Chapter 1Numbers and Functions 1
Chapter 1Numbers and Functions
Warm-up Exercise1.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 x1, find the values of the following expressions.
(a)x2 8x 16(b)3x2 – 10x – 24
Simplify the following expressions, and express your answers in positive indices. (4–7)
4.(a)a7 × a3 × a2(b)a4 × a5 × a3
5.(a)(b)
6.(a)(b)
7.(a)(4a2b3)1 × (2ab)2(b)
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)
Build-up Exercise[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
Elementary Set
Level 1
1.Determine whether each of the following is an even number or an odd number where N is an integer.
(a)N (N 2)(b)
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 expressions.
(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.
(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)(b)(c)
Level 2
11.(a)Simplify (N 1)2–N2.
(b)It is known that N is an integer. Prove that (N 1)2–N2 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.
Intermediate Set
Level 1
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 expressions.
(a)(–1)N 3N 5N 7N(b)(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)
16.It is known that N is an integer. Show that 9(N 1) – (N 5) is an even number.
17.The sum of two consecutive even numbers is 38. Find the numbers.
18.The sum of three consecutive numbers is 93. Find the numbers.
19.Convert the following recurring decimals into fractions.
(a)(b)(c)
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 N2 (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)(ii)
(iii)(iv)
(b)(i)Is it true that?
(ii)Is it true that?
23.Find the value of x if
(a).(b)
Advanced Set
Level 1
24.Given that N is an integer, find the values of the following expressions.
(a)
(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)
(b)
(c)
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)Convertinto fraction.
(b)Using the result of (a), convertinto fraction.
32.(a)Convert the following recurring decimals into fractions.
(i)(ii)
(iii)(iv)
(b)(i)Is it true that?
(ii)Is it true that?
33.Find the value of x if
(a).
(b).
(c).
Exercise 1B
Elementary Set
Level 1
Simplify the following surds. (14)
1.(a)(b)(c)
2.(a)(b)(c)
3.(a)(b)(c)
4.(a)(b)(c)
5.(a)Express the following as entire surds.
(i)(ii)(iii)
(iv)(v)(vi)
(b)Arrange the above surds in ascending order.
Simplify the following. (69)
6.(a)(b)(c)
7.(a)(b)
8.(a)(b)
(c)(d)
9.(a)(b)
(c)(d)
10.Rationalize the following.
(a)(b)
(c)(d)
11.Rationalize.
Simplify the following and rationalize the results. (1214)
12.(a)(b)
(c)(d)
13.(a)(b)
14.(a)(b)
Level 2
Simplify the following. (1516)
15.(a)(b)
16.(a)(b)
17.(a)Simplify.
(b)Hence rationalize.
18.(a)Simplify.
(b)Hence rationalize.
Intermediate Set
Level 1
Simplify the following surds. (1920)
19.(a)(b)(c)
20.(a)(b)(c)
(d)(e)(f)
21.(a)Express the following as entire surds.
(i)(ii)(iii)
(iv)(v)(vi)
(b)Arrange the above surds in ascending order.
Simplify the following. (2225)
22.(a)(b)(c)
23.(a)(b)
24.(a)(b)
(c)(d)
25.(a)(b)
(c)(d)
26.Rationalize the following.
(a)(b)(c)
27.Rationalize.
Simplify the following and rationalize the results. (2830)
28.(a)(b)
(c)(d)
29.(a)(b)
30.(a)(b)
Level 2
Simplify the following. (3132)
31.(a)(b)(c)
32.(a)(b)(c)
33.(a)Simplify.
(b)Hence rationalize.
34.(a)Simplify .
(b)Hence rationalize.
35.(a)Simplify.
(b)Hence rationalize.
36.Simplify .
Advanced Set
Level 1
Simplify the following surds. (3738)
37.(a)(b)(c)
38.(a)(b)(c)
Simplify the following. (3941)
39.(a)(b)
(c)(d)
40.(a)(b)
41.(a)(b)
(c)(d)
42.Rationalize the following.
(a)(b)(c)
43.Rationalize.
Simplify the following and rationalize the results. (4446)
44.(a)(b)(c)
45.(a)(b)
46.(a)(b)
Level 2
47.(a)Simplify the following and rationalize the results.
(i)(ii)(iii)
(iv)(v)(vi)
(b)Arrange the above surds in descending order.
Simplify the following. (4849)
48.(a)(b)(c)
49.(a)(b)(c)
50.(a)Simplify.
(b)Hence rationalize.
51.(a)Simplify.
(b)Hence rationalize.
52.(a)Simplify.
(b)Hence rationalize.
53.(a)Simplify.
(b)Hence rationalize .
54.Simplify.
55.Simplify.
Exercise 1C
Elementary Set
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) x2 1, find the values of the following.
(a)g(4)(b)g(3)(c)
3.If, find the values of the following.
(a)f(1)(b)f(6)(c)
4.If, find the values of the following.
(a)f(2)(b)f(1)(c)
5.If, find the values of the following.
(a)f(5)(b)f(2)(c)
6.If f(x) 5x2x, find the values of the following.
(a)2 – f(2)(b)(c)
7.If g(x) 3x 1, find the values of the following.
(a)g(5)(b)g(5)
(c)g(5) g(5)(d) g(5) g(5)
8.If h(x) x(5 –x), find the values of the following.
(a)5 h(1)(b)h(4) 2
(c)(d) h(2) h(1)
9.If f(x) 2x 1, find the values of the following.
(a)f(2)(b)2f(2)(c)[f(2)]2
10.If, find the values of the following.
(a)f(1)(b)(c)[f(2)]2
11.Given that f(x) 2x, g(x) 5x3 and F(x) g(x) – f(x), find the values of the following.
