1.
Port A is defined to be the origin of a set of coordinate axes and port B is located at the point (70, 30), where distances are measured in kilometres. A ship S1 sails from port A at 10:00 in a 10 t 20 straight line such that its position t hours after 10:00 is given by r = . A speedboat S2 is capable of three times the speed of S1 and is to meet S1 by travelling the shortest possible distance. What is the latest time that S2 can leave port B? (Total 7 marks)
2.
The equations of three planes, are given by ax + 2y + z = 3 –x + (a + 1)y + 3z = 1 –2x + y + (a + 2)z = k where a
(a)
.
Given that a = 0, show that the three planes intersect at a point. (3)
(b)
Find the value of a such that the three planes do not meet at a point. (5)
(c)
Given a such that the three planes do not meet at a point, find the value of k such that the planes meet in one line and find an equation of this line in the form x x0 l y y0 m z z n 0 . (6) (Total 14 marks)
1 3.
3
Consider the functions f(x) = x + 1 and g(x) = x 1 . The graphs of y = f(x) and y = g(x) meet at the point (0, 1) and one other point, P. 3
(a)
Find the coordinates of P. (1)
(b)
Calculate the size of the acute angle between the tangents to the two graphs at the point P. (4) (Total 5 marks)
4.
The points P(–1, 2, –3), Q(–2, 1, 0), R(0, 5, 1) and S form a parallelogram, where S is diagonally opposite Q. (a)
Find the coordinates of S. (2)
(b)
The vector product
13 7 PQ PS m
. Find the value of m. (2)
(c)
Hence calculate the area of parallelogram PQRS. (2)
(d)
Find the Cartesian equation of the plane, Π1, containing the parallelogram PQRS. (3)
(e)
Write down the vector equation of the line through the origin (0, 0, 0) that is perpendicular to the plane Π1. (1)
(f)
Hence find the point on the plane that is closest to the origin. (3)
(g)
A second plane, Π2, has equation x – 2y + z = 3. Calculate the angle between the two planes. (4) (Total 17 marks)
5.
The system of equations 2x – y + 3z = 2 3x + y + 2z = –2
–x + 2y + az = b is known to have more than one solution. Find the value of a and the value of b. (Total 5 marks)
6.
A plane π has vector equation r = (–2i + 3j – 2k) + λ(2i + 3j + 2k) + μ(6i – 3j + 2k). (a)
Show that the Cartesian equation of the plane π is 3x + 2y – 6z = 12. (6)
(b)
The plane π meets the x, y and z axes at A, B and C respectively. Find the coordinates of A, B and C. (3)
(c)
Find the volume of the pyramid OABC. (3)
(d)
Find the angle between the plane π and the x-axis. (4)
(e)
Hence, or otherwise, find the distance from the origin to the plane π. (2)
(f)
Using your answers from (c) and (e), find the area of the triangle ABC. (2) (Total 20 marks)
7.
The three planes 2x – 2y – z = 3 4x + 5y – 2z = –3 3x + 4y – 3z = –7 intersect at the point with coordinates (a, b, c).
(a)
Find the value of each of a, b and c. (2)
(b)
The equations of three other planes are 2x – 4y – 3z = 4 –x + 3y + 5z = –2 3x – 5y – z = 6. Find a vector equation of the line of intersection of these three planes. (4) (Total 6 marks)
8.
Consider the vectors a = sin(2α)i – cos(2α)j + k and b = cos α i – sin α j – k, where 0 < α < 2π. Let θ be the angle between the vectors a and b. (a)
Express cos θ in terms of α. (2)
(b)
Find the acute angle α for which the two vectors are perpendicular. (2)
(c)
7π For α = 6 , determine the vector product of a and b and comment on the geometrical significance of this result. (4) (Total 8 marks)
9.
The diagram shows a cube OABCDEFG.
Let O be the origin, (OA) the x-axis, (OC) the y-axis and (OD) the z-axis. Let M, N and P be the midpoints of [FG], [DG] and [CG], respectively. The coordinates of F are (2, 2, 2).
