NDA & NA Exam (I)- 2017 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2017 - Mathematics

NDA & NA Exam (I)- 2017 - Mathematics Defence Questions

31.
The equations
\( x + 2y + 3z = 1 \)
\( 2x + y + 3z = 2 \)
\( 5x + 5y + 9z = 4 \)
A.
have the unique solution
B.
have infinitely many solutions
C.
are inconsistent
D.
None of the above
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ANSWER :
A. have the unique solution
32.
If \( A = \begin{bmatrix} x+y & y \\ x & x-y \end{bmatrix} \), \( B = \begin{bmatrix} 3 \\ -2 \end{bmatrix} \) and \( C = \begin{bmatrix} 4 \\ -2 \end{bmatrix} \). If AB = C, then what is \( A^{2} \) equal to?
A.
\( \begin{bmatrix} 4 & 8 \\ -4 & -16 \end{bmatrix} \)
B.
\( \begin{bmatrix} 4 & -4 \\ 8 & -16 \end{bmatrix} \)
C.
\( \begin{bmatrix} -4 & -8 \\ 4 & 12 \end{bmatrix} \)
D.
\( \begin{bmatrix} -4 & -8 \\ 8 & 12 \end{bmatrix} \)
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ANSWER :
D. \( \begin{bmatrix} -4 & -8 \\ 8 & 12 \end{bmatrix} \)
33.
What is the value of the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+xyz & 1 \\ 1 & 1 & 1+xyz \end{vmatrix} \)?
A.
1 + x + y + z
B.
2xyz
C.
\( x^{2}y^{2}z^{2} \)
D.
\( 2x^{2}y^{2}z^{2} \)
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ANSWER :
C. \( x^{2}y^{2}z^{2} \)
34.
If \( \begin{vmatrix} x & y & 0 \\ 0 & x & y \\ y & 0 & x \end{vmatrix} = 0 \), then which one of the following is correct?
A.
\( \frac{x}{y} \) is one of the cube roots of unity
B.
x is one of the cube roots of unity
C.
y is one of the cube roots of unity
D.
\( \frac{x}{y} \) is one of the cube roots of −1
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ANSWER :
D. \( \frac{x}{y} \) is one of the cube roots of −1
35.
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?
A.
C is empty
B.
B has as many elements as C
C.
\( A = B \cup C \)
D.
B has thrice as many elements as C
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ANSWER :
B. B has as many elements as C
36.
If \( A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \), then what is \( A^{3} \) equal to?
A.
\( \begin{bmatrix} \cos 3\theta & \sin 3\theta \\ -\sin 3\theta & \cos 3\theta \end{bmatrix} \)
B.
\( \begin{bmatrix} \cos^{3}\theta & \sin^{3}\theta \\ -\sin^{3}\theta & \cos^{3}\theta \end{bmatrix} \)
C.
\( \begin{bmatrix} \cos 3\theta & -\sin 3\theta \\ \sin 3\theta & \cos 3\theta \end{bmatrix} \)
D.
\( \begin{bmatrix} \cos^{3}\theta & -\sin^{3}\theta \\ \sin^{3}\theta & \cos^{3}\theta \end{bmatrix} \)
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ANSWER :
A. \( \begin{bmatrix} \cos 3\theta & \sin 3\theta \\ -\sin 3\theta & \cos 3\theta \end{bmatrix} \)
37.
What is the order of \( \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} a & h & g \\ h & b & f \\ g & f & c \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \)?
A.
\( 3 \times 1 \)
B.
\( 1 \times 1 \)
C.
\( 1 \times 3 \)
D.
\( 3 \times 3 \)
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ANSWER :
B. \( 1 \times 1 \)
38.
If \( A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \), then what is \( A^{4} \) equal to?
A.
\( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
B.
\( \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \)
C.
\( \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} \)
D.
\( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
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ANSWER :
A. \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
39.
If \( \sin A = \frac{3}{5} \), where \( 450^{\circ} < A < 540^{\circ} \), then \( \cos\frac{A}{2} \) is equal to
A.
\( \frac{1}{\sqrt{10}} \)
B.
\( \frac{3}{\sqrt{10}} \)
C.
\( \frac{\sqrt{3}}{\sqrt{10}} \)
D.
None of the above
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ANSWER :
D. None of the above
40.
What is \( \frac{1}{\sin 10^{\circ}} - \frac{\sqrt{3}}{\cos 10^{\circ}} \) equal to?
A.
0
B.
1
C.
2
D.
4
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ANSWER :
D. 4
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