# 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**.