Matrices & Determinants · NDA previous year questions

2016 22 questions

1. Q292016 · 18 Sept 2016

   If $A=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 3\\ -1& -2\end{array}\right]$, then which of the following is/are correct ? 1. $AB(A^{-1}{B}^{-1})$ is a unit matrix. 2. $(AB{)}^{-1}=A^{-1}{B}^{-1}$. Select the correct answer using the code given below :

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

   Answer: D

2. Q302016 · 18 Sept 2016

   Consider the following for the next two (02) items that follow : For the system of linear equations $2x+3y+5z=9$, $7x+3y-2z=8$ and $2x+3y+\lambda z=\mu$ Under what condition does the above system of equations have infinitely many solutions ?

   1. A.$\lambda =5$ and $\mu \ne 9$
   2. B.$\lambda =5$ and $\mu =9$
   3. C.$\lambda =9$ and $\mu =5$
   4. D.$\lambda =9$ and $\mu \ne 5$
   Show answer

   Answer: B

3. Q312016 · 18 Sept 2016

   Consider the following for the next two (02) items that follow : For the system of linear equations $2x+3y+5z=9$, $7x+3y-2z=8$ and $2x+3y+\lambda z=\mu$ Under what condition does the above system of equations have unique solutions ?

   1. A.$\lambda =5$ and $\mu =9$
   2. B.$\lambda \ne 5$ and $\mu =7$ only
   3. C.$\lambda \ne 5$ and $\mu$ has any real value
   4. D.$\lambda$ has any real value and $\mu \ne 9$
   Show answer

   Answer: C

4. Q322016 · 18 Sept 2016

   If A is a square matrix of order 3 and $\det A=5$, then what is $\det [(2A{)}^{-1}]$ equal to ?

   1. A.1/10
   2. B.2/5
   3. C.8/5
   4. D.1/40
   Show answer

   Answer: D

5. Q332016 · 18 Sept 2016

   Consider the following for the next two (02) items that follow : Let $a{x}^3+b{x}^2+cx+d=\left|\begin{array}{ccc}x+1& 2x& 3x\\ 2x+3& x+1& x\\ 2-x& 3x+4& 5x-1\end{array}\right|$, then What is the value of c ?

   1. A.$-1$
   2. B.34
   3. C.35
   4. D.50
   Show answer

   Answer: C

6. Q342016 · 18 Sept 2016

   Consider the following for the next two (02) items that follow : Let $a{x}^3+b{x}^2+cx+d=\left|\begin{array}{ccc}x+1& 2x& 3x\\ 2x+3& x+1& x\\ 2-x& 3x+4& 5x-1\end{array}\right|$, then What is the value of a + b + c + d ?

   1. A.62
   2. B.63
   3. C.65
   4. D.68
   Show answer

   Answer: B

7. Q352016 · 18 Sept 2016

   If $m=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $n=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, then what is the value of the determinant of $m\cos \theta -n\sin \theta$ ?

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

   Answer: C

8. Q362016 · 18 Sept 2016

   If $f(x)=\left[\begin{array}{ccc}\sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$, then which of the following are correct ? 1. $f(\theta )\times f(\varphi )=f(\theta +\varphi )$. 2. The value of the determinant of the matrix $f(\theta )\times f(\varphi )$ is 1. 3. The determinant of $f(x)$ is an even function. 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

9. Q372016 · 18 Sept 2016

   Which of the following are correct in respect of the system of equations $x+y+z=8$, $x-y+2z=6$ and $3x-y+5z=k$ ? 1. They have no solution, if $k=15$. 2. They have infinitely many solutions, if $k=20$. 3. They have unique solution, if $k=25$. 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: A

10. Q382016 · 18 Sept 2016

    $\sin A+2\sin 2A+\sin 3A$ is equal to which of the following? 1. $4\sin 2A{\cos }^2\left(\frac{A}{2}\right)$ 2. $2\sin 2A{\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right)}^2$ 3. $8\sin A\cos A{\cos }^2\left(\frac{A}{2}\right)$ 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: C

11. Q392016 · 18 Sept 2016

    If $x=\sin 70^{\circ }\cdot \sin 50^{\circ }$ and $y=\cos 60^{\circ }\cdot \cos 80^{\circ }$, then what is xy equal to?

