Matrices & Determinants · NDA previous year questions

2025 26 questions

1. Q102025 · 14 Sept 2025

   What is the value of the determinant of the inverse of the matrix $\left[\begin{array}{cc}-4& -5\\ 2& 2\end{array}\right]$?

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

   Answer: A

2. Q112025 · 14 Sept 2025

   In a class of 45 students, 34 like to play cricket and 26 like to play football. Further, each student likes to play at least one of the two games. How many students like to play exactly one game?

   1. A.45
   2. B.30
   3. C.25
   4. D.15
   Show answer

   Answer: B

3. Q122025 · 14 Sept 2025

   The system of equations 15y - 10x + 50 = 0, 2x - 3y - 5 = 0

   1. A.has a unique solution
   2. B.has infinitely many solutions
   3. C.is inconsistent
   4. D.is consistent and has exactly two solutions
   Show answer

   Answer: C

4. Q132025 · 14 Sept 2025

   If $(\frac{1-i}{1+i}{)}^{2m}(\frac{1+i}{1-i}{)}^{2n}=1$ where $i=\sqrt{-1}$ then what is the smallest positive value of (m - n)?

   1. A.1
   2. B.2
   3. C.4
   4. D.8
   Show answer

   Answer: B

5. Q142025 · 14 Sept 2025

   In obtaining the solution of the system of equations x + y + z = 7, x + 2y + 3z = 16 and x + 3y + 4z = 22 by Cramer's rule, the value of y is obtained by dividing D by $D_2$ where $D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 4\end{array}\right|$ What is the value of the determinant $D_2$ ?

   1. A.-13
   2. B.-3
   3. C.3
   4. D.13
   Show answer

   Answer: B

6. Q152025 · 14 Sept 2025

   Consider the following in respect of non-singular matrices A and B: I. $(AB{)}^{-1}=A^{-1}{B}^{-1}$ II. $(BA)(AB{)}^{-1}=I$ where I is the identity matrix III. $(AB{)}^T=A^T{B}^T$ How many of the above are correct?

   1. A.None
   2. B.One
   3. C.Two
   4. D.All three
   Show answer

   Answer: A

7. Q162025 · 14 Sept 2025
   
   1. A.$\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ l& m& n\end{array}\right|$
   2. B.$\left|\begin{array}{ccc}l& m& n\\ a& b& c\\ p& q& r\end{array}\right|$
   3. C.$\left|\begin{array}{ccc}p& q& r\\ a& b& c\\ l& m& n\end{array}\right|$
   4. D.$\left|\begin{array}{ccc}a& p& l\\ b& q& m\end{array}\right|$
   Show answer

   Answer: C

8. Q392025 · 14 Sept 2025

   For the following two (02) items: Let $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$ What is the value of the determinant of the matrix $A^4$

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

   Answer: B

9. Q402025 · 14 Sept 2025

   For the following two (02) items: Let $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$ What is $[adjA{]}^{-1}$ equal to?

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

   Answer: C

10. Q442025 · 14 Sept 2025

    Consider the following statements: Statement-I: If X is an nn matrix, then $\det (mX)=m^n\det (X)$, where m is a scalar. Statement-II: If Y is a matrix obtained from X by multiplying any row or column by a scalar m, then $\det (Y)=m\det (X)$. Which one of the following is correct in respect of the above statements?

    1. A.Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
    2. B.Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-1
    3. C.Statement-I is correct but Statement-II is not correct
    4. D.Statement-I is not correct but Statement-II is correct
    Show answer

    Answer: A

11. Q452025 · 14 Sept 2025

    Consider the following statements about the matrix $M=\left[\begin{array}{ccc}71& 23& 48\\ 57& 28& 29\\ 65& 17& 48\end{array}\right]$ Statement-I: The inverse of M does not exist. Statement-II: M is non-singular. Which one of the following is correct in respect of the above statements?

    1. A.Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
    2. B.Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
    3. C.Statement-I is correct but Statement-II is not correct
    4. D.Statement-I is not correct but Statement-II is correct
    Show answer

    Answer: C

12. Q462025 · 14 Sept 2025

    What is ${\cot }^{-1}9+{\mathrm{cosec}}^{-1}\left(\frac{\sqrt{41}}{4}\right)$ equal to?

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

    Answer: A

13. Q472025 · 14 Sept 2025

    How many values of $\theta$, where $-\pi <\theta <\pi$ satisfy both the equations $\cot \theta =-\sqrt{3}$ and $\csc \theta =-2$ simultaneously?

    1. A.4
    2. B.2
    3. C.1
    4. D.None
    Show answer

    Answer: C

14. Q482025 · 14 Sept 2025

    If $x+\frac{1}{x}=2\cos \theta$ then what is $x^3+\frac{1}{x^3}$ equal to?

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

    Answer: C

15. Q492025 · 14 Sept 2025

    If $0\le x\le \frac{\pi }{2}$ then what is the number of values of x satisfying the equation $\tan x+\sec x=2\cos x$

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

    Answer: B

16. Q502025 · 14 Sept 2025

    What is the value of $\tan \left[\frac{1}{2}{\sec }^{-1}\left(\frac{2}{\sqrt{3}}\right)\right]$?

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

    Answer: A

17. Q92025 · 13 Apr 2025

    . Consider the following statements in respect of the determinant $$\Delta =\left|\begin{array}{ccc}k(k+2)& 2k+1& 1\\ 2k+1& k+2& 1\\ 3& 3& 1\end{array}\right|$$ I. $\Delta$ is positive if k>0. II. $\Delta$ is negative if k<0. III. $\Delta$ is zero if k=0. How many of the statements given above are correct?

    1. A.None
    2. B.One
    3. C.Two
    4. D.All three
    Show answer

    Answer: B

18. Q102025 · 13 Apr 2025
    
    1. A.-10
    2. B.-6
    3. C.0
    4. D.6
    Show answer

    Answer: B

19. Q112025 · 13 Apr 2025

    If $A^2+B^2+C^2=0$, then what is the value of the following? $$\left|\begin{array}{ccc}1& \cos C& \cos B\\ \cos C& 1& \cos A\\ \cos B& \cos A& 1\end{array}\right|$$

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

    Answer: B

20. Q242025 · 13 Apr 2025

    If $A=\left[\begin{array}{ccc}y& z& x\\ z& x& y\\ x& y& z\end{array}\right]$ where $x,y,z$ are integers, is an orthogonal matrix, then what is the value of $x^2+y^2+z^2$?

    1. A.0
    2. B.1
    3. C.4
    4. D.14
    Show answer

    Answer: B

21. Q252025 · 13 Apr 2025

    Consider the following in respect of a non-singular matrix $M$: I. $|M^2|=|M{|}^2$ II. $|M|=|M^{-1}|$ III. $|M|=|M^T|$ How many of the above are correct?

    1. A.None
    2. B.One
    3. C.Two
    4. D.All three
    Show answer

    Answer: C

22. Q262025 · 13 Apr 2025
    
    1. A.$\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
    2. B.$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
    3. C.$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$
    4. D.$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
    Show answer

    Answer: D

23. Q272025 · 13 Apr 2025
    
    1. A.$-5I_3$
    2. B.$-I_3$
    3. C.$I_3$
    4. D.$5I_3$
    Show answer

    Answer: D

24. Q282025 · 13 Apr 2025

    If the number of selections of r as well as (n+r) things from 5n different things are equal, then what is the value of r?

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

    Answer: B

25. Q292025 · 13 Apr 2025

    What is the number of selections of at most 3 things from 6 different things?

    1. A.20
    2. B.22
    3. C.41
    4. D.42
    Show answer

    Answer: D