NDA & NA Exam (I)- 2016 - Mathematics Defence Question Paper

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NDA & NA Exam (I)- 2016 - Mathematics

NDA & NA Exam (I)- 2016 - Mathematics Defence Questions

11.
Consider the following statements:
1. \( f(\theta) = 2 \) has no solution.
2. \( f(\theta) = \frac{7}{2} \) has a solution.
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
12.
For the next two (2) items that follow:
Consider the curves \( f(x) = x|x| - 1 \) and \( g(x) = \begin{cases} \frac{3x}{2}, & x > 0 \\ 2x, & x \leq 0 \end{cases} \)

Where do the curves intersect?
A.
At (2, 3) only
B.
At (−1, −2) only
C.
At (2, 3) and (−1, −2)
D.
Neither at (2, 3) nor at (−1, −2)
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ANSWER :
C. At (2, 3) and (−1, −2)
13.
What is the area bounded by the curves?
A.
\( \frac{17}{6} \) square units
B.
\( \frac{8}{3} \) square units
C.
2 square units
D.
\( \frac{1}{3} \) square unit
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ANSWER :
B. \( \frac{8}{3} \) square units
14.
For the next two (2) items that follow:
Consider the function \( f(x) = \frac{27\left(x^{\frac{2}{3}} - x\right)}{4} \).

How many solutions does the equation \( f(x) = 1 \) have?
A.
One
B.
Two
C.
Three
D.
Four
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ANSWER :
B. Two
15.
How many solutions does the equation \( f(x) = -1 \) have?
A.
One
B.
Two
C.
Three
D.
Four
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ANSWER :
A. One
16.
For the next two (2) items that follow:
Consider the functions \( f(x) = xg(x) \) and \( g(x) = \left[\frac{1}{x}\right] \), where [.] is the greatest integer function.

What is \( \int_{\frac{1}{3}}^{\frac{1}{2}} g(x) dx \) equal to?
A.
\( \frac{1}{6} \)
B.
\( \frac{1}{3} \)
C.
\( \frac{5}{18} \)
D.
\( \frac{5}{36} \)
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ANSWER :
B. \( \frac{1}{3} \)
17.
What is \( \int_{\frac{1}{3}}^{1} f(x) dx \) equal to?
A.
\( \frac{37}{72} \)
B.
\( \frac{2}{3} \)
C.
\( \frac{17}{72} \)
D.
\( \frac{37}{144} \)
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ANSWER :
A. \( \frac{37}{72} \)
18.
For the next five (5) items that follow:
Consider the function \( f(x) = |x - 1| + x^{2} \), where \( x \in R \).

Which one of the following statements is correct?
A.
f(x) is continuous but not differentiable at x = 0
B.
f(x) is continuous but not differentiable at x = 1
C.
f(x) is differentiable at x = 1
D.
f(x) is not differentiable at x = 0 and x = 1
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ANSWER :
B. f(x) is continuous but not differentiable at x = 1
19.
Which one of the following statements is correct?
A.
f(x) is increasing in \( \left(\frac{1}{2}, \infty\right) \) and decreasing in \( \left(-\infty, \frac{1}{2}\right) \)
B.
f(x) is decreasing in \( \left(\frac{1}{2}, \infty\right) \) and increasing in \( \left(-\infty, \frac{1}{2}\right) \)
C.
f(x) is increasing in \( (-\infty, 1) \) and decreasing in \( (1, \infty) \)
D.
f(x) is decreasing in \( (-\infty, 1) \) and increasing in \( (1, \infty) \)
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ANSWER :
A. f(x) is increasing in \( \left(\frac{1}{2}, \infty\right) \) and decreasing in \( \left(-\infty, \frac{1}{2}\right) \)
20.
Which one of the following statements is correct?
A.
f(x) has local minima at more than one point in \( (-\infty, \infty) \)
B.
f(x) has local maxima at more than one point in \( (-\infty, \infty) \)
C.
f(x) has local minimum at one point only in \( (-\infty, \infty) \)
D.
f(x) has neither maxima nor minima in \( (-\infty, \infty) \)
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ANSWER :
C. f(x) has local minimum at one point only in \( (-\infty, \infty) \)
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