Download S01e

Chapter 1 Numbers and Functions
Chapter 1
Numbers and Functions
WARM - UP E XERCISE
1. Evaluate the following expressions.
(a) 3(7  2)
(b) 4  2[3  (5)]
2. When x  2, find the values of the following expressions.
(a) 3x  4
(b) 5x  15
3. When x  1, find the values of the following expressions.
(a) x 2  8x  16
(b) 3x 2 – 10x – 24
Simplify the following expressions, and express your answers in positive indices. ( 4 – 7)
4. (a) a 7 × a 3 × a 2
5. (a)
6. (a)
a10
a2
( a 2 )(a 5 )
a
7. (a) (4a 2 b 3 ) 1 × (2ab) 2
(b) a 4 × a 5 × a 3
(b)
(b)
(b)
a3
a14
(2a 4 )(4a 3 )
a2
(a 4 b)  (3a 3 b 2 ) 2
3a 2 b 3
Simplify the following expressions. (8 – 9)
8. (a) 7  (2x  5)
(b) 3  (3x  4)
9. (a) 8x  (3x  5)
(b) x – 2(3 – 5x)
Solve the following equations. (10 – 11)
10. (a) x  3  5 – x
(b) 2x – 4  –2(x  5)
11. (a) (5 – 5x)  (3 – x)  2
(b) (6x – 2) – (3x  5)  –10
Expand the following expressions. (12 – 13)
12. (a) (2x – 1) 2
(b) (3x  2) 2
13. (a) (2x – 3)(2x  3)
(b) (3x – 4)(3x  4)
1
2
New Trend Mathematics S4A — Supplement
B UILD - UP E XERCISE
[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set,
Intermediate Set and Advanced Set. You may choose to complete any ONE set according to your need. ]
Exercise 1A
  El em en tar y S et
 
Level 1
1. Determine whether each of the following is an even number or an odd number where N is
an integer.
8N  10
(a) N  (N  2)
(b)
2
2. Determine whether the expression N  (N  1) + (N  2) is an even number or an odd
number if N is
(a) an odd number.
(b) an even number.
3. Given that N is an integer, find the values of the following expressi ons.
(a) (–1) 4N  1
(b) (–1) N  3N
(c) (1) N(N  3)
4. Given that N is an integer, find the values of the following expressions.
Ex.1A Elementary Set
(a) (–1) N  (–1) N  1
(b) (–1) N  (–1) N  1
5. Given that N is an integer, find the value of the expression ( –1) N  (–1) 3N  1 .
6. It is known that N is an integer. Show that 4(3N  1)  11 is divisible by 3.
7. The sum of two consecutive numbers is 49. Find the numbers.
8. The sum of two consecutive odd numbers is 112. Find the numbers.
9. The sum of three consecutive even numbers is 150. Find the numbers.
10. Convert the following recurring decimals into fractions.
(a) 0.3 2
(b) 0.05
(c) 0.41
Level 2
11. (a) Simplify (N  1) 2 – N 2 .
(b) It is known that N is an integer. Prove that (N  1) 2 – N 2 is an odd number.
12. (a) It is known that 3k is a multiple of 3 where k is an integer. Write down the next two
multiples of 3 following 3k.
(b) Prove that the sum of any three consecutive multiples of 3 is a multiple of 9.
Chapter 1 Numbers and Functions
 Intermediate Set
Level 1
3

13. Find the value of the expression (–1) N  (–1) N  2 if N is
(a) an odd number.
(b) an even number.
14. Given that N is an integer, find the values of the following expres sions.
(a) (–1) N  3N  5N  7N
(b) ( 1)
6N 3
3
(c) (–1) (2N  3) – (N  1)
15. Given that N is an integer, find the values of the following expressions.
(a) (–1) 2N + 1  (–1) 2N + 3
(b) (–1) 2N  (–1) N
(c) (1) 3N  (1) N 1
2
16. It is known that N is an integer. Show that 9(N  1) – (N  5) is an even number.
18. The sum of three consecutive numbers is 93. Find the numbers.
19. Convert the following recurring decimals into fractions.
(a) 0.8 5
(b) 4.27
(c) 0.32 1
Level 2
20. (a) Expand (3N – 1)(3N  2).
(b) It is known that N is an integer.
(i) Prove that (3N – 1)(3N  2) is an even number.
(ii) Prove that (3N – 1)(3N  2) – 1 is a multiple of 3.
21. (a) Simplify N 2  (N  1) 2 .
(b) Hence prove that the sum of the squares of any two consecutive numbers is an odd
number.
22. (a) Convert the following recurring decimals into fractions.
(i) 0.01
(ii) 0.07
(iii) 0.08
(iv) 0.1 5
(b) (i) Is it true that 0.07  0.08  0.1 5 ?
(ii) Is it true that 0.08  0.07  0.01 ?
23. Find the value of x if
(a) 0.3 x  0.5 4 .
(b) 0.4 x  0.1 6
Ex.1A Intermediate Set
17. The sum of two consecutive even numbers is 38. Find the numbers.
4
New Trend Mathematics S4A — Supplement
 Advanced Set

