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

1.
Let X be the set of all persons living in a city. Persons x, y in X are said to be related as \( x < y \) if y is at least 5 years older than x. Which one of the following is correct?
A.
The relation is an equivalence relation on X
B.
The relation is transitive but neither reflexive nor symmetric
C.
The relation is reflexive but neither transitive nor symmetric
D.
The relation is symmetric but neither transitive nor reflexive
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ANSWER :
B. The relation is transitive but neither reflexive nor symmetric
2.
Which one of the following matrices is an elementary matrix?
A.
\( \begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix} \)
B.
\( \begin{bmatrix}1 & 5 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \)
C.
\( \begin{bmatrix}0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix} \)
D.
\( \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 5 & 2\end{bmatrix} \)
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ANSWER :
B. \( \begin{bmatrix}1 & 5 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \)
3.
Consider the following statements in respect of the given equation:
\( (x^{2} + 2)^{2} + 8x^{2} = 6x(x^{2} + 2) \)
1. All the roots of the equation are complex.
2. The sum of all the roots of the equation is 6.
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
B. 2 only
4.
In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds −9 and −1 for roots. The correct equation is
A.
\( x^{2} - 10x + 9 = 0 \)
B.
\( x^{2} + 10x + 9 = 0 \)
C.
\( x^{2} - 10x + 16 = 0 \)
D.
\( x^{2} - 8x - 9 = 0 \)
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ANSWER :
A. \( x^{2} - 10x + 9 = 0 \)
5.
If \( A = \begin{bmatrix} 2 & 7 \\ 1 & 5 \end{bmatrix} \), then what is \( A + 3A^{-1} \) equal to, where I is the identity matrix of order 2?
A.
3I
B.
5I
C.
7I
D.
None of the above
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ANSWER :
C. 7I
6.
In a class of 60 students, 45 students like music, 50 students like dancing, 5 students like neither. Then the number of students in the class who like both music and dancing is
A.
35
B.
40
C.
50
D.
55
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ANSWER :
B. 40
7.
If \( \log_{10}2 \), \( \log_{10}(2^{x} - 1) \) and \( \log_{10}(2^{x} + 3) \) are three consecutive terms of an AP, then the value of x is
A.
1
B.
\( \log_{5} 2 \)
C.
\( \log_{2} 5 \)
D.
\( \log_{10} 5 \)
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ANSWER :
C. \( \log_{2} 5 \)
8.
The matrix \( \begin{bmatrix} 0 & -4+i \\ 4+i & 0 \end{bmatrix} \) is
A.
symmetric
B.
skew-symmetric
C.
Hermitian
D.
skew-Hermitian
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ANSWER :
D. skew-Hermitian
9.
Let Z be the set of integers and aRb, where \( a, b \in Z \), if and only if \( (a - b) \) is divisible by 5. Consider the following statements:
1. The relation R partitions Z into five equivalence classes.
2. Any two equivalence classes are either equal or disjoint.
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
C. Both 1 and 2
10.
If \( z = \frac{-2(1+2i)}{3+i} \), where \( i = \sqrt{-1} \), then the argument \( \theta (-\pi < \theta \leq \pi) \) of z is
A.
\( \frac{3\pi}{4} \)
B.
\( \frac{\pi}{4} \)
C.
\( \frac{5\pi}{6} \)
D.
\( -\frac{3\pi}{4} \)
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ANSWER :
D. \( -\frac{3\pi}{4} \)
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