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Complex number

1.       Let z = x + yi. Find the values of x and y if (1 – i)z = 13i.

Working:

 

 

Answer:

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(Total 4 marks)

 

 

2.       (a)     Evaluate (1 + i)2, where i = .

(2)

(b)     Prove, by mathematical induction, that (1 + i)4n = (–4)n, where n Î *.

(6)

(c)     Hence or otherwise, find (1 + i)32.

(2)

(Total 10 marks)

 


 

3.       Let z1 = , and z2 = 1 – i.

(a)     Write z1 and z2 in the form r(cos θ + i sin θ), where r > 0 and  £ θ £ .

(6)

(b)     Show that  = cos  + i sin .

(2)

(c)     Find the value of  in the form a + bi, where a and b are to be determined exactly in radical (surd) form. Hence or otherwise find the exact values of cos  and sin .

(4)

(Total 12 marks)

 

 

4.       Let z1 = a  and z2 = b

          Express  in the form z = x + yi.

Working:

 

 

Answer:

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(Total 3 marks)

 


 

5.       If z is a complex number and |z + 16| = 4 |z + l|, find the value of | z|.

Working:

 

 

Answer:

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(Total 3 marks)

 

 

6.       Find the values of a and b, where a and b are real, given that (a + bi)(2 – i) = 5 – i.

Working:

 

 

Answer:

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(Total 3 marks)

 


 

7.       Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z = 60°.

Working:

 

 

Answer:

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(Total 3 marks)

 

 

8.       The complex number z satisfies i(z + 2) = 1 – 2z, where . Write z in the form z = a + bi, where a and b are real numbers.

Working:

 

 

Answer:

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(Total 3 marks)

 


 

9.       The complex number z satisfies the equation

         =  + 1 – 4i.

          Express z in the form x + iy where x, y Î .

Working:

 

 

Answer:

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(Total 6 marks)

 

 

10.     Consider the equation 2(p + iq) = q – ip – 2 (1 – i), where p and q are both real numbers. Find p and q.

Working:

 

 

Answer:

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(Total 6 marks)

 


 

11.     Let the complex number z be given by

z = 1 + .

          Express z in the form a +bi, giving the exact values of the real constants a, b.

Working:

 

 

Answer:

.........................................................................

(Total 6 marks)

 

 

12.     A complex number z is such that .

(a)     Show that the imaginary part of z is .

(2)

(b)     Let z1 and z2 be the two possible values of z, such that 3.

(i)      Sketch a diagram to show the points which represent z1 and z2 in the complex plane, where z1 is in the first quadrant.

(ii)     Show that arg z1 = .

(iii)    Find arg z2.

(4)

(c)     Given that arg = π, find a value of k.

(4)

(Total 10 marks)

 

 

13.     Given that (a + i)(2 – bi) = 7 – i, find the value of a and of b, where a, b Î .

Working:

 

 

Answer:

.........................................................................

(Total 6 marks)

 

 

14.     Given that z Î , solve the equation z3 – 8i = 0, giving your answers in the form z = r (cosq + i sinq).

Working:

 

 

Answer:

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(Total 6 marks)

 


 

15.     Given that z = (b + i)2, where b is real and positive, find the exact value of b when arg z = 60°.

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(Total 6 marks)

 


 

16.     Given that | z | = 2, find the complex number z that satisfies the equation

(Total 6 marks)

 

 

17.     The two complex numbers z1 =  and z2 =  where a, bÎ , are such that z1 + z2 = 3. Calculate the value of a and of b.

(Total 6 marks)

 


 

18.     The complex numbers z1 and z2 are z1 = 2 + i, z2 = 3 + i.

(a)     Find z1z2, giving your answer in the form a + ib, a, bÎ .

(1)

(b)     The polar form of z1 may be written as .

(i)      Express the polar form of z2, z1 z2 in a similar way.

(ii)     Hence show that  = arctan + arctan .

(5)

(Total 6 marks)

 


 

19.     Let z1 = r  and z2 = 1 +  i.

(a)     Write z2 in modulus-argument form.

(b)     Find the value of r if  = 2.

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(Total 6 marks)

 


 

20.     Let z1 and z2 be complex numbers. Solve the simultaneous equations

                       2z1 + z2 = 7, z1 + iz2 = 4 + 4i

Give your answers in the form z = a + bi, where a, bÎ .

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(Total 6 marks)

 


 

21.     The complex number z is defined by

          z = 4

(a)     Express z in the form reiq, where r and q have exact values.

(b)     Find the cube roots of z, expressing in the form reiq, where r and q have exact values.

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(Total 6 marks)

 


 

22.     The polynomial P(z) = z3 + mz2 + nz −8 is divisible by (z +1+ i), where zΠand m, nÎ. Find the value of m and of n.

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(Total 6 marks)

 


 

23.     Let u =1+  and v =1+ i where i2 = −1.

(a)     (i)      Show that

(ii)     By expressing both u and v in modulus-argument form show that .

(iii)    Hence find the exact value of tan  in the form  where a, bÎ .

(15)

(b)     Use mathematical induction to prove that for nÎ +,

(7)

(c)     Let z =

Show that Re z = 0.

(6)

(Total 28 marks)

 

 

24.     (a)          Express the complex number 1+ i in the form , where a, bÎ +.

(2)

(b)     Using the result from (a), show that , where nÎ , has only eight distinct values.

(5)

(c)     Hence solve the equation z8 −1 = 0.

(2)

(Total 9 marks)

 


 

25.     Find, in its simplest form, the argument of (sinq + i (1− cosq ))2 where q is an acute angle.

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(Total 7 marks)

 


 

 

+.

