Linear Functions and Lines Solutions
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Linear functions...
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Linear Functions and Lines T eac h
y
er
Book - Ser ie s L 2 -
= m x + c
Mathlecs Instant Workbooks
Copyright ©
Linear functions and lines Topic Test Instructions
PART A 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
Marks
1
The equation of a line with gradient –1 and y -intercept 2 is A
2
B y = 2 x – 1
y =
C
2 – x
D
none of these
1
D
(1, 1)
1
Which point lies on the line 2 x + 3 y – 5 = 0? A
3
y = – x – 2
(–1, 1)
B
(–1, –1)
(1, –1)
C
y
The gradient of the line joining A to B is A
−2
5 4
−3
B
3
A 3
2
2
B
C
2
D
3
4
1
2
3
4
1
5 x
x =
3
B
x =
–5
C
y =
3
D
y =
1
–5
0
–1
B
C
1
D
45
D
−
1
The gradient of any line parallel to 4 x – 2 y + 3 = 0 is A
7
0
–1
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
6
–1
2
A line, parallel to the x -axis, passes through the point (3, –5). Its equation is A
5
1
3
2
–2
B
C
1 2
1
1
2
y
The shaded region is where
4
4 x – 6 y – 3 = 0
3
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
2
1
1
–2
–1
0
1
2
3
4 x
–1
8
The gradient of any line perpendicular to A
1 −
3
B
1 3
–2
1 y
=
3
C
x
+
2 is
–3
D
1
3
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Linear functions and lines Topic Test
PART A
near unc ons an 9
2 x – 5 y = 0
5 units
2 x + 5 y = 0
C
5 x + 2 y = 0
D
5 x – 2 y = 0
1
B
6 units
C
7 units
D
8 units
1
Which point lies within the region determined by the inequalities 2 x and 3 x – 4 y + 5 ≥ 0? (–4, 2)
A
12
B
The distance between the points ( –1, 5) and (7, 5) is A
11
Marks
Which line passes through the point ( –2, 5)? A
10
nes
B
(–1, –3)
C
Which diagram shows the region where y
A
B
x
x ≤
y
(2, 6)
0 and C
D
+ y <
0 1
(5, 2)
y ≥ 0? y
x
D
x
y
x
1
Total marks achieved for PART A
12
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Linear functions and lines Topic Test Instructions
PART B
This section consists of 18 questions Show all necessary working
Time allowed: 1 hour
Total marks = 88
Marks
13 Draw, on the number plane provided, the graph of: a
y = – x +
2
b y = 2 x – 3
y 6
y 6
5
5
4
4
3
3
2
2
1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
c
x =
–2
1 0 1 2 3 4
5 6
x
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
d y =
y 6
1
5 6
x
0 1 2 3 4
5 6
x
y 6
5
5
4
4
3
3
2
2
1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
0 1 2 3 4
1 0 1 2 3 4
5 6
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
x
8
14 Write down the gradient and y -intercept of each line. a y = 2 x – 5
b y =
c x + y =
–3 x
_____________________
___________________
___________________
_____________________
___________________
___________________
15 For the line 2 x + 3 y – 6 = 0 a
c
6 5 4 3 2 1
________________________________
find the y -intercept ______________________________
6
y
find the gradient ________________________________
b
4
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x –1 –2 –3 –4 –5 –6
______________________________
graph the line on the number plane provided .
6
16 Write the equation: a
y = 2 x – 7
in general form
b x – 3 y +
9 = 0 in gradient-intercept form
________________________________
__________________________________
________________________________
__________________________________
________________________________
__________________________________
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Linear functions and lines Topic Test
PART B Marks
17 A line makes an angle of 135 ° with the positive x -axis and passes through the point (0, 3). Find: a
b
the gradient
the y -intercept
c
the equation of the line
_____________________
___________________
___________________
_____________________
___________________
___________________
_____________________
___________________
___________________
18 The line l has gradient
1 2
6
. Find, to the nearest degree, the angle the line makes with the positive
direction of the x -axis. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________
2
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
4
. Find the value of x .
______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________
4
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Linear functions and lines Topic Test
PART B Marks
21 Find the equation of the line, in general form, which passes through: a
b
the point (–3, 5) with gradient 2
the points (–1, 6) and (2, –3)
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
4
22 Find the equation of the line through ( –1, 4): a
b
parallel to 2 x – 3 y + 7 = 0
perpendicular to y = –3 x + 5
________________________________
__________________________________
________________________________
__________________________________
________________________________
__________________________________
________________________________
__________________________________
________________________________
__________________________________
________________________________
__________________________________
4
23 Find the point of intersection of the lines y = 3 x – 2 and x + 3 y – 5 = 0
______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 4
______________________________________________________________________________ 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
______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________
4
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Linear functions and lines Topic Test
PART B Marks
25 The graph shows the lines x + 2 y – 8 = 0 and 2 x – y – 1 = 0
y
2 x – y – 1 = 0
6 5
Write down the point of intersection of the lines.
a
4 3 2
_________________________________________ b
1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
Shade the region where x + 2 y – 8 ≤ 0 and 2 x – y – 1 ≥ 0 hold simultaneously.
x + 2 y – 8 = 0 0 1 2 3 4 5 6
x
4
26 Find the distance between the points (–2, 7) and (6, 1)
______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________
4
27 Find the perpendicular distance from the point (2, 5) to the line 3 x – 4 y + 1 = 0
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
4
28 Find the midpoint of the interval joining (–7, 2) to (3, 9)
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
4
y 1) and Q(1, 4). Find the coordinates of P. 29 (6, –2) is the midpoint of P( x , 1
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
_____________________________________
4
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Linear functions and lines Topic Test 30 a
c
e
PART B arks
b
Find the gradient of the line k : y = –3 x + 2
Find the gradient of the line l: 6 x + 2 y – 9 = 0
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
What conclusion can be drawn about lines k and l?
d
P lies on the line y = –3 x + 2 and also on the line y = –1. Find the coordinates of P.
