1.
Find the non-unique solution for the following system of simultaneous equations
x−y−z=3
x − 2y + z = 2
2x − y − 4z = 7
2.
The lines l1 and l2 have equations
(Total 6 marks)
 4
 1 
 
 
r1 =  3   λ  5  and r2 =
 0
  2
 
 
 2
 0 
 
 
  1  μ  2 
 3
  3
 
 
respectively, where  and  are parameters.
(a) Show that l1 passes through the point (2, − 7, 4).
(2)
(b)
Determine whether the lines l1 and l2 intersect.
(4)
(Total 6 marks)
3.
 2
 
x 4 y  2 z 6


.
A plane Π has equation r •   1 = 16 and a line l has equations
1
2
4
 1 
 
Show that the line l lies in the plane Π.
(Total 6 marks)
4.
The lines L1 and L2 have parametric equations
L1 : x = 1 + 2, y = 1 + 3, z = 1 − 
L2 : x = 2 – , y = 3 + 4, z = 4 + 2
Find the angle between L1 and L2.
5.
Consider the four points A(1, 4, –1), B(2, 5, –2), C(5, 6, 3) and D(8, 8, 4). Find the point of
intersection of the lines (AB) and (CD).
6.
Let P be the point (1, 0, – 2) and Π be the plane x + y − 2z + 3 = 0. Let P′ be the reflection of P in
the plane Π. Find the coordinates of the point P′.
(Total 6 marks)
(Total 6 marks)
(Total 6 marks)
7.
The line
y2
z2
x 1
=
=
is reflected in the plane x + y + z = 1. Calculate the angle
1
1
1
between the line and its reflection. Give your answer in radians.
(Total 6 marks)
8.
9.
The parallelogram ABCD has vertices A (3, 2, 0), B (7, –1, 1), C (10, –3, 0) and D (6, 0, 1).
Calculate the area of the parallelogram.
(a)
y 1
z 5
x 1
=
=
and the point (1, −2, 3).
6
3
2
Show that the equation of  is 6x + 2y – 3z = –7.
(b)
Calculate the distance of the plane  from the origin.
(Total 6 marks)
The plane  contains the line
(7)
(4)
(Total 11 marks)
10.
Consider the system of equations
(a)
x + 2y + kz = 0
x + 3y + z = 3
kx + 8y + 5z = 6
Find the set of values of k for which this system of equations has a unique solution.
(b)
For each value of k that results in a non-unique solution, find the solution set.
(6)
(8)
(Total 14 marks)
1
11.
(a)
The line l1 passes through the point A (0, 1, 2) and is perpendicular to the plane x − 4y − 3z
= 0. Find the Cartesian equations of l1.
(b)
The line l2 is parallel to l1 and passes through the point P(3, –8, –11). Find the vector
equation of the line l2.
(2)
(2)
(c)
(i)
(ii)
The point Q is on the line l1 such that PQ is perpendicular to l1 and l2. Find the
coordinates of Q.
Hence find the distance between l1 and l2.
(10)
(Total 14 marks)
12.
x2 y z 9
 
.
3
1 2
Let M be a point on l1 with parameter μ. Express the coordinates of M in terms of μ.
A line l1 has equation
(a)
(1)
(b)
The line l2 is parallel to l1 and passes through P(4, 0, –3).
(i)
Write down an equation for l2.
(ii)
Express PM in terms of μ.
(4)
(c)
The vector PM is perpendicular to l1.
(i)
Find the value of μ.
(ii) Find the distance between l1 and l2.
(d)
The plane π1 contains l1 and l2. Find an equation for π1, giving your answer in the form
Ax + By + Cz = D.
(e)
The plane π2 has equation x – 5y – z = –11. Verify that l1 is the line of intersection of the
planes π1 and π2.
(5)
(4)
(2)
(Total 16 marks)
13.
Consider the points A(1, 2, –4), B(l, 5, 0) and C(6, 5, –12). Find the area of ABC.
14.
Find the angle between the plane 3x – 2y + 4z = 12 and the z-axis. Give your answer to the nearest
degree.
15.
The point A (2, 5, –1) is on the line L, which is perpendicular to the plane with equation
x + y + z – 1 = 0.
(a) Find the Cartesian equation of the line L.
(Total 6 marks)
(Total 6 marks)
(2)
(b)
Find the point of intersection of the line L and the plane.
(c)
The point A is reflected in the plane. Find the coordinates of the image of A.
(d)
Calculate the distance from the point B(2, 0, 6) to the line L.
(4)
(2)
(4)
(Total 12 marks)
16.
Consider the following system of equations where b is a constant.
3x + y + z = 1
2x + y – z = 4
5x + y + bz = 1
(a) Solve for z in terms of b.
(4)
(b)
Hence write down, with a reason, the range of values of b for which this system of
equations has a unique solution.
(2)
(Total 6 marks)
2
17.




