Correct Answer: C. 1\(\frac{1}{3}\)

Explanation

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

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

= \(\frac{\sqrt{6} + 3}{3}\)

= \(\frac{3 + \sqrt{6}}{3}\)

= Hence, (m + n) = 1 + \(\frac{1}{3}\)

= 1\(\frac{1}{3}\)

Correct Answer: A. 4

Explanation

\(\frac{1}{4}\)p + 3p = 10...(1)

2p - \(\frac{1}{3}\)q = 7...(2)

Multiply equation (2) by 3 to clear fraction

3 x 2p - 3 x \(\frac{1}{3}\)q = 3 x 7

6p - q = 21

6p - 21 = q....(3)

substituting 6p - 21 for q in (1)

\(\frac{1}{4}\)p + 3(6p - 21) = 10...(4)

Multiply equation (4) by 4 to clear fraction

4 x \(\frac{1}{4}\)p + 4 x 3(6p - 21) = 4 x 10

p + 12(6p - 21) = 40

p + 72p - 252

73p = 292

p = \(\frac{292}{73}\)

= 4

Correct Answer: A. 35°

Explanation

In the diagram, QOR + 2m(vertically opposite angles)

So, m + 90° + 2m = 180°

(angles on str. line)

3m = 180° - 90°

3m = 90°

m = \(\frac{90^o}{3}\)

= 30°

substituting 30° for m in

2m + 4n = 200° gives

2 x 30° + 4n = 200°

60° + 4n = 200°

4n = 200° - 60°

= 140°

n = \(\frac{140°}{4}\)

= 35°

Correct Answer: C. 81°

Explanation

\(\overrightarrow{x} . \overrightarrow{y} = |\overrightarrow{x}||\overrightarrow{y}|\cos\theta\)

\(\overrightarrow{x} . \overrightarrow{y} = (i - 3j) . (6i + j) = 6 - 3 = 3\)

\(|\overrightarrow{x}| = \sqrt{1^{2} + (-3)^{2}} = \sqrt{10}\)

\(|\overrightarrow{y}| = \sqrt{6^{2} + 1^{2}} = \sqrt{37}\)

\(\therefore 3 = (\sqrt{10})(\sqrt{37})\cos \theta\)

\(\cos\theta = \frac{3}{\sqrt{370}} = 0.1559\)

\(\theta = \cos^{-1} 0.1559 \approxeq 81°\)

Correct Answer: B. \(\frac{2}{5}\)

Explanation

Let x represent the entire farmland

then, \(\frac{2}{5}\)x + \(\frac{1}{3}\)[x - \(\frac{2}{3}x\)] + M = x

Where M represents the part of the farmland used for growing maize, continuing

\(\frac{2}{5}\)x + \(\frac{1}{3}\)x [1 - \(\frac{2}{3}x\)] + M = x

\(\frac{2}{5}x + \frac{1}{3}\)x [\(\frac{3}{5}\)] + M = x

\(\frac{2}{5}\)x + \(\frac{1x}{5}\) + M = x

\(\frac{3x}{5} + M = x\)

M = x - \(\frac{2}{5}\)x

= x[1 - \(\frac{3}{5}\)]

= x[\(\frac{2}{5}\)] = \(\frac{2x}{5}\)

Hence the part of the land used for growing maize is

\(\frac{2}{5}\)

Correct Answer: C. 12

Explanation

If logx 64 = 3, then \(64 = x^3\)

\(4^3 = x^3\)

Since the indices are equal,

x = 4

Hence, x log\(_2\)8

= 4(3.log\(_2\)2)

= where log\(_2\)2 = 1(1)

= 12 X 1

= 12

Correct Answer: A. K = \(\frac{H(T^2 - 1)}{T^2 - T}\)

Explanation

T = \(\sqrt{\frac{TK - H}{K - H}}\)

Taking the square of both sides, give

T² = \(\frac{TK - H}{K - H}\)

T²(K - H) = TK - H

T²K - T²H = TK - H

T²K - TK = T²H - H

K(T² - T) = H(T² - 1)

K = \(\frac{H(T^2 - 1)}{T^2 - T}\)

Correct Answer: B. 12

Explanation

The total number of candidates = 40, therefore, the median occurs as the average between the 20th and 21st candidates. = \(\frac{3+3}{2} = 3\)

Number of candidates that scored above 3 : 9 + 3 = 12.

Correct Answer: C. 14m

Explanation

\(a = \frac{v - u}{t} = \frac{9 - 5}{2} = 2 ms^{-2}\)

\(s = ut + \frac{1}{2}at^{2} = 5(2) + \frac{1}{2}(2 \times 2^{2})\)

= \(14m\)

Explanation

Spearman's rank correlation coefficient = \(1 - \frac{6 \sum d^{2}}{n(n^{2} - 1)}\)

= \(1 - \frac{6(10)}{5(5^{2} - 1)}\)

= \(1 - \frac{60}{120}\)

= \(0.5\)