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Mathematics QM016
Topic 1 : Number System – Tutorial
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TUTORIAL: NUMBER SYSTEM
1.
Determine whether each statement is true or false
(a) Every counting number is an integer
(b) Zero is a counting number
(c) Negative six is greater than negative three
(d) Some of the integers is natural numbers
2.
List the number describe and graph them on the number line
(a) The counting number smaller than 6
(b) The integer between -3 and 3
3.
Given S = {-3, 0, 7 ,
(a) natural numbers
(d) rational numbers
4.
1
, e, , 4, 8…}, identify the set of
3
(b) whole numbers
(e) irrational numbers
(c) integers
(f) real numbers
a
b
(b) 2.7181818….
Express each of the numbers as a quotient
(a) 0.7777……
5.
Write each of the following inequalities in interval notation and show them on the
real number line.
(a) 2 < x < 6
(b) 5 < x < 1
(c) 3  x  7
(d) 2 < x  0
(e) x < 3
(f) x  1
(g) x  2
(h) 3  x < 2
6.
Show each of the following intervals on the real number line.
(a) [2, 3]
(b) (4, 4)
(c) (, 5]
(d) [1, )
(e) (3, 6]
(f) [2, 3)
(g) (2, 0)  (3, 6)
(h) [6, 2)  [3, 7)
7.
Evaluate
1
(a)  
2
2
(e) (0.36)
(b) 27
1
2
3
1
3
(f) (2.56)
 100  2
(c) 

 9 
1
2
1
(g)
1
1
9 2 .8 2
2
2
 27  3
(d)  
 8 
1
2
1
(h)
9 3 .27
1
6
3 .3

1
2
2
3
Mathematics QM016
Topic 1 : Number System – Tutorial
________________________________________________________________________________________________
8.
Simplify the following expressions:
2
3 n  2 9 n  27 n
(a)
(c)
9.
9
n
2
3
n 3
 32
1
n
(b) 4 n 8 3  16 4

1
5
n
(d) 5 n 1 10 n  20 2 n  2 3n
Solve the following equations:
9 x = 27
(a)
(d)
4 x  2  1281 x
(g)
7 x - 49 6 2 x = 0
2
(b) 3 3 = 243
(e) x

2
5

(c) 2 x 1 =
1
16
(f) 4x
(h) (4 x ) x = 4  8 x
10. Solve the equations:
(a) 2(2 2 x )  5(2 x )  2  0
(b) 32 x1  26(3 x )  9  0
(b) 4 x  6(2 x )  16  0
(d) e 2 x  e x  2  0
11. Express in terms of the simplest possible surds:
(a)
8
(c)
180
(b)
75
(d) 125
12. Simplify:
2 (3  2 2 )
(a)
(b) ( 2  1)( 2  1)
(c) ( 3  2)( 3  1)
(d) (2 5  3)(3 5  2)
13. Rationalise the denominators and simplify in the form a + b :
(a)
(c)
(e)
1
(b)
8
1
(d)
3 2 5
1
2 1

1
2 1
2
3
2 3
1
6 5
2
3

1
64
1
9
Mathematics QM016
Topic 1 : Number System – Tutorial
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14. Solve the equations:
(a)
3x  1  x  1
(b)
4x  9  1  2 x
(c)
4 x  13  x  1  12  x
15. Express in logarithmic form:
5
(a)
5 3 = 125
(b) 7 2 = 49
(d)
e2t = 3 – x
(e) e0.1t = x + 2
16. Express in index form:
(a)
log 5 625 = 4
(c) 4 2 = 32
(b) log 3 27 = 3
(c) log 1 4 = -2
2
(d)
ln 100 = 4.6052
17. Evaluate
(a)
log 4 64
(e) ln 3 =1.0986
(b) log 10000
(c) log 1 4
2
(d)
log 125 25
(g)
log 9 9 2
(e) log 0.1
(f) log 8 0.5
(h) ln e2
(i) e2 + ln3
1
18. Simplify
(a) log 3 + log 4
(b) log 2 + log 6 – log 4
1
(d)
log 25 – 2 log 3 + 2 log 6
2
1
(f) ln y3 + ln(x3y6) – 5 lny
3
(c) 2 log 3 + log 2
(e)
log 9
log 3
1
(g) 2 ln x - 4 ln   - 3 ln (xy)
 y
19. Solve the equations:
(a) 3 x = 6
(d) (5 x )(5 x 1 ) = 10
(g) 3e-2x = 75
(b) 2 2 x = 5
(e) e2lnx = 9
(h) e2.5x = 40
3
(c) 3 3 x 1 = 7
(f) ex ln3 = 27
Mathematics QM016
Topic 1 : Number System – Tutorial
________________________________________________________________________________________________
20. Solve these equations:
(a) 2 log (x – 2) = log (2x – 5)
(c) log2 (2x – 4) = 2 + log2 (x2- 6)
(e) 9logx 5 = log5 x
(g) log16 x = (log 4 x)2 , x > 1
(i) ln x = 2 + ln (1-x)
(b)
(d)
(f)
(h)
(j)
log (x2 + 6) – log (x – 1) = 1
log3x – 2 logx3 = 1
log2 2 x 1  32 2 = x
2 ln x = ln 3 + ln (6 – x)
eln(1-x) = 2x


21. If log 2 x + log x 2 = 2, find x.
22. Simplify
(a)
( 2 + i )(3 – 4i)
(c)
(a + bi)2
(b) ( 3 + 4i )(3 – 4i)
(d) i(1 + i)( 2 + i)
23. Rationalise the denominator of each of the following fractions and express them in
the form a + bi.
2
3i
4i
x  yi
(a)
(b)
(c)
(d)
1 i
4  3i
4  3i
x  yi
Solve the following equations for x and y.
(a)
(d)
x + yi = ( 3 + i)(2 – 3i)
2  5i
 x  yi
1 i
(b) 3 + 4i = (x + yi )(1 + i )
(d) x + yi = 2
24. Find the modulus and argument of the following complex numbers.
1  7i
(a) ( 1 – i )(4 + 3i )
(b)
1 i
4