NDA & NA Exam (II)- 2015 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2015 - Mathematics

NDA & NA Exam (II)- 2015 - Mathematics Defence Questions

1.
Let X be the set of all persons living in Delhi. The persons a and b in X are said to be related if the difference in their ages is at most 5 years. The relation is
A.
an equivalence relation
B.
reflexive and transitive but not symmetric
C.
symmetric and transitive but not reflexive
D.
reflexive and symmetric but not transitive
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ANSWER :
D. reflexive and symmetric but not transitive
2.
The matrix \( A = \begin{bmatrix} 1 & 3 & 2 \\ 1 & x-1 & 1 \\ 2 & 7 & x-3 \end{bmatrix} \) will have inverse for every real number x except for
A.
\( x = \frac{11 \pm \sqrt{5}}{2} \)
B.
\( x = \frac{9 \pm \sqrt{5}}{2} \)
C.
\( x = \frac{11 \pm \sqrt{3}}{2} \)
D.
\( x = \frac{9 \pm \sqrt{3}}{2} \)
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ANSWER :
A. \( x = \frac{11 \pm \sqrt{5}}{2} \)
3.
If the value of the determinant \( \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix} \) is positive, where \( a \neq b \neq c \), then the value of abc
A.
cannot be less than 1
B.
is greater than −8
C.
is less than −8
D.
must be greater than 8
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ANSWER :
B. is greater than −8
4.
Consider the following statements in respect of the determinant \( \begin{vmatrix} \cos^{2}\frac{\alpha}{2} & \sin^{2}\frac{\alpha}{2} \\ \sin^{2}\frac{\beta}{2} & \cos^{2}\frac{\beta}{2} \end{vmatrix} \), where \( \alpha, \beta \) are complementary angles.
1. The value of the determinant is \( \frac{1}{\sqrt{2}}\cos\left(\frac{\alpha - \beta}{2}\right) \).
2. The maximum value of the determinant is \( \frac{1}{\sqrt{2}} \).
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
5.
What is \( (1000000001)_{2} - (0.0101)_{2} \) equal to?
A.
\( (512.6775)_{10} \)
B.
\( (512.6875)_{10} \)
C.
\( (512.6975)_{10} \)
D.
\( (512.0909)_{10} \)
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ANSWER :
B. \( (512.6875)_{10} \)
6.
If \( A = \begin{bmatrix} 1 & 0 & -2 \\ 2 & -3 & 4 \end{bmatrix} \), then the matrix X for which \( 2X + 3A = 0 \) holds true is
A.
\( \begin{bmatrix} -\frac{3}{2} & 0 & -3 \\ -3 & -\frac{9}{2} & -6 \end{bmatrix} \)
B.
\( \begin{bmatrix} \frac{3}{2} & 0 & -3 \\ 3 & -\frac{9}{2} & -6 \end{bmatrix} \)
C.
\( \begin{bmatrix} \frac{3}{2} & 0 & 3 \\ 3 & \frac{9}{2} & 6 \end{bmatrix} \)
D.
\( \begin{bmatrix} -\frac{3}{2} & 0 & 3 \\ -3 & \frac{9}{2} & -6 \end{bmatrix} \)
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ANSWER :
D. \( \begin{bmatrix} -\frac{3}{2} & 0 & 3 \\ -3 & \frac{9}{2} & -6 \end{bmatrix} \)
7.
If \( z_{1} \) and \( z_{2} \) are complex numbers with \( |z_{1}| = |z_{2}| \), then which of the following is/are correct?
1. \( z_{1} = z_{2} \)
2. Real part of \( z_{1} \) = Real part of \( z_{2} \)
3. Imaginary part of \( z_{1} \) = Imaginary part of \( z_{2} \)
Select the correct answer using the code given below.
A.
1 only
B.
2 only
C.
3 only
D.
None
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ANSWER :
D. None
8.
If \( A = \begin{bmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{bmatrix} \), then which of the following is/are correct?
1. A and B commute.
2. AB is a null matrix.
Select the correct answer using the code given below.
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
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ANSWER :
B. 2 only
9.
The number of real roots of the equation \( x^{2} - 3|x| + 2 = 0 \) is
A.
4
B.
3
C.
2
D.
1
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ANSWER :
A. 4
10.
If the sum of the roots of the equation \( ax^{2} + bx + c = 0 \) is equal to the sum of their squares, then
A.
\( a^{2} + b^{2} = c^{2} \)
B.
\( a^{2} + b^{2} = a + b \)
C.
\( ab + b^{2} = 2ac \)
D.
\( ab - b^{2} = 2ac \)
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ANSWER :
C. \( ab + b^{2} = 2ac \)
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