By David Martin, Catherine Berry, Terry Heard

This sequence, renowned for accessibility and for a student-friendly procedure, has a wealth of gains: labored examples, actions, investigations, graded workouts, Key issues summaries and dialogue issues. to make sure examination luck there are many up to date examination query, plus symptoms to point universal pitfalls. MEI provide complete help to high schools via their community with newsletters, education days and an annual convention. additional natural arithmetic 1 is the center part within the AS additional arithmetic qualification.

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**MEI AS Further Pure Mathematics (3rd Edition)**

This sequence, renowned for accessibility and for a student-friendly strategy, has a wealth of beneficial properties: labored examples, actions, investigations, graded routines, Key issues summaries and dialogue issues. to make sure examination luck there are many up to date examination query, plus symptoms to point universal pitfalls.

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**Extra resources for MEI AS Further Pure Mathematics (3rd Edition)**

**Example text**

49 EXERCISE 2A 1 Complex numbers 2 Express the following in the form x + yj. (i) (8 + 6j) + (6 + 4j) (iii) (2 + 7j) – (5 + 3j) (v) 3(4 + 6j) + 9(1 – 2j) (vii) (9 + 2j)(1 + 3j) (ix) (7 + 3j)2 (xi) (1 + 2j)(3 – 4j)(5 + 6j) (ii) (iv) (vi) (viii) (x) (xii) (9 – 3j) + (–4 + 5j) (5 – j) – (6 – 2j) 3j(7 – 4j) (4 – j)(3 + 2j) (8 + 6j)(8 – 6j) (3 + 2j)3 2 Solve each of the following equations, and check the roots in each case. (i) z 2 + 2z + 2 = 0 (ii) z 2 – 2z + 5 = 0 2 (iii) z – 4z + 13 = 0 (iv) z 2 + 6z + 34 = 0 (v) 4z 2 – 4z + 17 = 0 (vi) z 2 + 4z + 6 = 0 3 Given that z = 2 + 3j and w = 6 – 4j, find the following.

6 b y . 1 0 0 1 1 () ( (a) (b) (c) )( ) Show that if a = –4 and b = 2 then (0, 5) is an invariant point of TM. Show that if a = 2 and b = 1 then TM has no invariant point. Find a relation between a and b that must be satisfied if TM is to have any invariant points. [SMP] 43 INVESTIGATIONS Matrices 1 (11 10 ). 1 Investigate the sequence of matrices I, M, M 2, M 3, ... where M = 2 Mendel’s genetic theory states that offspring inherit characteristics from their parents through their genes. Each characteristic is controlled by a pair of genes, which may be of two types: G or g.

X + yj) – (u + vj) = (x – u) + (y – v)j Multiplication: multiply out the brackets in the usual way and simplify, remembering that j2 = –1. (x + yj)(u + vj) = xu + xvj + yuj + yvj2 = (xu – yv) + (xv + yu)j Division of complex numbers is dealt with later in the chapter. ● What are the values of j3, j4, j5? Explain how you would work out the value of jn for any positive integer value of n. Complex conjugates The complex number x – yj is called the complex conjugate, or just the conjugate, of x + yj.