(a)
Find the position vectors OM, ON and OP in component form. (3)
(b)
Find MP MN . (4)
(c)
Hence, (i)
calculate the area of the triangle MNP;
(ii)
show that the line (AG) is perpendicular to the plane MNP;
(iii)
find the equation of the plane MNP. (7)
(d)
Determine the coordinates of the point where the line (AG) meets the plane MNP. (6) (Total 20 marks)
10.
x 1 Find the angle between the lines 2 = 1 – y = 2z and x = y = 3z. (Total 6 marks)
11.
Consider the planes defined by the equations x + y + 2z = 2, 2x – y + 3z = 2 and 5x – y + az = 5 where a is a real number. (a)
If a = 4 find the coordinates of the point of intersection of the three planes. (2)
(b)
(i)
Find the value of a for which the planes do not meet at a unique point.
(ii)
For this value of a show that the three planes do not have any common point. (6) (Total 8 marks)
12.
The position vector at time t of a point P is given by
OP = (1 + t)i + (2 – 2t)j + (3t – 1)k, t ≥ 0.
(a)
Find the coordinates of P when t = 0. (2)
(b)
Show that P moves along the line L with Cartesian equations
y 2 z 1 3 . x – 1 = 2 (2)
(c)
(i)
Find the value of t when P lies on the plane with equation 2x + y + z = 6.
(ii)
State the coordinates of P at this time.
(iii)
Hence find the total distance travelled by P before it meets the plane. (6)
The position vector at time t of another point, Q, is given by
OQ
(d)
t2 1 t 1 t 2
, t ≥ 0.
(i)
Find the value of t for which the distance from Q to the origin is minimum.
(ii)
Find the coordinates of Q at this time. (6)
(e)
Let a, b and c be the position vectors of Q at times t = 0, t = 1, and t = 2 respectively. (i)
Show that the equation a – b = k(b – c) has no solution for k.
(ii)
Hence show that the path of Q is not a straight line. (7) (Total 23 marks)
13.
Given that a = 2 sin θ i + (1 – sin θ)j, find the value of the acute angle θ, so that a is perpendicular to the line x + y = 1. (Total 5 marks)
14.
x 1 1 y 3 2 z 6 1 . The vector equation of line l is given as Find the Cartesian equation of the plane containing the line l and the point A(4, –2, 5). (Total 6 marks)
15.
Two lines are defined by 3 3 x4 y7 4 2 and l 2 : 3 4 6 2 l1 : r = = –(z + 3).
(a)
Find the coordinates of the point A on l1 and the point B on l2 such that AB is perpendicular to both l1 and l2. (13)
(b)
Find │AB│. (3)
(c)
Find the Cartesian equation of the plane Π that contains l1 and does not intersect l2. (3) (Total 19 marks)
16.
A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector meets the plane π : x + 3y + 2z − 24 = 0.
4 1 2
and
Find the angle that the ray of light makes with the plane. (Total 6 marks)
17.
Find the vector equation of the line of intersection of the three planes represented by the following system of equations. 2x − 7y + 5z =1 6x + 3y – z = –1 −14x − 23y +13z = 5 (Total 6 marks)
18.
The angle between the vector a = i – 2j + 3k and the vector b = 3i – 2j + mk is 30°. Find the values of m. (Total 6 marks)
19.
(a)
Find the set of values of k for which the following system of equations has no solution. x + 2y – 3z = k 3x + y + 2z = 4 5x + 7z = 5 (4)
(b)
Describe the geometrical relationship of the three planes represented by this system of equations. (1) (Total 5 marks)
20.
(a)
Write the vector equations of the following lines in parametric form. 3 2 2 m 1 7 2 r1 = 1 4 4 n 1 2 1 r2 = (2)
(b)
Hence show that these two lines intersect and find the point of intersection, A. (5)
(c)
Find the Cartesian equation of the plane Π that contains these two lines. (4)
(d)
8 3 3 8 0 2 . Let B be the point of intersection of the plane Π and the line r = Find the coordinates of B. (4)
(e)
If C is the mid-point of AB, find the vector equation of the line perpendicular to the plane Π and passing through C. (3) (Total 18 marks)
21.
A triangle has its vertices at A(–1, 3, 2), B(3, 6, 1) and C(–4, 4, 3). (a)
Show that AB AC = –10. (3)
(b)
Find BAC . (5) (Total 8 marks)
