(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 \(x = \cos 30^\circ\) and \(y = \sin 30^\circ\), evaluate without using a mathematical table or calculator : \(\frac{x^{2} + y^{2}}{y^{2} - x^{2}}\).
Verified Correct Answer: .
Step-by-Step Mathematical Solution
(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) \(\frac{x^{2} + y^{2}}{y^{2} - x^{2}} = \frac{(\cos 30)^{2} + (\sin 30)^{2}}{(\sin 30)^{2} - (\cos 30)^{2}}\)
= \(\frac{(\frac{\sqrt{3}}{2})^{2} + (\frac{1}{2})^{2}}{(\frac{1}{2})^{2} - (\frac{\sqrt{3}}{2})^{2}}\)
= \(\frac{\frac{3}{4} + \frac{1}{4}}{\frac{1}{4} - \frac{3}{4}}\)
= \(\frac{1}{-\frac{1}{2}}\)
= \(-2\).
Source: Myschool Past Questions Archive (Record #53072)