Correct Answer: A. 11cm

Explanation

volume of rectangular container = L x B x H

1 litre = 1000cm3

96 litres = \(\frac{1000cm^3}{1 litres}\) x 96 litres

volume = 96000cm3

96000 = 220 x 40

96--- = 8800H

H = \(\frac{96000}{8800}\)

= 10.97cm

= 11cm

Explanation

(a) \(\frac{\frac{1}{2} of \frac{1}{4} \div \frac{1}{3}}{\frac{1}{6} - \frac{3}{4} + \frac{1}{2}}\)

Numerator - \(\frac{1}{2} \times \frac{1}{4} \div \frac{1}{3} = (\frac{1}{2} \times \frac{1}{4}) \div \frac{1}{3}\)

\(\frac{1}{8} \times 3 = \frac{3}{8}\)

Denominator - \(\frac{1}{6} - \frac{3}{4} + \frac{1}{2} = \frac{1}{6} - (\frac{3}{4} - \frac{1}{2})\)

\(\frac{1}{6} - (\frac{3 - 2}{4}) = \frac{1}{6} - \frac{1}{4}\)

\(\frac{2 - 3}{12} = \frac{-1}{12}\)

\(\therefore \frac{\frac{1}{2} \times \frac{1}{4} \div \frac{1}{3}}{\frac{1}{6} - \frac{3}{4} + \frac{1}{2}} = \frac{3}{8} \div \frac{-1}{12}\)

\(\frac{3}{8} \times -12 = \frac{-9}{2}\)

= \(-4.5\)

(b) \(\sqrt{x} = 10^{\bar{1}.6741}}\)

Squaring both sides, we have

\(x = (10^{\bar{1}.6741}})^{2}\)

\(x = 10^{\bar{1}.3482}\) (checking antilogarithm)

\(x = 0.2229\)

Correct Answer: D. 10

Explanation

let the number of question = n

n(n + 50) = 600

n² + 50n - 600 - 0 by quadratic formular

x = \(\frac{b \pm \sqrt{b^2 - 4ac}}{2}\)

n = \(\frac{-50 \pm \sqrt{50^2 - 4(1) (-600)}}{2(1)}\)

= \(\frac{-50 \pm \sqrt{4900}}{2}\)

= \(\frac{-50 + 70}{2}\)

= \(\frac{20}{2}\)

= 10

Correct Answer: A. \(\frac{-3}{4}\)

Explanation

\(9^{2x + 1} = 81^{3x + 2}\)

\(9^{2x + 1} = (9^{2})^{3x + 2}\)

\(9^{2x + 1} = 9^{6x + 4}\)

Equating powers,

\(2x + 1 = 6x + 4 \implies -3 = 4x\)

\(\therefore x = \frac{-3}{4}\)

Correct Answer: C. 128°

Explanation

x + 52^o = 90

x = 90 - 52

x = 38^o

k = opposite angle Z

k = 38^o

y + k = 90^o

y + 38^o = 90^o

y = 90^o - 38^o

y = 52^o

y = x = 180^o(sum of angles on straight line)

52 + x = 180^o

x = 180 - 52

x = 128

Correct Answer: C. (n + q)

Explanation

mn - nq - n\(^2\)

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

Explanation

27^x = 9^y

3^3x = 3^2y

3x = 2y

Correct Answer: B. \(x^{2} + y^{2} + 6x + 16y - 23 = 0\)

Explanation

Equation of a circle: \((x - a)^{2} + (y - b)^{2} = r^{2}\)

where (a, b) and r are the coordinates of the centre and radius respectively.

Given : \((a, b) = (-3, -8); r = 4\sqrt{6}\)

\((x - (-3))^{2} + (y - (-8))^{2} = (4\sqrt{6})^{2}\)

\(x^{2} + 6x + 9 + y^{2} + 16y + 64 = 96\)

\(x^{2} + y^{2} + 6x + 16y + 9 + 64 - 96 = 0\)

\(\implies x^{2} + y^{2} + 6x + 16y - 23 = 0\)

Correct Answer: D. \(10 ms^{-2}\)

Explanation

\(v(t) = (3t^{2} - 2t) ms^{-1}\)

\(a(t) = \frac{\mathrm d v}{\mathrm d t} = (6t - 2) ms^{-2}\)

\(a(2) = 6(2) - 2 = 12 - 2 = 10 ms^{-2}\)

Explanation

(a) The formula for the first principles method of calculating derivatives is

\(\frac{\mathrm d y}{\mathrm d x} = \lim \limits_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}\)

\(f(x) = (2x + 3)^{2} = 4x^{2} + 12x + 9\)

\(f(x + \Delta x) = 4(x + \Delta x)^{2} + 12(x + \Delta x) + 9\)

= \(4x^{2} + 8x \Delta x + 4 (\Delta x)^{2} + 12x + 12\Delta x + 9\)

\(\lim \limits_{\Delta x \to 0} \frac{(4x^{2} + 8x \Delta x + 4(\Delta x)^{2} + 12x + 12\Delta x + 9) - (4x^{2} + 12x + 9)}{\Delta x}\)

= \(\lim \limits_{\Delta x \to 0} \frac{4(\Delta x)^{2} + (8x + 12)\Delta x}{\Delta x}\)

= \(\lim \limits_{\Delta x \to 0} 4 \Delta x + (8x + 12)\)

\(\frac{\mathrm d y}{\mathrm d x} = 8x + 12\)

(b) \(\int_{1} ^{2} \frac{(x + 1)(x^{2} - 2x + 2)}{x^{2}} \mathrm {d} x\)

= \(\int_{1} ^{2} \frac{x^{3} - 2x^{2} +2x + x^{2} - 2x + 2}{x^{2}} \mathrm {d} x\)

= \(\int_{1} ^{2} \frac{x^{3} - x^{2} + 2}{x^{2}} \mathrm {d} x\)

= \(\int_{1} ^{2} (x - 1 + 2x^{2}) \mathrm {d} x\)

= \([\frac{x^{2}}{2} - x - \frac{2}{x}]_{1} ^{2} \)

= \((\frac{2^{2}}{2} - 2 - \frac{2}{2}) - (\frac{1^{2}}{2} - 1 - \frac{2}{1})\)

= \((2 - 2 - 1) - (\frac{1}{2} - 1 - \frac{2}{1})\)

= \(-1 - (-\frac{5}{2}) = \frac{3}{2}\)