11 questions · 100 marks · ≈ 2 hours
By drawing suitable straight lines on the grid below, show, by shading, the region R
defined by all of the inequalities
y ≤ x x + y ≤ 8 y ≥ 2
Label the region R.
y
x
The point P with coordinates (x, y) is a point of the region R.
Given that x and y are integers and that P lies on the straight line with equation
x − 2y + 2 = 0
write down the coordinates of all possible points P.
Question 1 continued
E
12 cm
D
C
10 cm
M
8 cm
A
B
Figure 1
Figure 1 shows a right pyramid ABCDE with horizontal rectangular base ABCD and
vertex E.
AB = 8 cm BC = 10 cm EA = EB = EC = ED = 12 cm
M is the midpoint of the base.
Calculate the total surface area, in cm² to 3 significant figures, of the pyramid.
The point P is the midpoint of AB and the point Q is the midpoint of BC.
E
Diagram NOT
accurately drawn
12 cm
D
C
10 cm
M
8 cm
A
B
Figure 1
Figure 1 shows a right pyramid ABCDE with horizontal rectangular base ABCD and
vertex E.
AB = 8 cm BC = 10 cm EA = EB = EC = ED = 12 cm
M is the midpoint of the base.
Calculate the size, in degrees to one decimal place, of ∠PEQ.
Cosine rule: a
b
c
bc
A
2
2
2
2
cos
Question 2 continued
The table below gives information about the lengths of time, in minutes, that 200 people
waited for a train.
Find the class interval that contains the median time.
The table below gives information about the lengths of time, in minutes, that 200 people
waited for a train.
Calculate an estimate for the mean time, in minutes, that these 200 people waited for
the train.
The table below gives information about the lengths of time, in minutes, that 200 people
waited for a train.
On the grid opposite, draw a histogram for the information in the table.
Question 3 continued
Frequency
density
0
0
10
20
30
40
50
60
70
80
Time in minutes
Ella sells jars of jam, jars of honey and jars of chutney from her market stall.
On Thursday, the total number of jars she sold was 320
55% of the jars were jars of jam.
The ratio of the number of jars of honey to the number of jars of chutney that Ella sold
on Thursday was 5 : 3
Calculate the number of jars of chutney that Ella sold on Thursday.
On Friday, Ella sold 99 jars of strawberry jam.
This was 9
20 of the total number of jars of jam that she sold on Friday.
The number of jars of jam that Ella sold increased from Thursday to Friday.
Given that this increase = 1
nthe number of jars of jam that Ella sold on Thursday,
Ella sells jars of jam, jars of honey and jars of chutney from her market stall.
On Thursday, the total number of jars she sold was 320
55% of the jars were jars of jam.
The ratio of the number of jars of honey to the number of jars of chutney that Ella sold
on Thursday was 5 : 3
find the value of n.
Ella increases the price of a jar of jam from 3.20 euros to 3.50 euros.
Ella sells jars of jam, jars of honey and jars of chutney from her market stall.
On Thursday, the total number of jars she sold was 320
55% of the jars were jars of jam.
The ratio of the number of jars of honey to the number of jars of chutney that Ella sold
on Thursday was 5 : 3
Calculate the percentage increase in the price of a jar of jam.
Ella increases the price of a jar of honey by 6.25%
The price of a jar of honey after the price increase is 5.10 euros.
Ella sells jars of jam, jars of honey and jars of chutney from her market stall.
On Thursday, the total number of jars she sold was 320
55% of the jars were jars of jam.
The ratio of the number of jars of honey to the number of jars of chutney that Ella sold
on Thursday was 5 : 3
Calculate the price of a jar of honey before the price increase.
Question 4 continued
On the grid opposite, trapezium B is the image of trapezium A under a single
transformation.
Describe fully the single transformation.
Trapezium C is the image of trapezium A under a reflection in the line with
equation x = −1
On the grid opposite, trapezium B is the image of trapezium A under a single
transformation.
On the grid opposite, draw and label trapezium C.
Trapezium A is transformed to trapezium D under the transformation with matrix M
where
2
0
M
0
1
On the grid opposite, trapezium B is the image of trapezium A under a single
transformation.
On the grid opposite, draw and label trapezium D.
Trapezium D is transformed to trapezium B under the transformation with matrix N.
On the grid opposite, trapezium B is the image of trapezium A under a single
transformation.
Find matrix N.
d
b
a
b
1
The inverse of matrix
is
c
a
c
d
ad
bc
Question 5 continued
y
x
−4
Turn over for a spare grid if you need to redraw your trapeziums.
Question 5 continued
Question 5 continued
Only use this grid if you need to redraw your trapeziums.
y
x
−4
The point A has coordinates (5, 4) and the point B has coordinates (−7, −1)
3
→ =
The point C is such that BC
4
Find the coordinates of the point C.