Level 1
24. Given that N is an integer, find the values of the following expressions.
(a) (1) 2 N
2
(b) (–1) 2N  1 – (–1) 2N  1
(c) (–1) N  (N  1)  (–1) 2N  (N  1)
25. It is known that N is an integer.
(a) Show that 4(3N  2)  3(N  1) is divisible by 5.
(b) Show that (N  5) 2 – (N – 1) 2 is a multiple of 12.
26. The sum of three consecutive numbers is equal to 2 times the smallest number. Find the
numbers.
27. Convert the following recurring decimals into fractions.
(a) 1.6 9
(b) 0.33 5
(c) 0.2 43
Ex.1A Advanced Set
Level 2
28. (a) Simplify (2x  1) 2  (2x – 1) 2 .
(b) Hence prove that the sum of the squares of any two consecutive odd numbers is an even
number.
29. Prove that the difference between the squares of any two consecutive odd numbers is a
multiple of 8.
30. Prove that the product of any three consecutive numbers is an even number.
31. (a) Convert 0.00 2 5 into fraction.
(b) Using the result of (a), convert 0.22 2 5 into fraction.
32. (a) Convert the following recurring decimals into fractions.
(i) 0.03
(ii) 0.06
(iii) 0.1 2
(iv) 0.1 8
(b) (i) Is it true that 0.06  0.1 2  0.1 8 ?
0.1 8
(ii) Is it true that
 0.03 ?
0.06
33. Find the value of x if
(a) 1.5 3 x  0.0 9 .
(b) 0.5  x  1.7 .
(c) 0.9 6  2 x  0.8 .
Chapter 1 Numbers and Functions
Exercise 1B
  El em en tar y S et
Level 1
Simplify the following surds. (1  4)
1. (a)
(b)
8
2. (a) 2 20
(b) 4 135
8
9
3. (a)
4. (a) 
108
(b)
27
4
54
25
(b)  2
 
(c)
960
(c) 5 32
(c)
8
9
5
125
4
(c)  5
32
25
5. (a) Express the following as entire surds.
(i) 2 3
(ii) 3 22
(iv) 5 7
(v) 7 6
39
2
(vi) 8 3
(iii) 4
Simplify the following. (6  9)
6. (a) 7 5  2 5
(b)
24  54
7. (a) 5 20  3 50  18
(b)
578  144  288
32
8

9
9
5
75
(c) 14
2
49
240
8. (a)
(b)
(d)
9. (a) 4 12  2 6
(c)
(b) 2 5 ( 5  10 )
(d) 5 3 ( 18  3 75 )
6 (2 3  3 2 )
10. Rationalize the following.
5
(a)
3
4 3
(c)
2
11. Rationalize
176
99
8
16
4
128
 450
36
3 5
3
.
(b)
(d)
2
45
5 3
2 20
(c) 2 8  18  32
Ex.1B Elementary Set
(b) Arrange the above surds in ascending order.
6
New Trend Mathematics S4A — Supplement
Simplify the following and rationalize the results. ( 12  14)
9
5
11
3


12. (a)
(b)
3
3
2 7
7
12
3
2
5

(c)
(d)

4
2
32
5
13. (a)
Ex.1B Elementary Set
14. (a)
30  3 5
5
1 3
3

2  3 1
(b)
3
2
(b)
5
2 11  1
2

5
2 2

88
2
Level 2
Simplify the following. (15  16)
15. (a) (2  3 ) 2
(b) ( 21  7 ) 2
16. (a) (2  3 )(3  2 3 )
(b) ( 3  3 2 )( 3  3 2 )
17. (a) Simplify ( 3  5)( 3  5) .
1
(b) Hence rationalize
.
3 5
18. (a) Simplify (2 5  1)(2 5  1) .
(b) Hence rationalize
5
2 5 1
.
 Intermediate Set
Level 1
Simplify the following surds. (19  20)
19. (a)
96
Ex.1B Intermediate Set
20. (a) 3 20
(d)
243
64
(b)

