NDA 2019 (II), Paper I, Mathematics question paper

1. A.
   12
2. B.
   10
3. C.
   7
4. D.
   6

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1. A.
   51
2. B.
   55
3. C.
   110
4. D.
   111

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1. A.
   $A \cup (A \cap B) = A \cup B$
2. B.
   $A \cap (A \cup B) = A$
3. C.
   $(A \cap B) \cup C = (A \cup C) \cap (B \cup C)$
4. D.
   $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$

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1. A.
   1
2. B.
   2
3. C.
   3
4. D.
   4

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1. A.
   -1
2. B.
   1
3. C.
   $\frac{3}{2}$
4. D.
   2

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1. A.
   $a$, $b$ and $c$ are in AP
2. B.
   $a$, $b$ and $c$ are in GP
3. C.
   $a$, $b$ and $c$ are in HP
4. D.
   $a$, $b$ and $c$ do not follow any regular pattern

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1. A.
   $x \leq 1$ or $x \geq 2$
2. B.
   $1 \leq x \leq 2$
3. C.
   $1 < x < 2$
4. D.
   $x$ is any real value except 3 and 4

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1. A.
   $m/n$
2. B.
   $n/m$
3. C.
   $m + (n/m)$
4. D.
   $n + (m/n)$

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1. A.
   3
2. B.
   6
3. C.
   8
4. D.
   9

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1. A.
   48
2. B.
   40
3. C.
   28
4. D.
   20

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1. A.
   0
2. B.
   12
3. C.
   24
4. D.
   36

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1. A.
   $-2 \pm \sqrt{3}$
2. B.
   $-1 \pm \sqrt{3}$
3. C.
   $-1 \pm \sqrt{6}$
4. D.
   $-2 \pm \sqrt{6}$

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1. A.
   0
2. B.
   $(a+b+c)^{2}$
3. C.
   $a^{2}+b^{2}+c^{2}$
4. D.
   $a+b+c-2$

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1. A.
   a null matrix
2. B.
   an identity matrix
3. C.
   equal to $A$
4. D.
   equal to $-A$

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1. A.
   29
2. B.
   30
3. C.
   32
4. D.
   35

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1. A.
   $2\sqrt{3} \leq m^{2} < 8$
2. B.
   $\sqrt{3} \leq m^{2} < 4$
3. C.
   $m^{2} \geq 8$
4. D.
   $m^{2} \leq \sqrt{3}$

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1. A.
   1
2. B.
   -1
3. C.
   $2i$
4. D.
   $-2i$

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1. A.
   8
2. B.
   6
3. C.
   4
4. D.
   2

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1. A.
   10%
2. B.
   15%
3. C.
   20%
4. D.
   25%

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1. A.
   $\varnothing$
2. B.
   $\{1\}$
3. C.
   $\{2\}$
4. D.
   $\{1, 2\}$

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1. A.
   $6i-3$
2. B.
   $-6i+3$
3. C.
   $-6i-3$
4. D.
   $6i+3$

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1. A.
   $\sqrt{2}-1$
2. B.
   $\sqrt{2}+1$
3. C.
   3
4. D.
   4

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1. A.
   7
2. B.
   14
3. C.
   28
4. D.
   56

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1. A.
   12
2. B.
   20
3. C.
   21
4. D.
   22

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1. A.
   10
2. B.
   8
3. C.
   5
4. D.
   4

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1. A.
   Both $AB$ and $BA$ exist
2. B.
   Neither $AB$ nor $BA$ exists
3. C.
   $AB$ exists but $BA$ does not exist
4. D.
   $AB$ does not exist but $BA$ exists

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1. A.
   95
2. B.
   85
3. C.
   80
4. D.
   No such value of $n$ exists

