Question 1: [1 mark]
Define ‘Parallel’
Answer type: Multiple choice type 2
A: Lines that meet at 90 \degree.
B: Lines that pass through 0.
C: Lines that have the same gradient.
D: Lines that have the same y-intercept.
ANSWER: C: Lines that have the same gradient.
Question 2(a): [1 mark]
Choose the line below which is parallel to y = 5x + 4.
Answer type: Multiple choice type 2
A: y + 5x = 3
B: y - 5x = 2
C: 2y = 5x + 10
D: y = 6x + 3
ANSWER: B: y - 5x = 2
WORKING:
y - 5x = 2 is equivalent to y = 5x + 2.
This line has the same gradient as y = 5x + 4, 5, so they are parallel.
Question 2(b): [1 mark]
Choose the line below which is parallel to y = 3x + 4.
Answer type: Multiple choice type 2
A: y + 3x = 3
B: y - \dfrac{1}{3}x = 2
C: 2y = 6x + 10
D: y = 6x + 4
ANSWER: C: 2y = 6x + 10
WORKING:
2y = 6x + 10 is equivalent to y = 3x + 5.
This line has the same gradient as y = 3x + 4, 3, so they are parallel.
Question 3(a): [2 marks]
Choose the equation of the line which passes through point (-1,5) and is parallel to 3x + y = - 12.
Answer type: Multiple choice type 2
A: 3y = -3x + 6
B: 3y = -3x + 7
C: y = -3x + 2
D: y = - 3x + 7
ANSWER: C: y = -3x + 2
WORKING:
3x + y = -12 is equivalent to y = -3x - 12
Both y = -3x + 2 and y = - 3x + 7 have the same gradient as 3x + y = -12, -3, so they are all parallel.
However, only y = -3x + 2 passes through the point (-1,5), since substituting in the x and y values of this point into the equation gives 5 = -3(-1) + 2 = 5.
Question 3(b): [2 marks]
Choose the equation of the line which passes through point (2,5) and is parallel to y = 4x - 10.
Answer type: Multiple choice type 2
A: 2y = -3x + 2
B: y = 4x - 3
C: 2 = -3x + 2y
D: 2y = -x + 12
ANSWER: B: y = 4x - 3
WORKING:
y = 4x - 3 has the same gradient as y = 4x - 10, 4, so they are parallel.
Further, y = 4x - 3 passes through the point (2,5), since substituting in the x and y values of this point into the equation gives 5 = 4(2) - 3 = 5.
Question 3(c): [2 marks]
Choose the equation of the line which passes through point (4,1) and is parallel to y = 2x - 2.
Answer type: Multiple choice type 2
A: 4y = x - 7
B: y = - 2x - 1
C: 1 = y - 2x - 7
D: y = 2x - 7
ANSWER: D: y = 2x - 7
WORKING:
1 = y - 2x - 7 is equivalent to y = 2x +8
Both y = 2x - 7 and y = 2x + 8 have the same gradient as y = 2x - 2, 4, so they are all parallel.
However, only y = 2x - 7 passes through the point (4,1), since substituting in the x and y values of this point into the equation gives 1 = 2(4) - 7 = 1.
Question 4:
Determine if each of the following lines are parallel to line y = 5x + 2 or not.
Question 4(a): [1 mark]
y = -5x + 4
Answer type: Multiple choice type 1
A: Yes
B: No
ANSWER: B: No
WORKING:
They have different gradients, so are not parallel.
Question 4(b): [1 mark]
5y = -x + 2
Answer type: Multiple choice type 1
A: Yes
B: No
ANSWER: B: No
WORKING: They have different gradients, so are not parallel.
Question 4(c): [1 mark]
y = 5x + 10
Answer type: Multiple choice type 1
A: Yes
B: No
ANSWER: A: Yes
WORKING: They have the same gradient, so are parallel.
Question 4(d): [1 mark]
y - 5x = 0
Answer type: Multiple choice type 1
A: Yes
B: No
ANSWER: A: Yes
WORKING: They have the same gradient, so are parallel.
Question 4(e): [1 mark]
y = \dfrac{1}{5}x + 4
Answer type: Multiple choice type 1
A: Yes
B: No
ANSWER: B: No
WORKING: They have different gradients, so are not parallel.
Question 5: (Changed slightly to ‘parallel’ or ‘not parallel’ answers)
For each part (b) to (g), choose whether the lines are parallel to CD, perpendicular to CD, or neither.
Question 5(a): [2 marks]
The line CD is defined by the points C(-2,1) and D(10,7).
Choose the correct equation of the line CD.