128
(c)
675
(b) 8 75
(c)
27
16
(e) 
25
81
21. (a) Express the following as entire surds.
2
1
(i)
(ii)
110
106
5
2
1
(iv) 4 5
(v) 5
2
(b) Arrange the above surds in ascending order.
54
16
(f)  3
(iii) 2 5
(vi) 6
3
2
Chapter 1 Numbers and Functions
7
Simplify the following. (22  25)
90  6 10
(b)
23. (a) 2 99  3 44  605
24. (a)
(c)
147
48

25
25
125
140 1 5


16
112 2 4
1
15  2 30
3
(c) 2 3 ( 12  2 72 )
25. (a)
3
26. Rationalize the following.
27
(a)
3
2 6 5
27. Rationalize
3
80  45
(c) 7 12  27  5 75
(b) 6 24  4 54  3 245
72
350
2
9
28
75
130
(d) 2
 18  5
12
65
(b)
(b)
5 ( 24  5 15 )
(d) 2 2 ( 48  5 18 )
(b)
20 3
(c)
50
2 8
3 10
.
Ex.1B Intermediate Set
22. (a)
Simplify the following and rationalize the results. (28  30)
7
5
10
1


28. (a)
(b)
3
3
3 2 2 2
21
6
12
12


(c)
(d)
9
5
20
3
29. (a)
30. (a)
32  6
(b)
3 3
6 2
3

21
12
(b)
2 5 5 6
3
2 3 5
3

2 18  6 8
27
Level 2
Simplify the following. (31  32)
2 2
)
3
31. (a) ( 4  15 ) 2
(b) ( 20  5 ) 2
(c) ( 6 
32. (a) (1  5 2 )(4  3 2 )
(b) (2 7  2 )(6 7  9 2 )
(c) (3 5  2 2 )(3 5  2 2 )
33. (a) Simplify ( 6  4)( 6  4) .
1
(b) Hence rationalize
.
64
8
New Trend Mathematics S4A — Supplement
Ex.1B Intermediate Set
34. (a) Simplify ( 2  11)( 2  11) .
3
(b) Hence rationalize
.
2  11
35. (a) Simplify (7  2 3 )(7  2 3 ) .
7
(b) Hence rationalize
72 3

3
72 3
.
36. Simplify [(4 3  5 )  2][(4 3  5 )  2] .
 Advanced Set
Level 1
Simplify the following surds. (37  38)
37. (a)
72
38. (a) 4 360

(b)
245
(c)
(b)
180
75
(c)  6
882
125
144
Simplify the following. (39  41)
39. (a) 3 75  2 48
(b)
(c) 3 363  2 108  6 12
Ex.1B Advanced Set
40. (a)
12 1 27

9 3 4
41. (a)
18 
(c)

28  5 42
2
20 ( 15  5 )
42. Rationalize the following.
10
(a)
12
43. Rationalize
12  10
3 5
72  50  200
(d) 2 84  4 6  4 189
169
117

52
4
(b)
208 
(b)
12 (2 6  3 8 )
(d)  3 3 ( 6  2 24 )
(b)
5 15
(c)
96
2 27
3 28
.
Simplify the following and rationalize the results. ( 44  46)
44. (a)
11
2

3
2
(b)
150
5

150
5
(c)
3
7

14
15

2
63
Chapter 1 Numbers and Functions
45. (a)
46. (a)
128  24
22 3
2

2  2 2  98
(b)
6
27
4 12
20  10
(b)
18
9
3

625  5 2
30
Level 2
47. (a) Simplify the following and rationalize the res ults.
(i) 2
45
2
(ii) 4
2
(iv) ( 250  2 10 )
5
5
8
90 
(v)
8
)(4  10 )
5
12
5

(vi)
250
40
(iii) (1 
12
10
(b) Arrange the above surds in descending order.
Simplify the following. (48  49)
(b) ( 22  3 11 ) 2
(c) ( 5 
49. (a) (2 5  3)(7 5  6)
(b) (4 8  5 5 )(2 8  3 5 )
(c) (4 7  5 )(4 7  5 )
50. (a) Simplify (3 5  10)(3 5  10) .
5
(b) Hence rationalize
.
3 5  10
51. (a) Simplify ( 12  8 )( 12  8 ) .
5
(b) Hence rationalize
.
12  8
52. (a) Simplify (2 5  1)(2 5  1) .
(b) Hence rationalize
3
2 5 1

