A LevelFurther Maths: Core PureEdexcelExam Questions1. Complex Numbers1.2 Exponential Form & de Moivre's Theorem

Exponential Form & de Moivre's Theorem (Edexcel A Level Further Maths: Core Pure)

Exam Questions

34 mins4 questions
1a
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A complex number z has modulus 1 and argument theta.

(a)
Show that

z to the power of n plus 1 over z to the power of n equals 2 cos space n theta comma      n element of straight integer numbers to the power of plus

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1b
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(b)
Hence, show that

cos to the power of 4 theta equals 1 over 8 left parenthesis cos 4 theta plus 4 cos 2 theta plus 3 right parenthesis

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2a
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(a)
Use de Moivre’s theorem to prove that

sin space 7 theta space equals space 7 space sin theta space minus space 56 space sin cubed space theta space plus space 112 space sin to the power of 5 space theta space minus space 64 space sin to the power of 7 space theta

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2b
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(b)
Hence find the distinct roots of the equation


1 space plus space 7 x space minus space 56 x cubed space plus space 112 x to the power of 5 space minus space 64 x to the power of 7 space equals space 0

giving your answer to 3 decimal places where appropriate.

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3a
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The infinite series C and S are defined by

C space equals space cos space theta space plus space 1 half space cos space 5 theta space plus space 1 fourth space cos space 9 theta space plus space 1 over 8 space cos space 13 theta space plus space...

S space equals space sin space theta space plus space 1 half space sin space 5 theta space plus space 1 fourth space sin space 9 theta space plus space 1 over 8 space sin space 13 theta space plus space...

Given that the series C and S are both convergent,

a)
show that

C plus straight i S equals fraction numerator 2 straight e to the power of straight i theta end exponent over denominator 2 minus straight e to the power of 4 straight i theta end exponent end fraction

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3b
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(b)
Hence show that

S equals fraction numerator 4 sin theta plus 2 sin 3 theta over denominator 5 minus 4 cos 4 theta end fraction

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4a
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In an Argand diagram, the points A, B and C are the vertices of an equilateral triangle with its centre at the origin. The point A represents the complex number 6 plus 2 straight i.

(a)
Find the complex numbers represented by the points B and C, giving your answers in the form x plus straight i y, where x and y are real and exact.

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4b
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The points D, E and F are the midpoints of the sides of triangle A B C.

(b)
Find the exact area of triangle D E F.

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