Linear Functions and Lines Solutions

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Linear Functions and Lines

Linear Functions and Lines

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I I I I I I I 1 _{The equation of a line with gradient}

–1 and y -intercept 2 is

A  _{y =}

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

 2 _{Which point lies on the line 2 x  + 3 y – 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 −2 3 B −3 2 C 2 3 D 3 2

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 of} 

the x -axis. The gradient of the line is

A ₀ B

–1 C 1 D 45

6 _{The gradient of any line parallel to 4 x} 

– 2 y  + 3 = 0 is A ₂ B –2 C 1 2 D − 1 2

7 _{The shaded region is where}

A _{4 x – 6 y – 3} ≤ ₀ B _4 x 

– 6 y – 3 ≥ 0

C _4 x 

– 6 y – 3 < 0 D 4 x – 6 y – 3 > 0

8 _{The gradient of any line perpendicular to}  y = x +

1 3 2 _is A − 1 3 B 1 3 C –3 D 3 Marks 1 1 1 1 1 1 1 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 4 x – 6 y – 3 = 0 ii

Linear functions and lines

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

Linear functions and lines

Topic Test

PART A

Instructions _{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

near unc ons an

nes

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

A 2 x _– 5 y  = 0 B 2 x  + 5 y  = 0 C 5 x  + 2 y  = 0 D 5 x _– 2 y  = 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 2 x + y  < 0} and 3 x _– 4 y  + 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  Marks 1 1 1 1 iii

Linear functions and lines

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

Linear functions and lines

Topic Test

PART A

12 Total marks achieved for PART A

Linear functions and lines

Topic Test

PART B

iv

Linear functions and lines

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

Instructions _{This section consists of 18 questions}

Show all necessary working

Time allowed: 1 hour Total marks = 88

I I I I I I I

13 _{Draw, on the number plane provided, the graph of:}

a  _{y =} – x  + 2  x   y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6 b  _{y  = 2 x}  – 3  x   y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6 c  _{x  = –2}  x   y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6 d  _{y  = 1}  x   y  6 5 4 3 2 1 –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  = 2 x} 

– 5 b  y  = –3 x  c  x + y  = 4

_____________________ ___________________ ___________________ _____________________ ___________________ ___________________ 15 _{For the line 2 x  + 3 y} 

– 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  = 2 x} 

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

________________________________ __________________________________ ________________________________ __________________________________ ________________________________ __________________________________  x   y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6 I I I I I I I 8 Marks 6 6 6

Linear functions and lines

Topic Test

PART B

v

Linear functions and lines

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

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} 1

2. 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 −

1

3 . Find the value of  x .

______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ Marks 6 4 4 2

Linear functions and lines

Topic Test

PART B

vi

Linear functions and lines

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

I I I I I I I

 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 2 x – 3 y  + 7 = 0} b _{perpendicular to y  = –3 x  + 5}

________________________________ __________________________________ ________________________________ __________________________________ ________________________________ __________________________________ ________________________________ __________________________________ ________________________________ __________________________________ ________________________________ __________________________________  23 _{Find the point of intersection of the lines y  = 3 x} 

– 2 and  x  + 3 y – 5 = 0 ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________  24 _{Find the equation of the line that passes through (0, 8) and through the point of intersection of} 

3 x _– y + 4 = 0 and 2 x + y _– 16 = 0

______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ I I I I I I I 4 4 4 4 Marks

Linear functions and lines

Topic Test

PART B

vii

Linear functions and lines

Mathletics Instant Workbooks – Series L 2 Copyright © 3P Learning  25 _{The graph shows the lines  x  + 2 y – 8 = 0 and 2 x – y – 1 = 0}

a Write down the point of intersection of the lines. _________________________________________ b Shade the region where x  + 2 y _– 8 ≤ 0

and 2 x– 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 3 x – 4 y  + 1 = 0}

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

_____________________________________ _____________________________________ _____________________________________ _____________________________________ _____________________________________ _____________________________________ _____________________________________ _____________________________________  29 _{(6, –2) is the midpoint of P( x}  1

 , y ₁) and Q(1, 4). Find the coordinates of P.

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

Linear functions and lines

Topic Test

PART B

viii

Linear functions and lines

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

88 Total marks achieved for PART B

I I I I I I I

30a _{Find the gradient of the line} b _{Find the gradient of the line}

k : y  = _–3 x  + 2 l: 6 x  + 2 y _– 9 = 0

________________________________ ___________________________________ ________________________________ ___________________________________ ________________________________ ___________________________________

c _{What conclusion can be drawn about} d _{P lies on the line  y  =–3 x  + 2 and also on}

lines k  and l? theline  y  =_–1. Find the coordinates of P. ________________________________ ___________________________________ ________________________________ ___________________________________ ________________________________ ___________________________________ ________________________________ ___________________________________ ________________________________ ___________________________________

e _{Find the shortest distance between}

lines k  and l. ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ I I I I I I I arks 10

f   y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6  y  = x  g  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6  y  = 3– x  h  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6  y  = 2 x – 1 2 a  y 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6  x  + y – 2 = 0 2 x – y  + 1 = 0 b  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6  x  + y – 2 = 0 2 x – y  + 1 = 0 c  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6  x  + y – 2 = 0 2 x – y  + 1 = 0 d  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6  x  + y – 2 = 0 2 x – y  + 1 = 0

3 a  y > ₂1x +1 and y  > 3 x _– 2 b 7 x + y _– 5≤ 0 and x  + 3 y  + 6 ≥ 0 and 2 x – y  + 2≥ 0

PAGE 88 1 a  y 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6  y  = x  + 2  y  = 3– x  _b  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6  y  = 2 x – 1  y  =1 x  2 c  y  6 5 4 3 2 1 –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  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 2 x  + y  = 5 3 x – y  = 2 e  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6  x  + 2 y – 5 = 0  y  = 2 x  _f   y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6 2 x  + 3 y  + 5 = 0 3 x – 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 17 17  units c 11 26 13  units PAGE 91 _{1 a (5, 3) b (–5, 1) c  (6,–3) d} 31 6 2,−

(

)

e 2 1 2 ,

( )

f 0 11 2 ,−

(

)

g

(

− ₁1 −₁

)

2, h (–6,–1) i 4 5 1 2 ,−

(

)

2  (11,–14)

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

73  units d 26 units 2 PAGE 93 _{1 a}  y 6 5 4 3 2 1 –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 −1 3 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}  y ₆ 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6  y  =– x  + 2  x  b  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6  y  = 2 x – 3  x  c  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6  x  =–2  x  d  y  6 5 4 3 2 1 –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

Answers – Linear functions and lines

x

Linear functions and lines

Mathletics Instant Workbooks – Series L 2 Copyright © 3P Learning 19–20 21–25 13 14 15 16 17 18

Answers – Linear functions and lines

xi

Linear functions and lines

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

I I I I I I I

c –1, 4 15 a −23 b 2 c see below 16 a 2 x – y – 7 = 0 b  y = x +

1

3 3 17 a –1 b 3 c  y  =– x  + 3 18 27° 19 −

1 8

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

25 a (2, 3) b see below 26 10 units 27 2.6 units 28

(

−_{2 5}1

)

2 , 29 (11,–8) 30 a–3 b–3 c  parallel d  (1,–1) e 10 4  units 15 c  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6 2 x  + 3 y – 6 = 0  x  25 b  y  6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6–5–4–3–2–1 0 1 2 3 4 5 6 2 x – y – 1 = 0  x  + 2 y – 8 = 0