Trigonometry · NDA previous year questions

2022 30 questions

1. Q422022 · 4 Sept 2022
   Passage

   ABC is a triangular plot with AB = 16 m, BC = 10 m and CA = 10 m. A lamp post is situated at the middle point of the side AB. The lamp post subtends an angle 45° at the vertex B.

   What is cos A + cos B + cos C equal to ?

   1. A.1
   2. B.$\frac{41}{25}$
   3. C.$\frac{37}{25}$
   4. D.$\frac{33}{25}$
   Show answer

   Answer: D

2. Q432022 · 4 Sept 2022
   Passage

   There are two points P and Q due south of a leaning tower, which leans towards north. P is at a distance x and Q is at distance y from the foot of the tower (x > y). The angles of elevation of the top of the tower from P and Q are 15° and 75° respectively.

   At what height is the top of the tower above the ground level ?

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

   Answer: A

3. Q442022 · 4 Sept 2022
   Passage

   There are two points P and Q due south of a leaning tower, which leans towards north. P is at a distance x and Q is at distance y from the foot of the tower (x > y). The angles of elevation of the top of the tower from P and Q are 15° and 75° respectively.

   If θ is the inclination of the tower to the horizontal, then what is cot θ equal to ?

   1. A.$2+\frac{\sqrt{3}(x-y)}{x+y}$
   2. B.$2-\frac{\sqrt{3}(x-y)}{x+y}$
   3. C.$2+\frac{\sqrt{3}(x+y)}{x-y}$
   4. D.$2-\frac{\sqrt{3}(x+y)}{x-y}$
   Show answer

   Answer: D

4. Q452022 · 4 Sept 2022
   Passage

   There are two points P and Q due south of a leaning tower, which leans towards north. P is at a distance x and Q is at distance y from the foot of the tower (x > y). The angles of elevation of the top of the tower from P and Q are 15° and 75° respectively.

   What is the length of the tower ?

   1. A.$\frac{x-y}{2\sqrt{3}}\sqrt{1+{\left\{2+\frac{\sqrt{3}(x+y)}{x-y}\right\}}^2}$
   2. B.$\frac{x-y}{2\sqrt{3}}\sqrt{1+{\left\{2-\frac{\sqrt{3}(x+y)}{x-y}\right\}}^2}$
   3. C.$\frac{x-y}{4\sqrt{3}}\sqrt{1+{\left\{2+\frac{\sqrt{3}(x+y)}{x-y}\right\}}^2}$
   4. D.$\frac{x-y}{4\sqrt{3}}\sqrt{1+{\left\{2-\frac{\sqrt{3}(x+y)}{x-y}\right\}}^2}$
   Show answer

   Answer: B

5. Q462022 · 4 Sept 2022

   What is the value of $\mathrm{cosec}(-\frac{73\pi }{3})$ ?

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

   Answer: B

6. Q472022 · 4 Sept 2022

   What is the value of $\cos \left(\frac{5\pi }{17}\right)+\cos \left(\frac{7\pi }{17}\right)+2\cos \left(\frac{11\pi }{17}\right)\cos \left(\frac{\pi }{17}\right)$ ?

   1. A.0
   2. B.1
   3. C.$4\cos \left(\frac{6\pi }{17}\right)\cos \left(\frac{\pi }{17}\right)$
   4. D.$4\cos \left(\frac{11\pi }{17}\right)\cos \left(\frac{\pi }{17}\right)$
   Show answer

   Answer: A

7. Q482022 · 4 Sept 2022

   What is the value of $\tan \left(\frac{3\pi }{8}\right)$ ?

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

   Answer: B

8. Q492022 · 4 Sept 2022

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

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

   Answer: D

9. Q502022 · 4 Sept 2022

   In a triangle ABC, a = 4, b = 3, c = 2. What is cos 3C equal to ?

