2 2018 Nov/Dec WASSCE Elective Maths Paper 2
Section A
48 marks
Answer all the questions in this section. All questions carry equal marks.
1. Two independent events K and L are such that \(p(K) = x, p(L) = (x + \frac15)\) and \(p(K \cap L) = \frac{3}{20}\). Find the value of \(x\).
2. Seven participants in an art contest were ranked by two judges as follows:
| Participant | A | B | C | D | E | F | G |
|---|---|---|---|---|---|---|---|
| 1st Judge | 3 | 4 | 1 | 6 | 5 | 7 | 2 |
| 2nd Judge | 3 | 6 | 2 | 5 | 7 | 4 | 1 |
- Calculate, correct to three decimal places, the Spearman’s rank correlation coefficient for the scores of the judges.
- Comment on your results.
3. \(\mathbf{F_1}(3 \text{N}, 030^\circ), \mathbf{F_2}(4 \text{N}, 090^\circ), \mathbf{F_3}( 6 \text{N}, 135^\circ)\) and \(\mathbf{F_4}(7 \text{N}, 240^\circ)\) act on a particle. Find, correct to two decimal places.
4. A uniform pole, \(\mathbf{PQ}\), 30 m long and of mass 4 kg is carried by a boy at \(P\) and a man 8 m away from \(Q\). Find the distance from \(P\) where a mass of 20 kg should be attached so that the man’s support is twice that of the boy, if the system is in equilibrium. [Take \(g = 10 \text{ ms}^{-2}\)]
5. Solve \(3x^{\frac12} + 5 - 2x^{-\frac12} = 0\).
6. A point P divides the straight line joining X(1, -2) and Y(5, 3) internally in a ratio 2:3. Find the
(a) coordinates of P.
(b) equation of the straight line that passes through N(3, -5) and P.
7. (a) Find the sum of the series 32 + 8 + 2 + …,
(b) Simplify: \(\displaystyle \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}} - \frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}}\).
8. Without using mathematical tables or calculator, find, in surd form (radicals), the value of \(\tan 22.5^\circ\).
Section B
52 marks
Answer four questions only from this section with at least one question from each part.
All questions carry equal marks.
Part I
Pure Mathematics
9. (a) Find the range of values of \(x\) for which \(2x^2 \geq 9x + 5\).
(b) (i) Write down in ascending powers of \(x\) the binomial expansion of \((2 + x)^6 - (2 - x)^6\).
(ii) Using the result in (b)(i), evaluate \((2.01)^6 - (1.99)^6\), correct to four decimal places.
10. A circle \(x^2 + y^2 - 2x - 4y - 5 = 0\) with centre O is cut by a line \(y = 2x + 5\) at points P and Q. Show that \(\overline{QO}\) is perpendicular to \(\overline{PO}\).
11. (a) Given that \(\pmatrix{ 3 & -5 \\ 4 & 2 }\), find:
(i) \(M^{-1}\), inverse of \(M\).
(ii) the image of \((1, -1)\) under \(M^{-1}\).
(b) Two linear transformations P and Q, are defined by \(P: (x, y) \to (5x + 3y, 6x + 4y)\) and \(Q:(x, y) \to (4x - 3y, -6x + 5y)\),
(i) Write down the matrices \(P\) and \(Q\).
(ii) Find the matrix \(R\) defined by \(R = PQ\).
(iii) Deduce \(Q^{-1}\), the inverse of \(Q\).
Part II
Statistics and Probability
12. A box contains 5 blue, 7 green and 4 red identical balls. Three balls are picked from teh box one after the other without replacement. Find, the probability of picking:
(a) two green balls and a blue ball.
(b) no blue ball.
(c) at least one green ball.
(d) three balls of the same colour.
13. The ages, \(x\) (in years), of a group of 18 adults have the following statistics, \(\sum x = 745\) and \(\sum x^2 = 33951\).
(a) Calculate the:
(i) mean age,
(ii) standard deviation of the ages of the adults, correct to two decimal places.
(b) One person leaves teh group and the mean age of the remaining 17 is 41 years. Find the
(i) age of the person who left;
(ii) standard deviation of the remaining 17 adults, correct to two decimal places.
Part III
Vectors and Mechanics
14.Three forces \(0\mathbf{i} - 63\mathbf{j}, 32.14\mathbf{i} + 38.3\mathbf{j}\) and \(14 \mathbf{i} - 24.25\mathbf{j}\) act on a body of mass 5 kg. Find, correct to the nearest whole number, the:
(a) magnitude of the resultant force;
(b) direction of teh resultant force;
(c) acceleration of the body.
15. Two particles P and Q move towards each other along a straight line \(MN\), 51 metres long. P starts from \(M\) with velocity \(5\ \text{ms}^{-1}\) and constant acceleration of \(1\ \text{ms}^{-2}\). Q starts from \(N\) at the same time with velocity \(6\ \text{ms}^{-1}\) and a constant acceleration of \(3\ \text{ms}^{-2}\). Find the time when the:
(a) particles are 30 metres apart;
(b) particles meet;
(c) velocity of P is \(\frac34\) of the velocity of Q.