1
1
B
NOT TO
SCALE
b
C
O
In the diagram, O is the origin,
= a and
a
A
Answer(a)
= ...................................................
= b.
C is on the line AB so that AC : CB = 1 : 2.
Find, in terms of a and b, in its simplest form,
(a)
,
[2]
(b) the position vector of C.
Answer(b) ...................................................
[2]
[Total: 4]
2
2
In the diagram, O is the origin,
(a) Find
and
.
, in terms of a and b, in its simplest form.
...................................................
[2]
(b)
Find the position vector of E, in terms of a and b, in its simplest form.
...................................................
[2]
[Total: 4]
3
3
(a) Find 3a – 2b.
[2]
(b) Find
.
...................................................
[2]
(c)
Write down two simultaneous equations and solve them to find the value of m and the value of n.
Show all your working.
m = ...................................................
n = ...................................................
[5]
[Total: 9]
4
4
OAB is a triangle and C is the mid-point of OB.
D is on AB such that AD : DB = 3 : 5.
OAE is a straight line such that OA : AE = 2 : 3.
= a and
= c.
(a) Find, in terms of a and c, in its simplest form,
(i)
(ii)
(iii)
(iv)
,
= ...................................................
[1]
= ...................................................
[1]
= ...................................................
[1]
= ...................................................
[2]
,
,
.
5
(b)
Find the value of k.
k = ...................................................
[1]
[Total: 6]
5
In the diagram, OABC is a parallelogram.
OP and CA intersect at X and CP : PB = 2 : 1.
= a and
(a) Find
(b)
= c.
, in terms of a and c, in its simplest form.
= ...................................................
[2]
= ...................................................
[2]
CX : XA = 2 : 3
(i)
Find
, in terms of a and c, in its simplest form.
6
(ii) Find OX : XP.
OX : XP = .................... : .................... [2]
[Total: 6]
6
ABCD is a parallelogram with
and
ABM is a straight line with AB : BM = 1 : 1.
ADN is a straight line with AD : DN = 3 : 2.
(a) Write
.
, in terms of p and q, in its simplest form.
= ...................................................
[2]
7
(b) The straight line NM cuts BC at X.
X is the midpoint of MN.
Find the value of k.
k = ...................................................
[2]
[Total: 4]
7
OABC is a parallelogram and O is the origin.
CK = 2KB and AL = LB.
M is the midpoint of KL.
and
.
Find, in terms of p and q, giving your answer in its simplest form
(a)
,
...................................................
[2]
8
(b) the position vector of M.
...................................................
[2]
[Total: 4]
8
The points P, Q, R and S lie on a circle with diameter PR.
Work out the size of angle PSQ, giving a geometrical reason for each step of your working.
....................................................................................................................................................................
....................................................................................................................................................................
....................................................................................................................................................................
[3]
[Total: 3]
9
9
A
B
x°
NOT TO
SCALE
D
O
44°
C
A, B and C are points on a circle, centre O.
DA and DC are tangents.
Angle ADC = 44°.
Work out the value of x.
x = ...................................................
[3]
[Total: 3]
10
10
O
142°
NOT TO
SCALE
C
A
y°
B
Points A, B and C lie on a circle, centre O.
Angle AOC = 142°.
Find the value of y.
y = ...................................................
[2]
[Total: 2]
11
11
P, R and Q are points on the circle.
AB is a tangent to the circle at Q.
QR bisects angle PQB.
Angle BQR = x° and x < 60.
Use this information to show that triangle PQR is an isosceles triangle.
Give a geometrical reason for each step of your work.
[3]
[Total: 3]
12
12
Points A, B, C, D, E and F lie on the circle, centre O.
Find the value of x and the value of y.
x = ...................................................
y = ...................................................
[2]
[Total: 2]
13
A curve has equation
.
Find the coordinates of its two stationary points.
( .................... , .................... ) and ( .................... , .................... ) [5]
13
[Total: 5]
14
A curve has the equation
.
(a) Work out the coordinates of the two turning points.
( .............................. , .............................. ) and ( .............................. , .............................. ) [6]
(b) Determine whether each of the turning points is a maximum or a minimum.
Give reasons for your answers.
[3]
[Total: 9]
14
15
, where
is the derived function.
Find the value of p and the value of q.
p = ...................................................
q = ...................................................
[2]
[Total: 2]
16
On the axes, sketch the graph of each of these functions.
(a)
y
O
x
[2]
15
(b)
y
O
x
[2]
[Total: 4]
17
On the diagram,
(a) sketch the graph of
(b) sketch the graph of
,
.
[2]
[2]
[Total: 4]
16
18
The graph of
is shown on the grid.
By drawing a suitable line on the grid, solve the equation
.
x = .............................. or x = .............................. [3]
[Total: 3]
19
The table shows some values for
x
−3.5
y
−4.1
−3
(a) Complete the table.
.
−2.5
−2
−1.5
−1
−0.5
5.1
6
5.4
4
2.6
0
1.5
2.9
1
1.5
12.1
[3]
17
(b) On the grid, draw the graph of
for
.
[4]
(c) Use your graph to solve the equation
for
.
x = ...................................................
(d) By drawing a suitable straight line, solve the equation
for
.
x = ...................................................
(e) For
, the equation
[1]
[2]
has three solutions and k is an integer.
Write down a possible value of k.
k = ...................................................
[1]
[Total: 11]
18
20
The diagram shows the graph of y = f(x) where
,
.
19
(a) Use the graph to find
(i)
f(1),
...................................................
[1]
...................................................
[2]
(ii) ff(−2).
(b) On the grid, draw a suitable straight line to solve the equation
for
.
x = .............................. or x = .............................. [4]
(c) By drawing a suitable tangent, find an estimate of the gradient of the curve at x = −2.
...................................................
(d)
(i)
for
Complete the table for y = g(x) where
x
y
−3
−2
[3]
.
−1
0
1
2
1
0.5
2
3
0.125
[3]
[3]
(ii) On the grid, draw the graph of y = g(x).
(iii) Use your graph to find the positive solution to the equation f(x) = g(x).
x = ...................................................
[1]
[Total: 17]
20
21
The table shows some values of
x
0.15
y
3.30
0.2
(a) Complete the table.
for
0.5
0.88
1
.
1.5
−0.04
2
2.5
3
3.5
−0.43
−0.58
−0.73
[3]
21
(b) On the grid, draw the graph of
for
.
The last two points have been plotted for you.
[4]
22
(c) Use your graph to solve the equation
for
.
x = ...................................................
(d)
(i) On the grid, draw the line
.
(ii) Write down the x co-ordinates of the points where the line
for
[1]
[2]
crosses the graph of
.
x = .............................. and x = .............................. [2]
(e) Show that the graph of
can be used to find the value of
for
.
[2]
[Total: 14]