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If log 7.5 = 0.8751 Then 2log75 + log750 = 2(1.8751) + 2.8751 = 3.7502 + 2.8751 = 6.6253

8x-2 = 2/25 x-2 = 2/25 x 1/8 x-2 = 2/200 x-2 = 1/100 1/x2 = 1/100 x2 = 100 x = 10

\(= \frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}\) \(= \frac{2\sqrt{2}(\sqrt{2}) + (2\sqrt{2})(-\sqrt{3})-\sqrt{3}(\sqrt{2})-\sqrt{3}(-\sqrt{3})}{(\sqrt{2})^2-(\sqrt{3})^2}\) \(= \frac{2 \times 2 - 2\sqrt{6} - \sqrt{6} + 3}{2 - 3}\) \(= \frac{4 - 3\sqrt{6} + 3}{-1}\) \(= \frac{7 - 3\sqrt{6}}{-1}\) \(= \frac{7}{-1} - \frac{3\sqrt{6}}{-1}\) \(= -7 + 3\sqrt{6}\) \(= 3\sqrt{6}-7\)

\( = log_2 \frac{8 \times 16}{4}\) \( = log_2 32\) \( = log_2 2^5\) \( = 5log_2 2\) \( = 5 \times 1\) \( = 5 \)

Q = (P \(\cup\) Q) - P {6,7}

gt2 - k - w = 0 gt2 = k + w \(g = \frac{k + w}{t^2}\)

2y2 - 15xy + 18x2 2y2 - 12xy - 3xy + 18x2 2y(y - 6x) - 3x(y - 6x) (2y - 3x) (y - 6x)

if y - 1 is a factor of y3 + 4y2 + ky - 6, then f(1) = (1)3 + 4(1)2 + k(1) - 6 = 0 (factor theorem) 1 + 4 + k - 6 = 0 5 - 6 + k = 0 -1 + k = 0 k = 1

\(y \propto w^2\) y = kw2 8 = k(2)2 8 = k(4) k = 8/4 k = 2 Thus y = 2w2 When w = 3, y = 2(3)2 y = 2 x 9 = 18

\(P \propto \frac{Q}{R}\) \(P = K \frac{Q}{R}\) When Q = 36, R = 16, P = 27 Then substitute into the equation \(27 = K \frac{36}{16}\) \(K = \frac{27 \times 16}{36}\) \(K = 12\) So the equation connecting P, Q and R is \(P = \frac{12Q}{R}\)

Consider the range -3 < x < -1 = { -2, -1, 0}, for instance When x = -2, \(\frac{-2 - 5}{-2 + 3} < -1\) \(\frac{-7}{1} < -1\) When x = -1, \(\frac{-1 - 5}{-1 + 3} < -1\) \(\frac{-6}{2} < -1\) = -3 < -1 When x = 0, \(\frac{0 - 5}{0 + 3} < -1\) \(\frac{- 5}{3} < -1\) Hence -3 < x < 1

\(\frac{x}{2} + \frac{3}{4} \leq \frac{5x}{6} - \frac{7}{12}\) \(12\frac{x}{2} + 12\frac{3}{4} \leq 12\frac{5x}{6} - 12\frac{7}{12}\) 6x + 9 \(\leq\) 10x - 7 6x - 10x \(\leq\) - 7 - 9 -4x \(\leq\) -16 -4x/-4 \(\geq\) -16/-4 x \(\geq\) 4

a + 3d = 13 .......... (1) a + 9d = 31 .......... (2) (2) - (1): 6d = 18 d = 18/6 = 3 From (1), a + 3(3) = 13 a + 9 = 13 a = 13 - 9 = 4 Hence, T24 = a + 23d T24 = 4 + 23(3) T24 = 4 + 69 T24 = 73

Common ratio r of the G.P is \(r = \frac{T_n + 1}{T_n} = \frac{T_2}{T_1}\) \(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}}\) \(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}} \times \frac{\sqrt{10} - \sqrt{5}}{\sqrt{10} - \sqrt{5}} \) \( = \frac{(\sqrt{10})(\sqrt{10}) + (\sqrt{10})(-\sqrt{5}) + (2\sqrt{5})(\sqrt{10}) + (2\sqrt{5})(-\sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}\) \(\frac{10 - \sqrt{50} + 2\sqrt{50} - 10}{10 - 5}\) \(\frac{\sqrt{50}}{5}\) \(\frac{\sqrt{25 \times 2}}{5}\) \(\frac{5\sqrt{2}}{5}\) \(\sqrt{2}\)

x * y = xy x * 2 = 12 - x Thus by comparison, x = x, y = 2 But x * y = x * 2 xy = 12 - x x2 = 12 - x x2 + x - 12 = 0 x2 + 4x - 3x - 12 = 0 x(x + 4) - 3(x + 4) = 0 (x - 3)(x + 4) = 0 x - 3 = 0 or x + 4 = 0 So x = 3 or x = -4