NDA & NA Exam (I)- 2023 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2023 - Mathematics

NDA & NA Exam (I)- 2023 - Mathematics Defence Questions

11.
If \( 2 - i\sqrt{3} \), where \( i = \sqrt{-1} \), is a root of the equation \( x^{2} + ax + b = 0 \), then what is the value of \( (a + b) \)?
A.
−11
B.
−3
C.
0
D.
3
😑 View Answer Error
ANSWER :
D. 3
12.
If \( z = \frac{1 + i\sqrt{3}}{1 - i\sqrt{3}} \), where \( i = \sqrt{-1} \), then what is the argument of \( z \)?
A.
\( \frac{\pi}{3} \)
B.
\( \frac{2\pi}{3} \)
C.
\( \frac{4\pi}{3} \)
D.
\( \frac{5\pi}{3} \)
😑 View Answer Error
ANSWER :
B. \( \frac{2\pi}{3} \)
13.
If \( a, b, c \) are in AP, then what is
\( \begin{vmatrix} x + 1 & x + 2 & x + 3 \\ x + 2 & x + 3 & x + 4 \\ x + a & x + b & x + c \end{vmatrix} \)
equal to?
A.
−1
B.
0
C.
1
D.
2
😑 View Answer Error
ANSWER :
B. 0
14.
If \( \log_{x} a, a^{\frac{x}{2}} \) and \( \log_{b} x \) are in GP, then what is \( x \) equal to?
A.
\( \log_{a}(\log_{b} a) \)
B.
\( \log_{b}(\log_{a} b) \)
C.
\( \frac{\log_{a}(\log_{b} a)}{2} \)
D.
\( \frac{\log_{b}(\log_{a} b)}{2} \)
😑 View Answer Error
ANSWER :
A. \( \log_{a}(\log_{b} a) \)
15.
If \( 2^{\frac{1}{a}}, 2^{\frac{1}{b}}, 2^{\frac{1}{c}} \) are in GP, then which one of the following is correct?
A.
\( a, b, c \) are in AP
B.
\( a, b, c \) are in GP
C.
\( a, b, c \) are in HP
D.
\( ab, bc, ca \) are in AP
😑 View Answer Error
ANSWER :
C. \( a, b, c \) are in HP
16.
The first and the second terms of an AP are \( \frac{5}{2} \) and \( \frac{23}{12} \) respectively. If \( n \)th term is the largest negative term, what is the value of \( n \)?
A.
5
B.
6
C.
7
D.
\( n \) cannot be determined
😑 View Answer Error
ANSWER :
B. 6
17.
For how many integral values of \( k \), the equation \( x^{2} - 4x + k = 0 \), where \( k \) is an integer has real roots and both of them lie in the interval \( (0, 5) \)?
A.
3
B.
4
C.
5
D.
6
😑 View Answer Error
ANSWER :
B. 4
18.
In an AP, the first term is \( x \) and the sum of the first \( n \) terms is zero. What is the sum of next \( m \) terms?
A.
\( \frac{mx(m + n)}{n - 1} \)
B.
\( \frac{mx(m + n)}{1 - n} \)
C.
\( \frac{nx(m + n)}{m - 1} \)
D.
\( \frac{nx(m + n)}{1 - m} \)
😑 View Answer Error
ANSWER :
B. \( \frac{mx(m + n)}{1 - n} \)
19.
Consider the following statements:
1. \( (25)! + 1 \) is divisible by 26
2. \( (6)! + 1 \) is divisible by 7
Which of the above statements is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
B. 2 only
20.
If \( z \) is a complex number such that \( \frac{z - 1}{z + 1} \) is purely imaginary, then what is \( |z| \) equal to?
A.
\( \frac{1}{2} \)
B.
\( \frac{3}{2} \)
C.
1
D.
2
😑 View Answer Error
ANSWER :
C. 1
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