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9(a)(i) p = \(\frac{1}{4}\), p = 1 - \(\frac{1}{4}\) p = \(\frac{3}{4}\) = \(\frac{3}{64}\) = 0.04687 ≈ 0.05 (ii) p(winning all) = \(\frac{3}{4}\) * \(\frac{3}{4}\) * \(\frac{3}{4}\) = \(\frac{27}{64}\) = 0.422 (iii) p(winning two) = (\(\frac{3}{4}\) * \(\frac{3}{4}\) * \(\frac{1}{4}\)) + (\(\frac{3}{4}\) * \(\frac{1}{4}\) * \(\frac{3}{4}\)) + (\(\frac{1}{4}\) * \(\frac{3}{4}\) * \(\frac{3}{4}\)) = \(\frac{9}{64}\) + \(\frac{9}{64}\) + \(\frac{9}{64}\) = \(\frac{27}{64}\) = 0.4218 = 0.42 (b) Volume of a cone = 1/3\pir\(^2\)h 1200 = \(\frac{1}{3}\) * \(\frac{22}{7}\) * r\(^2\) * 24 r\(^2\) = \(\frac{1200 * 3 * 7}{22 * 24}\) r\(^2\) = \(\frac{25200}{528}\) r\(^2\) = 47.727 r = \(\sqrt 47.727\) r = 6.9085cm Volume of a cone with same radius and height 84cm: v = \(\frac{1}{3}\) * \(\frac{22}{7}\) * (6.9085)\(^2\) * 84 = \(\frac{1}{3}\) * \(\frac{22}{7}\) * 47.727 * 84 volume = 4200cm\(^3\)

209 = 2 * \(\frac{22}{7}\) * \((\frac{7}{2})^2\) + 2 * \(\frac{22}{7}\) * \(\frac{7}{2}\) * h 209 = 77 + 22h 132 = 22h h = 6cm (b) tan 43º = \(\frac{h}{y}\) 0.9125y = h ...[i] tan 38º = \(\frac{h}{y+19}\) 0.7813(y+19) = h ...[ii] 0.7813(y+19) = 0.9125y 0.7813y + 14.8447 = 0.93257 14.8447 = 0.1512y y= 98.1792 = 98.2m h = 0.9325 x 98.1792 = 91.55 =91.6m

Average speed = \(\frac{total Distance}{Total Time}\) from Calabar to Enugu in time t1, hence t1 = \(\frac{P}{U}\) Also from Enugu to Benin t2 \(\frac{q}{w}\) Av. speed = \(\frac{p + q}{t_1 + t_2}\) = \(\frac{p + q}{p/u + q/w}\) = p + q x \(\frac{uw}{pw + qu}\) = \(\frac{uw(p + q)}{pw + qu}\)

Using the trial expansion of each option (m\(^2\) + 2) (m - 1) OR Group the terms (\(m^3 - m^2\)) + ( 2m - 2) \(m^2\)(m - 1) + 2(m - 1) (m-1)(\(m^2\) + 2)

81a\(^4\) - 16b\(^4\) = (9a\(^2\))\(^2\) - (4b\(^2\))\(^2\) = (9a\(^2\) + 4b\(^2\))(9a\(^2\) - 4b\(^2\)) N:B 9a\(^2\) - 4b\(^2\) = (3a - 2b)(3a + 2b)

Let the three items be M, Y and P. n{M ∩ Y} only = 4-3 = 1 n{M ∩ P) only = 5-3 = 2 n{ Y ∩ P} only = 2 n{M} only = 12-(1+3+2) = 6 n{Y} only = 10-(1+2+3) = 4 n{P} only = 14-(2+3+2) = 7 n{M∩P∩Y} = 3 Number of women in the group = 6+4+7+(1+2+2+3) as above =25 women.

Given (x + 2) and (x - 1), i.e. x = -2 or +1 when x = -2 L(-2) + 2k(-2)\(^2\) + 24 = 0 f(-2) = -2L + 8k = -24...(i) And x = 1 L(1) + 2k(1) + 24 = 0 f(1):L + 2k = -24...(ii) Subst, L = -24 - 2k in eqn (i) -2(-24 - 2k) + 8k = -24 +48 + 4k + 8k = -24 12k = -24 - 48 = -72 k = \(frac{-72}{12}\) k = -6 where L = -24 - 2k L = -24 - 2(-6) L = -24 + 12 L = -12 That is; K = -6 and L = -12

Using elimination method: 4x+2y=16 * 6 --> 24x +12y=96 ... eqn iii 6x-2y=4 * 4 --> 24x - 8y = 16 ... eqn iv Subtract eqn iv from iii 20y = 80 y = 4 Subst. y for 4 in 4x + 2y = 16 --> 4x + 2(4) = 16 --> 4x = 16 - 8 --> 4x = 8 --> x = 2 : The value of (y-x) is 4 - 2 = 2

A for Adults C for children a + c = 500 c = 500 - a 4.5a + 3c = 1987.50 where c = 500 - a 4.5a + 3(500 - a) = 1987.50 4.5a - 3a = 1987.50 - 1500 1.5a = 487.5 a = 487.5 / 1.5 a = 325 tickets c = 500 - a c = 500 - 325 c = 175 tickets Check: from 4.5a + 3c 4.5 (325) = 1,462.5 + 3(175) = 525

A man will be (x+10) years old in 8years time. As at today, he is x + 2 years of age. The man was 63 years old 2 years ago, so he is 63+2=65 now. 8 years from now, he will be 65+8=73. He will be (x+10) years old when he is 73. So x+10=73 x=73-10=63

A rational expression is undefined when the denominator is equal to zero. when x = -1 The denominator in this equation : x + 1 --> -1 + 1 = 0 This expression is undefined when x = -1

PQ\(^2\) = (x2 - x1)\(^2\) + (y2 - y1)\(^2\) = 12\(^2\) + 5\(^2\) = 144 + 25 = 169 PQ = \sqrt169 = 13 But PQ = diameter = 2r, r = PQ/2 = 6.5 units

Vertically opp angles are equal ∠CDE = 70^\circ Sum of interior angles in a quadilateral = 360^\circ (90 + 90 + 70 + t)^\circ = 360^\circ t^\circ = 360^\circ - 250^\circ t^\circ = 110^\circ

tan\theta = \(frac{3}{4}\) → tan\theta = 0.75 \theta = tan\(^{-1}\)[0.75] → 36.8698^\circ cos\theta = cos[36.8698^\circ] → 0.800 or \(frac{4}{5}\) However; in the third quadrant Cosine is negative i.e -\(frac{4}{5}\)

Isosceles Triangle. Two equal sides. Two equal angles ; ∠ X = ∠ Z ∠Y + ∠X + Z = 180 --> 40^\circ + ∠X + ∠X = 180^\circ ∠2X = 180^\circ - 40^\circ = 140^\circ ∠X = 70^\circ, i.e ∠X = 70 and ∠Z = 70^\circ. The exterior angle of triangle is the sum of two opposite internal angles : ∠X + ∠Y = ∠XZT =70^\circ + 40^\circ = 110^\circ