NDA & NA Exam (I)- 2015 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2015 - Mathematics

NDA & NA Exam (I)- 2015 - Mathematics Defence Questions

21.
\( x^{3} - 1 \) can be factorised as (where \( \omega \) is one of the cube roots of unity)
A.
\( (x - 1)(x - \omega)(x + \omega^{2}) \)
B.
\( (x - 1)(x - \omega)(x - \omega^{2}) \)
C.
\( (x - 1)(x + \omega)(x + \omega^{2}) \)
D.
\( (x - 1)(x + \omega)(x - \omega^{2}) \)
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ANSWER :
B. \( (x - 1)(x - \omega)(x - \omega^{2}) \)
22.
What is \( \left[\frac{\sin\frac{\pi}{6} + i\left(1 - \cos\frac{\pi}{6}\right)}{\sin\frac{\pi}{6} - i\left(1 - \cos\frac{\pi}{6}\right)}\right]^{3} \), where \( i = \sqrt{-1} \), equal to?
A.
1
B.
−1
C.
i
D.
−i
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ANSWER :
C. i
23.
Let \( A = \begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix} \), \( B = \begin{bmatrix} 2 \\ -1 \end{bmatrix} \) and \( C = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \). If \( AB = C \), then what is \( A^{2} \) equal to?
A.
\( \begin{bmatrix}6 & -10 \\ 4 & 26\end{bmatrix} \)
B.
\( \begin{bmatrix}-10 & 5 \\ 4 & 24\end{bmatrix} \)
C.
\( \begin{bmatrix}-5 & -6 \\ -4 & -20\end{bmatrix} \)
D.
\( \begin{bmatrix}-5 & -7 \\ -5 & 20\end{bmatrix} \)
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ANSWER :
A. \( \begin{bmatrix}6 & -10 \\ 4 & 26\end{bmatrix} \)
24.
The value of \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{vmatrix} \) is
A.
x + y
B.
x − y
C.
xy
D.
1 + x + y
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ANSWER :
C. xy
25.
If \( A = \{x : x is a multiple of 3\} \), \( B = \{x : x is a multiple of 4\} \) and \( C = \{x : x is a multiple of 12\} \), then which one of the following is a null set?
A.
\( (A \setminus B) \cup C \)
B.
\( (A \setminus B) \setminus C \)
C.
\( (A \cap B) \cap C \)
D.
\( (A \cap B) \setminus C \)
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ANSWER :
D. \( (A \cap B) \setminus C \)
26.
If \( (11101011)_{2} \) is converted to decimal system, then the resulting number is
A.
235
B.
175
C.
160
D.
126
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ANSWER :
A. 235
27.
What is the real part of \( (\sin x + i\cos x)^{3} \), where \( i = \sqrt{-1} \)?
A.
\( -\cos 3x \)
B.
\( -\sin 3x \)
C.
\( \sin 3x \)
D.
\( \cos 3x \)
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ANSWER :
B. \( -\sin 3x \)
28.
If \( E(\theta) = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \), then \( E(\alpha)E(\beta) \) is equal to
A.
\( E(\alpha\beta) \)
B.
\( E(\alpha - \beta) \)
C.
\( E(\alpha + \beta) \)
D.
\( -E(\alpha + \beta) \)
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ANSWER :
C. \( E(\alpha + \beta) \)
29.
Let \( A = \{x, y, z\} \) and \( B = \{p, q, r, s\} \). What is the number of distinct relations from B to A?
A.
4096
B.
4094
C.
128
D.
126
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ANSWER :
A. 4096
30.
If \( 2p + 3q = 18 \) and \( 4p^{2} + 4pq - 3q^{2} - 36 = 0 \), then what is \( (2p + q) \) equal to?
A.
6
B.
7
C.
10
D.
20
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ANSWER :
C. 10
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