Linear Functions and Lines Solutions

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Linear Functions and Lines

Mathletics Instant

Workbooks

y = mx+c

Teacher Book - Series L-2

94 EXCEL ESSENTIAL SKILLS: PRELIMINARY MATHEMATICS REVISION AND EXAM WORKBOOK

Linear functions and lines

TOPIC TEST SECTION I

1 The equation of a line with gradient –1 and y-intercept 2 is

A y = –x – 2 B y = 2x – 1 C y = 2 – x D none of these

2 Which point lies on the line 2x + 3y – 5 = 0?

A (–1, 1) B (–1, –1) C (1, –1) D (1, 1)

3 The gradient of the line joining A to B is

A−23

B −32

C 23

D 32

4 A line, parallel to the x-axis, passes through the point (3, –5). Its equation is

A x = 3 B x = –5 C y = 3 D y = –5

5 A line passes through the origin and makes an angle of 45° with the positive direction ofthe x-axis. The gradient of the line is

A 0 B –1 C 1 D 45

6 The gradient of any line parallel to 4x – 2y + 3 = 0 is

A 2 B –2 C 12

D − 12

7 The shaded region is where

A 4x – 6y – 3 ≤ 0 B 4x – 6y – 3 ≥ 0

C 4x – 6y – 3 < 0 D 4x – 6y – 3 > 0

8 The gradient of any line perpendicular to y x= +13

2 is

A − 13

B 13

C –3 D 3

Marks

1

1

1

1

1

1

1


Instructions • This section consists of 12 multiple-choice questions

• Each question is worth 1 mark• Fill in only ONE CIRCLE for each question• Calculators may be used

Time allowed: 12 minutes Total marks = 12

1

x

y

–1 0 1 2 3 4 5

5

4

3

2

1

–1

B

A

x

y

–2 –1 0 1 2 3 4

4

3

2

1

–1

–2

4x – 6y – 3 = 0

iiLinear functions and lines

Mathletics Instant Workbooks – Series L 2 Copyright © 3P Learning

Linear functions and linesTopic Test PART AInstructions This part consists of 12 multiple-choice questions Each question is worth 1 mark Calculators may be used Fill in only ONE CIRCLE for each question

Time allowed: 12 minutes Total marks = 12

95CHAPTER 3 – Linear functions and lines

Linear functions and lines Marks

1

1

1

1

9 Which line passes through the point (–2, 5)?

A 2x – 5y = 0 B 2x + 5y = 0 C 5x + 2y = 0 D 5x – 2y = 0

10 The distance between the points (–1, 5) and (7, 5) is

A 5 units B 6 units C 7 units D 8 units

11 Which point lies within the region determined by the inequalities 2x + y < 0and 3x – 4y + 5 ≥ 0?

A (–4, 2) B (–1, –3) C (2, 6) D (5, 2)

12 Which diagram shows the region where x ≤ 0 and y ≥ 0?

A

x

y B

x

y C

x

y D

x

y

Total marks achieved for SECTION I 12



1

1

1

1

9 Which line passes through the point (–2, 5)?

A 2x – 5y = 0 B 2x + 5y = 0 C 5x + 2y = 0 D 5x – 2y = 0

10 The distance between the points (–1, 5) and (7, 5) is

A 5 units B 6 units C 7 units D 8 units

11 Which point lies within the region determined by the inequalities 2x + y < 0and 3x – 4y + 5 ≥ 0?

A (–4, 2) B (–1, –3) C (2, 6) D (5, 2)

12 Which diagram shows the region where x ≤ 0 and y ≥ 0?

A

x

y B

x

y C

x

y D

x

y

Total marks achieved for SECTION I 12

iiiLinear functions and lines


Linear functions and linesTopic Test PART A

12Total marks achieved for PART A

Linear functions and linesTopic Test PART B

ivLinear functions and lines


Instructions This section consists of 18 questions Show all necessary working

Time allowed: 1 hour Total marks = 88



8

13 Draw, on the number plane provided, the graph of:

a y = –x + 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

b y = 2x – 3

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

c x = –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

d y = 1

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

14 Write down the gradient and y-intercept of each line.

a y = 2x – 5 b y = –3x c x + y = 4

_____________________ ___________________ ___________________

_____________________ ___________________ ___________________

15 For the line 2x + 3y – 6 = 0

a find the gradient ________________________________

________________________________

b find the y-intercept ______________________________

______________________________

c graph the line on the number plane provided.

