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    FP 1 Cass

    Uploaded by

    Tianlu Wang

    Copyright:

    © All Rights Reserved
    Download as DOC, PDF, TXT or read online from Scribd

    FP 1 Cass

    Uploaded by

    Tianlu Wang
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    54 views2 pages

    FP 1 Cass

    Uploaded by

    Tianlu Wang

    Copyright:

    © All Rights Reserved
    Download as DOC, PDF, TXT or read online from Scribd
    Download as doc, pdf, or txt
    of 2

    Further Pure Mathematics 1

    Chapter Assessment
    Complex Numbers

    1. The complex number α is given by α = –2 + 5j.


    (i) Write down the complex conjugate α*.
    [1]
    (ii) Find the modulus and argument of α.
    [3]
      *
    (iii) Find in the form a + bj.

    [3]

    2. (i) Given that w = 1 + 2j, express w², w³ and w4 in the form a + bj.
    [5]
    (ii) Given that w is a root of the equation z  pz  qz  6 z  65  0 , find the
    4 3 2

    values of p and q.
    [5]
    (iii) Write down a second root of the equation.
    [1]
    (iv) Find the other two roots of the equation.
    [6]

    3. (i) Show that z1 = 2 + j is one of the roots of the equation z² – 4z + 5 = 0.


    Find the other root, z2.
    [3]
    1 1 4
    (ii) Show that   .
    z1 z2 5
    [3]
    (iii) Show also that Im (z1² + z2²) = 0 and find Re (z1² – z2²).
    [4]
    (iv) Find in the form r (cos   jsin  ) , the complex numbers z1, z1², and z1³.
    [7]
    (v) Plot the three complex numbers z1, z1², and z1³ on an Argand diagram.
    [3]

    © MEI, September 2001


    Further Pure Mathematics 1

    4. (a) Solve the equation z² + 2z + 10 = 0.


    Find the modulus and argument of each root.
    [4]
    (b) Complex numbers α and β are given by
        5 5 
      2  cos  jsin  ,   4 2  cos  jsin 
     8 8  8 8 
    (i) Write down the modulus and argument of each of the complex
    numbers α and β.
    Illustrate these two complex numbers on an Argand diagram.
    [3]
    (ii) Indicate a length on your diagram which is equal to    , and show
    that     6 .
    [3]
    (iii) On your diagram, draw and label
    (A) the locus L of points representing complex numbers z such that
    z   6 ,
    [3]
    (B) the locus M of points representing complex numbers z such that
    5
    arg( z   )  .
    8
    [3]

    © MEI, September 2001

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