Trigonometry · NDA previous year questions

Consider the following statements: I. The triangle is right-angled. II. One of the sides of the triangle is 3 times the other. III. The angles A, C and B of the triangle are in AP. Which of the statements given above is/are correct?

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

If a line in 3 dimensions makes angles α, β and γ with the positive directions of the coordinate axes, then what is cos(α+β)cos(α−β) equal to?

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

Consider the following for the two (02) items that follow: Let the function $y=(1-\cos x{)}^{-1}$ where $x\ne 2n\pi$ and n is an integer What is the range of the function?

1. A.$[0,\infty )$
2. B.$[0.5,\infty )$
3. C.$[1,\infty )$
4. D.$(-\infty ,0.5]$

Consider the following for the two (02) items that follow: Let $y={\sin }^{-1}\left(x-\frac{4x^3}{27}\right)$. What is y equal to?

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

Consider the following for the two (02) items that follow: Let $y={\sin }^{-1}\left(x-\frac{4x^3}{27}\right)$. What is $\frac{dy}{dx}$ euqal to ?

1. A.$\frac{1}{\sqrt{9-x^2}}$
2. B.$\frac{1}{\sqrt{3-x^2}}$
3. C.$\frac{3}{\sqrt{9-x^2}}$
4. D.$\frac{9}{\sqrt{9-x^2}}$

If $4{\sin }^{-1}x+{\cos }^{-1}x=\pi$, then what is ${\sin }^{-1}x+4{\cos }^{-1}x$ equal to?

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

What is ${\cot }^2({\sec }^{-1}2)+{\tan }^2({\mathrm{cosec}}^{-1}3)$ equal to ?

1. A.11/12
2. B.11/24
3. C.7/24
4. D.1/24

In a triangle ABC $\frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}$. What is the area of the triangle if $a=6\mathrm{cm}$?

1. A.$9\sqrt{3}\mathrm{square}\mathrm{cm}$
2. B.$12\mathrm{square}\mathrm{cm}$
3. C.$18\sqrt{3}\mathrm{square}\mathrm{cm}$
4. D.$24\mathrm{square}\mathrm{cm}$

The roots of the equation $7x^2-6x+1=0$ are $\tan \alpha$ and $\tan \beta$, where $2\alpha$ and $2\beta$ are the angles of a triangle. Which one of the following is correct?

1. A.The triangle is equilateral
2. B.The triangle is isosceles but not right-angled
3. C.The triangle is right-angled
4. D.The triangle is right-angled isosceles

In a triangle ABC, ∠A = 75° and ∠B = 45°. What is 2a - b equal to?

1. A.c
2. B.√2c
3. C.2c
4. D.2√2c

What is the number of solutions of the equation cot2x cot3x = 1 for 0 < x < π?

1. A.Only one
2. B.Only two
3. C.Only five
4. D.More than five

What is the general solution of ${\cos }^{100}x-{\sin }^{100}x=1$?

1. A.$n\pi$
2. B.$(2n+1)\pi$
3. C.$2n\pi$
4. D.$(2n+1)\pi /2$

In a triangle ABC tan A + tan B + tan C = k  What is the value of cot A cot B cot C?

1. A.0.5 k
2. B.1/k
3. C.3/k
4. D.$1/k^3$

What is sin12° sin48° equal to?

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

What is $\frac{\cos 17^{\circ }-\sin 17^{\circ }}{\cos 17^{\circ }+\sin 17^{\circ }}$ equal to

1. A.$\tan 34^{\circ }$
2. B.$\cot 34^{\circ }$
3. C.$\tan 62^{\circ }$
4. D.$\cot 62^{\circ }$

Consider the following numbers : I. tan 22.5° II. cot 22.5° III. tan 22.5° - cot 22.5° How many of the above are irrational numbers?

1. A.None
2. B.Only one
3. C.Only two
4. D.All three

If $\frac{x}{\cos \theta }=\frac{y}{\cos \left(\frac{2\pi }{3}-\theta \right)}=\frac{z}{\cos \left(\frac{2\pi }{3}+\theta \right)}$ then what is $x+y+z$ equal to?

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

If p tan (θ - 30°) = q tan (θ + 120°), then what is (p + q) / (p - q) equal to?

1. A.sin 2θ
2. B.cos 2θ
3. C.2 sin 2θ
4. D.2 cos 2θ

Direction : Consider the following for the items that follow : Let $f\circ g(x)={\cos }^2\sqrt{x}$ and $g\circ f(x)=|\cos x|$. Which one of the following is f(x)?

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

Direction : Consider the following for the items that follow : Let $f\circ g(x)={\cos }^2\sqrt{x}$ and $g\circ f(x)=|\cos x|$. Which one of the following is g(x)?

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

Direction : Consider the following for the items that follow : Let $f(x)=|x{|}^2-|x^2|$ What is $f(0.999)+f(1.001)$ equal to

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

Direction : Consider the following for the items that follow : Let $f(t)=\ln (t+\sqrt{1+t^2})$ and $g(t)=\tan (f(t))$. Consider the following statements : I. f(t) is an odd function. II. g(t) is an odd function. Which of the statements given above is/are correct?

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

Direction : Consider the following for the items that follow : Let $f(t)=\ln (t+\sqrt{1+t^2})$ and $g(t)=\tan (f(t))$. What is ${\int }_{-\pi }^{\pi }g(t)dt$ equal to

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

Direction : Consider the following for the items that follow : Let $I={\int }_0^{\pi /2}\frac{f(x)}{g(x)}dx$, where $f(x)=\sin x$ and $g(x)=\sin x+\cos x+1$ What is ${\int }_0^{\pi /2}\frac{dx}{g(x)}$ equal to

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

Direction : Consider the following for the items that follow : Let $I={\int }_0^{\pi /2}\frac{f(x)}{g(x)}dx$, where $f(x)=\sin x$ and $g(x)=\sin x+\cos x+1$ What is ${\int }_0^{\pi /2}\frac{dx}{g(x)}$ equal to What is I equal to?

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