16 Write the equation:

a y = 2x – 7 in general form b x – 3y + 9 = 0 in gradient-intercept form

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

TOPIC TEST SECTION II


Instructions • This section consists of 18 questions

• Show all necessary working

Marks


6

6

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

6



8

13 Draw, on the number plane provided, the graph of:

a y = –x + 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

b y = 2x – 3

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

c x = –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

d y = 1

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

14 Write down the gradient and y-intercept of each line.

a y = 2x – 5 b y = –3x c x + y = 4

_____________________ ___________________ ___________________

_____________________ ___________________ ___________________

15 For the line 2x + 3y – 6 = 0

a find the gradient ________________________________

________________________________

b find the y-intercept ______________________________

______________________________

c graph the line on the number plane provided.

16 Write the equation:

a y = 2x – 7 in general form b x – 3y + 9 = 0 in gradient-intercept form

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

TOPIC TEST SECTION II


Instructions • This section consists of 18 questions

• Show all necessary working

Marks


6

6

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

6


vLinear functions and lines




6

17 A line makes an angle of 135° with the positive x-axis and passes through the point (0, 3). Find:

a the gradient b the y-intercept c the equation of the line

_____________________ ___________________ ___________________

_____________________ ___________________ ___________________

_____________________ ___________________ ___________________

18 The line l has gradient 12

. Find, to the nearest degree, the angle the line makes with the positive

direction of the x-axis.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

19 Find the gradient of the line joining (5, 7) to (–3, 8).

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

20 The gradient of the line joining P(3, –2) to Q(x, 4) is − 13

. Find the value of x.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________ 4

4

2



6

17 A line makes an angle of 135° with the positive x-axis and passes through the point (0, 3). Find:

a the gradient b the y-intercept c the equation of the line

_____________________ ___________________ ___________________

_____________________ ___________________ ___________________

_____________________ ___________________ ___________________

18 The line l has gradient 12

. Find, to the nearest degree, the angle the line makes with the positive

direction of the x-axis.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

19 Find the gradient of the line joining (5, 7) to (–3, 8).

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

20 The gradient of the line joining P(3, –2) to Q(x, 4) is − 13

. Find the value of x.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________ 4

4

2


viLinear functions and lines




4

4

4

4

Marks

21 Find the equation of the line, in general form, which passes through:

a the point (–3, 5) with gradient 2 b the points (–1, 6) and (2, –3)

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

22 Find the equation of the line through (–1, 4):

a parallel to 2x – 3y + 7 = 0 b perpendicular to y = –3x + 5

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

23 Find the point of intersection of the lines y = 3x – 2 and x + 3y – 5 = 0

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

24 Find the equation of the line that passes through (0, 8) and through the point of intersection of3x – y + 4 = 0 and 2x + y – 16 = 0

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________



4

4

4

4

Marks

21 Find the equation of the line, in general form, which passes through:

a the point (–3, 5) with gradient 2 b the points (–1, 6) and (2, –3)

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

22 Find the equation of the line through (–1, 4):

a parallel to 2x – 3y + 7 = 0 b perpendicular to y = –3x + 5

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

________________________________ __________________________________

23 Find the point of intersection of the lines y = 3x – 2 and x + 3y – 5 = 0

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

24 Find the equation of the line that passes through (0, 8) and through the point of intersection of3x – y + 4 = 0 and 2x + y – 16 = 0

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________


viiLinear functions and lines




4

Marks

25 The graph shows the lines x + 2y – 8 = 0 and 2x – y – 1 = 0

a Write down the point of intersection of the lines.

_________________________________________

b Shade the region where x + 2y – 8 ≤ 0and 2x – y – 1 ≥ 0 hold simultaneously.

26 Find the distance between the points (–2, 7) and (6, 1)

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

27 Find the perpendicular distance from the point (2, 5) to the line 3x – 4y + 1 = 0

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

28 Find the midpoint of the interval joining (–7, 2) to (3, 9)

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

29 (6, –2) is the midpoint of P(x1, y1) and Q(1, 4). Find the coordinates of P.

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

4

4

4

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6x + 2y – 8 = 0

2x – y – 1 = 0

4



4

Marks

25 The graph shows the lines x + 2y – 8 = 0 and 2x – y – 1 = 0

a Write down the point of intersection of the lines.

