Download OCR FP1 Complex Numbers

OCR Further Pure 1
Complex Numbers
Chapter assessment
1. Find the complex number z which satisfies (2  i) z  (3  2i) z*  32 .
2. The complex number α is given by α = –2 + 5i.
(i) Write down the complex conjugate α*.
(ii) Find the modulus and argument of α.
  *
(iii)Find
in the form a + bi.

3. (i) Show that z1 = 2 + i is one of the roots of the equation z² – 4z + 5 = 0.
Find the other root, z2.
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(ii) Show that   .
z1 z2 5
(iii)Show also that Im (z1² + z2²) = 0 and find Re (z1² – z2²).
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4. The cubic equation z 3  z 2  4 z  48  0 has one real root α and two complex
roots β and γ.
(i) Verify that α = 3 and find β and γ in the form a + bi. Take β to be the root with
positive imaginary part, and give your answers in an exact form.
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(ii) Find the modulus and argument of each of the numbers α, β, γ, giving the
arguments in radians between –π and π.
Illustrate the three numbers on an Argand diagram.
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(iii)On your Argand diagram, draw the locus of points representing complex
numbers z such that
arg( z   )  arg  .
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5. You are given that the complex number   1  4i satisfies the cubic equation
z 3  5 z 2  kz  m  0 ,
where k and m are real constants.
(i) Find α² and α³ in the form a + bi.
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(ii) Find the value of k, and show that m = 119.
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(iii)Find the other two roots of the cubic equation. Give the arguments of all three
roots.
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(iv) Verify that there is a constant c such that all three roots of the cubic equation
satisfy
z2 c.
Draw an Argand diagram showing the locus of points representing all complex
numbers z for which z  2  c . Mark the points corresponding to the three
roots of the cubic equation.
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OCR Further Pure 1
6. (i) On an Argand diagram, mark the points A and B representing the complex
numbers 0 + 9i and 12 + 0i respectively.
Describe in words the locus L1 of points representing complex numbers w
which satisfy
w  9i  w 12 .
Draw L1 on your diagram.
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The locus L2 consists of the points representing complex numbers z for which
z  9i  2 z  12 .
(ii) By writing z = x + yi, where x and y are real, obtain an equation relating
x and y, and hence show that L2 is a circle. Give the centre and radius of this
circle.
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(iii)Hence write down an equation for L2 in which z occurs only once.
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Total 70 marks
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