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## 5 theory further maths ss3 mock questions

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MOCK EXAMINATION, 2010/2011 SESSION.
SUBJECT: FURTHER MATHEMATICS 2
CLASS: SSS 3   	         	        TIME ALLOWED: 2½HRS

SECTION A
[48 marks]										       Answer all the questions in this section								            All questions carry equal marks
1. A binary operation  is defined on the set R, of real numbers by , where x, y R.               (a) Evaluate (2 3)  5.								                                        (b) If (x + 1)  (x + 2) =  find the value of x.
2. If the first of the first n terms of the series 4 + 7 + 10 + ... is 209, find n.
3. Find the equation of the tangent to the curve y = x2 – 3x + 4 at the point where the tangent makes an angle of 135° with the positive x – axes.
4. Find the equation whose roots are the squares of the roots of the equation x2 – mx + n = 0
5. A committee of four persons is to be formed from 7 girls and 5 boys. Calculate, correct to two decimal places, the probability that the committee will consist of:                                                                                 (a) 2 boys and 2 girls;                                                                                                                                               (b) at least 3 boys.
6. The table below gives the distribution of heights, in metres, of 100 students of the same age group.
Height
1.40 -1.42
1.43-1.45
1.46-1.48
1.49-1.51
1.52-1.54
1.55-1.57
1.58-1.60
1.61-1.63

Frequency
2
4
19
30
24
14
6
1


(a) Calculate the mean height of the distribution.                                                                                                     (b) What is the probability that the height of a student selected at random is greater than the mean height of the distribution?

7. Vectors p, q and r are given by  and . Find:                                                     (a) 4p – 2q + 5r;                                                                                                                                                  (b) the position vector which divides p and q in the ratio 2:3.
8. A body of mass 6kg is hung from a fixed point by a light inextensible string. A horizontal force is applied   to the body such that the body is in equilibrium when the string is inclined at 35° to the vertical. Find, correct to one decimal place, the:                                                                               (a) horizontal force;                                                                                                                                            (b) tension in the string.                                           [Take g = 10ms-1

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 1
PURE MATHEMATICS.
9. Points A(2, 1) and B(4, -5) lie on a circle. If line 2x – y – 13 = 0 is a tangent to the circle at B, find the;                                                                                                                                                       (a) coordinates of the centre of the circle;                                                                                                         (b) equation of the circle.
10. (a) If x2 + y2 = py (1 + x2), where p is a constant, find .                                                                                                (b) A curve cuts the x- axis at the points where x = 0 and x = 3. If find the equation of the curve.
11.   (a) Find the area enclosed by the x – axis and curve y = 3x2 + 2x – 1.                                                     (b) If 2x2 – x – 3 = P(x + Q)2  + R where P, Q and R are constants, find the:                                                                   (i) values of P, Q and R;                                                                                                                                    (ii) minimum value of 2x2 – x – 3.
12. (a) Given that   m > 0, determine the value of m.                                                     (b) If (x – y)2 = 3xy + 1, find the gradient at point (1, 0).
‘

PART II  (STATISTICS AND PROBABILITY
13. The table below shows the marks obtained by 40 students in a test.
Marks
1-10
11-20
21-30
31-40
41-50
51-60
61-70
71-80
81-90
91-100

Frequency
2
4
4
6
8
6
5
2
2
1


Using an assumed mean of 45.5, calculate, correct to two decimal places, the:                                    (i) mean;      (ii) variance  of the distribution.
If a student is selected at random from the distribution, find the probability that he obtained at most 40.
14. (a) A box P contains 3 white and 5 green identical balls. Another box Q contains 6 white and 4 green identical balls.  A ball is drawn at random from P and dropped into Q. A ball is then drawn at random from Q. Find the probability that the ball drawn from Q is green.
(b) A fair die is thrown five times. Find, correct to three decimal places, the probability of obtaining at least two sixies.
(c) If find x.
15. (a) A team of 3 students is to be selected from 5 boys and 4 girls to represent a school in a quiz.             (i) If there is no restriction, in how many ways can the team be selected?                                                       (ii) Calculate correct to three decimal places, the probability that there will be at least two girls in the team.
(b) The probability that a player will score a penalty kick is 0.8. If he takes 5 penalty kicks, calculate, correct to four decimal places, the probability that he will score:                                                                                  (i) no goal;                      (ii) at most two goals.

PART III (VECTORS AND MECHANICS)
16. The position vectors of two points P and Q are    p= 5i + 3j  and q = 4i +7j . Find:                                    (a) |6p – 5q|;          (b) the scalars m and n such that mp + nq = 14i + 13j;                                                          (c) the acute anglw between p and q.

17. (a) A particle Q is acted upon by three coplanar forces 40N, 15N and 20N, as shown in   the below diagram.

40N
15N

20N
Find the force required to prevent Q from moving.
Abody of mass 5kg at rest is acted upon by forces F1 = (10N, 090°), F2 = (20N, 210°) and F3 = (4N, 330°). Find correct to one decimal place, the magnitude and direction of the resultant force.
18. (a) Two forces (3i + 5j) and ( - 2i + 3j) act on a body of mass 2kg and cause it to move. Find, correct to two decimal places, the magnitude of the:                                                                                                          (i) resultant force acting on the body;                                                                                                                       (ii) acceleration of the body;                                                                                                                             (iii) change in velocity, if the forces acted on the body for 5 seconds.
(b) A unifrom beam of length 100cm and mass 1.5kg is placed on a pivot which is 20cm from one end. A vertical force T, is applied upwards 5cm from the other end to keep the beam in equilibrium.
Caculate the:     (i) magnitude of T;       (ii) reeaction at the pivot    [Take g = 10ms-2]

```

### 5 theory further maths ss3 mock guide

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