Matrices & Determinants · NDA previous year questions

2009 21 questions

1. Q242009 · 19 Apr 2009

   Consider the following statements in respect of symmetric matrices $A$ and $B$: 1. $AB$ is symmetric. 2. $A^2+B^2$ is symmetric. Which of the above statements is/are correct?

   1. A.1 only
   2. B.2 only
   3. C.Both 1 and 2
   4. D.Neither 1 nor 2
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   Answer: B

2. Q252009 · 19 Apr 2009

   If $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ and $C=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$, then which one of the following is correct?

   1. A.$C=A\cos \alpha -B\sin \alpha$
   2. B.$C=A\sin \alpha +B\cos \alpha$
   3. C.$C=A\sin \alpha -B\cos \alpha$
   4. D.$C=A\cos \alpha +B\sin \alpha$
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   Answer: D

3. Q262009 · 19 Apr 2009
   
   1. A.$\frac{-1+i\sqrt{3}}{2}$
   2. B.$\frac{1-i\sqrt{3}}{2}$
   3. C.$\frac{-1-i\sqrt{3}}{2}$
   4. D.$\frac{1+i\sqrt{3}}{2}$
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   Answer: B

4. Q272009 · 19 Apr 2009

   If $\omega$ is an imaginary cube root of unity, then what is the value of the determinant of the matrix $\left[\begin{array}{ccc}1+\omega & {\omega }^2& \omega \\ 1+{\omega }^2& \omega & {\omega }^2\\ \omega +{\omega }^2& \omega & {\omega }^2\end{array}\right]$ ?

   1. A.$0$
   2. B.$-2\omega$
   3. C.$-2{\omega }^2$
   4. D.$3{\omega }^2$
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   Answer: D

5. Q372009 · 19 Apr 2009
   
   1. A.A implies B, but B does not imply A
   2. B.B implies A, but A does not imply B
   3. C.Neither A implies B nor B implies A
   4. D.A implies B as well as B implies A
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   Answer: D

6. Q382009 · 19 Apr 2009

   If $2^x+3^y=17$ and $2^{x+2}-3^{y+1}=5$, then what is the value of $x$?

   1. A.$3$
   2. B.$2$
   3. C.$1$
   4. D.$0$
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   Answer: A

7. Q392009 · 19 Apr 2009

   If $P(32,6)=kC(32,6)$, then what is the value of $k$?

   1. A.$6$
   2. B.$32$
   3. C.$120$
   4. D.$720$
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   Answer: D

8. Q402009 · 19 Apr 2009

   What is $(\sqrt{3}+i)/(1+\sqrt{3}i)$ equal to?

   1. A.$1+i$
   2. B.$1-i$
   3. C.$\sqrt{3}(1-i)/2$
   4. D.$(\sqrt{3}-i)/2$
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   Answer: D

9. Q502009 · 19 Apr 2009
   
   1. A.$A^{-1}$ does not exist
   2. B.$A=(-1)I$
   3. C.$A$ is a unit matrix
   4. D.$A^2=I$
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   Answer: D

10. Q512009 · 19 Apr 2009

    The formula ${\sin }^{-1}\{2x(1-x^2)\}=2{\sin }^{-1}x$ is true for all values of $x$ lying in the interval

    1. A.$[-1,1]$
    2. B.$[0,1]$
    3. C.$[-1,0]$
    4. D.$[-1/\sqrt{2},1/\sqrt{2}]$
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    Answer: Dropped in the final answer key (full marks to everyone)

11. Q522009 · 19 Apr 2009

    What is the value of $1-\sin 10^{\circ }\sin 50^{\circ }\sin 70^{\circ }$?

    1. A.$1/8$
    2. B.$3/8$
    3. C.$5/8$
    4. D.$7/8$
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    Answer: D

12. Q532009 · 19 Apr 2009

    The sines of two angles of a triangle are equal to $5/13$ and $99/101$. What is the cosine of the third angle?

    1. A.$255/1313$
    2. B.$265/1313$
    3. C.$275/1313$
    4. D.$770/1313$
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    Answer: A

13. Q542009 · 19 Apr 2009

    After subtending an angle of $1000^{\circ }$ from its initial position, the revolving line will be situated in which one of the following quadrants?

    1. A.First quadrant
    2. B.Second quadrant
    3. C.Third quadrant
    4. D.Fourth quadrant
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    Answer: D