(a) Two fair die are thrown once. Find the probabitlity of getting : (i) the same digit ; (ii) a total score greater than 5.
(b) Given that \(x = \cos 30°\) and \(y = \sin 30°\), evaluate without using a mathematical table or calculator : \(\frac{x^{2} + y^{2}}{y^{2} – x^{2}}\).
Explanation
(a) Sample - space
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | 1, 1 | 1, 2 | 1, 3 | 1, 4 | 1, 5 | 1, 6 |
| 2 | 2, 1 | 2, 2 | 2, 3 | 2, 4 | 2, 5 | 2, 6 |
| 3 | 3, 1 | 3, 2 | 3, 3 | 3, 4 | 3, 5 | 3, 6 |
| 4 | 4, 1 | 4, 2 | 4, 3 | 4, 4 | 4, 5 | 4, 6 |
| 5 | 5, 1 | 5, 2 | 5, 3 | 5, 4 | 5, 5 | 5, 6 |
| 6 | 6, 1 | 6, 2 | 6, 3 | 6, 4 | 6, 5 | 6, 6 |
(i) Same digit = {(1,1), (2,2),(3,3), (4,4),(5,5), (6,6)}
P(same digit) = \(\frac{6}{36}\)
(ii) P(total score greater than 5) = \(\frac{26}{36}\)
= \(\frac{13}{18}\)
(b) \(\frac{x^{2} + y^{2}}{y^{2} - x^{2}} = \frac{(\cos 30)^{2} + (\sin 30)^{2}}{(\sin 30)^{2} - (\cos 30)^{2}}\)
= \(\frac{(\frac{\sqrt{3}}{2})^{2} + (\frac{1}{2})^{2}}{(\frac{1}{2})^{2} - (\frac{\sqrt{3}}{2})^{2}}\)
= \(\frac{\frac{3}{4} + \frac{1}{4}}{\frac{1}{4} - \frac{3}{4}}\)
= \(\frac{1}{-\frac{1}{2}}\)
= \(-2\).