Explanation

a. Let the number of sides of polygon Q be x

therefore, No. of sides of polygon P = 2x

then, \(\frac{360}{x} - \frac{360}{2x}\) = 45,

\(\frac{360}{x} - \frac{180}{x}\) = 45

L. C. M = x

360 - 180 = 45x

x = \(\frac{180}{4}\) = 4

therefore, the number of P polygon = 2x = 2 x 4 = 8

b. Area of semi-circle = \(\frac{\pi r^2}{2}\)

but Area of the semi-circle = 32π

\(\frac{\pi r^2}{2}\) = 32π

\(r^2 = 32\times2 = 64\)

\(r^2 = \sqrt{64}\) = 8cm.

Circumference of semi-circle = πr + 2r

= π(8) + 2(8)

= 8π + 16

= 8(π + 2 ) cm

Correct Answer: A. 4.7m²

Explanation

Area of floor = 3.5m x 2.2m = 7.7m²

Area of carpet = (3.5 - 0.5 - 0.5)m x (2.2 - 0.5 - 0.5)m

2.5m x 1.2m = 3m²

hence, area of margine

= area of floor - area of carpet

= (7.7 - 3)m²

= 4.7m²

Explanation

(a) \(\sqrt{\frac{0.18 \times 12.5}{0.05 \times 0.2}}\)

\(\frac{0.18 \times 12.5}{0.05 \times 0.2} = \frac{18 \times 10^{-2} \times 125 \times 10^{-1}}{5 \times 10^{-2} \times 2 \times 10^{-1}}\)

= \(9 \times 25 \times 10^{-3 - (-3)}\)

= \(9 \times 25\)

= \(225\)

\(\therefore \sqrt{\frac{0.18 \times 12.5}{0.05 \times 0.2}} = \sqrt{225}\)

= \(15\).

(b) \(\frac{8 - 4\sqrt{18}}{\sqrt{50}}\)

\(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\)

\(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\)

\(\frac{8 - 4(3\sqrt{2})}{5\sqrt{2}} = \frac{8 - 12\sqrt{2}}{5\sqrt{2}}\)

Rationalising, we have

= \(\frac{8 - 12\sqrt{2}}{5\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\)

= \(\frac{8\sqrt{2} - 24}{10}\)

= \(0.8\sqrt{2} - 2.4\)

= \(-2.4 + 0.8\sqrt{2}\)

(c) \(x \propto \frac{y^{3}}{z^{2}}\)

\(\implies x = \frac{ky^{3}}{z^{2}}\)

\(108 = \frac{k \times 3^{3}}{4^{2}}\)

\(k = \frac{108 \times 16}{27}\)

\(k = 64\)

\(\therefore x = \frac{64y^{3}}{z^{2}}\)

\(\therefore 4000 = \frac{64 \times 10^{3}}{z^{2}}\)

\(4 = \frac{64}{z^{2}}\)

\(z^{2} = \frac{64}{4} = 16\)

\(z = \sqrt{16} = \pm 4\)

Correct Answer: B. 90°

Explanation

\(a . b = |a||b|\cos \theta\)

\(\cos \theta = \frac{a . b}{|a||b|}\)

= \( \frac{(5i + 3j).(3i - 5j)}{(\sqrt{5^2 + 3^2})(\sqrt{3^{2} + (-5)^{2}})}\)

= \(\frac{0}{34} = 0\)

\(\theta = \cos^{-1} 0 = 90°\)

Correct Answer: A. 1.43

Explanation

(a) Cost price of a radio = N6,000.00

\(\therefore\) 30% profit = \(\N6000 \times \frac{30}{100} = N1800\)

\(\therefore\) Marked price = N(6000 + 1800) = N7800

Final Selling price = \(\frac{100 - 10}{100} \times N7800 = N7020\)

Profit = N(7020 - 6000) = N1020.

\(\therefore\) % profit = \(\frac{1020}{6000} \times 100% = 17%\)

(b) \(2^{(2x + 1)} - 9(2^{x}) + 4 = 0\)

\((2^{x})^{2} \times 2 - 9(2^{x}) + 4 = 0\)

Let \(2^{x} = y\)

\(2y^{2} - 9y + 4 = 0\)

\(2y^{2} - 8y - y + 4 = 0\)

\(2y(y - 4) - 1(y - 4) = 0\)

\((2y - 1)(y - 4) = 0\)

\(2y = 1 \implies y = \frac{1}{2}; y = 4\)

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

\(x = -1\)

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

\(x = 2\)

\(\therefore x = -1, 2\)

Correct Answer: B. 42

Correct Answer: C. 15

Explanation

The first 6 questions can only be selected in 1 way.

The remaining 4 questions can be selected in \(^{6}C_{4}\) ways.

= \(\frac{6!}{(6 - 4)! 4!} = 15\)

Explanation

\(\int^3_1(3x - 2)^5 dx\)

Let u = 3x - 2

\(\frac{du}{dx}\) = 3

dx = \(\frac{dy}{3}\)

\(\int^3_1(3x - 2)^5 dx\) = \(\int^7_1 \frac{u^5}{3} du\)

= \(\frac{u^{5 + 1}}{3(5 + 1)}|^7_1\)

= \(\frac{u^5}{18}|^7_1\)

= \(\frac{7^6}{18} = \frac{1^6}{18}\)

= \(\frac{7^6 - 1}{18}\)

= \(\frac{117649}{18}\)

= 6536