_________________________________________

b Shade the region where x + 2y – 8 ≤ 0and 2x – y – 1 ≥ 0 hold simultaneously.

26 Find the distance between the points (–2, 7) and (6, 1)

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

27 Find the perpendicular distance from the point (2, 5) to the line 3x – 4y + 1 = 0

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

28 Find the midpoint of the interval joining (–7, 2) to (3, 9)

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

29 (6, –2) is the midpoint of P(x1, y1) and Q(1, 4). Find the coordinates of P.

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

_____________________________________ _____________________________________

4

4

4

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6x + 2y – 8 = 0

2x – y – 1 = 0

4


viiiLinear functions and lines


88Total marks achieved for PART B



88Total marks achieved for SECTION II

10

30 a Find the gradient of the line b Find the gradient of the linek: y = –3x + 2 l: 6x + 2y – 9 = 0

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

c What conclusion can be drawn about d P lies on the line y = –3x + 2 and also onlines k and l? the line y = –1. Find the coordinates of P.

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

e Find the shortest distance betweenlines k and l.

________________________________

________________________________

________________________________

________________________________

________________________________

________________________________

________________________________

________________________________



88Total marks achieved for SECTION II

10

30 a Find the gradient of the line b Find the gradient of the linek: y = –3x + 2 l: 6x + 2y – 9 = 0

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

c What conclusion can be drawn about d P lies on the line y = –3x + 2 and also onlines k and l? the line y = –1. Find the coordinates of P.

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

________________________________ ___________________________________

e Find the shortest distance betweenlines k and l.

________________________________

________________________________

________________________________

________________________________

________________________________

________________________________

________________________________

________________________________


Answers

PAGE 76 1 a

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2

b

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –4

c

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –3

d

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = 1

2 a –1, 1, 3

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2x + 1

b 5, 4, 3 c –3, 0, 3 d 4, 2, 0 e 2.5, 3, 3.5 f 0, –1, –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 4 – x

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3x

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –2x + 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 1 x + 3 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y + 1 = 0

PAGE 77 1 a negative b positive c positive d negative 2 a 23 b −3

4 c 4 3 a x-axis b y-axis c y-intercept

4 a 1 b –1 5 a 63° b 60° c 34° d 153°

PAGE 78 1 a 3, 2 b 2, –3 c –4, 1 d 5, 0 e 0. –2 f –5, 6 g 1, 4 h 23

13, i − −1

41

2, 2 a y = 5x + 3 b y = –2x + 1 c

y = x – 2 d y x= 12 e y x= −3

4 3 f y x= − − 12 3 a 1

3132 2, , y x= + b –2, –1, y = –2x – 1

4 a 3, –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3x – 2

b − 12 , 1

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –1 x + 1 2

PAGE 79 1 a 32 b –2 c −1

3 d 15 e –1 f 11

5 g 1 h 0 2 56° 3 y = –9

PAGE 80 1 a 2x – y – 3 = 0 b 5x + y – 4 = 0 c 6x – y – 1 = 0 d x – 2y + 8 = 0 e x + 3y + 15 = 0 f x + 4y – 2 = 0

g 2x – 3y + 15 = 0 h 9x – 12y – 8 = 0 i 10x + 12y – 3 = 0 2 a y = –2x + 7 b y = 3x + 8 c y = 4x – 2 d y x= − +3 12

e y x= − +43 2 f y x= +2

313 g y x= +2 3

4 h y x= +25

15 i y x= − +7

3 1 3 a 5, 6 b –3, 7 c 3, 43 d − 3

412,

PAGE 81 1 a y = 2x + 1 b y = 4x + 17 c y = –x + 3 d y = –3x – 9 e y x= −12 1 f y x= − −2

3233 2 a 2x – y + 3 = 0

b 2x + y – 5 = 0 c 3x – y + 10 = 0 d x – 3y + 17 = 0 e 3x + 4y – 36 = 0 f 5x – 3y – 6 = 0 3 0 4 34,−( )

PAGE 82 1 a 4x – y – 1 = 0 b 7x + 3y – 15 = 0 c x + y – 4 = 0 d y + 1 = 0 e 5x + y = 0 f x + 2y – 3 = 0 2 a y x= 12

b y = x + 2 c y = –x + 4

PAGE 83 1 a gradient b –1 2 a 3 b − 12 c 4 3 a − 1

2 b 34 c –3 4 both gradients = − 5

3

5 m m m m159 2

95 1 2 1= = − = −, ,

PAGE 84 1 a 2x – 3y + 13 = 0 b 4x – 5y – 22 = 0 2 a 3x + y – 7 = 0 b 3x – 5y + 22 = 0 3 both gradients = 25

