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

11.
If m and n are the roots of the equation \( (x + p)(x + q) - k = 0 \), then the roots of the equation \( (x - m)(x - n) + k = 0 \) are
A.
p and q
B.
\( \frac{1}{p} \) and \( \frac{1}{q} \)
C.
−p and −q
D.
p + q and p − q
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ANSWER :
C. −p and −q
12.
What is the sum of the series \( 0.5 + 0.55 + 0.555 + \ldots \) to n terms?
A.
\( \frac{5}{9}\left[n - \frac{2}{9}\left(1 - \frac{1}{10^{n}}\right)\right] \)
B.
\( \frac{1}{9}\left[5 - \frac{2}{9}\left(1 - \frac{1}{10^{n}}\right)\right] \)
C.
\( \frac{1}{9}\left[n - \frac{5}{9}\left(1 - \frac{1}{10^{n}}\right)\right] \)
D.
\( \frac{5}{9}\left[n - \frac{1}{9}\left(1 - \frac{1}{10^{n}}\right)\right] \)
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ANSWER :
D. \( \frac{5}{9}\left[n - \frac{1}{9}\left(1 - \frac{1}{10^{n}}\right)\right] \)
13.
If \( 1, \omega, \omega^{2} \) are the cube roots of unity, then the value of \( (1 + \omega)(1 + \omega^{2})(1 + \omega^{4})(1 + \omega^{8}) \) is
A.
−1
B.
0
C.
1
D.
2
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ANSWER :
C. 1
14.
Let \( A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \). Then the number of subsets of A containing exactly two elements is
A.
20
B.
40
C.
45
D.
90
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ANSWER :
C. 45
15.
What is the square root of i, where \( i = \sqrt{-1} \)?
A.
\( \frac{1+i}{2} \)
B.
\( \frac{1-i}{2} \)
C.
\( \pm\frac{1+i}{\sqrt{2}} \)
D.
None of the above
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ANSWER :
C. \( \pm\frac{1+i}{\sqrt{2}} \)
16.
The decimal number \( (127.25)_{10} \), when converted to binary number, takes the form
A.
\( (1111111.11)_{2} \)
B.
\( (1111110.01)_{2} \)
C.
\( (1110111.11)_{2} \)
D.
\( (1111111.01)_{2} \)
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ANSWER :
D. \( (1111111.01)_{2} \)
17.
Consider the following in respect of two non-singular matrices A and B of same order:
1. \( \det(A + B) = \det A + \det B \)
2. \( (A + B)^{-1} = A^{-1} + B^{-1} \)
Which of the above is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
D. Neither 1 nor 2
18.
If \( X = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \), \( B = \begin{bmatrix} 5 & 2 \\ -2 & 1 \end{bmatrix} \) and \( A = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \) satisfy the equation \( AX = B \), then the matrix A is equal to
A.
\( \begin{bmatrix}-7 & 26 \\ 1 & -5\end{bmatrix} \)
B.
\( \begin{bmatrix}7 & 26 \\ 4 & 17\end{bmatrix} \)
C.
\( \begin{bmatrix}-7 & -4 \\ 26 & 13\end{bmatrix} \)
D.
\( \begin{bmatrix}-7 & 26 \\ -6 & 23\end{bmatrix} \)
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ANSWER :
A. \( \begin{bmatrix}-7 & 26 \\ 1 & -5\end{bmatrix} \)
19.
What is \( \sum_{r=0}^{1} {}^{n+r}C_{n} \) equal to?
A.
\( {}^{n+2}C_{1} \)
B.
\( {}^{n+2}C_{n} \)
C.
\( {}^{n+3}C_{n} \)
D.
\( {}^{n+3}C_{n+2} \)
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ANSWER :
A. \( {}^{n+2}C_{1} \)
20.
How many words can be formed using all the letters of the word ‘NATION’ so that all the three vowels should never come together?
A.
354
B.
348
C.
288
D.
None of the above
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ANSWER :
C. 288
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