NDA & NA Exam (II)- 2018 - Mathematics Defence Question Paper

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NDA & NA Exam (II)- 2018 - Mathematics

NDA & NA Exam (II)- 2018 - Mathematics Defence Questions

1.
What is the value of \( \log_{7}\log_{7}\sqrt{7\sqrt{7\sqrt{7}}} \)?
A.
\( 3\log_{2}7 \)
B.
\( 1 - 3\log_{2}7 \)
C.
\( 1 - 3\log_{7}2 \)
D.
\( \frac{7}{8} \)
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ANSWER :
C. \( 1 - 3\log_{7}2 \)
2.
If an infinite GP has the first term x and the sum 5, then which one of the following is correct?
A.
\( x < -10 \)
B.
\( -10 < x < 0 \)
C.
\( 0 < x < 10 \)
D.
\( x > 10 \)
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ANSWER :
C. \( 0 < x < 10 \)
3.
Consider the following expressions:
1. \( x + x^{2} - \frac{1}{x} \)
2. \( \sqrt{ax^{2} + bx + x - c + \frac{d}{x} - \frac{e}{x^{2}}} \)
3. \( 3x^{2} - 5x + ab \)
4. \( \frac{2}{x^{2} - ax + b^{3}} \)
5. \( \frac{1}{x} - \frac{2}{x + 5} \)
Which of the above are rational expressions?
A.
1, 4 and 5 only
B.
1, 3, 4 and 5 only
C.
2, 4 and 5 only
D.
1 and 2 only
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ANSWER :
B. 1, 3, 4 and 5 only
4.
A square matrix A is called orthogonal if (where A′ is the transpose of A)
A.
\( A = A^{2} \)
B.
\( A' = A^{-1} \)
C.
\( A = A^{-1} \)
D.
\( A = A' \)
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ANSWER :
B. \( A' = A^{-1} \)
5.
If A, B and C are subsets of a Universal set, then which one of the following is not correct? (where A′ is the complement of A)
A.
\( A \cup (B \cap C) = (A \cap B) \cap (A \cup C) \)
B.
\( A' \cup (A \cup B) = (B' \cap A)' \cup A \)
C.
\( A' \cup (B \cup C) = (B' \cap C')' \cup A' \)
D.
\( (A \cap B) \cup C = (A \cup C) \cap (B \cup C) \)
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ANSWER :
A. \( A \cup (B \cap C) = (A \cap B) \cap (A \cup C) \)
6.
Let x be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to
A.
2480
B.
2481
C.
2482
D.
2483
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ANSWER :
B. 2481
7.
The sum of the series \( 3 - 1 + \frac{1}{3} - \frac{1}{9} + \ldots \) is equal to
A.
\( \frac{20}{9} \)
B.
\( \frac{9}{20} \)
C.
\( \frac{9}{4} \)
D.
\( \frac{4}{9} \)
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ANSWER :
C. \( \frac{9}{4} \)
8.
Consider the following information for the next two (02) items that follow:
A survey was conducted among 300 students. It was found that 125 students like to play cricket, 145 students like to play football and 90 students like to play tennis. 32 students like to play exactly two games out of the three games.

How many students like to play all the three games?
A.
14
B.
21
C.
28
D.
35
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ANSWER :
A. 14
9.
How many students like to play exactly only one game?
A.
196
B.
228
C.
254
D.
268
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ANSWER :
C. 254
10.
If \( \alpha \) and \( \beta \) \( (\neq 0) \) are the roots of the quadratic equation \( x^{2} + \alpha x - \beta = 0 \), then the quadratic expression \( -x^{2} + \alpha x + \beta \), where \( x \in R \), has
A.
least value \( -\frac{1}{4} \)
B.
least value \( -\frac{9}{4} \)
C.
greatest value \( \frac{1}{4} \)
D.
greatest value \( \frac{9}{4} \)
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ANSWER :
D. greatest value \( \frac{9}{4} \)
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