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
________________________________
___________________________________
Find the shortest distance between lines k and l. ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ 10
________________________________
Total marks achieved for PART B
88
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Answers – Linear functions and lines y
f
6 5 4 3 2 1
g
y = x
y =
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6
c
6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
d
0 1 2 3 4 5 6
y =
6 5 4 3 2 1
3 – x
3 a
y = x +
2
6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
y
x + 2 y – 5 = 0
PAGE 89 14 1 a 3 units b 3 units
3 a
5 units
y = 2 x – 1
y
b
6 5 4 3 2 1
5 5
y =
1 2
x
0 1 2 3 4 5 6
3
2 17 17
units
and x + 3 y + 6 ≥ 0 and 2 x
– y +
2 ≥ 0
–1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
3 x – y = 2
y
d
6 5 4 3 2 1
0 1 2 3 4 5 6
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6 2 x + y =
x + y = 3
5
2 x + 3 y + 5 = 0
c 6 units d 7 units e 10 units f 7 units 2 a 5 units b 13 units c 10 units
c
11 26 13
b P is equidistant from Q and R
2 a 1.2 units b 1.5 units c
12 89 89
units
units e (2, 21 ) f
(3 21 , −6) 52 73 73
(0, −1 21 )
g
( −1 21 , −1)
h (–6, –1) i
( 4, −5 21 )
2 (11, –14)
units d 26 units2
b (3, 1) c (1, 0) d − 13
y 6 5 4 3 2 1
x + y – 2 = 0
0 1 2 3 4 5 6
73 units b 3 x + 8 y + 9 = 0 c
PAGE 93 18 1 a
0 1 2 3 4 5 6
3 x – y – 4 = 0
PAGE 91 16 1 a (5, 3) b (–5, 1) c (6, –3) d PAGE 92 17 1 a
x – y =
6 5 4 3 2 1
b 2 2 units c 2 26 units 4 a i 5 2 units ii 5 2 units
units b
+ y – 5 ≤ 0
y
c
15 1 a 5 units b 4 units c 1 unit d 3 units e 15 units PAGE 90
3 a
b 7 x
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
0
2 x – y + 1 = 0
6 5 4 3 2 1
6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
0 1 2 3 4 5 6
y > 21 x + 1 and y > 3 x – 2
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
f
6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 x + y – 2 = –6
y
b
x + y – 2 = 0
y
y = 2 x
y
6 5 4 3 2 1
0 1 2 3 4 5 6
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6
e
2 x – y + 1 = 0
y
2 a
6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 y = 2 x – 1 –3 –4 –5 –6
2 x – y + 1 = 0
y
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
x + y – 2 = 0
13 1 a PAGE 88
6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –1 –2 –3 –4 –5 –6
2 x – y + 1 = 0
y
3 – x
y
h
y
e gradient of DC = − 13
f gradient of BC = 3
A(2,3) B(5,2)
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x –1 –2 C(4,–1) –3 –4 –5 –6
19–20 1 C PAGES 94-95
2 D
PAGES 96-100 21–25 13 a
3 A
4 D
y
5 C b
6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x –1 –2 –3 –4 –5 y = – x + 2 –6
6 A
7 B
y 6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
8 C
y = 2 x – 3
0 1 2 3 4 5 6 x
c
9 C
10 D
11 B
y
d
6 5 4 3 2 1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 x = –2
0 1 2 3 4 5 6
x
12 B 14 a 2, –5 b –3, 0
y 6 5 4 3 2 1
y =
1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x –1 –2 –3 –4 –5 –6
Linear functions and lines Mathletics Instant Workbooks – Series L 2 Copyright © 3P Learning
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Answers – Linear functions and lines c –1, 4
15 a
20 x = –15
− 23
b 2
c see below
16 a 2 x – y – 7 = 0 b
21 a 2 x – y + 11 = 0 b 3 x + y – 3 = 0
25 b
y 6 5
2
3
x + 3
17 a –1 b 3 c y = – x + 3
18 27°
19
− 18
23 (1.1, 1.3) 24 4 x – 3 y + 24 = 0
29 (11, –8) 30 a –3 b –3 c parallel d (1, –1) e
10 4
units
x + 2 y – 8 = 0
6 4
2 x + 3 y – 6 = 0
1
–6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6
1
5
4 3
y
=
22 a 2 x – 3 y + 14 = 0 b x – 3 y + 13 = 0
25 a (2, 3) b see below 26 10 units 27 2.6 units 28 ( −2,5 21 ) 15 c
y
3 2 1
0 1 2 3 4 5 6
x
–6 –5 –4 –3 –2 –1 –1 –2 2 x – y – 1 = 0 3 – –4 –5 –6
0 1 2 3 4 5 6
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