2002 Pure Maths Paper 2

Formulas For Reference

Section A (40 marks)

  1. Find the indefinite integral dx.

Hence evaluate the improper integral dx. (6 marks)

  1. Let f(x)

If f is continuous at , show that .

Furthermore, if f is differentiable at , find the values of a and b. (5 marks)

  1. Let f : R→R be a continuous function satisfying the following conditions:

(i) ;

(ii) f(x + y) = f(x)f(y) for all x and y.

(a)Prove that exists and for every R.

(b)By considering the derivative of , show that .

(5 marks)

  1. The equation of a straight linelin R³ is

, R.

Let A and B be two points on l with OA = OB = r, where O is the origin.

(a)Express the distance between A and B in terms of r.

(b)If is an equilateral triangle, find the value of r.

(5 marks)

  1. (a) If the function g : R→R is both even and odd, show that g(x) = 0 for all

R.

(b) For any function f : R→R, define and .

(i) Show that F is an even function and G is an odd function.

(ii) If for all R, where Mis even and Nis odd, show that M(x) = F(x) and N(x) = G(x) for all R.

(6 marks)

  1. (a) Find.

(b) Let

Find .

(6 marks)

  1. Let be the curve with polar equation , .

(a)Find the polar coordinates of all the points on that are farthest from

the pole O.

(b) Sketch the curve .

(c) Find the area enclosed by .

(7 marks)

Section B (60 marks)

  1. Let .

(a) Find and .

(2 marks)

(b) Determine the range of values of x for each of the following cases:

(i)> 0,

(ii)< 0,

(iii)> 0,

(iv)< 0.

(3 marks)

(c) Find the relative extreme point(s) and point(s) of inflexion of f(x).

(2 marks)

(d) Find the asymptote(s) of the graph of f(x).

(1 mark)

(e) Sketch the graph of f(x).

(2 marks)

(f) Let .

(i)Is g(x) differentiable at x = 0? Why?

(ii)Sketch the graph of g(x).

(5 marks)

  1. (a) Find .

(3 marks)

(b) Let f : R→be a periodic function with period T.

(i) Prove that for any positive

integer k.

(ii) Let .

Prove that for any positive integer n.

(iii) If l is a positive number and n is a positive integer such that

l, prove that

Hence find the improper integral in terms of and T.

(9 marks)

(c) Using the results of (a) and (b)(iii), evaluate .

(3 marks)

  1. Let f and g be continuous functions defined on such that f is decreasing and for all .

For , define and .

(a)(i) Prove that . Hence prove that for all .

(ii) Evaluate and hence prove that .

(7 marks)

(b) Let for all .

(i) Prove that G(1) + H(1) = 1.

(ii) Using (a)(ii), prove that .

(5 marks)

(c) Using the results of (a)(ii) and b(ii), prove that , where n is a positive integer.

Hence show that .

(3 marks)

  1. Consider the parabolas and .

Let be a point on . The two tangents drawn from P to touch at the points and .

(a)Find the equations of PS and PT and hence show that , .

(4 marks)

(b) is a point on the arc ST of . Prove that the area of is a maximum if and only if .

(6 marks)

(c) Let Q be the point in (b) where the area of is a maximum. If the straight line PQ cuts the chord ST at M, find the equation of the locus of M as P moves along .

(5 marks)

  1. (a) Let g(x) be a function continuous on , differentiable in ,

with decreasing on and g(a) = g(b) = 0. Using Mean

Value Theorem, show that there exists such that g is

increasing on and decreasing on .

Hence show that for all .

(5 marks)

(b) Let f be a twice differentiable function and on an open interval I.

Supposea, b, I with .

By considering the function or otherwise, show that .

Hence, or otherwise, prove that for all I, where with .

(5 marks)

(c) Let and be positive numbers.

(i) If with , prove that .

(ii) If , are positive numbers, prove that .

(5 marks)

  1. (a) (i) Let , where n is a non-negative integer and

.

Show that for all .

(ii) Using the substitution , or otherwise, show that for any

positive integer n.

(5 marks)

(b) (i) Let and n be a positive integer. Prove that .

(ii) Using (a) or otherwise, show that.

(iii) Suppose that . Evaluate and , and show that .

Hence prove that .

(10 marks)