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

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NDA & NA Exam (II)- 2023 - Mathematics

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

21.
The Cartesian product \( A \times A \) has 16 elements among which are \( (0, 2) \) and \( (1, 3) \). Which of the following statements is/are correct?
1. It is possible to determine set A.
2. \( A \times A \) contains the element \( (3, 2) \).
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
😑 View Answer Error
ANSWER :
C. Both 1 and 2
22.
Let \( A = \{1, 2, 3, \ldots, 20\} \). Define a relation R from A to A by \( R = \{(x, y): 4x - 3y = 1\} \), where \( x, y \in A \). Which of the following statements is/are correct?
1. The domain of R is \( \{1, 4, 7, 10, 13, 16\} \).
2. The range of R is \( \{1, 5, 9, 13, 17\} \).
3. The range of R is equal to codomain of R.
Select the correct answer using the code given below:
A.
1 only
B.
2 only
C.
1 and 2
D.
2 and 3
😑 View Answer Error
ANSWER :
B. 2 only
23.
Consider the following statements:
1. The relation f defined by \( f(x) = \begin{cases} x^{3}, & 0 \leq x \leq 2 \\ 4x, & 2 \leq x \leq 8 \end{cases} \) is a function.
2. The relation g defined by \( g(x) = \begin{cases} x^{2}, & 0 \leq x \leq 4 \\ 3x, & 4 \leq x \leq 8 \end{cases} \) is a function.
Which of the statements given above is/are correct?
A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2
😑 View Answer Error
ANSWER :
A. 1 only
24.
Consider the following statements:
1. \( A = (A \cup B) \cup (A - B) \)
2. \( A \cup (B - A) = (A \cup B) \)
3. \( B = (A \cup B) - (A - B) \)
Which of the statements given above are correct?
A.
1 and 2 only
B.
2 and 3 only
C.
1 and 3 only
D.
1, 2 and 3
😑 View Answer Error
ANSWER :
B. 2 and 3 only
25.
A function satisfies \( f(x - y) = \frac{f(x)}{f(y)} \), where \( f(y) \neq 0 \). If \( f(1) = 0.5 \), then what is \( f(2) + f(3) + f(4) + f(5) + f(6) \) equal to?
A.
\( \frac{15}{32} \)
B.
\( \frac{31}{32} \)
C.
\( \frac{29}{64} \)
D.
\( \frac{31}{64} \)
😑 View Answer Error
ANSWER :
D. \( \frac{31}{64} \)
26.
What is \( 2 \cot\left(\frac{1}{2} \cos^{-1} \frac{\sqrt{5}}{3}\right) \) equal to?
A.
−1
B.
1
C.
\( 3 + \sqrt{5} \)
D.
\( 3 - \sqrt{5} \)
😑 View Answer Error
ANSWER :
C. \( 3 + \sqrt{5} \)
27.
If \( \sec^{-1} p - cosec^{-1} q = 0 \), where \( p > 0, q > 0 \); then what is the value of \( p^{-2} + q^{-2} \)?
A.
1
B.
2
C.
\( \frac{1}{2} \)
D.
\( \frac{1}{2\sqrt{2}} \)
😑 View Answer Error
ANSWER :
A. 1
28.
What is \( 1 + \sin^{2}\left(\cos^{-1} \frac{3}{\sqrt{17}}\right) \) equal to?
A.
\( \frac{25}{17} \)
B.
\( \frac{8}{17} \)
C.
\( \frac{9}{17} \)
D.
\( \frac{41}{17} \)
😑 View Answer Error
ANSWER :
A. \( \frac{25}{17} \)
29.
If \( \tan(\pi \cos \theta) = \cot(\pi \sin \theta) \), \( 0 < \theta < \frac{3\pi}{4} \), then what is the value of \( 8 \sin^{2}\left(\theta + \frac{\pi}{4}\right) \)?
A.
16
B.
2
C.
1
D.
\( \frac{1}{2} \)
😑 View Answer Error
ANSWER :
C. 1
30.
If \( \tan \alpha = \frac{1}{7} \), \( \sin \beta = \frac{1}{\sqrt{10}} \); \( 0 < \alpha, \beta < \frac{\pi}{2} \), then what is the value of \( \cos(\alpha + 2\beta) \)?
A.
\( -\frac{1}{2} \)
B.
\( -\frac{1}{\sqrt{2}} \)
C.
\( \frac{1}{\sqrt{2}} \)
D.
\( \frac{1}{2} \)
😑 View Answer Error
ANSWER :
C. \( \frac{1}{\sqrt{2}} \)
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