|$. 2. The length of the perpendicular from the point $(\alpha ,\beta )$ to the line $\frac{x}{a}+\frac{y}{b}=1$ is $\left|\frac{a\alpha +b\beta -ab}{\sqrt{a^2+b^2}}\right|$. 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

If in a right-angled triangle ABC, hypotenuse AC = p, then what is $\overrightarrow{AB}\cdot \overrightarrow{AC}+\overrightarrow{BC}\cdot \overrightarrow{BA}+\overrightarrow{CA}\cdot \overrightarrow{CB}$ equal to?

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

1. A.The triangle ABC is isosceles
2. B.The triangle ABC is equilateral
3. C.The triangle ABC is scalene
4. D.No conclusion can be drawn with regard to the nature of the triangle

What is $\frac{2\tan \theta }{1+{\tan }^2\theta }$ equal to?

1. A.$\cos 2\theta$
2. B.$\tan 2\theta$
3. C.$\sin 2\theta$
4. D.$\mathrm{cosec}2\theta$

A is an angle in the fourth quadrant. It satisfies the trigonometric equation $3(3-{\tan }^{2}A-\cot A{)}^2=1$. Which one of the following is a value of A?

1. A.$300^{\circ }$
2. B.$315^{\circ }$
3. C.$330^{\circ }$
4. D.$345^{\circ }$

What is/are the solution(s) of the trigonometric equation $\mathrm{cosec}x+\cot x=\sqrt{3}$, where $0<x<2\pi$?

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

If $\theta =\frac{\pi }{8}$, then what is the value of $(2\cos \theta +1{)}^{10}(2\cos 2\theta -1{)}^{10}(2\cos \theta -1{)}^{10}(2\cos 4\theta -1{)}^{10}$?

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

A balloon is directly above one end of a bridge. The angle of depression of the other end of the bridge from the balloon is $48^{\circ }$. If the height of the balloon above the bridge is 122 m, then what is the length of the bridge?

1. A.$122\sin 48^{\circ }$ m
2. B.$122\tan 42^{\circ }$ m
3. C.$122\cos 48^{\circ }$ m
4. D.$122\tan 48^{\circ }$ m

The top of a hill observed from the top and bottom of a building of height h is at angles of elevation $\frac{\pi }{6}$ and $\frac{\pi }{3}$ respectively. What is the height of the hill?

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

In a triangle ABC, if taken in order, consider the following statements: 1. $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=\vec{0}$ 2. $\overrightarrow{AB}+\overrightarrow{BC}-\overrightarrow{CA}=\vec{0}$ 3. $\overrightarrow{AB}-\overrightarrow{BC}+\overrightarrow{CA}=\vec{0}$ 4. $\overrightarrow{BA}-\overrightarrow{BC}+\overrightarrow{CA}=\vec{0}$ How many of the above statements are correct?

1. A.One
2. B.Two
3. C.Three
4. D.Four

A spherical balloon of radius $r$ subtends an angle $\alpha$ at the eye of an observer, while the angle of elevation of its centre is $\beta$. What is the height of the centre of the balloon (neglecting the height of the observer)?

1. A.$\frac{r\sin \beta }{\sin \left(\frac{\alpha }{2}\right)}$
2. B.$\frac{r\sin \beta }{\sin \left(\frac{\alpha }{4}\right)}$
3. C.$\frac{r\sin \left(\frac{\beta }{2}\right)}{\sin \alpha }$
4. D.$\frac{r\sin \alpha }{\sin \left(\frac{\beta }{2}\right)}$

In a triangle ABC if $a=2,b=3$ and $\sin A=2/3$, then what is angle B equal to?

1. A.$\pi /4$
2. B.$\pi /2$
3. C.$\pi /3$
4. D.$\pi /6$

What is the distance between the points which divide the line segment joining $(4,3)$ and $(5,7)$ internally and externally in the ratio $2:3$?