(2)

(b)     Using the result from (a), show that 26.          Consider w =  where z = x + iy, y ¹ 0 and z2 + 1 ¹ 0.

Given that Im w = 0, show that  = 1.

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(Total 7 marks)

 

 

27.     Let x and y be real numbers, and w be one of the complex solutions of the equation
z3 = 1. Evaluate:

(a)     1 + w + w2;

(2)

(b)     (w x + w2y)(w2x + w y).

(4)

(Total 6 marks)

 

 

28.     (a)     Express z5 – 1 as a product of two factors, one of which is linear.

(2)

(b)     Find the zeros of z5 – 1, giving your answers in the form
r(cos θ + i sin θ) where r > 0 and –π < θ £ π.

(3)

(c)     Express z4 + z3 + z2 + z + 1 as a product of two real quadratic factors.

(5)

(Total 10 marks)

 

 

 

õ @29.        (a) Express the complex number 8i in polar form.

(b)     The cube root of 8i which lies in the first quadrant is denoted by z. Express z

(i)      in polar form;

(ii)     in cartesian form.

Working:

 

 

Answers:

(a)       ..................................................................

(b)   (i)   ...........................................................

       (ii)   ...........................................................

(Total 6 marks)

 


 

30.     Consider the complex number z = .

(a)     (i)      Find the modulus of z.

(ii)     Find the argument of z, giving your answer in radians.

(4)

(b)     Using De Moivre’s theorem, show that z is a cube root of one, ie z = .

(2)

(c)     Simplify (l + 2z)(2 + z2), expressing your answer in the form a + bi, where a and b are exact real numbers.

(5)

(Total 11 marks)

 

 

31.     (a)     Prove, using mathematical induction, that for a positive integer n,

(cosq + i sinq)n = cos nq + i sin nq  where i2 = –1.

(5)

(b)     The complex number z is defined by z = cosq + i sinq.

(i)      Show that  = cos (–q) + i sin (–q).

(ii)     Deduce that zn + z–n = 2 cos .

(5)

(c)     (i)      Find the binomial expansion of (z + z–l)5.

(ii)     Hence show that cos5q =  (a cos 5q + b cos 3q + c cos q),
where a, b, c are positive integers to be found.

(5)

(Total 15 marks)

 


 

32.     (a)     Use mathematical induction to prove De Moivre’s theorem
     (cosq + i sinq)n = cos (nq) + i sin (nq), n Î +.

(7)

(b)     Consider z5 – 32 = 0.

(i)      Show that z1 = 2  is one of the complex roots of this equation.

(ii)     Find z12, z13, z14, z15, giving your answer in the modulus argument form.

(iii)    Plot the points that represent z1, z12, z13, z14 and z15, in the complex plane.

(iv)    The point z1n is mapped to z1n+1 by a composition of two linear transformations, where n = 1, 2, 3, 4. Give a full geometric description of the two transformations.

(9)

(Total 16 marks)

 

 

33.     Consider the complex number z = cosq + i sinq.

(a)     Using De Moivre’s theorem show that

zn +  = 2 cos nq.

(2)

(b)     By expanding  show that

cos4q = (cos 4q + 4 cos 2q + 3).

(4)

(c)     Let g (a) = .

(i)      Find g (a).

(ii)     Solve g (a) = 1

(5)

(Total 11 marks)

 

 

34.     Let z = cos q + i sin q, for – < q < .

(a)     (i)      Find z3 using the binomial theorem.

(ii)     Use de Moivre’s theorem to show that

          cos 3q = 4 cos3q – 3 cosq and sin 3q = 3 sinq – 4 sin3q.

(10)

(b)     Hence prove that  = tanq.

(6)

(c)     Given that sinq = , find the exact value of tan 3q.

(5)

(Total 21 marks)

 

 

35.     Let y = cosq + i sinq.

(a)     Show that  = iy.

          [You may assume that for the purposes of differentiation and integration, i may be treated in the same way as a real constant.]

(3)

(b)     Hence show, using integration, that y = eiq.

(5)

(c)     Use this result to deduce de Moivre’s theorem.

(2)

(d)     (i)      Given that  = a cos5q + b cos3q + c cosq, where sinq  0, use de Moivre’s theorem with n = 6 to find the values of the constants a, b and c.

(ii)     Hence deduce the value of .

(10)

(Total 20 marks)

 


 

36.     Prove by induction that 12n + 2(5n−1) is a multiple of 7 for n Î +.

(Total 10 marks)

 

 

37.     Prove that  is real, where nÎ +.

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(Total 6 marks)

 


 

38.     Express  in the form  where a, bÎ.

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(Total 5 marks)

 

 

39.     Let w = cos

(a)     Show that w is a root of the equation z5 − 1 = 0.

(3)

(b)     Show that (w − 1) (w4 + w3 + w2 + w + 1) = w5 − 1 and deduce that w4 + w3 + w2 + w + 1 = 0.

(3)

(c)     Hence show that cos

(6)

(Total 12 marks)

 


 

40.     z1 =  and z2 =

(a)     Find the modulus and argument of z1 and z2 in terms of m and n, respectively.

(6)

(b)     Hence, find the smallest positive integers m and n such that z1 = z2.

(8)

(Total 14 marks)

 

 

41.     (z + 2i) is a factor of 2z3–3z2 + 8z – 12. Find the other two factors.

Working:

 

 

Answer:

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(Total 3 marks)

 


 

42.     Let P(z) = z3 + az2 + bz + c, where a, b, and c Î . Two of the roots of P(z) = 0 are –2 and
(–3 + 2i). Find the value of a, of b and of c.

(Total 6 marks)

 

 

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