Matrices & Determinants · NDA previous year questions

2017 27 questions

1. Q232017 · 10 Sept 2017

   The matrix $A$ has $x$ rows and $x+5$ columns. The matrix $B$ has $y$ rows and $11-y$ columns. Both $AB$ and $BA$ exist. What are the values of $x$ and $y$ respectively?

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

   Answer: C

2. Q242017 · 10 Sept 2017

   If $A$ is a square matrix, then the value of $\mathrm{adj}{A}^T-(\mathrm{adj}A{)}^T$ is equal to

   1. A.$A$
   2. B.$2|A|I$, where $I$ is the identity matrix
   3. C.null matrix whose order is same as that of $A$
   4. D.unit matrix whose order is same as that of $A$
   Show answer

   Answer: C

3. Q252017 · 10 Sept 2017

   The adjoint of the matrix is

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

   Answer: B

4. Q262017 · 10 Sept 2017

   If $a,b,c$ are non-zero real numbers, then the inverse of the matrix $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$ is equal to

   1. A.$\left[\begin{array}{ccc}{a}^{-1}& 0& 0\\ 0& b^{-1}& 0\\ 0& 0& c^{-1}\end{array}\right]$
   2. B.$\frac{1}{abc}\left[\begin{array}{ccc}{a}^{-1}& 0& 0\\ 0& b^{-1}& 0\\ 0& 0& c^{-1}\end{array}\right]$
   3. C.$\frac{1}{abc}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
   4. D.$\frac{1}{abc}\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$
   Show answer

   Answer: A

5. Q272017 · 10 Sept 2017

   The system of equations $kx+y+z=1$, $x+ky+z=k$ and $x+y+kz=k^2$ has no solution if $k$ equals

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

   Answer: D

6. Q282017 · 10 Sept 2017

   The value of the determinant $\left|\begin{array}{ccc}1-\alpha & \alpha -{\alpha }^2& {\alpha }^2\\ 1-\beta & \beta -{\beta }^2& {\beta }^2\\ 1-\gamma & \gamma -{\gamma }^2& {\gamma }^2\end{array}\right|$ is equal to

   1. A.$(\alpha -\beta )(\beta -\gamma )(\alpha -\gamma )$
   2. B.$(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )$
   3. C.$(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )(\alpha +\beta +\gamma )$
   4. D.0
   Show answer

   Answer: B

7. Q292017 · 10 Sept 2017

   If $p+q+r=a+b+c=0$, then the determinant $\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ equals

   1. A.0
   2. B.1
   3. C.$pa+qb+rc$
   4. D.$pa+qb+rc+a+b+c$
   Show answer

   Answer: A

8. Q302017 · 10 Sept 2017

   The value of $\tan 9^{\circ }-\tan 27^{\circ }-\tan 63^{\circ }+\tan 81^{\circ }$ is equal to

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

   Answer: D

9. Q312017 · 10 Sept 2017

   The value of $\sqrt{3}\mathrm{cosec}20^{\circ }-\sec 20^{\circ }$ is equal to

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

   Answer: A

10. Q322017 · 10 Sept 2017

    Angle $\alpha$ is divided into two parts $A$ and $B$ such that $A-B=x$ and $\tan A:\tan B=p:q$. The value of $\sin x$ is equal to

    1. A.$\frac{(p+q)\sin \alpha }{p-q}$
    2. B.$\frac{p\sin \alpha }{p+q}$
    3. C.$\frac{p\sin \alpha }{p-q}$
    4. D.$\frac{(p-q)\sin \alpha }{p+q}$
    Show answer

    Answer: D

11. Q1082017 · 10 Sept 2017
    
    1. A.$\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$
    2. B.$\left[\begin{array}{cc}-1& -1\\ -1& 2\end{array}\right]$
    3. C.$\left[\begin{array}{cc}1& -1\\ 1& 2\end{array}\right]$
    4. D.$\left[\begin{array}{cc}-1& -1\\ 1& 2\end{array}\right]$
    Show answer

    Answer: B

12. Q1092017 · 10 Sept 2017

    If $|a|$ denotes the absolute value of an integer, then which of the following are correct? 1. $|ab|=|a||b|$ 2. $|a+b|\le |a|+|b|$ 3. $|a-b|\ge ||a|-|b||$ Select the correct answer using the code given below.

    1. A.1 and 2 only
    2. B.2 and 3 only
    3. C.1 and 3 only
    4. D.1, 2 and 3
    Show answer

    Answer: D

13. Q1122017 · 10 Sept 2017

    The value of the determinant $$\left|\begin{array}{cc}{\cos }^2\frac{\theta }{2}& {\sin }^2\frac{\theta }{2}\\ {\sin }^2\frac{\theta }{2}& {\cos }^2\frac{\theta }{2}\end{array}\right|$$ for all values of $\theta$, is

    1. A.1
    2. B.$\cos \theta$
    3. C.$\sin \theta$
    4. D.$\cos 2\theta$
    Show answer

    Answer: B

14. Q252017 · 23 Apr 2017

    If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$ then what is $A{A}^T$ equal to (where $A^T$ is the transpose of $A$)?