5
2 5 1
.
53. (a) Simplify (5 6  2)(5 6  2) .
(b) Hence rationalize
2
5 6 2

3
5 6 2
.
54. Simplify [( 5  2 )  1][( 5  2 )  1] .
55. Simplify [(2 5  13 )  3][(2 5  13 )  3] .
Ex.1B Advanced Set
1 2
)
15
48. (a) (5  13 ) 2
10
New Trend Mathematics S4A — Supplement
Exercise 1C
  El em en tar y S et
Level 1
 
1. If f (x)  x  5, find the values of the following.
(a) f (0)
(b) f (4)
(c) f (4)
2. If g(x)  x 2  1, find the values of the following.
(a) g(4)
(b) g(3)
3
, find the values of the following.
x
(a) f (1)
(b) f (6)
1
(c) g ( )
2
3. If f ( x) 
4. If f ( x)  10 
(a) f (2)
5. If f ( x) 
1
, find the values of the following.
2x
(b) f (1)
1
Ex.1C Elementary Set
x 4
2
(a) f (5)
3
(c) f ( )
2
3
(c) f ( )
2
, find the values of the following.
(b) f (2)
6. If f (x)  5x 2  x, find the values of the following.
1
1
(a) 2 – f (2)
(b)  f ( )
2
2
1
(c) f ( )
2
(c)
12
f (1)
7. If g(x)  3x  1, find the values of the following.
(a) g(5)
(c) g(5)  g(5)
(b) g(5)
(d) g(5)  g(5)
8. If h(x)  x(5 – x), find the values of the following.
(a) 5  h(1)
h(10)
(c)
10
(b) h(4)  2
(d) h(2)  h(1)
9. If f (x)  2x  1, find the values of the following.
(a) f (2)
(b) 2 f (2)
x2  1
, find the values of the following.
2
1
(a) f (1)
(b)  f (3)
5
(c) [f (2)] 2
10. If f ( x) 
(c) [f (2)] 2
Chapter 1 Numbers and Functions
11
11. Given that f (x)  2x, g(x)  5x 3 and F(x)  g(x) – f (x), find the values of the following.
1
(a) F(1)
(b) F(1)
(c) F ( )
2
12. Given that f (x)  2x – 5, g(x)  x  1 and G(x)  f (x)  2g(x), find the values of the
following.
1
(a) G(0)
(b) G(5)
(c) G ( )
2
13. If f (x)  3(x – 5), find
(a) f (n).
(b) f (n  1).
(c) f (n)  f (n  1).
14. If g(x)  (x  4) 2 , find
(a) g(n).
(b) g(n  1).
(c) g(n)  g(n  1).
Level 2
16. If f (x) = kx + 8 and f (9) = 44, find the value of k.
1
17. If g(x) = k(2x 2 + 1) and g ( )  3 , find the value of k.
2
18. It is given that f (x)  kx  2k and f (10)  36.
(a) Find the value of k.
(b) Hence find the value of f (5).
19. Let f (x)  4x – 5 and g(x)  8x  13.
(a) If f (x)  g(x), find the value of x.
(b) If H(x)  g(x) – f (x) and H(x)  2, find the value of x.
20. It is given that f (x)  kx – 7 and f (3)  1.
(a) Find the value of k.
(b) Find f (a) and f (a  1).
(c) Hence find the value of a such that 3f (a)  f (a  1).
21. Let f (x)  2x – 1.
(a) Find f (2x).
(b) Find f (x – 1).
(c) Hence find the value of x such that f (2x)  f (x  1).
1
(c) g[g ( )]
2
Ex.1C Elementary Set
15. If g(x)  x – 2, find the values of the following.
1
1
(a) g ( )
(b) [g ( )]2
2
2
12
New Trend Mathematics S4A — Supplement
Ex.1C Elementary Set
22. Let f (x  1)  2x  2.
(a) Find the value of f (5).
(b) Find the function f (x).
(c) Find the value of x such that 2f (x)  f (x  1).
23. The length and width of a rectangular plot are 4 m and x m respectively. The perimeter of
the rectangular plot is y m. It is given that y is a function of x, denoted by f (x).
(a) Find f (x).
(b) Find the perimeter of the rectangular plot if the width of the plot is 5 m.
(c) Find the width of the plot if the perimeter is 14 m.
 Intermediate Set
Level 1
24. If g(x)  3x – 2, find the values of the following.
(a) g(2)
(b) g(2)

