Trigonometry · NDA previous year questions

2025 30 questions

1. Q192025 · 14 Sept 2025

   If a, b, c are the sides of a triangle ABC and p is the perimeter of the triangle, then what is equal to? $\det \left|\begin{array}{ccc}p+c& a& b\\ c& p+a& b\\ c& a& p+b\end{array}\right|$

   1. A.$p^3$
   2. B.$2p^3$
   3. C.$3p^3$
   4. D.$4p^3$
   Show answer

   Answer: B

2. Q252025 · 14 Sept 2025

   For the following two (02) items: Let sin A + sin B = p and cos A + cos B = q What is p/q equal to?

   1. A.$\tan \left(\frac{A-B}{2}\right)$
   2. B.$\cot \left(\frac{A-B}{2}\right)$
   3. C.$\tan \left(\frac{A+B}{2}\right)$
   4. D.$\cot \left(\frac{A+B}{2}\right)$
   Show answer

   Answer: C

3. Q262025 · 14 Sept 2025

   What is $\frac{p^2-q^2}{p^2+q^2}$ equal to?

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

   Answer: D

4. Q272025 · 14 Sept 2025

   For the following two (02) items: Let p = cosec $20^{\circ }$ and q = cosec $70^{\circ }$. What is $(\frac{\sqrt{3}p}{4}-\frac{q}{4})$ equal to?

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

   Answer: C

5. Q282025 · 14 Sept 2025

   For the following two (02) items: Let p = cosec $20^{\circ }$ and q = cosec $70^{\circ }$. What is $\frac{p^2+q^2}{p^2{q}^2}$ equal to?

   1. A.1/2
   2. B.1
   3. C.3/2
   4. D.2
   Show answer

   Answer: B

6. Q292025 · 14 Sept 2025

   For the following two (02) items: Let $\cos (2x+3y)=\frac{1}{2}$ and $\cos (3x+2y)=\frac{\sqrt{3}}{2}$ where $-\pi <(2x+3y)<\pi$ and $-\pi <(3x+2y)<\pi$. How many values does (x + y) have?

   1. A.Two
   2. B.Three
   3. C.Four
   4. D.More than four
   Show answer

   Answer: C

7. Q302025 · 14 Sept 2025

   For the following two (02) items: Let $\cos (2x+3y)=\frac{1}{2}$ and $\cos (3x+2y)=\frac{\sqrt{3}}{2}$ where $-\pi <(2x+3y)<\pi$ and $-\pi <(3x+2y)<\pi$. How many values does (y-x) have?

   1. A.Two
   2. B.Three
   3. C.Four
   4. D.More than four
   Show answer

   Answer: C

8. Q632025 · 14 Sept 2025

   What is the value of $\sin \theta$ if $\theta$ is the acute angle between the lines whose equations are px + qy = p + q and p(x - y) + q(x + y) = 2q

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

   Answer: D

9. Q802025 · 14 Sept 2025

   For the following three (03) items: Let $y=f(x)=\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}+\ln \sqrt{1-x^2}$. If $x=\sin \theta$ then what is $\frac{dy}{dx}$ equal to?

   1. A.$\theta \sec \theta$
   2. B.$\theta {\sec }^2\theta$
   3. C.$\theta {\sec }^3\theta$
   4. D.$2\tan \theta +\theta {\sec }^2\theta$
   Show answer

   Answer: C

10. Q942025 · 14 Sept 2025

    If k is an arbitrary constant, then what is the general solution of the equation $(x+y{)}^2\frac{dy}{dx}=k^2$

    1. A.$y+x=\tan (x+c)+k$
    2. B.$x+y=k\tan \left(\frac{y-c}{k}\right)$
    3. C.$x-y=k\tan \left(\frac{y-c}{k}\right)$
    4. D.$y-x=\tan (x+c)+k$
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    Answer: B

11. Q952025 · 14 Sept 2025

    What is $\int \frac{dx}{10^x+10^{-x}}$ equal to?

    1. A.${\tan }^{-1}(10^x)+c$
    2. B.$(\ln 10){\tan }^{-1}(10^x)+c$
    3. C.$\frac{1}{\ln 10}{\tan }^{-1}(10^x)+c$
    4. D.$\ln (10^x+10^{-x})+c$
    Show answer

    Answer: C

12. Q962025 · 14 Sept 2025

    A wire of length 20 cm is to be bent into a rectangle. Which of the following statements is/are correct? I. The rectangle of the largest area is the square. II. It is possible to form a rectangle of an area of $27\mathrm{c}{\mathrm{m}}^2$ Select the answer using the code given below.

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

    Answer: A

13. Q972025 · 14 Sept 2025

    If $I_1={\int }_e^{e^2}\frac{dx}{\ln x}$ and $I_2={\int }_1^2\frac{e^x}{x}dx$ then which one of the following is correct?