   1. A.$\frac{7}{128}$
   2. B.$\frac{11}{128}$
   3. C.$\frac{7}{64}$
   4. D.$\frac{11}{64}$
   Show answer

   Answer: A

10. Q512022 · 4 Sept 2022

    What is cos 36° − cos 72° equal to ?

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

    Answer: C

11. Q522022 · 4 Sept 2022

    If $\sec x=\frac{25}{24}$ and x lies in the fourth quadrant, then what is the value of $\tan x+\sin x$ ?

    1. A.$-\frac{625}{168}$
    2. B.$-\frac{343}{600}$
    3. C.$\frac{625}{168}$
    4. D.$\frac{343}{600}$
    Show answer

    Answer: B

12. Q532022 · 4 Sept 2022

    What is the value of ${\tan }^{2}165^{\circ }+{\cot }^{2}165^{\circ }$ ?

    1. A.7
    2. B.14
    3. C.$4\sqrt{3}$
    4. D.$8\sqrt{3}$
    Show answer

    Answer: B

13. Q542022 · 4 Sept 2022

    What is the value of $\sin \left(2n\pi +\frac{5\pi }{6}\right)\sin \left(2n\pi -\frac{5\pi }{6}\right)$ where $n\in Z$ ?

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

    Answer: A

14. Q552022 · 4 Sept 2022

    If 1 + 2(sin x + cos x)(sin x − cos x) = 0 where 0 < x < 360°, then how many values does x take ?

    1. A.Only one value
    2. B.Only two values
    3. C.Only three values
    4. D.Four values
    Show answer

    Answer: D

15. Q872022 · 4 Sept 2022

    What is ${\lim }_{h\to 0}\frac{{\sin }^2(x+h)-{\sin }^{2}x}{h}$ equal to ?

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

    Answer: C

16. Q962022 · 4 Sept 2022

    What is ${\int }_{-\pi /2}^{\pi /2}(e^{\cos x}\sin x+e^{\sin x}\cos x)\mathrm{dx}$ equal to ?

    1. A.$\frac{e^2-1}{e}$
    2. B.$\frac{e^2+1}{e}$
    3. C.$\frac{1-e^2}{e}$
    4. D.0
    Show answer

    Answer: A

17. Q272022 · 10 Apr 2022

    In a triangle ABC, sin A - cos B - cos C = 0. What is angle B equal to?

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

    Answer: D

18. Q282022 · 10 Apr 2022

    If $\alpha +\beta =\frac{\pi }{4}$ and $2\tan \alpha =1$, then what is $\tan 2\beta$ equal to?

    1. A.$\frac{1}{3}$
    2. B.$\frac{2}{3}$
    3. C.$\frac{3}{4}$
    4. D.$\frac{3}{5}$
    Show answer

    Answer: C

19. Q292022 · 10 Apr 2022

    If $\tan (45^{\circ }+\theta )=1+\sin 2\theta$, where $-\frac{\pi }{4}<\theta <\frac{\pi }{4}$, then what is the value of $\cos 2\theta$?

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

    Answer: C

20. Q302022 · 10 Apr 2022

    Let $\sin 2\theta =\cos 3\theta$, where $\theta$ is acute angle. What is the value of $1+4\sin \theta$? (given that $\sin 18^{\circ }=\frac{\sqrt{5}-1}{4}$)

    1. A.$\sqrt{3}$
    2. B.2
    3. C.$\sqrt{5}$
    4. D.3
    Show answer

    Answer: C

21. Q312022 · 10 Apr 2022

    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
    Show answer

    Answer: C

22. Q322022 · 10 Apr 2022

    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.$\frac{3}{16}$
    Show answer

    Answer: B

23. Q332022 · 10 Apr 2022

    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.$2\cos \theta$
    Show answer

    Answer: A

24. Q352022 · 10 Apr 2022

    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.$60^{\circ }<\theta <90^{\circ }$
    Show answer

    Answer: B

25. Q362022 · 10 Apr 2022

    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
    Show answer

    Answer: A