NDA & NA Exam (II)- 2021 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2021 - Mathematics

NDA & NA Exam (II)- 2021 - Mathematics Defence Questions

21.
For what values of \( k \) is the system of equations
\( 2k^{2}x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 \)
consistent?
A.
\( \frac{3 \pm \sqrt{11}}{10} \)
B.
\( \frac{21 \pm \sqrt{161}}{10} \)
C.
\( \frac{3 \pm \sqrt{7}}{10} \)
D.
\( \frac{4 \pm \sqrt{11}}{10} \)
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ANSWER :
B. \( \frac{21 \pm \sqrt{161}}{10} \)
22.
The inverse of a matrix \( A \) is given by
\( \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix} \)
What is \( A \) equal to?
A.
\( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
B.
\( \begin{bmatrix} 1 & -2 \\ -3 & 4 \end{bmatrix} \)
C.
\( \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix} \)
D.
\( \begin{bmatrix} -1 & 2 \\ 3 & 4 \end{bmatrix} \)
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ANSWER :
A. \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
23.
What is the period of the function \( f(x) = \ln(2 + \sin^{2} x) \)?
A.
\( \frac{\pi}{2} \)
B.
\( \pi \)
C.
\( 2\pi \)
D.
\( 3\pi \)
😑 View Answer Error
ANSWER :
B. \( \pi \)
24.
If \( \sin(A + B) = 1 \) and \( 2 \sin(A - B) = 1 \), where \( 0 < A, B < \frac{\pi}{2} \), then what is \( \tan A : \tan B \) equal to?
A.
1 : 2
B.
2 : 1
C.
1 : 3
D.
3 : 1
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ANSWER :
D. 3 : 1
25.
Consider a regular polygon with 10 sides. What is the number of triangles that can be formed by joining the vertices which have no common side with any of the sides of the polygon?
A.
25
B.
50
C.
75
D.
100
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ANSWER :
B. 50
26.
Consider all the real roots of the equation \( x^{4} - 10x^{2} + 9 = 0 \). What is the sum of the absolute values of the roots?
A.
4
B.
6
C.
8
D.
10
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ANSWER :
C. 8
27.
Consider the expansion of \( (1 + x)^{n} \). Let \( p, q, r \) and \( s \) be the coefficients of first, second, \( n \)th and \( (n + 1) \)th terms respectively. What is \( (ps + qr) \) equal to?
A.
\( 1 + 2n \)
B.
\( 1 + 2n^{2} \)
C.
\( 1 + n^{2} \)
D.
\( 1 + 4n \)
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ANSWER :
C. \( 1 + n^{2} \)
28.
Let
\( \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2} \)
for \( 0 \leq x, y, z \leq 1 \). What is the value of \( x^{1000} + y^{1001} + z^{1002} \)?
A.
0
B.
1
C.
3
D.
6
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ANSWER :
C. 3
29.
Let \( \sin x + \sin y = \cos x + \cos y \) for all \( x, y \in \mathbb{R} \). What is \( \tan\left(\frac{x+y}{2}\right) \) equal to?
A.
1
B.
2
C.
\( \sqrt{2} \)
D.
\( 2\sqrt{2} \)
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ANSWER :
A. 1
30.
Let
\( A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} \) and \( (mI + nA)^{2} = A \)
where \( m, n \) are positive real numbers and \( I \) is the identity matrix. What is \( (m + n) \) equal to?
A.
0
B.
\( \frac{1}{2} \)
C.
1
D.
\( \frac{3}{2} \)
😑 View Answer Error
ANSWER :
D. \( \frac{3}{2} \)
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