(a) Given that \(3 \times 9^{1 + x} = 27^{-x}\), find x.
(b) Evaluate \(\log_{10} \sqrt{35} + \log_{10} \sqrt{2} - \log_{10} \sqrt{7}\)
(a) Evaluate, without using mathematical tables, \(17.57^{2} - 12.43^{2}\).
(b) Prove that angles in the same segment of a circle are equal.
(a) A tower and a building stand on the same horizontal level. From the point P at the bottom of the building, the angle of elevation of the top, T of the tower is 65°. From the top Q of the building, the angle of elevation of the point T is 25°. If the building is 20m high, calculate the distance PT.
(b) Hence or otherwise, calculate the height of the tower. [Give your answers correct to 3 significant figures].
(a) Divide \(11111111_{two}\) by \(101_{two}\)
(b) A sector of radius 6 cm has an angle of 105° at the centre. Calculate its:
(i) perimeter ; (ii) area . [Take \(\pi = \frac{22}{7}\)]
The table below gives the frequency distribution of the marks obtained by some students in a scholarship examination.
| Scores (x) | 15 | 25 | 35 | 45 | 55 | 65 | 75 |
| Freq (f) | 1 | 4 | 12 | 24 | 18 | 8 | 3 |
(a) Calculate, correct to 3 significant figures, the mean mark.
(b) Find the : (i) mode ; (ii) range of the distribution.