    1. A.Null matrix
    2. B.Identity matrix
    3. C.$A$
    4. D.$-A$
    Show answer

    Answer: B

15. Q262017 · 23 Apr 2017
    
    1. A.$\left[\begin{array}{cc}4& 8\\ -4& -16\end{array}\right]$
    2. B.$\left[\begin{array}{cc}4& -4\\ 8& -16\end{array}\right]$
    3. C.$\left[\begin{array}{cc}-4& -8\\ 4& 12\end{array}\right]$
    4. D.$\left[\begin{array}{cc}-4& -8\\ 8& 12\end{array}\right]$
    Show answer

    Answer: D

16. Q272017 · 23 Apr 2017

    Consider the set $A$ of all matrices of order $3\times 3$ with entries 0 or 1 only. Let $B$ be the subset of $A$ consisting of all matrices whose determinant is 1. Let $C$ be the subset of $A$ consisting of all matrices whose determinant is $-1$. Then which one of the following is correct?

    1. A.$C$ is empty
    2. B.$B$ has as many elements as $C$
    3. C.$A=B\cup C$
    4. D.$B$ has thrice as many elements as $C$
    Show answer

    Answer: B

17. Q282017 · 23 Apr 2017
    
    1. A.$\left[\begin{array}{cc}\cos 3\theta & \sin 3\theta \\ -\sin 3\theta & \cos 3\theta \end{array}\right]$
    2. B.$\left[\begin{array}{cc}{\cos }^3\theta & {\sin }^3\theta \\ -{\sin }^3\theta & {\cos }^3\theta \end{array}\right]$
    3. C.$\left[\begin{array}{cc}\cos 3\theta & -\sin 3\theta \\ \sin 3\theta & \cos 3\theta \end{array}\right]$
    4. D.$\left[\begin{array}{cc}{\cos }^3\theta & -{\sin }^3\theta \\ {\sin }^3\theta & {\cos }^3\theta \end{array}\right]$
    Show answer

    Answer: A

18. Q292017 · 23 Apr 2017

    What is the order of $[xyz]\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$?

    1. A.$3\times 1$
    2. B.$1\times 1$
    3. C.$1\times 3$
    4. D.$3\times 3$
    Show answer

    Answer: B

19. Q302017 · 23 Apr 2017

    If $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, then the value of $A^4$ is

    1. A.$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
    2. B.$\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$
    3. C.$\left[\begin{array}{cc}0& 0\\ 1& 1\end{array}\right]$
    4. D.$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
    Show answer

    Answer: A

20. Q312017 · 23 Apr 2017

    If $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$ and $\det (A^3)=125$, then $\alpha$ is equal to

    1. A.$\pm 1$
    2. B.$\pm 2$
    3. C.$\pm 3$
    4. D.$\pm 5$
    Show answer

    Answer: C

21. Q322017 · 23 Apr 2017

    If $B$ is a non-singular matrix and $A$ is a square matrix, then the value of $\det (B^{-1}AB)$ is equal to

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

    Answer: B

22. Q332017 · 23 Apr 2017

    If $a\ne b\ne c$, then one value of $x$ which satisfies the equation $$\left|\begin{array}{ccc}0& x-a& x-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|=0$$ is given by

    1. A.$a$
    2. B.$b$
    3. C.$c$
    4. D.0
    Show answer

    Answer: D

23. Q342017 · 23 Apr 2017

    The equations $x+2y+3z=1$ $2x+y+3z=2$ $5x+5y+9z=4$

    1. A.have the unique solution
    2. B.have infinitely many solutions
    3. C.are inconsistent
    4. D.None of the above
    Show answer

    Answer: A

24. Q352017 · 23 Apr 2017

    What is the value of the determinant $$\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+xyz& 1\\ 1& 1& 1+xyz\end{array}\right|$$ ?

    1. A.$1+x+y+z$
    2. B.$2xyz$
    3. C.$x^2{y}^2{z}^2$
    4. D.$2x^2{y}^2{z}^2$
    Show answer

    Answer: C

25. Q362017 · 23 Apr 2017
    
    1. A.$\frac{x}{y}$ is one of the cube roots of unity
    2. B.$x$ is one of the cube roots of unity
    3. C.$y$ is one of the cube roots of unity
    4. D.$\frac{x}{y}$ is one of the cube roots of $-1$
    Show answer

    Answer: D