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

41.
Directions (Qn. 41-42): Consider the following for the next two (02) items that follow:
Let \( f(x) = x^{2} - 1 \) and \( g \circ f(x) = x - \sqrt{x} + 1 \).

Which one of the following is a possible expression for \( g(x) \)?
A.
\( \sqrt{x + 1} - \sqrt[4]{x + 1} \)
B.
\( \sqrt{x + 1} - \sqrt[4]{x + 1} + 1 \)
C.
\( \sqrt{x + 1} + \sqrt[4]{x + 1} \)
D.
\( x + 1 - \sqrt{x + 1} + 1 \)
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ANSWER :
B. \( \sqrt{x + 1} - \sqrt[4]{x + 1} + 1 \)
42.
What is \( g(15) \) equal to?
A.
1
B.
2
C.
3
D.
4
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ANSWER :
C. 3
43.
Directions (Qn. 43-44): Consider the following for the next two (02) items that follow:
Let a function \( f \) be defined on \( \mathbb{R} - \{0\} \) and \( 2f(x) + f\left(\frac{1}{x}\right) = x + 3 \).

What is \( f(0.5) \) equal to?
A.
\( \frac{1}{2} \)
B.
\( \frac{2}{3} \)
C.
1
D.
2
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ANSWER :
B. \( \frac{2}{3} \)
44.
If \( f \) is differentiable, then what is \( f'(0.5) \) equal to?
A.
\( \frac{1}{4} \)
B.
\( \frac{2}{3} \)
C.
2
D.
4
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ANSWER :
C. 2
45.
Directions (Qn. 45-46): Consider the following for the next two (02) items that follow:
A function is defined by
\( f(x) = \begin{vmatrix} x + 1 & 2 & 3 \\ 2 & x + 4 & 6 \\ 3 & 6 & x + 9 \end{vmatrix} \)

The function is decreasing on:
A.
\( \left[-\frac{28}{3}, 0\right] \)
B.
\( \left[0, \frac{28}{3}\right] \)
C.
\( \left[0, \frac{56}{3}\right] \)
D.
\( \left[-\frac{56}{3}, 0\right] \)
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ANSWER :
A. \( \left[-\frac{28}{3}, 0\right] \)
46.
The function attains local minimum value at:
A.
\( x = -\frac{28}{3} \)
B.
\( x = -1 \)
C.
\( x = 0 \)
D.
\( x = \frac{28}{3} \)
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ANSWER :
C. \( x = 0 \)
47.
Directions (Qn. 47-48): Consider the following for the next two (02) items that follow:
Given that \( 4x^{2} + y^{2} = 9 \).

What is the maximum value of \( y \)?
A.
\( \frac{3}{2} \)
B.
3
C.
4
D.
6
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ANSWER :
B. 3
48.
What is the maximum value of \( xy \)?
A.
\( \frac{9}{4} \)
B.
9
C.
\( \frac{4}{9} \)
D.
\( \frac{9}{2} \)
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ANSWER :
A. \( \frac{9}{4} \)
49.
Directions (Qn. 49-50): Consider the following for the next two (02) items that follow:
A function is defined by \( f(x) = \pi + \sin^{2} x \).

What is the range of the function?
A.
\( [0, 1] \)
B.
\( [\pi, \pi + 1] \)
C.
\( [\pi - 1, \pi + 1] \)
D.
\( (\pi - 1, \pi + 1) \)
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ANSWER :
B. \( [\pi, \pi + 1] \)
50.
What is the period of the function?
A.
\( 2\pi \)
B.
\( \pi \)
C.
\( \frac{\pi}{2} \)
D.
The function is non-periodic
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ANSWER :
B. \( \pi \)
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