(a) (i) (3 - x), 6, (7 - 5x). First term = a = (3 - x) common ratio = \(\frac{6}{3 - x} = \frac{7 - 5x}{6}\) \(36 = (3 - x)(7 - 5x)\) \(36 = 21 - 15x - 7x + 5x^{2}\) \(36 = 21 - 22x + 5x^{2}\) \(5x^{2} - 22x + 21 - 36 = 0\) \(5x^{2} - 22x - 15 = 0\) \(5x^{2} - 25x + 3x - 15 = 0 \implies 5x(x - 5) + 3(x - 5) = 0\) \((x - 5)(5x + 3) = 0\) \(x = -\frac{3}{5} \text{ or = } 5\) (ii) Constant ratio = \(\frac{6}{3 - x} = \frac{7 - 5x}{6}\) Using x = 5, \(r = \frac{7 - 5(5)}{6} = \frac{-18}{6}\) = \(-3\) (b) Considering \(\Delta\) AOC in the above diagram. \(b^{2} = 4^{2} + 3^{2}\) \(b^{2} = 16 + 9 = 25\) \(b = \sqrt{25} = 5 cm\) Considering \(\Delta\) ACD, Using cosine rule, \(\cos D = \frac{a^{2} + b^{2} - c^{2}}{2ab} = \frac{a^{2} + c^{2} - b^{2}}{2ac}\) = \(\frac{6^{2} + 7^{2} - 5^{2}}{2 (6)(7)}\) = \(\frac{36 + 49 - 25}{84}\) \(\cos D = \frac{60}{84}\) \(\cos D = 0.714\) \(D = \cos^{-1} (0.714)\) \(D = 44.4^\circ \approxeq 44^\circ\) (to the nearest degree)