Essential Mathematical Methods 3 & 4 CAS
Chapter 4 Polynomial functions: SAC Revision
Multiple-choice questions
1 The expression (2x + 3)2 + 5x - 1 is equal to:
A 2x2 + 5x + 8
B 2x2 +17x + 8
C 4x2 + 5x + 8
D 4x2 + 11x + 8
E 4x2+ 17x + 8
2 The expression 6x2 + 7x - 20 is equal to:
A (3x - 2)(2x + 10)
B (3x - 4)(2x + 5)
C (3x - 5)(2x + 4)
D (3x + 2)(2x - 10)
E (3x + 4)(2x - 5)
3 The value(s) of x for which 2x2 + x - 1 < 0 are:
A {x : x < -1}
B {x : x < } È { x : x > 1}
C {x : x < -1} È { x : x > }
D {x : < x < 1}
E {x : -1 < x < }
4 The coordinates of the turning point of the graph with equation y = x2 + 4x + 5 are:
A (2, 1)
B (1, -2)
C (2, -1)
D (-2, 1)
E (-2, -1)
5 The equation of the curve shown is most likely to be:A y – 5 = (x + 2)2
B y + 5 = (x – 2)2
C y = (x + 2)2 – 5
D y + 2 = (x – 5)2
E y + 5 = (x + 2)2 /
6 If the graph of y = 2x2 – kx + 3 touches the x-axis then the possible values of k are:
A k = 2 or k = –3
B k = 1
C k = – 3 or k = –
D k = 1 or k = 3
E k = 2or k = –2
7 The curve with equation y = x3 is transformed under a dilation of factor 4 from the y-axis followed by a translation of 6 units in the positive direction of the x-axis. The equation of the image is:
A y = 4(x – 3)3
B y = 4x3 + 3
C y =
D y =
E y =
8 The x-axis intercepts of the graph with equation y = (x + a)2(x2 – b2), where a and b are positive constants, are:
A a, –a, b, –b
B a, b, –b
C a, –a, b
D –a, b, –b
E a, –a, –b
9 The curve with equation y = x4 is transformed under a dilation of factor 5 from the y-axis, followed by a translation of 3 units in the positive direction of the y-axis. The equation of the image is:
A y = 5(x – 3)4
B y = 625x3 + 3
C y =
D y = + 3
E y = + 3
10 The equation of the graph shown could be:
A y = x(x - 1)(x + 3)
B y = x(x + 1)(x - 3)
C y = (x + 1)2(x - 3)
D y = (x + 1)(x - 3)2
E y = (x + 1)2(x - 3)2
11 If, when the polynomial P(x) = x3 + 2x2 – 10x + d is divided by x - 1, the remainder is 6, the value of d is:
A 10
B 14
C –10
D 2
E 13
12 The function with rule y = f(x) is shown below.
Which one of the following could be the graph of the function with rule y = f(–x) + 1?
A / / B /C / / D /
E /
13 If 3(x - 1)2 = a(x + 1)2 + bx, the values of a and b are:
A a = 3, b = 12
B a = 3, b = -12
C a = 3, b = 0
D a = 3, b = 6
E a = 3, b = - 6
14 The maximum value of the function f: [-1, 1] ®R, f(x) = | x2 - 1| + 3 is:
A 0
B 2
C 3
D 4
E 5
15 If ax3 + x2 + 6 is exactly divisible by x + 1, the value of a is
A 1
B 7
C –1
D 4
E –7
16 If the solutions of x2 + bx + 15 = 0 are integers less than 10 but greater than –10, the possible values of b are:
A 5 and 7
B –5
C –3 and –5
D ±5 and ±3
E ±8
17 For the line y = x to be tangent to the curve with equation y = :
A k = 1
B k > 1
C k > –
D k = –
E 4k + 1 < 0
18 The equation x2 – bx – c = 0 has two distinct solutions if and only if:
A b2 > 4c
B b2 > –4c
C b2 = 4c
D b2 + 4c < 0
E b < c
Short-answer questions (technology-free)
1 Solve the equation x2 -3x - 7 = 0, giving exact solutions.
2 Find the distance between the x-axis intercepts of the parabola with equation
y = x2 + 2x - 5
3 A parabola has turning point (2, -3) and passes through the point (3, 6). Find its equation.
4 A parabola has x intercepts of -1 and 2 and passes through the point (3, 6). Find the equation of the parabola.
5 A farmer has a straight, fenced road along the boundary of his property. He wishes to fence an enclosure and has enough materials to erect 500 m of fence. What would be the dimensions to enclose the largest possible rectangular area, assuming that he uses the existing boundary fence as one of the sides?
6 Sketch the graph of f: R R, f(x) = –3(x + 1)3 + 2
7 Give the sequence of transformations which takes the graph of y = x3 to the graph of
y = -3(x - 1)3 + 2
8 a Find in terms of p the remainder when x3 - 2x2 + px - 6 is divided by x - 2.
b Find the value of p for which x3 - 2x2 + px - 6 is exactly divisible by x - 2.
9 The graph of of y = f(x) has rule of the form f(x) = a(x + b)3 + 2.
Find the values of a and b.
10 For the quadratic with rule y = 2x2 + mx + 4, find the values of m for which there is:
a one solution
b two solutions
11 Find the coordinates of the points of intersection of the line with equation y = x + k and the parabola with equation y = x2 – 4x, where k > 0.
12 Find the coordinates of the points of intersection of the line with equation y = kx + 1 and the circle with equation x2 + y2 = 9
Extended-response questions
A cuboid (rectangular prism) has dimensions x metres, hmetres and 5x metres as shown on the diagram. The cuboid is made of 640 m of wire. /1 Find h in terms of x.
2 Find the volume, V m3, of the cuboid in terms of x.
3 Find V when x = 11
4 Find the possible values of x for the cuboid to exist.
5 Find the possible values of x when V = 60 000, correct to two decimal places.
6 Find the maximum volume of the cuboid and the value of x for which it occurs, correct to two decimal places.
Answers to Chapter 4 Test A
Answers to multiple-choice questions
1 E
2 B
3 E
4 D
5 B
6 E
7 C
8 D
9 E
10 C
11 E
12 D
13 B
14 D
15 B
16 E
17 D
18 B
Answers to short-answer (technology-free) questions
1 x =
2 2
3 y = 9(x – 2)2 – 3
4 y =(x +1)(x – 2)
5 125 m ´ 250 m
6
7 dilation of factor 3 from the x-axis, reflection in the x-axis, translation 1 unit in the positive direction of the x-axis, 2 units in the positive direction of the y-axis
8 a 2p – 6
b p = 3
9 a = , b = –1
10 a m = ± 4
b m > 4 or m < –4
11 (, )
12 (,) or
(,)
Answers to extended-response questions
1 h =160 – 6x
2 V = 5x2(160 – 6x)
3 V =56 870
4 0 < x <
5 x = 11.47 or x = 22.83
6 maximum = 84 279.84 m3 when x = 17.78 m
2