(c) g(5)
25. If f (x)  (x  1)(x  3) , find the values of the following.
(a) f (2)
26. If f ( x) 
(b) f (3)
x2
2  x2
(a) f (1)
1
(c) f ( )
2
, find the values of the following.
Ex.1C Intermediate Set
1
(b) f ( )
2
(c) f ( 5 )
27. If g(x) = 8x, find the values of the following.
3
(a) 10  g(1)
(b) 2  g ( )
4
12
(c) 5  g(1)
(d)
g ( 34 )
28. If f (x)  x 2 – 2x, find the values of the following.
(a) f (0)
(b) f (3)
f ( 0)
(c) f (0)  f (3)
(d)
f (3)
5
, find the values of the following.
2
(b) f (2)  f (5)
2  f (3)
29. If f ( x)  x 2 
(a)
30. If f (x) = 2x 2 – 1, find the values of the following.
1
1
(a) 4  f ( )
(b) f ( )  f (2)
2
2
(c) f (1)  f (1)
1
(c) [ f ( )]2
2
Chapter 1 Numbers and Functions
13
31. Given that f (x)  2x, g(x)  4x 2  2x and G(x)  f (x)  g(x), find the values of the
following.
1
(a) G(2)
(b) G(2)
(c) G ( )
2
32. Given that f (x)  3x 2  1, g ( x) 
1
1
and H ( x)  f ( x)  g ( x) , find the values of the
x2
2
following.
(b) H (2)
1
(c) H ( )
3
33. If f (x)  8x 2 , find
(a) f (n).
(b) f (n  1).
(c) f (n)  f (n  1).
34. If h(x)  2x 2  x, find
(a) h(n).
(b) h(n  1).
(c) h(n  1)  h(n).
(a) H (1)
(a) g(1)
(b) [g(1)] 2
(c) g[g(1)]
Level 2
36. If f (x)  x 2  k and f (2)  9, find the value of k.
37. If g(x)  x 2  kx  4 and g(2)  0, find the value of k.
1
38. If f (x)  9x 2  k and f ( )  10 , find the value of k.
3
1
39. It is given that f (x)  x 2  3kx – 1 and f (1)  f ( ) .
3
(a) Find the value of k.
(b) Hence find the value of f (3).
40. It is given that f (x) = 4kx  k and f (5)  42.
(a) Find the value of k.
(b) Hence find the value of f (2) – f (1).
41. Let f (x) = 3x – 2 and g(x)  3 – 2x.
1
(a) If 2 f ( x)   g ( x) , find the value of x.
2
(b) If H(x)  3f (x) – 2g(x) and H(x)  1, find the value of x.
42. It is given that f (x)  k(x  1)(x  2) and f (3)  1.
(a) Find the value of k.
(b) Hence find the value of a such that f (a)  f (a  5).
Ex.1C Intermediate Set
35. If g(x)  x 2  x – 5, find the values of the following.
14
New Trend Mathematics S4A — Supplement
43. Let g(x)  2x 2 – 1.
(a) Find g(3x).
(b) Find g(x  1).
(c) Hence find the value of x such that g(3x)  9g(x  1).
Ex.1C Intermediate Set
44. Let f (2x)  8x  1.
(a) Find the value of f (5).
(b) Find the function f (x).
(c) Find the value of x such that f (2x)  f (x)  1.
45. Peter needs to prepare class picnic. It is known that the cost for hiring a coach is $600 and
that for the food per student is $30. The total expenditure is $ y if there are x students
joining the picnic.
(a) Express y in terms of x. Explain whether y is a function of x.
(b) (i) When x  40, find the value of y.
(ii) When y  1 650, find the value of x.
46. Vincent deposits $50 000 in a bank at an interest rate of 0.2% p.a. for x years on simple
interest. The interest he will earn is $y. It is given that y is a function of x, denoted by f (x).
(a) Find f (x).
(b) Calculate the interest he will earn after 2 years from the deposit date.
(c) How long does Vincent take to earn $500 simple interest?
 Advanced Set
Level 1