4 m1 = –4, m2 = 14 , m1m2 = –1

PAGE 85 1 a (2, 4) b (1, 1) 2 a (5, 21) b (12, 9) c (1, 1) d (6, 11)

PAGE 86 1 a (9, 29) b passes through P 2 concurrent 3 3x + y – 8 = 0

PAGE 87 1 a

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = 2

b

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –1

c

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –2

d y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 y = 0

e y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = x + 1


Answers

PAGE 76 1 a

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2

b

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –4

c

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –3

d

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = 1

2 a –1, 1, 3

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2x + 1

b 5, 4, 3 c –3, 0, 3 d 4, 2, 0 e 2.5, 3, 3.5 f 0, –1, –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 4 – x

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3x

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –2x + 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 1 x + 3 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y + 1 = 0



4 a 1 b –1 5 a 63° b 60° c 34° d 153°

PAGE 78 1 a 3, 2 b 2, –3 c –4, 1 d 5, 0 e 0. –2 f –5, 6 g 1, 4 h 23

13, i − −1

41

2, 2 a y = 5x + 3 b y = –2x + 1 c

y = x – 2 d y x= 12 e y x= −3

4 3 f y x= − − 12 3 a 1

3132 2, , y x= + b –2, –1, y = –2x – 1

4 a 3, –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3x – 2

b − 12 , 1

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –1 x + 1 2

PAGE 79 1 a 32 b –2 c −1

3 d 15 e –1 f 11

5 g 1 h 0 2 56° 3 y = –9



e y x= − +43 2 f y x= +2

313 g y x= +2 3

4 h y x= +25

15 i y x= − +7

3 1 3 a 5, 6 b –3, 7 c 3, 43 d − 3

412,


3233 2 a 2x – y + 3 = 0



b y = x + 2 c y = –x + 4



3

5 m m m m159 2

95 1 2 1= = − = −, ,


4 m1 = –4, m2 = 14 , m1m2 = –1

PAGE 85 1 a (2, 4) b (1, 1) 2 a (5, 21) b (12, 9) c (1, 1) d (6, 11)


PAGE 87 1 a

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = 2

b

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –1

c

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –2

d y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 y = 0

e y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = x + 1


Answers

PAGE 76 1 a

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2

b

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –4

c

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –3

d

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = 1

2 a –1, 1, 3

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2x + 1

b 5, 4, 3 c –3, 0, 3 d 4, 2, 0 e 2.5, 3, 3.5 f 0, –1, –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 4 – x

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3x

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –2x + 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 1 x + 3 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y + 1 = 0



4 a 1 b –1 5 a 63° b 60° c 34° d 153°

PAGE 78 1 a 3, 2 b 2, –3 c –4, 1 d 5, 0 e 0. –2 f –5, 6 g 1, 4 h 23

13, i − −1

41

2, 2 a y = 5x + 3 b y = –2x + 1 c

y = x – 2 d y x= 12 e y x= −3

4 3 f y x= − − 12 3 a 1

3132 2, , y x= + b –2, –1, y = –2x – 1

4 a 3, –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3x – 2

b − 12 , 1

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –1 x + 1 2

PAGE 79 1 a 32 b –2 c −1

3 d 15 e –1 f 11

5 g 1 h 0 2 56° 3 y = –9



e y x= − +43 2 f y x= +2

313 g y x= +2 3

4 h y x= +25

15 i y x= − +7

3 1 3 a 5, 6 b –3, 7 c 3, 43 d − 3

412,


3233 2 a 2x – y + 3 = 0



b y = x + 2 c y = –x + 4



3

5 m m m m159 2

95 1 2 1= = − = −, ,


4 m1 = –4, m2 = 14 , m1m2 = –1

PAGE 85 1 a (2, 4) b (1, 1) 2 a (5, 21) b (12, 9) c (1, 1) d (6, 11)


