11 questions · 100 marks · ≈ 2 hours
The functions f and g are defined as
f(x) = 3x
15
g(x) =
x −
2
3
Write down the value of x for which f(x) = 81
The functions f and g are defined as
f(x) = 3x
15
g(x) =
x −
2
3
Write down the value of x that must be excluded from any domain of g
The functions f and g are defined as
f(x) = 3x
15
g(x) =
x −
2
3
Find fg
Question 1 continued
Find the set of values of x for which
–2x - 4 < 3x + 6 ≤ 21 - 2x
Solve the inequality 4x2 - 8x < 21
Hence represent on the number line on the opposite page, the set of values of x for
which
–2x - 4 < 3x + 6 ≤ 21 - 2x and 4x2 - 8x < 21
Question 2 continued
x
–5
–4
–3
–2
–1
0
1
2
3
4
5
Peter has two boxes of bricks, box A and box B.
Each box contains only red bricks, green bricks and yellow bricks.
In box A there are 15 bricks of which
5 are red,
7 are green,
and 3 are yellow.
In box B there are 20 bricks of which
8 are red,
9 are green,
and 3 are yellow.
Ami is going to take a brick at random from box A and a brick at random from box B.
She is going to put these two bricks on a table.
Complete the probability tree diagram.
box B
box A
red
green
red
5
yellow
15
red
green
green
yellow
red
green
yellow
yellow
Question 3 continued
Peter has two boxes of bricks, box A and box B.
Each box contains only red bricks, green bricks and yellow bricks.
In box A there are 15 bricks of which
5 are red,
7 are green,
and 3 are yellow.
In box B there are 20 bricks of which
8 are red,
9 are green,
and 3 are yellow.
Ami is going to take a brick at random from box A and a brick at random from box B.
She is going to put these two bricks on a table.
Calculate the probability that the two bricks on the table are different colours.
Peter has a third box of bricks, box C.
This box also contains only red bricks, green bricks and yellow bricks.
After taking a brick from box A and a brick from box B, Ami then takes at random a
brick from box C.
Ami then puts this brick with the other two bricks on the table.
The probability that there are three yellow bricks on the table is 19
1000
Peter has two boxes of bricks, box A and box B.
Each box contains only red bricks, green bricks and yellow bricks.
In box A there are 15 bricks of which
5 are red,
7 are green,
and 3 are yellow.
In box B there are 20 bricks of which
8 are red,
9 are green,
and 3 are yellow.
Ami is going to take a brick at random from box A and a brick at random from box B.
She is going to put these two bricks on a table.
Find the least number of bricks that were in box C before Ami took a brick from box C.
Question 3 continued
Question 3 continued
3
7
2
C =
4 A =
B =
10
5
7
5
4
1
Find D such that 2A - B = 3D
3
7
2
C =
4 A =
B =
10
5
7
5
4
1
Find A2
3
7
2
C =
4 A =
B =
10
5
7
5
4
1
Find B–1
3
7
2
C =
4 A =
B =
10
5
7
5
4
1
Find CA
a
b
d
b
1
Inverse of matrix
c
d
c
a
ad
bc
Question 4 continued
The graph of the straight line with equation 3x + y = 11 is drawn on the grid opposite
for 0 ≤ x ≤ 4
On the grid, draw the graph of the straight line with equation 4x = 3 + 2y
for 0 ≤ x ≤ 4
The graph of the straight line with equation 3x + y = 11 is drawn on the grid opposite
for 0 ≤ x ≤ 4
Use your graphs to find estimates, to one decimal place, of the value of x and the
value of y that satisfy the simultaneous equations
3x + y = 11
4x = 3 + 2y
The graph of the straight line with equation 3x + y = 11 is drawn on the grid opposite
for 0 ≤ x ≤ 4
Solve the simultaneous equations
4x + xy - x2 = –10
3y + x = 7
Show clear algebraic working.
Question 5 continued
y
11
10
9
8
7
6
5
4
3
2
1
0
–1
–2
Question 5 continued
Question 5 continued
Triangle A is drawn on the grid opposite.
Triangle A is reflected in the line with equation x = –1 to give triangle B.
On the grid, draw and label triangle B.
Triangle A is transformed to triangle C under a rotation of 90° clockwise about the point
with coordinates (1, 0)
Triangle A is drawn on the grid opposite.
Triangle A is reflected in the line with equation x = –1 to give triangle B.
On the grid, draw and label triangle C.
2
Triangle B is transformed to triangle D under the translation
5
Triangle A is drawn on the grid opposite.
Triangle A is reflected in the line with equation x = –1 to give triangle B.
On the grid, draw and label triangle D.
Triangle A is transformed to triangle E under the transformation with matrix M where
1
0
M
0
1
Triangle A is drawn on the grid opposite.
Triangle A is reflected in the line with equation x = –1 to give triangle B.
On the grid, draw and label triangle E.
Triangle E is the image of triangle C under a single transformation.
Triangle A is drawn on the grid opposite.
Triangle A is reflected in the line with equation x = –1 to give triangle B.
Describe fully this transformation.
Question 6 continued
y
x
Turn over for a spare grid if you need to redraw your triangles.
Question 6 continued
Question 6 continued
Only use this grid if you need to redraw your triangles.
y
x
There are 480 people in an airport departure lounge.
30% of these people are catching a plane to Dubai.
Show that 336 of these people are not catching a plane to Dubai.
