12 questions · 100 marks · ≈ 2 hours
D
E
Diagram NOT
C
accurately drawn
B
A
Figure 1
Figure 1 shows part of a regular 12‑sided polygon.
The vertices B and E of the polygon are joined with a straight line.
Calculate the size, in degrees, of ∠ABE.
Give reasons for each stage of your working.
Solve the simultaneous equations
3x - 2y = 7
x + 6y = 15
Show clear algebraic working.
The original price of each 6‑day ski pass is reduced by 15% in a sale.
In the sale the price of each 6‑day ski pass is $272
Calculate the original price of each 6‑day ski pass.
The price of each 3‑day ski pass is £110
The exchange rate is £1 = $1.70
The original price of each 6‑day ski pass is reduced by 15% in a sale.
In the sale the price of each 6‑day ski pass is $272
Calculate how much Andrew will save by buying one 6‑day ski pass in the sale rather
than two 3‑day ski passes.
Express 56 as the product of its prime factors.
Trains to Watson leave Denby station every 56 minutes.
Trains to Barbe leave Denby station every 24 minutes.
A train to Watson and a train to Barbe both leave Denby station at 12 00.
Find the next time that a train to Watson and a train to Barbe leave Denby station at
the same time.
On the grid opposite, draw the graph of y = 3x + 2 for the values of x from 0 to 5
Show, by shading on the grid, the region R defined by all of the inequalities
y ≤ 3x + 2 and y ≥ 4 and 8 ≤ 4x ≤ 18
Label the region R.
Question 5 continued
y
18
16
14
12
10
8
6
4
2
x
O
1
2
3
4
5
One day, 80 customers at a health spa were asked if they used any of the gym (G), the pool (P)
and the sauna (S).
Of these 80 customers
34 had used the gym
60 had used the pool
30 had used the sauna
20 had used the gym and the pool but not the sauna
17 had used the pool and the sauna but not the gym
6 had used the gym, the pool and the sauna
no one had used the gym and the sauna but not the pool.
Using this information, complete the Venn diagram to show the number of elements
in each appropriate subset.
E
P
G
S
One day, 80 customers at a health spa were asked if they used any of the gym (G), the pool (P)
and the sauna (S).
Of these 80 customers
34 had used the gym
60 had used the pool
30 had used the sauna
20 had used the gym and the pool but not the sauna
17 had used the pool and the sauna but not the gym
6 had used the gym, the pool and the sauna
no one had used the gym and the sauna but not the pool.
Find (i) n([G ∪ P ∪ S]' )
One day, 80 customers at a health spa were asked if they used any of the gym (G), the pool (P)
and the sauna (S).
Of these 80 customers
34 had used the gym
60 had used the pool
30 had used the sauna
20 had used the gym and the pool but not the sauna
17 had used the pool and the sauna but not the gym
6 had used the gym, the pool and the sauna
no one had used the gym and the sauna but not the pool.
n(G ∪ S)
One day, 80 customers at a health spa were asked if they used any of the gym (G), the pool (P)
and the sauna (S).
Of these 80 customers
34 had used the gym
60 had used the pool
30 had used the sauna
20 had used the gym and the pool but not the sauna
17 had used the pool and the sauna but not the gym
6 had used the gym, the pool and the sauna
no one had used the gym and the sauna but not the pool.
n(P ∩ S' )
Question 6 continued
Vincent plays a game in which he can score 0, 1, 2, 3 or 4 each time he plays the game.
The score he gets in a game is independent of the score he got in the previous games.
The table gives information about the probability of getting each score in a game.
Vincent plays the game 150 times.
Calculate the number of times he would expect to score 1
Vincent plays the game another two times and records his two scores.
He adds these two scores together to get his Total.
Vincent plays a game in which he can score 0, 1, 2, 3 or 4 each time he plays the game.
The score he gets in a game is independent of the score he got in the previous games.
