TUTORIAL 1

SM015 CHAPTER 1: NUMBER SYSTEM

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TUTORIAL 1: NUMBER SYSTEM

1.1 REAL NUMBERS

1. Graph the following intervals and solve each of the following

a) (− 3,2) (0,1) b) (− ,0) (2,3

c) (2,6) (−1,3 d) (− 4,−1)−1,3

e) (− ,2) (−1,0)0,3 f) (− 3,−1) (0,3) (2,4)

1.2 COMPLEX NUMBER

2. Given z1 = 1 + 2i , z2 = 4 − 3i and z3 = −2i − 5. Find the following in the form of a + bi

a) z1 + z3 b) z2 − z3 c) z * d) (z1* )(z3* )
2

3. Rationalise the following and express them in the form a + bi

2 4i 3+i d) 2 − i
a) 1 − i b) 4 + 3i c) i − 2 3 + 2i

(3 + i)(1− 2i)
e) (1− 3i)2

4. Find the values of x and y of the following equations

a) (2 + i)( x + 3yi) = 1− i b) 1+ 2i − (4 + i) = x + yi

2 − 3i

5. Plot each of the following complex numbers on separate Argand diagrams, and obtain the

modulus and argument in each case. Hence, express in polar form

a) (3 − i)3 b) 2 − i c) (1+ i)(3 − i)
1+ 3i (1− 2i)(1− i)

6. Given z1 = 3 − 2i and z2 = 1+ 2i , determine z in the form of a + bi . Hence, express z in polar

form

( )a) z1* + z2* 2 b) 2 − z2 c) −z1* z2*
z1* + i z1* − z2

1.3 INDICES, SURDS AND LOGARITHMS
7. Simplify

1 11 c) (16)−43 12

a) (−0.008)3 (3)3(9)3 81 (25)2(8)3
b) 6 d) 2

e) 4n 2n 1n f) 5n+1 10n  202n  23n (125)3

 83 164 d) 6 + 2 − 2 2
3+ 2
8. Simplify
d) log64 8
a) 1 b) 1−√5 c) 3√3−√2
2+√3 1+√5 2√3+√2

9. Evaluate the following logarithms without using table or calculator.

a) log1 8 ( )b) log5 125 5 c) log 18
32
2

e) log 27 + log 8 − log 125
log6 − log5

10. Simplify

a) − 2log4 5 + 2 b) logb 32 c) logc 9c2
logb 2 logc 3c

PAST SEMESTER EXAM (PSE)

11. a) Given the complex number z and its conjugate z satisfy the equation zz + 2z = 12 + 6i .

Find the possible values of z . [6m]

b) An equation in a complex number system is given by

z = (z1 1 ) + 1 where z1 = 1 + 2i and z2 = 2−i. Find
− z1
z2

i. the value of z in the Cartesian form a + ib [3m]
ii. the modulus and argument of z . [3m]

12. If z1 =4−i and z2 = 1 − 2i , find z1 − 5 . Express the answer in polar form . [6m]
z2

13. a) Given z1 = 1 − i and z2 = 4 + 2i . Express z12 in the form of a + bi , where a and b are
z1 − z2

real numbers. Hence, determine z12 . [5m]
z1 − z2

b) Given that z = x + iy , where x and y are the real numbers and z is the complex conjugate

of z . Find the positive values of x and y so that 1 + 2 = 3 − i . [6m]

zz [6m]
14. Given a complex number z = a + bi which satisfy the equation z2 = 8 + 6i . [6m]

a) Find all the possible values of z .

b) Hence, express z in polar form.
15. a) Given two complex numbers z1= 2 + i and z2= 1 – 2i .

i. Express z12 + 1 in the form x + iy , where x and y are real numbers and z2 is the

z2

conjugate of z2 . [4m]

ii. Hence, find the modulus of z12 + 1 . [2m]
z2

16. Given two complex numbers 1 = 5 + 3 and 2 = 2 −

a) State ̅ 1 and ̅ 2. [1m]
[3m]
b) Determine the value of k if 1 = ̅ 1. [6m]
1

c) Find 1 2. Hence, show that ̅ 1 ̅ 2 = ̅ ̅1̅̅ ̅ 2̅ .

