(a) Total number of fruits = 16 limes + 14 apples = 30 (i) P(both are good limes) = \(\frac{10}{30} \times \frac{9}{29} = \frac{3}{29}\) (ii) P(both are bad fruits) = \(\frac{12}{30} \times \frac{11}{29} = \frac{22}{145}\) (iii) P(one is a good apple and the other a bad lime) = \((\frac{8}{30} \times \frac{6}{29}) + (\frac{6}{30} \times \frac{8}{29})\) = \(\frac{8}{145} + \frac{8}{145}\) = \(\frac{16}{145}\) Note : The arrangement is not specific. Hence, it can be a good apple first and then a bad lime or a bad lime first and then a good apple. (b) \(\log_{3} (4x + 1) - \log_{3} (3x - 5) = 2\) \(\log_{3} (\frac{4x + 1}{3x - 5}) = 2\) \((\frac{4x + 1}{3x - 5}) = 3^{2}\) \(\frac{4x + 1}{3x - 5} = 9 \implies 4x + 1 = 9(3x - 5)\) \(4x + 1 = 27x - 45 \implies 23x = 46\) \(x = 2\)