47. If f (x)  2x 2  1, find the values of the following.
(a) f (0)
(b) f (5)
1
, find the values of the following.
5x
1
(a) f (1)
(b) 4  f ( )
10
1
(c) f ( )
2
48. If f ( x) 
Ex.1C Advanced Set
(c)
1
f ( 2 )
49. If g(x)  2x 2 – x – 1, find the values of the following.
(a) g(1)
(c) g(1)  g(2)
(b) g(2)
g (1)
(d)
g ( 2)
x
, find the values of the following.
1 x
(a) f (3)
(b) f (3)  f (3)
50. If f ( x) 
(c) [f (3)] 2
(d) f (3)  f (3)
51. Given that f (x)  2x, g(x)  (x  1) 2 and H(x)  f (x)  g(x), find the values of the following.
(a) H(0)
(b) H(3)
(c) H(4)
Chapter 1 Numbers and Functions
15
52. Given that f (x)  3x  1 and g(x)  x 2  1, find the values of the following.
1
(a) f (4)  g(4)
(b) 2 f ( 5 )  3g ( 5 )
(c)
 f (6)  g (6)
14
53. If k(x)  x 2  2x, find
(a) k(n).
(b) k(n  2).
2x
, find the values of the following.
x 1
1
1
(a) g ( )
(b) [g ( )]2
2
2
(c) k(n)  k(n  2).
54. If g ( x) 
1
(c) g[g ( )]
2
Level 2
55. If f (x)  x 2  kx  1 and f ( 2 )  5 , find the value of k.
1
1
57. It is given that f (x)  3x 2  x – k and f ( )  .
2
2
(a) Find the value of k.
(b) Hence find the value of f (0).
58. It is given that g(x)  kx 2  1 and g(1)  4  g(3).
(a) Find the value of k.
(b) Hence find the value of g(4)  g(4).
59. Let f (x)  (x – 7)(x  2) and g(x)  14 – 5x.
(a) If f (x) = g(x) – 12, find the values of x.
(b) If H(x)  5f (x)  xg(x) and H(x)  7, find the value of x.
60. It is given that f (x)  x 2  k and f (2)  7.
(a) Find the value of k.
(b) Hence find the value of a such that f (a)  f (a  1).
61. It is given that g(x)  kx 2  8x and g(5)  15.
(a) Find the value of k.
(b) Hence find the value of a such that g(a)  2g(a – 1)  a 2  2.
62. Let f (x)  5x – 4.
(a) Find f (3x).
(b) Find f (x – 4).
(c) Hence find the value of x such that f (3x) = f (x – 4).
Ex.1C Advanced Set
56. If f (x)  x(x  k) and f (5)  0, find the value of k.
16
New Trend Mathematics S4A — Supplement
63. Let f (x)  (x – 1)(2x  1).
(a) Find f (2x).
(b) Find f (x – 1).
(c) Hence find the value of x such that f (2x) – 4f (x – 1) = 0.
64. Let f (x  1)  x 2 – 1.
(a) Find the value of f (–1).
(b) Find the function f (x).
(c) Find the value of x such that f (x)  f (x  1).
x
x2
 3x  1.
65. Let f ( ) 
2
2
Ex.1C Advanced Set
1
(a) Find the value of f ( ) .
2
(b) Find the function f (x).
(c) Find the value of x such that f (x) = f (x – 1).
66. A company is planning for an annual dinner. It is known that the rent of a function room is
$3 000 and the cost of food for each person is $100. The total expenditure is $y if there are
x participants.
(a) Express y in terms of x. Explain whether y is a function of x.
(b) (i) When x  250, find the value of y.
(ii) When y  50 000, find the value of x.
67. The base radius of a cylinder is r cm and the height of the cylinder is 3 times its base
radius. The total surface area of the cylinder is A cm2 . It is given that A is a function of r,
denoted by f (r).
(a) Find f (r).
(b) Find the total surface area of the cylinder if its base radius is 10 cm. (Express your
answer in terms of .)
(c) Find the base radius of the cylinder if its total surface area is 50  cm2 .
C HAPTER T EST
(Time allowed: 1 hour)
Section A
1. (a) Simplify (2n  1) 2  1.
(b) Given that n is an integer, find the value of (1)
2. Simplify ( 24  2 3 )( 24  2 3 ) .
(1 mark)
( 2n1)2 1
 (1) .
2
(2 marks)
(3 marks)
Chapter 1 Numbers and Functions
3. (a) Rationalize
4
6
.
(b) Hence solve the equation 2 6 ( x  1)  8 .
17
(1 mark)
(3 marks)
4. It is given that f (x)  3x  2. Find the value of k such that f (k  1)  k  15.
(4 marks)
5. Let f (x)  2x and g (x)  x 2 .
(a) Find the values of f (2) and g(2).
(b) Find the values of x such that g[f (x)] – f [g(x)]  16.
(2 marks)
(3 marks)
6. (a) Convert the following recurring decimals into fractions.
(i) 0.2 0
(ii) 0.5 0
(b) Hence solve the equation 0.2 0 x  6  2.5 0 x .
(2 marks)
(4 marks)
Section B
7. It is given that g(x)  4x  5.
(a) Find [g(x  1)] 2 .
2
(b) Find g[g(x )].
(c) Hence solve the equation [g(x  1)] 2  g[g(x 2 )].
(4 marks)
(4 marks)
(2 marks)
8. The height of a rectangular box is 15 cm and the length of each side of its square base is
x cm.
15 cm
x cm
x cm
(a) Let the volume of the rectangular box be V cm3 , where V is a function of x, denoted by
f (x).
(i) Find the function f (x).
(ii) Find the value of V when x  12.
(iii) Find the value of x when V  960.
(5 marks)
(b) Let the total surface area of the rectangular box be A cm2 , where A is a function of x,
denoted by g(x).
(i) Find the function g(x).
(ii) Find the value of A when x  12.
(iii) If the painting cost is $0.2 per cm 2 , find the cost for painting a rectangular box with
the sides of the square base of 10 cm each.
(5 marks)
18
New Trend Mathematics S4A — Supplement
Multiple Choice Questions (3 marks each)
9. If n is a positive integer, which of the
following must be true?
I. 3 2n is even.
II. 3
2n
14.
3 2
1
32