1. A.$\frac{12\sqrt{17}}{5}$
2. B.$\frac{13\sqrt{17}}{5}$
3. C.$\frac{\sqrt{17}}{5}$
4. D.$\frac{6\sqrt{17}}{5}$

$\sqrt{1+\sin A}=-\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right)$ is true if

1. A.$\frac{3\pi }{2}<A<\frac{5\pi }{2}$ only
2. B.$\frac{\pi }{2}<A<\frac{3\pi }{2}$ only
3. C.$\frac{3\pi }{2}<A<\frac{7\pi }{2}$
4. D.$0<A<\frac{3\pi }{2}$

In triangle $ABC$, if $\frac{{\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C}{{\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C}=2$ then the triangle is

1. A.right-angled
2. B.equilateral
3. C.isosceles
4. D.obtuse-angled

The angle of elevation of a stationary cloud from a point $25\text{ m}$ above a lake is $15^{\circ }$ and the angle of depression of its image in the lake is $45^{\circ }$. The height of the cloud above the lake level is

1. A.$25\text{ m}$
2. B.$25\sqrt{3}\text{ m}$
3. C.$50\text{ m}$
4. D.$50\sqrt{3}\text{ m}$

The angles of elevation of the top of a tower from the top and foot of a pole are respectively $30^{\circ }$ and $45^{\circ }$. If $h_T$ is the height of the tower and $h_P$ is the height of the pole, then which of the following are correct? 1. $\frac{2h_P{h}_T}{3+\sqrt{3}}=h_P^2$ 2. $\frac{h_T-h_P}{\sqrt{3}+1}=\frac{h_P}{2}$ 3. $\frac{2(h_P+h_T)}{h_P}=4+\sqrt{3}$ Select the correct answer using the code given below.

1. A.1 and 3 only
2. B.2 and 3 only
3. C.1 and 2 only
4. D.1, 2 and 3

In a triangle $ABC$, $a-2b+c=0$. The value of $\cot \left(\frac{A}{2}\right)\cot \left(\frac{C}{2}\right)$ is

1. A.$\frac{9}{2}$
2. B.3
3. C.$\frac{3}{2}$
4. D.1

From the top of a lighthouse, 100 m high, the angle of depression of a boat is ${\tan }^{-1}\left(\frac{5}{12}\right)$. What is the distance between the boat and the lighthouse?

1. A.120 m
2. B.180 m
3. C.240 m
4. D.360 m

If a vertex of a triangle is $(1,1)$ and the midpoints of two sides of the triangle through this vertex are $(-1,2)$ and $(3,2)$, then the centroid of the triangle is

1. A.$\left(-\frac{1}{3},\frac{7}{3}\right)$
2. B.$\left(-1,\frac{7}{3}\right)$
3. C.$\left(\frac{1}{3},\frac{7}{3}\right)$
4. D.$\left(1,\frac{7}{3}\right)$

A moving boat is observed from the top of a cliff of 150 m height. The angle of depression of the boat changes from $60^{\circ }$ to $45^{\circ }$ in 2 minutes. What is the speed of the boat in metres per hour?

1. A.$\frac{4500}{\sqrt{3}}$
2. B.$\frac{4500(\sqrt{3}-1)}{\sqrt{3}}$
3. C.$4500\sqrt{3}$
4. D.$\frac{4500(\sqrt{3}+1)}{\sqrt{3}}$

In a right-angled triangle ABC, if the hypotenuse AB = p, then what is $\overrightarrow{AB}\cdot \overrightarrow{AC}+\overrightarrow{BC}\cdot \overrightarrow{BA}+\overrightarrow{CA}\cdot \overrightarrow{CB}$ equal to?

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

A force $\vec{F}=3\hat{i}+2\hat{j}-4\hat{k}$ is applied at the point $(1,-1,2)$. What is the moment of the force about the point $(2,-1,3)$?

1. A.$\hat{i}+4\hat{j}+4\hat{k}$
2. B.$2\hat{i}+\hat{j}+2\hat{k}$
3. C.$2\hat{i}-7\hat{j}-2\hat{k}$
4. D.$2\hat{i}+4\hat{j}-\hat{k}$

$f(xy)=f(x)+f(y)$ is true for all

1. A.Polynomial functions $f$
2. B.Trigonometric functions $f$
3. C.Exponential functions $f$
4. D.Logarithmic functions $f$