The people in the airport lounge who are not catching a plane to Dubai are catching a
plane to Sweden or a plane to Greece or a plane to Brazil.
Of these 336 people
the number catching a plane to Sweden is s
the number catching a plane to Greece is g
the number catching a plane to Brazil is b where
s : g : b = 6 : 7 : 8
There are 480 people in an airport departure lounge.
30% of these people are catching a plane to Dubai.
Calculate the value of s
Pablo went by plane from Canada to Brazil in February 2020 and in February 2021
In February 2020, the cost of his ticket was $680
In February 2021, the cost of his ticket was $730
There are 480 people in an airport departure lounge.
30% of these people are catching a plane to Dubai.
Calculate the percentage increase, to one decimal place, in the cost of the ticket
from February 2020 to February 2021
Pablo bought a ticket to go by plane to Sweden in June 2021
The cost of his ticket was $468
The cost of this ticket was 4% greater than the cost of his ticket the last time he went by
plane to Sweden, which was in December 2020
There are 480 people in an airport departure lounge.
30% of these people are catching a plane to Dubai.
Calculate the cost of Pablo’s ticket to Sweden in December 2020
At the end of his trip, Pablo had 320 Swedish krona left.
He changed the 320 Swedish krona into Canadian dollars.
Using the following exchange rates,
1 Canadian dollar = 0.57 euros
1 Swedish krona = 0.094 euros
There are 480 people in an airport departure lounge.
30% of these people are catching a plane to Dubai.
calculate the number, to 2 decimal places, of Canadian dollars that Pablo should
have received.
Question 7 continued
Question 7 continued
Question 7 continued
The equation of a curve C is y
x
x
2
1
2
The curve C has a minimum at the point A
Show that the coordinates of A are (0.75, –1.5625)
3
The equation of a curve C is y
x
x
2
1
2
The curve C has a minimum at the point A
Complete the table of values for y
x
x
2
1
2
The point A has been plotted on the grid opposite.
3
1 for values
The equation of a curve C is y
x
x
2
1
2
The curve C has a minimum at the point A
On the grid opposite, draw the curve with equation y
x
x
2
2
of x from –3 to 4
The equation of a curve C is y
x
x
2
1
2
The curve C has a minimum at the point A
Using your curve, find an estimate, to one decimal place, for the range of values
3
of x for which x
x
−≤ 3
2
−
1
2
Show your working clearly.
The equation of a curve C is y
x
x
2
1
2
The curve C has a minimum at the point A
By drawing a suitable straight line on the grid, find estimates, to one decimal place,
7
1
of the solutions of the equation x
x
2
2
2
Show your working clearly.
Question 8 continued
y
x
Turn over for a spare grid if you need to redraw your graph.
Question 8 continued
Question 8 continued
Only use this grid if you need to redraw your graph.
y
x
A
C
Diagram NOT
O
B
accurately drawn
Figure 1
Figure 1 shows a trapezium OACB in which OA is parallel to BC and OA : BC = 2 : 1
The point P lies on AC such that AP : PC = 3 : 1
→
®
= λ OP
where λ > 1 and such that BCD is a straight line.
The point D is such that OD
→
→
= 6a
b
Given that OA
OB
and that
= 8
®
use a vector method to find and simplify an expression, in terms of a and b only, for AD
Show your working clearly.
Question 9 continued
Q
Diagram NOT
accurately drawn
51°
M
x
R
P
T
Figure 2
In Figure 2, P, Q, R and T are points on a circle with centre M such that Ð PQR = 51°
Find the size, in degrees, of the angle marked x in Figure 2
Q
51°
M
x
R
P
T
Figure 2
In Figure 2, P, Q, R and T are points on a circle with centre M such that Ð PQR = 51°
Find the size, in degrees, of the obtuse angle PTR.
B
S
O
118°
C
A
Figure 3
In Figure 3, A, B and C are points on a circle with centre O such that Ð AOC = 118°
and BA = BC.
The area of the region S, shown shaded in Figure 3, is 70 cm²
Q
51°
M
x
R
P
T
Figure 2
In Figure 2, P, Q, R and T are points on a circle with centre M such that Ð PQR = 51°
Calculate the total area, in cm² to 3 significant figures, of the unshaded region inside
the circle.
1
Area of triangle
2 ab
C
sin
Question 10 continued
Diagram NOT
accurately drawn
3r cm
10 cm
r cm
Figure 4
Figure 4 shows a solid right circular cone of radius r cm and height 3r cm and a sphere
of radius 10 cm.
The total surface area of the cone is equal to the surface area of the sphere.
Calculate the value, to one decimal place, of r
Curved surface area of cone
rl
Surface area of sphere =
r
2
4
Question 11 continued
Question 11 continues on page 34
V
Diagram NOT
accurately drawn
15 cm
B
C
A
D
E
F
Figure 5
Figure 5 shows a right pyramid.
The base ABCDEF of the pyramid is a regular hexagon and the vertex V of the pyramid
is such that
AV = BV = CV = DV = EV = FV = 15 cm and ÐAVD = 40°
3r cm
10 cm
r cm
Figure 4
Figure 4 shows a solid right circular cone of radius r cm and height 3r cm and a sphere
of radius 10 cm.
The total surface area of the cone is equal to the surface area of the sphere.
Calculate the volume, in cm³ to 3 significant figures, of the pyramid.
1
Area of triangle
ab
C
sin
2
1
Volume of pyramid
base area
height
3
Question 11 continued
Question 11 continued