NDA & NA Exam (II)- 2024 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2024 - Mathematics

NDA & NA Exam (II)- 2024 - Mathematics Defence Questions

11.
Let x > 1, y > 1 and z > 1 be in GP. Then \( \frac{1}{1 + \ln x} \), \( \frac{1}{1 + \ln y} \) and \( \frac{1}{1 + \ln z} \) are
A.
in AP.
B.
in GP.
C.
in HP.
D.
neither in AP nor in GP nor in HP.
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ANSWER :
C. in HP.
12.
If \( \omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \), then what is \( \begin{vmatrix} 1 + \omega & 1 + \omega^{2} & \omega + \omega^{2} \\ 1 & \omega & \omega^{2} \\ 1 & \frac{1}{\omega} & \frac{1}{\omega^{2}} \end{vmatrix} \) equal to?
A.
0
B.
\( \omega \)
C.
\( \omega^{2} \)
D.
\( 1 - \omega^{2} \)
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ANSWER :
A. 0
13.
If the sum of the first n terms of a series is n(2n + 1), then what is the \( n^{th} \) term?
A.
4n − 1
B.
4n
C.
4n + 1
D.
4n + 3
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ANSWER :
A. 4n − 1
14.
In how many ways can the letters of the word INDIA be permuted such that in each arrangement, vowels occupy odd positions?
A.
3
B.
6
C.
9
D.
12
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ANSWER :
B. 6
15.
The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?
A.
240
B.
720
C.
1440
D.
1620
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ANSWER :
C. 1440
16.
If n is a root of the equation \( x^{2} + px + m = 0 \) and m is a root of the equation \( x^{2} + px + n = 0 \), where \( m \neq n \), then what is the value of p + m + n?
A.
−1
B.
0
C.
1
D.
2
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ANSWER :
C. 1
17.
In how many ways can a student choose (n − 2) courses out of n courses if 2 particular courses are compulsory (n > 4)?
A.
(n − 3)(n − 4)
B.
(n − 1)(n − 2)
C.
\( \frac{(n - 3)(n - 4)}{2} \)
D.
\( \frac{(n - 2)(n - 3)}{2} \)
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ANSWER :
D. \( \frac{(n - 2)(n - 3)}{2} \)
18.
If \( D_{n} = \begin{vmatrix} n & 20 & 30 \\ n^{2} & 40 & 50 \\ n^{3} & 60 & 70 \end{vmatrix} \), then what is the value of \( \sum_{n=1}^{4} D_{n} \)?
A.
−10000
B.
−10
C.
10
D.
10000
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ANSWER :
A. −10000
19.
Consider the following in respect of the matrices \( P = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix} \) and \( Q = \begin{bmatrix} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{bmatrix} \):
(I) PQ is null matrix.
(II) QP is an identity matrix of order 3.
(III) PQ = QP.
Which of the above is/are correct?
A.
I only
B.
II only
C.
I and III
D.
II and III
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ANSWER :
C. I and III
20.
If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?
A.
−1
B.
0
C.
1
D.
3
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ANSWER :
B. 0
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