NDA & NA Exam (I) & (II)- 2020 - Mathematics Defence Question Paper

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NDA & NA Exam (I) & (II)- 2020 - Mathematics

NDA & NA Exam (I) & (II)- 2020 - Mathematics Defence Questions

11.
Let \( S = \{1, 2, 3, \ldots\} \). A relation R on \( S \times S \) is defined by xRy if \( \log_{a} x > \log_{a} y \) when \( a = \frac{1}{2} \). Then the relation is
A.
reflexive only
B.
symmetric only
C.
transitive only
D.
both symmetric and transitive
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ANSWER :
C. transitive only
12.
What is the value of the determinant \( \begin{vmatrix} i & i^{2} & i^{3} \\ i^{4} & i^{6} & i^{8} \\ i^{9} & i^{12} & i^{15} \end{vmatrix} \), where \( i = \sqrt{-1} \)?
A.
0
B.
−2
C.
\( 4i \)
D.
\( -4i \)
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ANSWER :
D. \( -4i \)
13.
Let \( A = \begin{bmatrix} a & h & g \\ h & b & f \\ g & f & c \end{bmatrix} \) and \( B = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \), then what is AB equal to?
A.
\( \begin{bmatrix} ax + hy + gz \\ y \\ z \end{bmatrix} \)
B.
\( \begin{bmatrix} ax + hy + gz \\ hx + by + fz \\ z \end{bmatrix} \)
C.
\( \begin{bmatrix} ax + hy + gz \\ hx + by + fz \\ gx + fy + cz \end{bmatrix} \)
D.
\( \begin{bmatrix} ax + hy + gz & hx + by + fz & gx + fy + cz \end{bmatrix} \)
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ANSWER :
C. \( \begin{bmatrix} ax + hy + gz \\ hx + by + fz \\ gx + fy + cz \end{bmatrix} \)
14.
What is the number of ways in which the letters of the word ‘ABLE’ can be arranged so that the vowels occupy even places?
A.
2
B.
4
C.
6
D.
8
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ANSWER :
B. 4
15.
What is the maximum number of points of intersection of 5 non-overlapping circles?
A.
10
B.
15
C.
20
D.
25
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ANSWER :
C. 20
16.

Directions for the following three (03) items:
Consider the following Venn diagram, where X, Y and Z are three sets. Let the number of elements in Z be denoted by n(Z) which is equal to 90.


If the number of elements in Y and Z are in the ratio 4 : 5, then what is the value of b?

A.

18

B.

19

C.

21

D.

23

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ANSWER :

C. 21

17.
What is the value of \( n(X) + n(Y) + n(Z) - n(X \cap Y) - n(Y \cap Z) - n(X \cap Z) + n(X \cap Y \cap Z) \)?
A.
\( a + b + 43 \)
B.
\( a + b + 63 \)
C.
\( a + b + 96 \)
D.
\( a + b + 106 \)
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ANSWER :
D. \( a + b + 106 \)
18.
If the number of elements belonging to neither X, nor Y, nor Z is equal to p, then what is the number of elements in the complement of X?
A.
\( p + b + 60 \)
B.
\( p + b + 40 \)
C.
\( p + a + 60 \)
D.
\( p + a + 40 \)
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ANSWER :
A. \( p + b + 60 \)
19.
Directions for the following two (02) items:
Read the following information and answer the two items that follow:
Let \( \frac{\tan 3A}{\tan A} = K \), where \( \tan A \neq 0 \) and \( K \neq \frac{1}{3} \).

What is \( \tan^{2} A \) equal to?
A.
\( \frac{K + 3}{3K - 1} \)
B.
\( \frac{K - 3}{3K - 1} \)
C.
\( \frac{3K - 3}{K - 3} \)
D.
\( \frac{K + 3}{3K + 1} \)
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ANSWER :
B. \( \frac{K - 3}{3K - 1} \)
20.
For real values of \( \tan A \), K cannot lie between
A.
\( \frac{1}{3} \) and 3
B.
\( \frac{1}{4} \) and 4
C.
\( \frac{1}{5} \) and 5
D.
\( \frac{1}{7} \) and 7
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ANSWER :
A. \( \frac{1}{3} \) and 3
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