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V = \(\pi r^2 h\) = 123.2 x 1000 = \(\frac{22}{7}\) x 28 x 28 x h = 123200 = 88 x 28h = 2464h ∴ h = \(\frac{123200}{2464}\) = 50cm N.B: 1 litre = 1 dm3 = 1000cm3

Taking moment about P, 2 x d = 0.5 x (50 - 30) 2x = 10 x = 5 cm

The horizontal (eastward) component of the 10N force is = 10 cos60 → 10 x 0.5 = 5N. The equilibrant force needed to stop the body moving eastward must act in exactly the opposite direction. Therefore, the needed force is 5N in the direction of West

In optics, parabolic mirrors are used to gather light in reflecting telescopes and solar furnaces, and project a beam of light in flashlights, searchlights, stage spotlights, and car headlights.

When a solid is heated, the particles are given more energy and start to vibrate faster. At a certain temperature, the particles vibrate so much that their ordered structure breaks down. At this point the solid melts into liquid.

Zinc oxide doesn't react to hydrogen gas because the reaction is not spontaneous.

\(1011_2 + X_7 = 25_{10} = 1011_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0\) = 8 + 0 + 2 + 1 = \(11_{10}\) \(X_7 = 25_{10} - 11_{10}\) = \(14_{10}\) \(\begin{array}{c|c} 7 & 14 \\ 7 & 2 R 0 \\ & 0 R 2 \end{array}\) X = \(20_7\)

[\(\frac{1}{0.03}\) + \(\frac{1}{0.024}\)] = [\(\frac{1}{0.03 \times 0.024}\)]-1 = [\(\frac{0.024}{0.003}\)]-1 = \(\frac{0.03}{0.024}\) = \(\frac{30}{24}\) = 1.25

\(c^{\frac{1}{3}}\) = \(a^{\frac{1}{2}}\)b b = \(a^{-2}\) and \(c^{\frac{1}{3}}\) = \(a^{\frac{1}{2}}\)b b = \(a^{-\frac{2}{3}}\) \(c^{\frac{1}{3}}\) = \(a^{\frac{1}{2}}\) x \(a^{-\frac{2}{3}}\) = \(a^{-\frac{1}{6}}\) take the cube of both sides c = (\(a^{-\frac{1}{6}})^3\) = \(a^{-\frac{1}{2}}\)

log4(y - 1) + log4(\(\frac{1}{2}\)x) = 1 log4(y - 1)(\(\frac{1}{2}\)x) \(\to\) (y - 1)(\(\frac{1}{2}\)x) = 4 ........(1) log2(y + 1) + log2x = 2 log2(y + 1)x = 2 \(\to\) (y + 1)x = 22 = 4.....(ii) From equation (ii) x = \(\frac{4}{y + 1}\)........(iii) put equation (iii) in (i) = y (y - 1)[\(\frac{1}{2}(\frac{4}{y - 1}\))] = 4 = 2y - 2 = 4y + 4 2y = -6 y = -3 x = \(\frac{4}{-3 + 1}\) = \(\frac{4}{-2}\) X = 2 therefore x = -2, y = -3

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

V\(\pi\)r2h = \(\pi\)(3)2(10) = 90\(\pi\)cm3 V = \(\pi\)(5)2 x 18 = 450\(\pi\)cm3 No of volume = \(\frac{450\pi}{90\pi}\) = 5 selling price = 5 x N15 = N75 profit = N75 - N50 = N25.00

The cost of normal work = 35r The cost of overtime = q x 2r = 2qr The man's total weekly earning = 35r + 2qr = r(35 + 2q)

\(\sqrt{x}\) - \(\sqrt{(x - 2)}\) - 1 = 0 = \(\sqrt{x}\) - \(\sqrt{(x - 2)}\) = 1 = (\(\sqrt{x}\) - \(\sqrt{(x - 2)}\))2 = 1 = x - 2 \(\sqrt{x(x - 2)}\) + x -2 = 1 = (2x - 3)2 = [2 \(\sqrt{x(x - 4)}\)]2 = 4x2 - 12x + 9 = 4(x2 - 2x) = 4x2 - 12x + 9 = 4x2 - 8x 4x = 9 x = \(\frac{9}{4}\)

\(\frac{x + a}{x - a}\) = m x + a = mx - ma a + ma = mx - x a(m + 1) = x(m - 1) \(\frac{a}{x}\) = \(\frac{m - 1}{m + 1}\)