Paper 1 [60 marks]

Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. When an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.SECTION AAnswer all questions in the boxes provided. Working may be continued below the lines if necessary.

1. [Maximum mark: 4]

Given that and (where )Find

(a)

[2 marks](b)

[2 marks]

2. [Maximum mark: 8]Solve the equations:(a)

[4 marks]

(b) [4 marks]

3.

[Maximum mark: 8]

Consider the function g, where .(a) Given that the domain of g is , find the least value of a such that g has an inverse function.

[1 mark](b) On the same set of axes, sketch

(i) the graph of g for this value of a;

(ii) the corresponding inverse, .

[4 marks]

(c) Find an expression for

[3 marks]

4.[Maximum mark: 10] (a) Prove with induction that

.

[6 marks](b) A geometric series has first term 1.2 and common ratio 0.5, how many terms

of the series need to be added before the sum exceeds 2.399? [4 marks]5.

[Maximum mark: 7]

Use the method of mathematical induction to prove that is divisible

by 576 for .

6.

[Maximum mark: 7]

The function is given by , for

(a) Show that for all

[3 marks] (b) Solve the equation .

[4 marks]

7.

[Maximum mark: 5]The sum of the first n terms of a series is given by:

, where . (a) Find the first three terms of the series.

[3 marks](b) Find an expression for the nth term of the series, giving your answer in terms

of n.

[2 marks]

8. [Maximum mark: 5]Solve the equation , expressing your answers in the form , where .

Do NOT write solutions on this page.

SECTION B

Answer all questions on the answer booklet provided. Please start each question on a new page.9. [Maximum mark: 8]

(a) Describe two transformations whose composition transforms the graph of

to the graph of .

[2 marks](b) Sketch the graph of .

[3 marks](c) Sketch the graph of marking clearly the positions of any

asymptotes and x-intercepts.

[3 marks]Do NOT write solutions on this page.10.[Maximum mark: 22]

The function f is defined by , with domain .

(a) Express f ( x ) in the form , where A and .[2 marks]

(b) Hence show that on D.

[2 marks]

(c) State the range of f.

[2 marks]

(d) (i) Find an expression for .

(ii) Sketch the graph of y = f ( x ), showing the points of intersection with

both axes.(iii) On the same diagramm, sketch the graph of .[8 marks]

(e)(i) On a different diagram, sketch the graph of where .

(ii) Find all solutions of the equation .

[8 marks]Do NOT write solutions on this page.11.[Maximum mark: 11]

(a)(i) Express the sum of the first n positive odd integers using sigma notation.

(ii) Show that the sum stated above is.

(iii) Deduce the value of the difference between the sum of the first 47

positive odd integers and the sum of the first 14 positive odd integers.

[4 marks]

(b)A number of distinct points are marked on the circumference of a circle, forming a polygon. Diagonals are drawn by joining all pairs of non-adjacent points.

(i) Show on a diagram all diagonals if there are 5 points.

(ii) Show that the number of diagonals is if there are n points, where

n > 2.

(iii) Given that there are more than one million diagonals, determine the least

number of points for which this is possible.

[7 marks]

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