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1. A.
   $A = \{x \mid x > a\}$ and $B = \{x \mid x > b\}$
2. B.
   $A = \{x \mid x < a\}$ and $B = \{x \mid x > b\}$
3. C.
   $A = \{x \mid x < a\}$ and $B = \{x \mid x < b\}$
4. D.
   $A = \{x \mid x > a\}$ and $B = \{x \mid x < b\}$

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1. A.
   $\pm 2$
2. B.
   $\pm 3$
3. C.
   $\pm 5$
4. D.
   $\pm 9$

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1. A.
   $C(47, 4)$
2. B.
   $C(52, 5)$
3. C.
   $C(52, 4)$
4. D.
   $C(47, 5)$

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1. A.
   1 and 2 only
2. B.
   2 and 3 only
3. C.
   1 and 3 only
4. D.
   1, 2 and 3

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1. A.
   21
2. B.
   22
3. C.
   24
4. D.
   25

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1. A.
   5
2. B.
   6
3. C.
   7
4. D.
   9

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1. A.
   $q^{2} = pr$
2. B.
   $r^{2} = pq$
3. C.
   $pqr = 1$
4. D.
   $2q = p + r$

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1. A.
   $x \geq -4,\, y \leq 0$
2. B.
   $x \leq 4,\, y \geq 0$
3. C.
   $x \leq -4,\, y = 0$
4. D.
   $x \geq -4,\, y = 0$

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1. A.
   $x \in (0, \infty)$
2. B.
   $x \in \left(\frac{1}{7}, 7\right)$
3. C.
   $x \in \left(0, \frac{1}{7}\right) \cup (7, \infty)$
4. D.
   $x \in \left(\frac{1}{7}, \infty\right)$

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1. A.
   Zero
2. B.
   One
3. C.
   Two
4. D.
   Four

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1. A.
   1 only
2. B.
   2 only
3. C.
   Both 1 and 2
4. D.
   Neither 1 nor 2

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1. A.
   10
2. B.
   20
3. C.
   512
4. D.
   1024

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1. A.
   1848
2. B.
   2048
3. C.
   2842
4. D.
   2872

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1. A.
   5
2. B.
   10
3. C.
   15
4. D.
   20

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1. A.
   1 only
2. B.
   2 only
3. C.
   Both 1 and 2
4. D.
   Neither 1 nor 2

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1. A.
   9 m
2. B.
   10·5 m
3. C.
   13·5 m
4. D.
   15 m

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1. A.
   $\sin\left(\dfrac{\theta}{2}\right)$
2. B.
   $\cos\left(\dfrac{\theta}{2}\right)$
3. C.
   $2\sin\left(\dfrac{\theta}{2}\right)$
4. D.
   $2\cos\left(\dfrac{\theta}{2}\right)$

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1. A.
   $\dfrac{2}{3}$
2. B.
   $\dfrac{3}{4}$
3. C.
   $\dfrac{3}{8}$
4. D.
   $\dfrac{1}{9}$

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1. A.
   $\dfrac{\sqrt{6}}{9}$
2. B.
   $\dfrac{19}{9}$
3. C.
   $\dfrac{9}{19}$
4. D.
   $\dfrac{\sqrt{6}}{19}$

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1. A.
   The vectors are parallel
2. B.
   The vectors are perpendicular
3. C.
   The vectors are anti-parallel
4. D.
   The vectors must be unit vectors

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1. A.
   1, 2 and 3
2. B.
   1 and 2 only
3. C.
   1 and 3 only
4. D.
   2 and 3 only

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1. A.
   1 only
2. B.
   2 only
3. C.
   Both 1 and 2
4. D.
   Neither 1 nor 2

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1. A.
   $\dfrac{\left|\vec{a}+\vec{b}\right|^{2}}{4}$
2. B.
   $\dfrac{\left|\vec{a}-\vec{b}\right|^{2}}{4}$
3. C.
   $\dfrac{\left|\vec{a}+\vec{b}\right|^{2}}{2}$
4. D.
   $\dfrac{\left|\vec{a}-\vec{b}\right|^{2}}{2}$