The table gives information about the probability of getting each score in a game.
Vincent plays the game 150 times.
Calculate the probability that Vincent’s Total is greater than 6
Given that for these two games Vincent has a Total greater than 6 points,
Vincent plays a game in which he can score 0, 1, 2, 3 or 4 each time he plays the game.
The score he gets in a game is independent of the score he got in the previous games.
The table gives information about the probability of getting each score in a game.
Vincent plays the game 150 times.
calculate the probability that he got a score of 4 in the first of these two games.
Question 7 continued
A
Diagram NOT
accurately drawn
C
P
O
D
B
Q
Figure 2
→ = 2a and OB
→ = 3b
Figure 2 shows the triangle OAB with OA
→ = 1
→
The point C lies on OA such that OC
3 OA
→ = 2
→
The point D lies on OB such that OD
3 OB
→ in terms of a and b.
Find CD
The point P is such that ODBP is a straight line and AP is parallel to CD.
→ in terms of b.
A
Diagram NOT
accurately drawn
C
P
O
D
B
Q
Figure 2
→ = 2a and OB
→ = 3b
Figure 2 shows the triangle OAB with OA
→ = 1
→
The point C lies on OA such that OC
3 OA
→ = 2
→
The point D lies on OB such that OD
3 OB
→ in terms of a and b.
Find OP
→ = DQ
→
The point Q is such that CD
A
Diagram NOT
accurately drawn
C
P
O
D
B
Q
Figure 2
→ = 2a and OB
→ = 3b
Figure 2 shows the triangle OAB with OA
→ = 1
→
The point C lies on OA such that OC
3 OA
→ = 2
→
The point D lies on OB such that OD
3 OB
→ in terms of a and b.
Show that A, B and Q are collinear.
Question 8 continued
Question 8 continued
Question 8 continued
The vertices of triangle A are the points with coordinates (0, 0), (1, 0) and (0, 2)
On the grid opposite, draw and label triangle A.
Triangle A is transformed to triangle B under the transformation with matrix M where
−
3
4
M =
4
3
The vertices of triangle A are the points with coordinates (0, 0), (1, 0) and (0, 2)
On the grid, draw and label triangle B.
The transformation with matrix M is equivalent to an enlargement with centre the origin,
with scale factor k followed by an anticlockwise rotation of θ° about the origin.
The vertices of triangle A are the points with coordinates (0, 0), (1, 0) and (0, 2)
Calculate the value of k.
The vertices of triangle A are the points with coordinates (0, 0), (1, 0) and (0, 2)
Calculate the value, to one decimal place, of θ.
Triangle B is transformed to triangle C under a reflection in the x‑axis.
The vertices of triangle A are the points with coordinates (0, 0), (1, 0) and (0, 2)
On the grid, draw and label triangle C.
Triangle A is transformed to triangle C under the transformation with matrix T.
The vertices of triangle A are the points with coordinates (0, 0), (1, 0) and (0, 2)
Find matrix T.
Question 9 continued
y
x
Turn over for a spare grid if you need to redraw your triangles.
Question 9 continued
Question 9 continued
Only use this grid if you need to redraw your triangles.
y
x
A rocket is launched from a point which is 5 m above horizontal ground.
The rocket moves vertically upwards so that at time t seconds, the height, h metres, of the
rocket above the ground is given by
h = 5 + 3t + 9t2 - t3
At time t seconds, the velocity of the rocket is v m/s and the acceleration of the rocket is a m/s2
Find an expression for v in terms of t.
A rocket is launched from a point which is 5 m above horizontal ground.
The rocket moves vertically upwards so that at time t seconds, the height, h metres, of the
rocket above the ground is given by
h = 5 + 3t + 9t2 - t3
At time t seconds, the velocity of the rocket is v m/s and the acceleration of the rocket is a m/s2
Find an expression for a in terms of t.
A rocket is launched from a point which is 5 m above horizontal ground.
