KOLEJ TINGKATAN ENAM SHAH ALAM

MATHEMATICS T

2021 TERM 1 TOPICAL TEST 4

FORM : LOWER SIX SCIENCE

TIME : 1 HOUR 30 MINUTES

Name : ………………………….. Class : ……………………

Instructions to candidates :

Answer all questions .

1.(a) Given that z = 2 − i , find the complex number z in the form a + bi where

z+2

a,b R [ 3 marks ]

( b) The complex number z is given by z = 1 − i 3 , using de Moivre’s theorem, show

that z 7 = 128(1 − i 3 ) and express z6 in the form x + iy when z is the

2 2 z

conjugate of z and x, y R [ 6 marks ]

2. (a) Given p(1+ 5i) − 2q = 3 + 7i , find the value of p and q if p and q are both real

numbers. [ 3 marks ]

b) Express the complex number 3 + i in the form of r(cos + i sin ) , where r is

5

( )the modulus and is the argument of the complex number. Hence, simplify 3 + i

. [ 5 marks ]

3. A complex number is given by z = 6 .

[5 marks]

3−i 3 [2 marks]

(a) Express in the polar form.

(b) Using de Moivre’s theorem, find 8.

4. Find the solution of 4 = 16 and plot them on an argand diagram. [8 marks]

5. Find all the cube roots of 3 + 3√3 . [8 marks]

6. a) The roots of the equation z3 − 8z2 + 22 z − 20 = 0 are z1 , z2 and z3 . [4 marks]

i) Given that z1 = 3 + i , find z2 and z 3 . [2 marks]

ii) Show on a single Argand diagram, the points representing z1 , z2 and z3 .

b) Express the complex numbers 3 + i and 2 − 2i in the form r(cos + isin ) , where r 0

Determine 3 + i 10 in polar form, hence , solve the equation z3 = 3+i 10 , giving each

2 − 2i 2 − 2i

answer in polar form. [14 marks]