PAGE 87 1 a

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = 2

b

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –1

c

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –2

d y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 y = 0

e y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = x + 1


Answers

PAGE 76 1 a

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2

b

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –4

c

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –3

d

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = 1

2 a –1, 1, 3

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2x + 1

b 5, 4, 3 c –3, 0, 3 d 4, 2, 0 e 2.5, 3, 3.5 f 0, –1, –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 4 – x

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3x

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –2x + 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 1 x + 3 2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y + 1 = 0



4 a 1 b –1 5 a 63° b 60° c 34° d 153°

PAGE 78 1 a 3, 2 b 2, –3 c –4, 1 d 5, 0 e 0. –2 f –5, 6 g 1, 4 h 23

13, i − −1

41

2, 2 a y = 5x + 3 b y = –2x + 1 c

y = x – 2 d y x= 12 e y x= −3

4 3 f y x= − − 12 3 a 1

3132 2, , y x= + b –2, –1, y = –2x – 1

4 a 3, –2

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3x – 2

b − 12 , 1

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –1 x + 1 2

PAGE 79 1 a 32 b –2 c −1

3 d 15 e –1 f 11

5 g 1 h 0 2 56° 3 y = –9



e y x= − +43 2 f y x= +2

313 g y x= +2 3

4 h y x= +25

15 i y x= − +7

3 1 3 a 5, 6 b –3, 7 c 3, 43 d − 3

412,


3233 2 a 2x – y + 3 = 0



b y = x + 2 c y = –x + 4



3

5 m m m m159 2

95 1 2 1= = − = −, ,


4 m1 = –4, m2 = 14 , m1m2 = –1

PAGE 85 1 a (2, 4) b (1, 1) 2 a (5, 21) b (12, 9) c (1, 1) d (6, 11)


PAGE 87 1 a

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = 2

b

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –1

c

x

y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –2

d y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 y = 0

e y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = x + 1

Answers – Linear functions and lines

ixLinear functions and lines


1

2

3

45

6

7

8

9

1011

12

227ANSWERS

Answers

f y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = x g y

654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 3 – x h y

654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2x – 1

2 a y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y – 2 = 0

2x – y + 1 = 0 b y

654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y – 2 = 0

2x – y + 1 = 0

c y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y – 2 = 0

2x – y + 1 = 0 d y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y – 2 = 0

2x – y + 1 = 0 3 a y x> +12 1 and y > 3x – 2 b 7x + y – 5 ≤ 0 and x + 3y + 6 ≥ 0 and 2x – y + 2 ≥ 0

PAGE 88 1 a y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = x + 2y = 3 – x b y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2x – 1

y = x 12

c y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + y = 3

x – y = –1 d y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

2x + y = 5

3x – y = 2

e y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x + 2y – 5 = 0

y = 2x f y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

2x + 3y + 5 = 0

3x – y – 4 = 0

PAGE 89 1 a 3 units b 3 units c 6 units d 7 units e 10 units f 7 units 2 a 5 units b 13 units c 10 units

3 a 5 units b 2 2 units c 2 26 units 4 a i 5 2 units ii 5 2 units b P is equidistant from Q and R

PAGE 90 1 a 5 units b 4 units c 1 unit d 3 units e 15 units 2 a 1.2 units b 1.5 units c 12 89

89 units

3 a 5 5

3 units b

2 1717

units c 11 26

13 units

PAGE 91 1 a (5, 3) b (–5, 1) c (6, –3) d 3 612 ,−( ) e 2 1

2,( ) f 0 1 12,−( ) g − −( )1 11

2 , h (–6, –1) i 4 5 12,−( ) 2 (11, –14)

PAGE 92 1 a 73 units b 3x + 8y + 9 = 0 c 52 73

73 units d 26 units2

PAGE 93 1 a y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

A(2,3)

B(5,2)

C(4,–1)x

b (3, 1) c (1, 0) d − 13 e gradient of DC = − 1

3 f gradient of BC = 3

PAGES 94-95 1 C 2 D 3 A 4 D 5 C 6 A 7 B 8 C 9 C 10 D 11 B 12 B

PAGES 96-100 13 a y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = –x + 2

x

b y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 2x – 3

x

c y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

x = –2

x

d y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

y = 1

x

14 a 2, –5 b –3, 0


xLinear functions and lines


19–20

21–25

13

14

15

16

17

18


xiLinear functions and lines

Mathletics Instant Workbooks – Series L 2 Copyright © 3P Learning228 EXCEL ESSENTIAL SKILLS: PRELIMINARY MATHEMATICS REVISION AND EXAM WORKBOOK