The rocket moves vertically upwards so that at time t seconds, the height, h metres, of the
rocket above the ground is given by
h = 5 + 3t + 9t2 - t3
At time t seconds, the velocity of the rocket is v m/s and the acceleration of the rocket is a m/s2
Find the time when the rocket stops accelerating upwards.
The rocket is instantaneously at rest when it is at point A.
A rocket is launched from a point which is 5 m above horizontal ground.
The rocket moves vertically upwards so that at time t seconds, the height, h metres, of the
rocket above the ground is given by
h = 5 + 3t + 9t2 - t3
At time t seconds, the velocity of the rocket is v m/s and the acceleration of the rocket is a m/s2
Show that the height, in metres to one decimal place, of A above the ground is 131.2 m.
The rocket now falls vertically downwards and hits the ground.
A rocket is launched from a point which is 5 m above horizontal ground.
The rocket moves vertically upwards so that at time t seconds, the height, h metres, of the
rocket above the ground is given by
h = 5 + 3t + 9t2 - t3
At time t seconds, the velocity of the rocket is v m/s and the acceleration of the rocket is a m/s2
Find the total distance, to the nearest metre, travelled by the rocket at the instant it
hits the ground.
b
b
ac
= −
±
−
(
4
)
2
ax
bx
c
x
=
Solutions of
0 are
2
a
2
Question 10 continued
Question 10 continued
Question 10 continued
f (x) = 4x3 - 13x - 6
Use the factor theorem to show that (2x + 1) is a factor of f (x).
f (x) = 4x3 - 13x - 6
Hence factorise f (x) fully.
The curve C has equation y = f (x)
f (x) = 4x3 - 13x - 6
Find the coordinates of the points of intersection of C with the x‑axis.
f (x) = 4x3 - 13x - 6
Find the coordinates, to 2 decimal places, of the turning points of C.
The table below gives the coordinates of three points on C.
f (x) = 4x3 - 13x - 6
On the grid opposite, draw the curve C for –2 ≤ x ≤ 2
Clearly label the coordinates of the turning points of C and the coordinates of the
points of intersection with the x‑axis and the y‑axis.
Question 11 continued
y
x
Turn over for a spare grid if you need to redraw your curve.
Question 11 continued
Question 11 continued
Only use this grid if you need to redraw your curve.
y
x
2a cm
b cm
b cm
Figure 3
Figure 3 shows a solid silver paperweight made from a cuboid and a half cylinder.
The cuboid is 2a cm wide, b cm long and b cm high.
The plane face of the half cylinder coincides with the top face of the cuboid.
The total surface area of the paper weight is A cm²
Find an expression for A in terms of , a and b.
Given that a = 6 5 and that the surface area of the paperweight can be written as
(2b2 + 6ab + 60 15) cm²
2a cm
b cm
b cm
Figure 3
Figure 3 shows a solid silver paperweight made from a cuboid and a half cylinder.
The cuboid is 2a cm wide, b cm long and b cm high.
The plane face of the half cylinder coincides with the top face of the cuboid.
The total surface area of the paper weight is A cm²
show that the exact value of b is 10 3 - 6 5
The paperweight is melted down to form a different cuboid.
This second cuboid is 2a cm wide, b cm long and h cm high, as shown in Figure 4.
F
G
h cm
E
H
B
C
b cm
A
2a cm
D
Figure 4
2a cm
b cm
b cm
Figure 3
Figure 3 shows a solid silver paperweight made from a cuboid and a half cylinder.
The cuboid is 2a cm wide, b cm long and b cm high.
The plane face of the half cylinder coincides with the top face of the cuboid.
The total surface area of the paper weight is A cm²
Calculate the size, to the nearest degree, of angle GAC.
Question 12 continued
r h
2
Volume of cylinder
rh
Curved surface area of cylinder 2
Question 12 continued
Question 12 continued