Correct Answer: B. 52

Explanation

22 35 41 63 74 82

Median = \(\frac{41 + 63}{2} = \frac{104}{2}\)

= 52

Correct Answer: C. \(\frac{\sqrt{3}}{2}\)

Explanation

\(a ♦ b = \frac{ab}{4}\)

\(\sqrt{2} ♦ \sqrt{6} = \frac{\sqrt{2} \times \sqrt{6}}{4} = \frac{\sqrt{12}}{4}\)

= \(\frac{2\sqrt{3}}{4} \)

= \(\frac{\sqrt{3}}{2}\)

Correct Answer: D. (13, 157.38°)

Explanation

\(p = (5i - 12j); |p| = \sqrt{5^{2} + (-12)^{2}}\)

= \(\sqrt{169} = 13\)

\(\tan\theta = \frac{-12}{5} = -2.4 \implies \theta = -67.38°\)

Direction = \(90° - (-67.38°) = 157.38°\)

Correct Answer: D. x < -4

Explanation

\(\frac{x+2}{4}-\frac{x+1}{3}>\frac{1}{2}\\

\frac{3(x+2)-4(x+1)>6}{12}; 3x + 6

Explanation

\(\log_{2} x - \log_{2} y = 2 \implies \log_{2} (\frac{x}{y}) = 2 \)

\(\frac{x}{y} = 2^{2} = 4 \implies x = 4y ... (1)\)

\(\log_{2} (x - 2y) = 3 \implies x - 2y = 2^{3} = 8 ... (2)\)

Putting (1) into (2),

\(4y - 2y = 8 \implies 2y = 8\)

\(y = 4\)

\(x = 4y \implies x = 4(4) = 16\)

\(x = 16 ; y = 4\)

Correct Answer: A. l = -12, k = -6

Correct Answer: A. 35/3

Explanation

Tn = a + (n-1)d

10th term = 32

32 = 3/2 + 9d

9d = 32 - 3/2

9d = 61/2

d = 61/18

: T4 = 3/2 + 3(61/18)

T4 = 3/2 + 61/6 → 35/3

Correct Answer: A. 6.0cm

Explanation

Volume of water in tank = l x b x h = 180 litres

but 1 litre = 1000cm²

250 x 120 x hw = 180 x 1000

hw = \(\frac{180 \times 1000}{250 \times 120}\)

= 6cm

Explanation

Position vector of A is (8i - 10j)

Position vector of B is (2i + 6j)

Position vector of C is (-10i + 4j).

Let position vector of N be (ai + bj).

ABCN is a parallelogram.

\(\therefore \overrightarrow{AB} = \overrightarrow{NC}\).

\(\begin{bmatrix} (2 - 8)i \\ (6 + 10)j \end{bmatrix} = \begin{bmatrix} (-10 - a)i \\ (4 - b)j \end{bmatrix}\)

Equating components,

\(2 - 8 = -10 - a \implies a = -4\)

\(6 + 10 = 4 - b \implies b = -12\)

The position vector of N is (-4i - 12j).

(b) \(\overrightarrow{AN} \cdot \overrightarrow{AB}\)

\(\overrightarrow{AN} = \bar{A} - \bar{N} \)

= \((8i - 10j) - (-4i - 12j)\)

= \(12i + 2j\)

\(\overrightarrow{AB} = \bar{A} - \bar{B}\)

= \((8i - 10j) - (2i + 6j)\)

= \(6i - 16j\)

\(\therefore \overrightarrow{AN} \cdot \overrightarrow{AB} = (12i + 2j) \cdot (6i - 16j)\)

= \(72 - 32 = 40\)

(c) Let \(\theta\) be the acute angle between \(\overrightarrow{AN}\) and \(\overrightarrow{AB}\).

\(\overrightarrow{AN} \cdot \overrightarrow{AB} = |AN||AB| \cos \theta\)

\(|AN| = |12i + 2j| = \sqrt{12^{2} + 2^{2}} = \sqrt{144 + 4} = \sqrt{148}\)

\(|AB| = |6i - 16j| = \sqrt{6^{2} + (-16)^{2}} = \sqrt{36 + 256} = \sqrt{292}\)

\(\therefore 40 = (\sqrt{148})(\sqrt{292}) \cos \theta\)

\(\cos \theta = \frac{40}{(\sqrt{148})(\sqrt{292})}\)

\(\cos \theta = \frac{40}{207.88} = 0.1924\)

\(\theta = \cos^{-1} (0.1924) = 78.907°\)

\(\approxeq 78.9°\)

Correct Answer: C. 10m

Explanation

H\(^2\) = 8\(^2\) + 6\(^2\) + 6\(^2\)

H\(^2\) = 64 + 36 = 100

H = \(\sqrt{100}\) = 10m