The equation of a circle is given as x² + y² + 4x -2y - 4 = 0. Determine the centre and the diameter of the circle.      (3mks)

Find the value of x given that log(x -1) + 2 = log(3x + 2) + log 25.            (3mks)

Two matrices P and Q are such that P = and Q = . Given that the determinant of QP is 44, find the value of k.   (3mks)

Without using a mathematical table or a calculator, evaluate leaving your answer in the form.  , where a, b and c are constants. (3mks)

The roots of the quadratic equation x² + px = q are x = -³/₇  and  x = 2. Find the values of p and q. (3mks)

Two variables x and y are related by the law y = -2 + bxⁿ where b and n are constants. The table below shows the variations between x and y

a) Write down the function y = -2 + bxⁿ in the linear form            (1 mark)

b) On the grid provided draw a suitable line graph to represent the relation y = -2 + bxⁿ .     (5 marks)

c) Find the values of b and n                    (3 marks)

d) Write a relationship connecting y and x (1 mark)

The table below shows some values of the function y = x² + 3

a) Complete the table above                (2marks)

b) Use the mid-ordinate rule with six strips to estimate the area bounded by the curve y = x² + 3, the y-axis, the x-axis and the line x = 6             (3marks)

c) Find the exact area in (b) above.         (3marks)

d) Calculate the percentage error in the approximated area from the exact area. (2 marks)

In the figure below OP = p, OQ = q, PQ = QR and OQ: QS = 3:1

a) Determine:-

 i. PQ                        (1 mark)

 ii. RS in terms of p and q                       (2 marks)

b) If RS:ST = 1:n and OP:PT = 1:m, determine:

i. ST in terms of p,q and m                 (1 mark)

ii. The values of m and n                       (4 marks)

iii. Show that R,S and T are collinear (2 marks)

From a watch tower M on a hill, N is 5km on a bearing of 078⁰ and a railway station 9km away on a bearing of 200⁰ .

a) Using a scale 1:100000, draw the relative positions of M, N and P. (4 marks)

b) Find;

i. The bearing of N from the railway station.     (1 mark)

ii. The distance between P and N    (2 marks)

iii. The shortest distance between M and the line PN. (3 marks)

In the figure below A,B,C and D are points on the circumference of the circle centre O. Line TDF is a tangent to the circle at D and BA produced meets the tangent at T. <ATD =38⁰ and <BAC = 28⁰.

Giving reasons in each case , find the size of:

(a) <AOD                                                               (2mks)

(b) <BDC                                                               (2mks)

(c) <ACB                                                               (2mks)

(d) <FDC                                                               (2mks)

(e) <ATD                                                               (2mks)