NDA & NA Exam (I)- 2018 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2018 - Mathematics

NDA & NA Exam (I)- 2018 - Mathematics Defence Questions

21.
What is the number of non-zero terms in the expansion of \( (1 + 2\sqrt{3}x)^{11} + (1 - 2\sqrt{3}x)^{11} \) (after simplification)?
A.
4
B.
5
C.
6
D.
11
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ANSWER :
C. 6
22.
What is the greatest integer among the following by which the number \( 5^{5} + 7^{5} \) is divisible?
A.
6
B.
8
C.
11
D.
12
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ANSWER :
D. 12
23.
If \( x = 1 - y + y^{2} - y^{3} + \ldots \) up to infinite terms, where \( |y| < 1 \), then which one of the following is correct?
A.
\( x = \frac{1}{1+y} \)
B.
\( x = \frac{1}{1-y} \)
C.
\( x = \frac{y}{1+y} \)
D.
\( x = \frac{y}{1-y} \)
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ANSWER :
A. \( x = \frac{1}{1+y} \)
24.
What is the inverse of the matrix \( A = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \)?
A.
\( \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
B.
\( \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix} \)
C.
\( \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{bmatrix} \)
D.
\( \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & \sin\theta \\ 0 & -\sin\theta & \cos\theta \end{bmatrix} \)
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ANSWER :
A. \( \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
25.
If A is a 2 × 3 matrix and AB is a 2 × 5 matrix, then B must be a
A.
3 × 5 matrix
B.
5 × 3 matrix
C.
3 × 2 matrix
D.
5 × 2 matrix
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ANSWER :
A. 3 × 5 matrix
26.
If \( A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \) and \( A^{2} - kA - I_{2} = O \), where \( I_{2} \) is the 2 × 2 identity matrix, then what is the value of k?
A.
4
B.
−4
C.
8
D.
−8
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ANSWER :
A. 4
27.
What is the number of triangles that can be formed by choosing the vertices from a set of 12 points in a plane, seven of which lie on the same straight line?
A.
185
B.
175
C.
115
D.
105
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ANSWER :
A. 185
28.
What is C(n, r) + 2C(n, r − 1) + C(n, r − 2) equal to?
A.
C(n + 1, r)
B.
C(n − 1, r + 1)
C.
C(n, r + 1)
D.
C(n + 2, r)
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ANSWER :
D. C(n + 2, r)
29.
Let [x] denote the greatest integer function. What is the number of solutions of the equation \( x^{2} - 4x + [x] = 0 \) in the interval [0, 2]?
A.
Zero (no solution)
B.
One
C.
Two
D.
Three
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ANSWER :
B. One
30.
A survey of 850 students in a University yields that 680 students like music and 215 like dance. What is the least number of students who like both music and dance?
A.
40
B.
45
C.
50
D.
55
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ANSWER :
B. 45
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