    1. A.1/16
    2. B.1/8
    3. C.1/4
    4. D.1/2
    Show answer

    Answer: A

12. Q402016 · 18 Sept 2016

    If $\sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3+\sin {\theta }_4=4$, then what is the value of $\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3+\cos {\theta }_4$?

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

    Answer: A

13. Q412016 · 18 Sept 2016

    What is the value of $(1+\cos \frac{\pi }{8})(1+\cos \frac{3\pi }{8})(1+\cos \frac{5\pi }{8})(1+\cos \frac{7\pi }{8})$?

    1. A.$\frac{1}{2}$
    2. B.$\frac{1}{2}+\frac{1}{2\sqrt{2}}$
    3. C.$\frac{1}{2}-\frac{1}{2\sqrt{2}}$
    4. D.$\frac{1}{8}$
    Show answer

    Answer: D

14. Q422016 · 18 Sept 2016

    If $x\cos \theta +y\sin \theta =z$, then what is the value of $(x\sin \theta -y\cos \theta {)}^2$?

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

    Answer: A

15. Q432016 · 18 Sept 2016

    If $\sin 18^{\circ }=\frac{\sqrt{5}-1}{4}$, then what is the value of $\sin 81^{\circ }$?

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

    Answer: A

16. Q442016 · 18 Sept 2016

    What is $\frac{1-\tan 2^{\circ }\cot 62^{\circ }}{\tan 152^{\circ }-\cot 88^{\circ }}$ equal to?

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

    Answer: B

17. Q452016 · 18 Sept 2016

    What is the value of $\cos (2{\cos }^{-1}0.8)$?

    1. A.0.81
    2. B.0.56
    3. C.0.48
    4. D.0.28
    Show answer

    Answer: D

18. Q462016 · 18 Sept 2016

    The top of a hill when observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. What is the height of the hill?

    1. A.$\frac{h\cot q}{\cot q-\cot p}$
    2. B.$\frac{h\cot p}{\cot p-\cot q}$
    3. C.$\frac{2h\tan p}{\tan p-\tan q}$
    4. D.$\frac{2h\tan q}{\tan q-\tan p}$
    Show answer

    Answer: B

19. Q172016 · 17 Apr 2016

    Consider the following in respect of the matrix $A=\left(\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right)$ : 1. $A^2=-A$ 2. $A^3=4A$ Which of the above is/are correct?

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

    Answer: B

20. Q182016 · 17 Apr 2016

    Which of the following determinants have value 'zero'? 1. $\left|\begin{array}{ccc}41& 1& 5\\ 79& 7& 9\\ 29& 5& 3\end{array}\right|$ 2. $\left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$ 3. $\left|\begin{array}{ccc}0& c& b\\ -c& 0& a\\ -b& -a& 0\end{array}\right|$ 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

21. Q192016 · 17 Apr 2016

    The system of linear equations $kx+y+z=1,x+ky+z=1$ and $x+y+kz=1$ has a unique solution under which one of the following conditions?

    1. A.$k\ne 1$ and $k\ne -2$
    2. B.$k\ne 1$ and $k\ne 2$
    3. C.$k\ne -1$ and $k\ne -2$
    4. D.$k\ne -1$ and $k\ne 2$
    Show answer

    Answer: A

22. Q1182016 · 17 Apr 2016

    If $A$ is a square matrix, then what is $\mathrm{adj}(A^{-1})-(\mathrm{adj}A{)}^{-1}$ equal to?

    1. A.$2|A|$
    2. B.Null matrix
    3. C.Unit matrix
    4. D.None of the above
    Show answer

    Answer: B