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

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NDA & NA Exam (II)- 2017 - Mathematics

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

11.
If \( A = \begin{bmatrix} 4i-6 & 10i \\ 14i & 6+4i \end{bmatrix} \) and \( k = \frac{1}{2i} \), where \( i = \sqrt{-1} \), then kA is equal to
A.
\( \begin{bmatrix} 2+3i & 5 \\ 7 & 2-3i \end{bmatrix} \)
B.
\( \begin{bmatrix} 2-3i & 5 \\ 7 & 2+3i \end{bmatrix} \)
C.
\( \begin{bmatrix} 2-3i & 7 \\ 5 & 2+3i \end{bmatrix} \)
D.
\( \begin{bmatrix} 2+3i & 5 \\ 7 & 2+3i \end{bmatrix} \)
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ANSWER :
A. \( \begin{bmatrix} 2+3i & 5 \\ 7 & 2-3i \end{bmatrix} \)
12.
The sum of all real roots of the equation \( |x - 3|^{2} + |x - 3| - 2 = 0 \) is
A.
2
B.
3
C.
4
D.
6
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ANSWER :
D. 6
13.
It is given that the roots of the equation \( x^{2} - 4x - \log_{3} P = 0 \) are real. For this, the minimum value of P is
A.
\( \frac{1}{27} \)
B.
\( \frac{1}{64} \)
C.
\( \frac{1}{81} \)
D.
1
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ANSWER :
C. \( \frac{1}{81} \)
14.
If A is a square matrix, then the value of \( adj A^{T} - (adj A)^{T} \) is equal to
A.
A
B.
2|A|I, where I is the identity matrix
C.
null matrix whose order is same as that of A
D.
unit matrix whose order is same as that of A
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ANSWER :
C. null matrix whose order is same as that of A
15.
The value of the product \( 6^{\frac{1}{2}} \times 6^{\frac{1}{4}} \times 6^{\frac{1}{8}} \times 6^{\frac{1}{16}} \times \ldots \) up to infinite terms is
A.
6
B.
36
C.
216
D.
512
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ANSWER :
A. 6
16.
The value of the determinant \( \begin{vmatrix} \cos^{2}\frac{\theta}{2} & \sin^{2}\frac{\theta}{2} \\ \sin^{2}\frac{\theta}{2} & \cos^{2}\frac{\theta}{2} \end{vmatrix} \) for all values of \( \theta \), is
A.
1
B.
\( \cos\theta \)
C.
\( \sin\theta \)
D.
\( \cos 2\theta \)
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ANSWER :
B. \( \cos\theta \)
17.
The number of terms in the expansion of \( (x + a)^{100} + (x - a)^{100} \) after simplification is
A.
202
B.
101
C.
51
D.
50
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ANSWER :
C. 51
18.
In the expansion of \( (1 + x)^{50} \), the sum of the coefficients of odd powers of x is
A.
\( 2^{26} \)
B.
\( 2^{49} \)
C.
\( 2^{50} \)
D.
\( 2^{51} \)
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ANSWER :
B. \( 2^{49} \)
19.
If a, b, c are non-zero real numbers, then the inverse of the matrix \( A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \) is equal to
A.
\( \begin{bmatrix} a^{-1} & 0 & 0 \\ 0 & b^{-1} & 0 \\ 0 & 0 & c^{-1} \end{bmatrix} \)
B.
\( \frac{1}{abc} \begin{bmatrix} a^{-1} & 0 & 0 \\ 0 & b^{-1} & 0 \\ 0 & 0 & c^{-1} \end{bmatrix} \)
C.
\( \frac{1}{abc} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
D.
\( \frac{1}{abc} \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \)
😑 View Answer Error
ANSWER :
A. \( \begin{bmatrix} a^{-1} & 0 & 0 \\ 0 & b^{-1} & 0 \\ 0 & 0 & c^{-1} \end{bmatrix} \)
20.
A person is to count 4500 notes. Let \( a_{n} \) denote the number of notes he counts in the \( n^{th} \) minute. If \( a_{1} = a_{2} = a_{3} = \ldots = a_{10} = 150 \), and \( a_{10}, a_{11}, a_{12}, \ldots \) are in AP with the common difference −2, then the time taken by him to count all the notes is
A.
24 minutes
B.
34 minutes
C.
125 minutes
D.
135 minutes
😑 View Answer Error
ANSWER :
B. 34 minutes
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