The vector n = 2 i – j +3 k is normal to a plane which passes through the point (2, 1, 2).
(a) Find an equation for the plane.
(b) Find a if the point (a, a – 1, a – 2) lies on the plane.
(Total 4 marks)
18.
The rectangle box shown in the diagram has dimensions 6 cm × 4 cm × 3 cm.
H
G
E
F
3cm
D
C
4cm
B
6cm
A
Find, correct to the nearest one-tenth of a degree, the size of the angle AHˆ C .
(Total 4 marks)
19.
Calculate the shortest distance from the point A(0, 2, 2) to the line







r = 5 i + 9 j + 6 k + t( i + 2 j + 2 k )
where t is a scalar.
20.
Consider the points A(l, 2, 1), B(0, –1, 2), C(1, 0, 2), and D(2, –1, –6).
(Total 4 marks)
(a)
Find the vectors AB and BC .
(2)
(b)
Calculate AB × BC .
(c)
Hence, or otherwise find the area of triangle ABC.
(d)
Find the equation of the plane P containing the points A, B, and C.
(e)
Find a set of parametric equations for the line through the point D and
perpendicular to the plane P.
(f)
Find the distance from the point D to the plane P.
(g)
Find a unit vector which is perpendicular to the plane P.
(h)
The point E is a reflection of D in the plane P. Find the coordinates of E.
(3)
(2)
(3)
(2)
(3)
(2)
(4)
(Total 21 marks)
21.
Find an equation for the line of intersection of the following two planes.
x + 2y – 3z = 2
2x + 3y – 5z = 3
(Total 6 marks)
22.
Let a be the angle between the vectors a and b, where a = (cos θ)i + (sin θ)j,
b = (sin θ)i + (cos θ)j and 0 < θ <
.
4
Express  in terms of θ.
(Total 3 marks)
23.
(a)
Given matrices A, B, C for which AB = C and det A  0, express B in terms of A and C.
(b)
1

Let A =  2
3

(2)
(i)
(ii)
2
1
3
3

2 , D =
2 
  4 13

7
 2
 3 9

 7

 4  and C =
5 
 5
 
 7.
10 
 
Find the matrix DA;
Find B if AB = C.
(3)
3
(c)
Find the coordinates of the point of intersection of the planes
x + 2y + 3z = 5, 2x – y + 2z = 7 and 3x – 3y + 2z = 10.
(2)
(Total 7 marks)
24.
(a)
(b)
If u = i +2j + 3k and v = 2i – j + 2k, show that
u × v = 7i + 4j – 5k.
Let w = u + μv where  and µ are scalars. Show that w is perpendicular to the line of
intersection of the planes x + 2y + 3z = 5 and 2x – y + 2z = 7 for all values of  and μ.
(2)
(4)
(Total 6 marks)
26.
Point A(3, 0, –2) lies on the line r = 3i – 2k + λ(2i – 2j + k), where λ is a real parameter. Find the
coordinates of one point which is 6 units from A, and on the line.
27.
(a)
Solve the following system of linear equations
x + 3y – 2z = –6
2x + y + 3z = 7
3x – y + z = 6.
(b)
Find the vector v = (i + 3j – 2k) × (2i + j + 3k).
(c)
If a = i + 3j – 2k, b = 2i + j + 3k and u = ma + nb where m, n are scalars, and u  0, show that
v is perpendicular to u for all m and n.
(d)
The line l lies in the plane 3x – y + z = 6, passes through the point (1, –1, 2) and is
perpendicular to v. Find the equation of l.
(Total 3 marks)
(3)
(3)
(3)
(4)
(Total 13 marks)
28.
Consider the points A (1, 3, –17) and B (6, – 7, 8) which lie on the line l.
(a) Find an equation of line l, giving the answer in parametric form.
(4)
(b)
The point P is on l such that OP is perpendicular to l.
Find the coordinates of P.
(3)
(Total 7 marks)
4