    1. A.$I_1-I_2=0$
    2. B.$I_1+I_2=0$
    3. C.$I_1-2I_2=0$
    4. D.$2I_1-I_2=0$
    Show answer

    Answer: A

14. Q312025 · 13 Apr 2025

    Consider the following for the three (03) items that follow: Let p=sin35°, q=sin25° and r=sin(-95°). What is (p+q+r) equal to?

    1. A.-1
    2. B.0
    3. C.$2\sin 5^{\circ }$
    4. D.$2\cos 5^{\circ }$
    Show answer

    Answer: B

15. Q322025 · 13 Apr 2025

    What is (pq+qr+rp) equal to?

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

    Answer: A

16. Q332025 · 13 Apr 2025

    Consider the following for the three (03) items that follow: Let $p=\sin 35^{\circ }$, $q=\sin 25^{\circ }$ and $r=\sin (-95^{\circ })$. What is $(p^2+q^2+r^2)$ equal to?

    1. A.1/2
    2. B.1
    3. C.3/2
    4. D.2
    Show answer

    Answer: C

17. Q362025 · 13 Apr 2025
    Passage

    Consider the following for the three (03) items that follow: The sides of a triangle ABC are AB=3cm, BC=5cm and CA=7cm. Consider the following statements: I. The triangle is obtuse-angled triangle. II. The sum of acute angles of the triangle is also acute.

    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
    Show answer

    Answer: C

18. Q372025 · 13 Apr 2025
    Passage

    Consider the following for the three (03) items that follow: The sides of a triangle ABC are AB=3cm, BC=5cm and CA=7cm. Consider the following statements: I. The triangle is obtuse-angled triangle. II. The sum of acute angles of the triangle is also acute.

    What is ∠B equal to?

    1. A.$60^{\circ }$
    2. B.$105^{\circ }$
    3. C.$120^{\circ }$
    4. D.$150^{\circ }$
    Show answer

    Answer: C

19. Q382025 · 13 Apr 2025
    Passage

    Consider the following for the three (03) items that follow: The sides of a triangle ABC are AB=3cm, BC=5cm and CA=7cm. Consider the following statements: I. The triangle is obtuse-angled triangle. II. The sum of acute angles of the triangle is also acute.

    What is the area of the triangle?

    1. A.$15\sqrt{3}/4$ square cm
    2. B.$15\sqrt{3}/2$ square cm
    3. C.$15\sqrt{3}$ square cm
    4. D.$30\sqrt{3}$ square cm
    Show answer

    Answer: A

20. Q412025 · 13 Apr 2025
    Passage

    Consider the following for the three (03) items that follow: Let p = tan 2α   - tanα and q = cotα - cot 2α

    What is (p/q) equal to?

    1. A.−tanα⋅tan2α
    2. B.−cotα⋅cot2α
    3. C.tanα⋅tan2α
    4. D.cotα⋅cot2α
    Show answer

    Answer: C

21. Q422025 · 13 Apr 2025
    Passage

    Consider the following for the three (03) items that follow: Let p = tan 2α   - tanα and q = cotα - cot 2α

    What is (p+q) equal to?

    1. A.sec4α
    2. B.cosec4α
    3. C.2sec4α
    4. D.2cosec4α
    Show answer

    Answer: D

22. Q432025 · 13 Apr 2025
    Passage

    Consider the following for the three (03) items that follow: Let p = tan 2α   - tanα and q = cotα - cot 2α

    What is ${\tan }^2\alpha$ equal to?

    1. A.(pq)/(p+q)
    2. B.(p+2q)/p
    3. C.p/(p+2q)
    4. D.p/(2p+q)
    Show answer

    Answer: C

23. Q442025 · 13 Apr 2025
    Passage

    Consider the following for the two (02) items that follow: Let 2sinα + cosα = 2 where 0 < α < 90°

    What is tanα equal to?

    1. A.1/2
    2. B.1
    3. C.3/4
    4. D.2
    Show answer

    Answer: C

24. Q452025 · 13 Apr 2025
    Passage

    Consider the following for the two (02) items that follow: Let 2sinα + cosα = 2 where 0 < α < 90°

    What is 2sin2α+cos2α equal to?

    1. A.11/10
    2. B.11/5
    3. C.12/5
    4. D.13/5
    Show answer

    Answer: B

25. Q462025 · 13 Apr 2025
    Passage

    Consider the following for the two (02) items that follow: In a triangle ABC, two sides BC and CA are in the ratio 2:1 and their opposite corresponding angles are in the ratio 3: 1.

    One of the angles of the triangle is

    1. A.$15^{\circ }$
    2. B.$30^{\circ }$
    3. C.$45^{\circ }$
    4. D.$75^{\circ }$
    Show answer

    Answer: B