Trigonometry · NDA previous year questions

Direction : Consider the following for the items that follow : Let $2\int \frac{x^2-1}{\sqrt{x^2+1}}dx=U(x)V(x)-3\ln \{U(x)+V(x)\}+c$ What is $|U^2(x)-V^2(x)|$ equal to?

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

Direction : Consider the following for the items that follow : Let $2\int \frac{x^2-1}{\sqrt{x^2+1}}dx=U(x)V(x)-3\ln \{U(x)+V(x)\}+c$ What is U(x) V(x) equal to?

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

What is $2\cot \left(\frac{1}{2}{\cos }^{-1}\frac{\sqrt{5}}{3}\right)$ equal to ?

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

If ${\sec }^{-1}p-{\mathrm{cosec}}^{-1}q=0$, where $p>0,q>0$; then what is the value of $p^{-2}+q^{-2}$ ?

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

What is $1+{\sin }^2\left({\cos }^{-1}\left(\frac{3}{\sqrt{17}}\right)\right)$ equal to ?

1. A.$\frac{25}{17}$
2. B.$\frac{8}{17}$
3. C.$\frac{9}{17}$
4. D.$\frac{47}{17}$

If $\tan (\pi \cos \theta )=\cot (\pi \sin \theta )$, $0<\theta <\frac{\pi }{2}$; then what is the value of $8{\sin }^2\left(\theta +\frac{\pi }{4}\right)$ ?

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

If $\tan \alpha =\frac{1}{7}$, $\sin \beta =\frac{1}{\sqrt{10}}$; $0<\alpha ,\beta <\frac{\pi }{2}$, then what is the value of $\cos (\alpha +2\beta )$ ?

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

Consider the following for the next items that follow: Consider the equation $(1-x{)}^4+(5-x{)}^4=82$. What is the number of real roots of the equation?

1. A.0
2. B.2
3. C.4
4. D.8

Consider the following for the next items that follow: Consider the equation $(1-x{)}^4+(5-x{)}^4=82$. What is the sum of all the roots of the equation?

1. A.24
2. B.12
3. C.10
4. D.6

If $f^{\prime }(x)=\cos (\ln x)$ and $y=f\left(\frac{2x-3}{x}\right)$, then what is $\frac{dy}{dx}$ equal to ?

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

1. A.2
2. B.4
3. C.8
4. D.16

Consider the following for the next items that follow: Let $\sin x+\sin y=\sqrt{3}(\cos y-\cos x)$; $x+y=\frac{\pi }{2}$, $0<x,y<\frac{\pi }{2}$ What is a value of $\sin 3x+\sin 3y$?

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

Consider the following for the next items that follow: Let $\sin x+\sin y=\sqrt{3}(\cos y-\cos x)$; $x+y=\frac{\pi }{2}$, $0<x,y<\frac{\pi }{2}$ What is a value of ${\cos }^{3}x+{\cos }^{3}y$?

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

Consider the following for the next items that follow: The angles A, B and C of a triangle ABC are in the ratio 3 ∶ 5 ∶ 4. What is the value of a + b + √2 c equal to ?

1. A.За
2. B.2b
3. C.3b
4. D.2c

What is the ratio of $a^2:b^2:c^2$ ?

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

Consider the following for the next two (02) items that follow : Let sin β be the GM of sin α and cos α; tan γ be the AM of sin α and cos α. What is cos 2β equal to ?

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

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

Consider the following for the next two (02) items that follow : A flagstaff 20 m long standing on a pillar 10 m high subtends an angle ${\tan }^{-1}(0.5)$ at a point P on the ground. Let $\theta$ be the angle subtended by the pillar at this point P. If x is the distance of P from the bottom of the pillar, then consider the following statements : 1. x can take two values which are in the ratio $1:3$ 2. x can be equal to the height of the flagstaff Which of the statements given above is/are correct?

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

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

What is the perimeter of the triangle ?

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

Consider the following statements : 1. ABC is right angled triangle 2. The angles of the triangle are in AP Which of the statements given above 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 for the next two (02) items that follow : In the triangle ABC, $a^2+b^2+c^2=ac+\sqrt{3}bc$ What is the nature of the triangle ?

1. A.Equilateral
2. B.Isosceles
3. C.Right angled triangle
4. D.Scalene but not right angled

1. A.4√3
2. B.6√3
3. C.8√3
4. D.12√3

Consider the following for the next two (02) items that follow : Let $f(x)=\sin [{\pi }^2]x+\cos [-{\pi }^2]x$ where [.] is a greatest ‘integer function What is $f\left(\frac{\pi }{2}\right)$ equal to ?

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

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