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1. A.
   for all real numbers $a$, $b$ and $c$
2. B.
   only when $a eq 0$
3. C.
   only when $b eq 0$
4. D.
   only when at least one of $a$ and $b$ is non-zero

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1. A.
   $\beta - \alpha$
2. B.
   $\pi + \beta - \alpha$
3. C.
   $\dfrac{\pi + 2\beta + 2\alpha}{2}$
4. D.
   $\dfrac{\pi - 2\beta + 2\alpha}{2}$

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1. A.
   $|2m\sin(\alpha-\beta)|$
2. B.
   $|2m\cos(\alpha-\beta)|$
3. C.
   $|m\sin(2\alpha-2\beta)|$
4. D.
   $|m\sin(2\alpha-2\beta)|$

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1. A.
   $\left(-\sqrt{2}, \sqrt{2}\right)$
2. B.
   $\left(\sqrt{2}, -\sqrt{2}\right)$
3. C.
   (1, 1)
4. D.
   (1, -1)

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1. A.
   $\dfrac{1}{2}$
2. B.
   $\dfrac{1}{\sqrt{2}}$
3. C.
   $\dfrac{\sqrt{3}}{2}$
4. D.
   $\dfrac{1}{2\sqrt{2}}$

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1. A.
   $\dfrac{1}{2},\, \dfrac{1}{2},\, \dfrac{1}{\sqrt{2}}$
2. B.
   $\dfrac{1}{\sqrt{2}},\, \dfrac{1}{2},\, \dfrac{1}{2}$
3. C.
   $\dfrac{1}{\sqrt{2}},\, \dfrac{1}{2},\, -\dfrac{1}{2}$
4. D.
   Cannot be determined due to insufficient data

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1. A.
   (3, 5, 4)
2. B.
   (2, 5, 5)
3. C.
   (-1, -1, 5)
4. D.
   (2, -1, 0)

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1. A.
   2
2. B.
   3
3. C.
   5
4. D.
   7

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1. A.
   90°
2. B.
   60°
3. C.
   45°
4. D.
   30°

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1. A.
   1, -1
2. B.
   -1, 1
3. C.
   0, 2
4. D.
   3, 4

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1. A.
   $(a+b)^{2}$
2. B.
   $(a-b)^{2}$
3. C.
   $a^{2}+b^{2}$
4. D.
   $|a^{2}+b^{2}|$

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1. A.
   30°
2. B.
   45°
3. C.
   60°
4. D.
   75°

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1. A.
   $\dfrac{1}{x} + \dfrac{1}{y}$
2. B.
   $\dfrac{1}{y} - \dfrac{1}{x}$
3. C.
   $\dfrac{xy}{x+y}$
4. D.
   $1 + \dfrac{1}{xy}$

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1. A.
   0
2. B.
   $2\sin\alpha$
3. C.
   $2\sin\beta$
4. D.
   $\sin\alpha + \sin\beta$

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1. A.
   $\dfrac{1}{5}$
2. B.
   $\dfrac{1}{6}$
3. C.
   $\dfrac{1}{7}$
4. D.
   $\dfrac{1}{9}$

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1. A.
   2
2. B.
   1
3. C.
   0
4. D.
   -19

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1. A.
   $\dfrac{a^{2}-b^{2}}{ab}$
2. B.
   $\dfrac{a^{2}}{bc}$
3. C.
   $\dfrac{b^{2}}{ca}$
4. D.
   $\dfrac{c^{2}}{ab}$

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1. A.
   $\tan A$
2. B.
   $\cot A$
3. C.
   $2\tan A$
4. D.
   $2\cot A$

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1. A.
   $\tan\left(\dfrac{A}{2}\right)$
2. B.
   $\cot\left(\dfrac{A}{2}\right)$
3. C.
   $2\tan\left(\dfrac{A}{2}\right)$
4. D.
   $2\cot\left(\dfrac{A}{2}\right)$

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