17. Given a complex number z = 1 − 3i . Determine the value of k if z2 = k 1 . [7m]

z

18. Find the value of p and q if p + q = 1+ 5 i. [6m]

4 − 2i 4 + 2i 2
19. Let z = a + bi be a nonzero complex number.

a) Show that 1 = z [4m]
z z2.

b) Show that if z = −z , then z is a complex number with only an imaginary part. [3m]

( )c) Find the value of a and b if z(2 − i) = z +1 (1+ i). [5m]

20. Given z1 = 3 − 3i and z2 = 3 + 2i .

a) Write z1 in polar form. [4m]
[8m]
( )b)  i3 
Express z1z2 + − z2 in the form a + bi , a,b .
13

21. Solve for p and q where p  q , such that

( )( p + qi) = 3 + −16 − i3 [6m]
3i
22. Given a complex number z = 2 + i

a) Express z − 1 in the form of a + bi , where a and b are real numbers. [4m]

z

b) Obtain z − 1 . Hence, determine the values of real numbers  and  if
z

 + i = z−1  z − 1 2 [8m]
z  z 
[2m]
23. Given the complex number z1 = −i and z2 = 2 + i 3 [7m]

a) Express z12 and z2 in the form of a + bi , where a,b  R

b) From part (a) find W = z12 + z2 . Hence, find W and argW
z1

Answer: b)  c) (2,3 d) (− 4,3
f) (2,4)
1. a) (− 3,2) c) 4 + 3i d) −1 +12i
b) 9 − i
e) (−1,3 d) 4 − 7

2. a) −4 13 13

3. a) 1+ i b) 12 + 16 i c) −1 − i
25 25

e) − 1 + 7
10 10

4. a) x = 1 ; y = − 1 b) x = − 56 ; y = − 6
55 13 13

5. a) = 10√10 (−0.9653) + (−0.9653)

b) z = 2 (cos(−1.7127) + i sin (−1.7127)) c) z = 2  cos 3 + i sin 3 
2  4 4 

6. a) = 16 0 + 0 b) z = 5 (cos(−1.8926) + i sin (−1.8926))
18

c) z = 65 (cos (2.6224) + i sin (2.6224))
2

7. a) −0.2 b) 1 c) 27 d) 4
2 8 5

e) 2n f) 5

8. a) 2 − √3 b) − 3 + 5 c) 2 − 6 d) − 2
22 2

9. a) -3 1

3 b) 7 c) 2 d) 2
e) 2
2

10. a) log  16  b) 5 c) 2
 25 
4

11. a) z = −3 + 3i , z =1+ 3i

z= 1 + 1i 2 , 0.7854rad
b) i. 10 10 ii. 10

12. z = 3 − 3i , 3 2cos(− 45 )+ i sin(− 45 )

13. a) z12 = 1 + 1 i , z12 = 2 b) x = 1 , y = 1
z1 − z2 3 3 z1 − z2 3 22

14. a) z = 3 + i or z = -3 - i b) z = 10(cos0.3218 + i sin 0.3218),

15. z = 10(cos(− 2.8198) + i sin(− 2.8198))

a) i. 16 + 18 i ii. 4.817
55
z2 = 2 + i b) k = 1
16. a) z1 = 5 − 3i 34

17. k = −8 18. p = 15, q = −10
19. c) a = 1,b = 1

( )20. a) z1 = 3 2 cos 45 + i sin 45 b) 5 + 12 i
13 13

21. p = −15, q = 9

22. a) 8 − 6 i b) 2 ;  = 56 ,  = − 192
55 25 25

23. a) −1; 2 − i 3 b) W = 3 + i; W = 2;arg (W ) = 

6