# Mock Exam Paper D P1 (Project Maths) Higher Level

Pre-Leaving Cert Examination 2014

Mathematics

(Projects Maths – Phase 3)

Paper I

Higher Level

Time hours

300 marks

 For Examiners use only Question Mark 1 2 3 4 5 6 7 8 9 Total

Instructions

There are two sections in this examination paper:

Section A Concept and skills 150 marks 6 questions

Section B Contexts and Applications 150 marks 3 questions

Write your answers in the spaces provided in this booklet. There is space for extra work at the back of the booklet. You may also ask the superintendent for more paper. Label any extra paper clearly with the question number and part.

The superintendent will give you a copy of the booklet of Formulae and Tables. You must return is at the end of the examination. You are not allowed to bring your won copy into the examination.

Marks will be lost if all necessary work is not clearly shown.

Answers should include the appropriate units of measurement, where relevant.

Answers should be given in simplest form, where relevant.

Write the mark and model of your calculator(s) here:

Section A Concepts and Skills 125 marks

Answer all four questions from this section.

Question 1 25 marks

(a) Sketch the region which satisfies

and

(b) Given that express the following in terms of k

(i) (ii)

(c) If

(i) Find the values of m and c where , and

(ii) Find a value for such that

(d) If, for all x, , find the value of c.

Question 2 25 marks

(a) The complex number z has modulus 4 and argument

The complex number w has modulus 2 and argument

(i) Express z and w in the form , where a and b are integers or surds.

(ii) Mark on an Argand diagram the complex numbers z and w.

(iii) Calculate

(b) Use De Moivre’s Theorem to prove that

Question 3 25 marks

(a) Prove that for all and

(b) Given that and that a and b are positive integers, find the value of a and b.

(c) Using a geometric series express in the form , where

Question 4 25 marks

(a) Using mathematical inductio...