A. 0.
 1 is even.
B. 2 3 .
2n  1
III. 3
is odd.
A. II only
B. I and II only
C. II and III only
D. I, II and III
C. 4.
D. 2 3  4 .
10. If x and y are two consecutive numbers,
which of the following must be true?
A. xy is odd.
B. (x  1)(y  2) is odd.
C. (2x  1)(2y  1) is odd.
15. Let f (x)  3x 2 – kx  5 and g(x)  3x – 7.
What is the value of k such that
f (1) – g(1)  0?
A. –12
B. 0
C. 12
D. 20
D. x 2  y 2 is even.
11. Which of the following
irrational number?
is
7 2
A.
B. ( 2 )( 3 )
6
5

4
4
C.
D. ( 2  3)( 2  3)
12. Simplify
300  2 24  192 .
A. 6 3
B. 2 3  8 6
C. 24 3  4 6
D. 2 3  4 6
13.
12  2
2

A.
6  2.
B.
6 2.
C.
12  2 .
D.
24  4 .
not
an
16. If f (x)  x  1 and g(x)  x  1, then
g[ f (x)] 
A. x.
B. x  2.
C. x  2.
D. x 2  1.
17. If f (x  1)  4x  2, then f (x) 
A. 4x – 2.
B. 4x  1.
C. 4x  3.
D. 4x  6.
1
1
 1  2 , then x 2  2 
x
x
A. 3  2 2 .
18. If x 
B. 6.
C. 1  2 2 .
10
D.
.
3
Chapter 1 Numbers and Functions
H INTS
(for questions with
19
in the textbook)
Revision Exercise 1
26. (a) Key information
 f (x)  2x 2  x  k where k is an integer.
 f (x) can be expressed as the product of two binomials with integral coefficients and
constant terms.
Analysis
This question can be tackled by trial and error or working backwards.
Method
 By trial and error: Simply assuming k to be a particular integer and check if
2x 2  x  k can be factorized.
 By working backwards: Let f (x)  (ax  b)(cx  d), expand (ax  b)(cx  d) and
compare the coefficients and the constant term of the expression with those of
2x 2  x  k.
28. (a) Key information
 APQR is a rectangular garden.
 ABC forms a right-angled triangle.
 AC  12 m
 AB  9 m
 AR  x m
Analysis
Since no congruent triangles can be observed in this question and the Pythagoras’
theorem does not help to find the length of AP, other methods such as using
trigonometry or similar triangles should be considered.
Method
Since APQR is a rectangular garden, AP  RQ.
By knowing that ABC ~ RQC or tanACB  tanRCQ, we can use
express RQ in terms of x.
QR BA
to

RC AC