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Equations of Lines
Question Paper 3
Level
Subject
Exam Board
Module
Topic
Sub Topic
Booklet
A Level
Mathematics (Pure)
AQA
Core 4
Vectors
Equations of lines
Question Paper 3
Time Allowed:
55 minutes
Score:
/46
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
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The coordinates of the points A and B are (3, –2, 1) and (5, 3, 0) respectively.
Q1.
The line l has equation r =
(a)
Find the distance between A and B.
(2)
(b)
Find the acute angle between the lines AB and l. Give your answer to the
nearest degree.
(5)
(c)
The points B and C lie on l such that the distance AC is equal to the distance
AB. Find the coordinates of C.
(5)
(Total 12 marks)
Q2.
The points A, B and C have coordinates (3, –2, 4), (5, 4, 0) and (11, 6, –4)
respectively.
(a)
(i)
Find the vector
.
(2)
(ii)
Show that the size of angle ABC is
(5)
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(b)
The line l has equation r =
(i)
Verify that C lies on l.
(2)
(ii)
Show that AB is parallel to l.
(1)
(c)
The quadrilateral ABCD is a parallelogram. Find the coordinates of D.
(3)
(Total 13 marks)
Q3.
The lines l1 and l2 have equations r =
respectively.
(a)
and r =
Show that l1 and l2 are perpendicular.
(2)
(b)
Show that l1 and l2 intersect and find the coordinates of the point of
intersection, P.
(5)
(c)
The point A ( – 4, 0, 11) lies on l2. The point B on l1 is such that AP = BP. Find
the length of AB.
(4)
(Total 11 marks)
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Q4.
The quadrilateral ABCD has vertices A (2, 1, 3), B (6, 5, 3), C (6, 1, –1) and D
(2, –3, –1).
The line l1 has vector equation r =
(a)
(i)
Find the vector
+λ
.
.
(2)
(ii)
Show that the line AB is parallel to l1.
(1)
(iii)
Verify that D lies on l1.
(2)
(b)
The line l2 passes through D (2, –3, –1) and M (4, 1, 1).
(i)
Find the vector equation of l2.
(2)
(ii)
Find the angle between l2 and AC.
(3)
(Total 10 marks)