(a)F(1)(b)F(1)(c)
12.Given that f(x) 2x – 5, g(x) x 1 and G(x) f(x) 2g(x), find the values of the following.
(a)G(0)(b)G(5)(c)
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).
15.If g(x) x – 2, find the values of the following.
(a)(b)(c)
Level 2
16.If f(x) = kx + 8 and f(9) = 44, find the value of k.
17.If g(x) = k(2x2 + 1) and, find the value of k.
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).
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)(b)f(3)(c)
26.If, find the values of the following.
(a)f(1)(b)(c)
27.If g(x) = 8x, find the values of the following.
(a)10 g(1)(b)
(c)5 g(1)(d)
28.If f(x) x2 – 2x, find the values of the following.
(a)f(0)(b)f(3)
(c)f(0) f(3)(d)
29.If, find the values of the following.
(a)(b)(c)
30.If f(x) = 2x2 – 1, find the values of the following.
(a)(b)(c)
31.Given that f(x) 2x, g(x) 4x2 2x and G(x) f(x) g(x), find the values of the following.
(a)G(2)(b)G(2)(c)
32.Given that f(x) 3x2 1,and, find the values of the following.
(a)H(1)(b)H(2)(c)
33.If f(x) 8x2, find
(a)f(n).(b)f(n 1).(c)f(n) f(n 1).
34.If h(x) 2x2x, find
(a)h(n).(b)h(n 1).(c)h(n 1) h(n).
35.If g(x) x2x – 5, find the values of the following.
(a)g(1)(b)[g(1)]2(c)g[g(1)]
Level 2
36.If f(x) x2k and f(2) 9, find the value of k.
37.If g(x) x2kx 4 and g(2) 0, find the value of k.
38.If f(x) 9x2k and, find the value of k.
39.It is given that f(x) x2 3kx – 1 and.
(a)Find the value of k.
(b)Hence find the value of f(3).
40.It is given that f(x) = 4kxk 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.
(a)If, find the value of x.
(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).
43.Let g(x) 2x2 – 1.
(a)Find g(3x).
(b)Find g(x 1).
(c)Hence find the value of x such that g(3x) 9g(x 1).
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) 2x2 1, find the values of the following.
(a)f(0)(b)f(5)(c)
48.If, find the values of the following.
(a)f(1)(b)(c)
49.If g(x) 2x2 – x – 1, find the values of the following.
(a)g(1)(b)g(2)
(c)g(1) g(2)(d)
50.If, find the values of the following.
(a)f(3)(b)f(3) f(3)
(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)
52.Given that f(x) 3x 1 and g(x) x2 1, find the values of the following.
(a)f(4) g(4)(b)(c)
53.If k(x) x2 2x, find
(a)k(n).(b)k(n 2).(c)k(n) k(n 2).
54.If, find the values of the following.
(a)(b)(c)
Level 2
55.If f(x) x2kx 1 and, find the value of k.
56.If f(x) x(xk) and f(5) 0, find the value of k.
57.It is given that f(x) 3x2x – k and.
(a)Find the value of k.
(b)Hence find the value of f(0).
58.It is given that g(x) kx2 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) x2k 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) kx2 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) a2 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).
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) x2 – 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).
65.Let.
(a)Find the value of.
(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.
Chapter Test / (Time allowed: 1 hour)Section A
1.(a)Simplify (2n 1)2 1.(1 mark)
(b)Given that n is an integer, find the value of.(2 marks)
2.Simplify.(3 marks)
3.(a)Rationalize.(1 mark)
(b)Hence solve the equation.(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) x2.
(a)Find the values of f(2) and g(2).(2 marks)
(b)Find the values of x such that g[f(x)] – f[g(x)] 16.(3 marks)
6.(a)Convert the following recurring decimals into fractions.
(i)
(ii)(2 marks)
(b)Hence solve the equation.(4 marks)
Section B
7.It is given that g(x) 4x 5.
(a)Find [g(x 1)]2.(4 marks)
(b)Find g[g(x2)].(4 marks)
(c)Hence solve the equation [g(x 1)]2g[g(x2)].(2 marks)
8.The height of a rectangular box is 15 cm and the length of each side of its square base is xcm.
(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 cm2, find the cost for painting a rectangular box with the sides of the square base of 10 cm each. (5 marks)
Multiple Choice Questions (3 marks each)
Chapter 1Numbers and Functions 1
9.If n is a positive integer, which of the following must be true?
I.32n is even.
II.32n 1 is even.
III.32n 1 is odd.
A.II only
B.I and II only
C.II and III only
D.I, II and III
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.
D.x2y2 is even.
11.Which of the following is not an irrational number?
A.
B.
C.
D.
12.Simplify.
A.
B.
C.
D.
13.
A..
B..
C..
D..
14.
A.0.
B..
C.4.
D..
15.Let f(x) 3x2 – 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
16. If f(x) x 1 and g(x) x 1, then g[f(x)]
A.x.
B.x 2.
C.x 2.
D.x2 1.
17.If f(x 1) 4x 2, then f(x)
A.4x – 2.
B.4x 1.
C.4x 3.
D.4x 6.
18.If, then
A..
B.6.
C..
D..
Chapter 1Numbers and Functions 1
Hints / (for questions with in the textbook)Revision Exercise 1
26.(a)Key information
f(x) 2x2xk 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 2x2xk can be factorized.
By working backwards: Let f(x) (axb)(cxd), expand (axb)(cxd) and compare the coefficients and the constant term of the expression with those of 2x2xk.
28.(a)Key information
APQR is a rectangular garden.
ABC forms a right-angled triangle.
AC 12 m
AB 9 m
ARx 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, APRQ.
By knowing that ABC ~ RQC or tanACB tanRCQ, we can useto express RQ in terms of x.