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

PRE-LEAVING CERTIFICATE EXAMINATION 2012

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MATHEMATICS – HIGHER LEVEL

PAPER 1 (300 marks)

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TIME: 2 Hours 30 Minutes

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Attempt SIX QUESTIONS. Each question carries 50 marks.

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

Answers should include the appropriate units of measurement,

where relevant.

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1 (a) Express in the form , where .

(b) Let, where.

(i) Given that is a factor of, find the value of m.

(ii) The cubic equation has one real root and two complex roots. Find the

three roots.

(i) Show that it has real roots for all.

(ii) Hence, find the two roots, one of which is independent of a and b and the other is

not.

2. (a) Solve the simultaneous equations,

(b) (i) Solve the inequality , where and

(ii) If for all integers n,, show that

(c) (i) If the roots of the equation differ by 3, show that:.

(ii) If and , prove

3. (a) Evaluate where.

(b) Given that and, find.

If , find

(i) The values of a and b where .

(ii)

(c) (i) Express in the form

(ii) Hence, solve the equation, giving your complex roots in the form , where

4. (a) Three consecutive terms of an arithmetic series are, where .

Three consecutive terms of a geometric series are.

Find the value of a and b.

(b) (i) Express in the form .

(ii) Hence, find

(iii) Evaluate,

(c) Given the sequence 5, 55, 555, 5555, ……

(i) The nth term, of this sequence can be written as a series. Find the series.

(ii) Hence, find the sum of the first n terms of the given sequence.

5. (a) Solve the equation, for.

(b) Prove by induction that 7 is a factor of for any possible integer n.

(c) (i) Solve for x,

(ii) Find the coefficients of and in the expansion of in ascending powers of x. If these coefficients are equal find the value of a.

6. (a) Differentiate with respect to x

(i) (ii)

(b) (i) Find the slope of the ...