1)Let f (x) = x^2

and g(x) = 3ln (x + 1), for x > 1.

(a) Solve f (x) = g(x).

(b) Find the area of the region enclosed by the graphs of f and g.

2) (a) Find the term in x^6

in the expansion of (x + 2)^9

(b) Hence, find the term in x^7

in the expansion of 5x(x + 2)^9

3) In a geometric sequence, the fourth term is 8 times the first term. The sum of the

first 10 terms is 2557.5. Find the 10th term of this sequence.

4)

3 6 The points A and B lie on a line L, and have position vectors 2 and 4 respectively. 2 1 Let 0 be the origin. This is shown on the following diagram. diagram not to scale L B A 9 (a) Find KB. [2] The point C also lies on L. such that KC = 2613. 3 (b) Show that CC = 2 . [3] 0 Let 9 be the angle between EB and 6C. (c) Find 9. [5] Let D be a point such that DD = kOC. where k > 1 . Let E be a point on L such that CED is a right angle. This is shown on the following diagram. diagram not to scale ) DE > (d) (i) Show that =(k1) 0C sin. (ii) The distance from D to line L is less than 3 units. Find the possible values of k. [6] A particle P moves along a straight line so that its velocity. v ms\". after 1' seconds. is given by v = cos 3: 25in: 0.5. for 0 s t 5 5. The initial displacement of P from a xed point 0 is 4 metres. (a) Find the displacement of P from 0 after 5 seconds. The following sketch shows the graph of v. (b) Find when P is rst at rest. (c) Write down the number of times P changes direction. (d) Find the acceleration of P after 3 seconds. (e) Find the maximum speed of P. Note: One decade is 10 years A population of rare birds. R, can be modelled by the equation I} = 3,6\" . where R, is the initial population, and t is measured in decades. After one decade. it is estimated that i = 0.9. 1% (a) (i) Find the value of 1:. (ii) Interpret the meaning of the value of k. P (b) Find the least number of whole years for which 3'