Answersc –1, 4 15 a − 2

3 b 2 c see below 16 a 2x – y – 7 = 0 b y x= +13 3 17 a –1 b 3 c y = –x + 3 18 27° 19 − 1

8

20 x = –15 21 a 2x – y + 11 = 0 b 3x + y – 3 = 0 22 a 2x – 3y + 14 = 0 b x – 3y + 13 = 0 23 (1.1, 1.3) 24 4x – 3y + 24 = 0

25 a (2, 3) b see below 26 10 units 27 2.6 units 28 −( )2 5 12, 29 (11, –8) 30 a –3 b –3 c parallel d (1, –1) e

104

units

15 c y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

2x + 3y – 6 = 0

x

25 b y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

2x – y – 1 = 0

x + 2y – 8 = 0

PAGE 101 1 a true b false c false d true 2 a 7 b 5 c 17 3 a 43 b –45 c 1 d 1.59 e 2 f 4 g 51 h 81

i 27 j − 14 k 0 l 35 35

36 4 a 18 b 2 c –2

PAGE 102 1 a x2 – 4x + 3 b x2 + 4x – c2 – 4c c x2 + 2xh + h2 + 4x + 4h + 3 3 a 11

+−

xx

b xx

−+

11

c −+x

x 2PAGE 103 1 a 2xh + h2 – 4h b –2xh – h2 2 a 7 b 2x + h + 1 3 a 4 b x + c – 1

PAGE 104 1 a domain b range 2 a all real numbers b all real numbers c x ≠ 0 d all real numbers e x ≥ 0 f all realnumbers g all real numbers h all real numbers i –3 ≤ x ≤ 3 3 a y ≥ 3 b all real numbers c y ≠ 0 d y > 0 e y ≥ 0

f all real numbers g y ≥ 0 h y ≤ 9 i 0 ≤ y ≤ 3 4 a –6 ≤ y ≤ 2 b 14 16≤ ≤y c –7 ≤ y ≤ 2 5 a all real numbers, all real

numbers b all real numbers, y > 0 c x ≥ –5, y ≥ 0

PAGE 105 1 a f(x) b –f(x) c odd d even 2 a odd b even c neither d odd 3 a even b neither c odd 4 a evenb even c odd d neither e even f neither g odd h neither i oddPAGE 106 1 a parabola b straight line c semi-circle d cubic curve e straight line f exponential curve g parabola h hyperbola

i straight line j hyperbola k parabola l semi-circle 2 a y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

y x= + 2

b y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

y x= −2 2 c y

654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

yx=

3

2

d y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

y x=

e y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

y x= −9 2

f y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

yx

= 6 g y

654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

x y+ − =6 0 h y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

y x= 2

i y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

y x= − +2 1

PAGE 107 1 a x ≠ 3 b − 13 c 0 d 0 e –∞ f ∞ g x = 3, y = 0 h see below 2 a all real numbers b 1 c even d ∞

e 1.0, 1.2, 1.5, 2.0, 3.0, 4.8, 8.4, 16.0 f y18161412108642

–2–4

–2 –1 0 1 2 x

y x= 22

1 h y654321

–1–2–3–4–5–6

–3 –2 –1 0 1 2 3 4 5 6 7 8 9 x

yx

=−1

3

PAGE 108 1 a (0, 0), 3 units b (0, 0), 5 units c (0, 0), 9 units d (0, 0), 14 units 2 a x2 + y2 = 16 b x2 + y2 = 49c x2 + y2 = 100 d x2 + y2 = 900 3 a (3, 5), 2 units b (–2, 1), 7 units c (4, –7), 6 units d (0, 6), 11 units e (–9, –3), 1 unitf (–8, 0), 12 units 4 a (x – 1)2 + (y – 2)2 = 9 b (x + 3)2 + (y – 3)2 = 64 c (x – 5)2 + (y + 1)2 = 100 d (x + 4)2 + (y + 5)2 = 169

e (x – 6)2 + y2 = 36 f (x + 2)2 + (y + 1)2 = 1 5 a y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

x y2 2 16+ = b y

654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

x y−( ) + −( ) =1 3 92 2

c y654321

–1–2–3–4–5–6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

x y+( ) + +( ) =3 1 42 2

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Linear Functions and Lines Solutions