Answer type: Multiple choice type 2
A: y = 2x - 13
B: y = \dfrac{1}{2} x - 2
C: y = \dfrac{3}{4} x - \dfrac{1}{2}
D: y = \dfrac{1}{2} x + 2
ANSWER: D: y = \dfrac{1}{2} x + 2
WORKING:
m = \dfrac{\text{change in} \, y}{\text{change in} \, x} = \dfrac{7 - 1}{10 - - 2} = \dfrac{6}{12} = \dfrac{1}{2}
7 = \dfrac{1}{2} \times 10 + c
7 = 5 + c
c = 2
y =\dfrac{1}{2} x + 2
Question 5(b): [1 mark]
y = - 2x
Answer type: Multiple choice type 2
A: Parallel
B: Perpendicular
C: Neither
ANSWER: B: Perpendicular
Question 5(c): [1 mark]
y = \dfrac{1}{2}x
Answer type: Multiple choice type 2
A: Parallel
B: Perpendicular
C: Neither
ANSWER: A: Parallel
Question 5(d): [1 mark]
12y = 6x + 7
Answer type: Multiple choice type 2
A: Parallel
B: Perpendicular
C: Neither
ANSWER: A: Parallel
Question 5(e): [1 mark]
2y = 2x + 2
Answer type: Multiple choice type 2
A: Parallel
B: Perpendicular
C: Neither
ANSWER: C: Neither
Question 5(f): [1 mark]
2(y - 3x) = 5 - 2x
Answer type: Multiple choice type 2
A: Parallel
B: Perpendicular
C: Neither
ANSWER: C: Neither
Question 5(g): [1 mark]
0 = \dfrac{2x + y}{2}
Answer type: Multiple choice type 2
A: Parallel
B: Perpendicular
C: Neither
ANSWER: B: Perpendicular
Question 6: [2 marks]
Line D is parallel to the line C.
Two points on C are (2,-2) and (11,4).
(3,2) is a point on D.
Choose which of the following is another point on D.
Answer type: Multiple choice type 2
A: (4,6)
B: (5,6)
C: (6,4)
D: (6,5)
ANSWER: C: (6,4)
WORKING:
Parallel lines have the same gradient, so \text{Gradient of} \, D = \text{Gradient of} \, C.
\text{Gradient of} \, C = \dfrac{\text{change in} \, y}{\text{change in} \, x} = \dfrac{4 - - 2}{11 - 2} = \dfrac{6}{9} = {2}{3}
\text{Gradient of} \, D = \dfrac{2}{3}
\text{points on} \, D = (3 + n, 2 + \dfrac{2}{3} n)
Correct point on D is (6,4).
Question 7:
The line A is shown below.
Choose the equation below which is parallel to line A.
Answer type: Multiple choice type 2
A: y = \dfrac{2}{3} x + 4
B: y = - \dfrac{2}{3} x + 1
C: y = \dfrac{3}{2}x - 1
D: y = 3x
ANSWER: C: y = \dfrac{3}{2}x - 1
WORKING:
Line A has a gradient of \dfrac{3}{2}, which is the same as y = \dfrac{3}{2}x - 1, so they are parallel.
Question 8(a): [2 marks]
Choose the correct equation of the line that passes through (9, 14) and is parallel to y = \dfrac{1}{3} x - 5.
Answer type: Multiple choice type 2
A: y = \dfrac{1}{3}x + 17
B: y = \dfrac{1}{3}x + 11
C: y = - 3x + 41
D: y = - 3x - 13
ANSWER: B: y = \dfrac{1}{3}x + 11
WORKING:
Parallel so m = \dfrac{1}{3}.
Substituting values for x and y:
14 = \dfrac{1}{3} \times 9 + c
c = 11
y = \dfrac{1}{3} x + 11
Question 8(b): [2 marks]
Choose the correct equation of the line that is parallel to 2y = 3(2 - 3x) and passes through the point of intersection of y = x+8 and y = -3x + 4.
Answer type: Multiple choice type 2
A: y = - \dfrac{9}{2} x + \dfrac{23}{2}
B: y = - \dfrac{9}{2} x + \dfrac{5}{2}
C: y = - \dfrac{9}{2} x + \dfrac{7}{2}
D: y = x
ANSWER: B: y = - \dfrac{9}{2} x + \dfrac{5}{2}
WORKING:
2y = 3(2 - 3x) \, ; \, 2y = 7 - 9x \, ; \, y = - \dfrac{</span></p> <p style="text-align: left;"><span style="color: #ff0000;">9}{2} x + \dfrac{7}{2} \,
Line is parallel, so m = - \dfrac{9}{2}.
y = x + 8 and y = - 3x + 4
x + 8 = - 3x + 4 \, ; \, 4x + 8 = 4 \, ; \, 4x - 4 \, ; \, x = - 1 \,
y = - 1 + 8 \, ; \, y = 7
Passes through the point (-1,7)
y = - \dfrac{9}{2} x + c \, ; \, 7 = - \dfrac{9}{2} \times -1 = c \, ; \, c = \dfrac{5}{2}
y = - \dfrac{9}{2} x + \dfrac{5}{2}