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

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**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.

**(c)** Given the quadratic equation,

**(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 n^{th} 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 ...