The point D is such that ABCD is a parallelogram with diagonals AC and BD.
The length of BC is 5 cm.
The point A has coordinates (5, 4) and the point B has coordinates (−7, −1)
3
→ =
The point C is such that BC
4
Find the area, in cm², of the parallelogram ABCD.
a
b
c
bc
A
2
2
2
Cosine rule:
2
cos
1
ab
C
Area of triangle
sin
2
Question 6 continued
Here is the equation of a curve y = x3 − 3x − 1
Complete the table of values for y = x3 − 3x − 1, giving your values of y to
2 decimal places where necessary.
Here is the equation of a curve y = x3 − 3x − 1
On the grid opposite, plot the points from your completed table and join them to form
a smooth curve.
Here is the equation of a curve y = x3 − 3x − 1
Use your graph to estimate, to one decimal place, the solutions in the interval
−2 ≤ x ≤ 2 of the equation
x3 − 3x − 1 = 0.5
Here is the equation of a curve y = x3 − 3x − 1
By drawing a suitable straight line on the grid, find an estimate, to one significant
figure, of the gradient of the curve at the point where x = 0.5
Question 7 continued
y
x
Turn over for a spare grid if you need to redraw your curve.
Question 7 continued
Question 7 continued
Only use this grid if you need to redraw your curve.
y
x
Some college students were each asked which of the subjects Mathematics (M),
Physics (P) and Chemistry (C) they were studying.
Here is some information about their answers.
n(M ∩ P) = 21
n(M ∩ C ) = 24
n(P ∩ C ) = 25
n(M ) = 43
n(P) = 50
n([M ∪ P ∪ C ]') = 25
n([M ∪ P]' ∩ C ) = 8
n(M ∩ P ∩ C ) = x , where x is a positive integer.
Use all the given information to complete the Venn diagram on the opposite page,
giving the number of elements in each appropriate subset, in terms of x where necessary.
Given that n(C ) = 40
Some college students were each asked which of the subjects Mathematics (M),
Physics (P) and Chemistry (C) they were studying.
Here is some information about their answers.
n(M ∩ P) = 21
n(M ∩ C ) = 24
n(P ∩ C ) = 25
n(M ) = 43
n(P) = 50
n([M ∪ P ∪ C ]') = 25
n([M ∪ P]' ∩ C ) = 8
n(M ∩ P ∩ C ) = x , where x is a positive integer.
find the total number of college students that were asked.
One of these college students is to be chosen at random.
Given that the college student studies Chemistry,
Some college students were each asked which of the subjects Mathematics (M),
Physics (P) and Chemistry (C) they were studying.
Here is some information about their answers.
n(M ∩ P) = 21
n(M ∩ C ) = 24
n(P ∩ C ) = 25
n(M ) = 43
n(P) = 50
n([M ∪ P ∪ C ]') = 25
n([M ∪ P]' ∩ C ) = 8
n(M ∩ P ∩ C ) = x , where x is a positive integer.
find the probability that this student also studies Physics.
Question 8 continued
E
P
M
x
C
B
Diagram NOT
C
accurately drawn
O
18°
120°
A
D
E
Figure 2
Figure 2 shows a circle ABCDE with centre O.
∠BAO = 18° ∠AOD = 120°
The area of segment BCD, shown shaded in Figure 2, is T cm²
Given that the perimeter of the sector AODE is 5(3 + ) cm,
calculate the value, to one decimal place, of T.
1
2 ab
C
Area of triangle
sin
Question 9 continued
A
P
O
Diagram NOT
accurately drawn
C
Q
B
T
Figure 3
→ = 4a and OB
→ = 6b
Figure 3 shows the triangle OAB in which OA
C is the point on AB such that AC : CB = 3 : 1
→ in terms of a and b.
Find and simplify an expression for OC
The point T is such that ABT is a straight line and AC = CT.
→ = 3
→
The point P is such that OP
4 OA
Q is the point on OB such that the points P, Q and T are collinear.
Given that OQ : QB = n : 1
A
P
O
Diagram NOT
accurately drawn
C
Q
B
T
Figure 3
→ = 4a and OB
→ = 6b
Figure 3 shows the triangle OAB in which OA
C is the point on AB such that AC : CB = 3 : 1
→ in terms of a and b.
find the value of n.
Question 10 continued
Question 10 continued
Question 10 continued
The three functions, f, g and h, are defined as
f : x ↦ 4x3 + 4x2 − 5x − 3
g : x ↦ x
−
7
x
−
3
2
h : x ↦ 2x + 1
Write down the value of x that must be excluded from any domain of g
The three functions, f, g and h, are defined as
f : x ↦ 4x3 + 4x2 − 5x − 3
g : x ↦ x
−
7
x
−
3
2
h : x ↦ 2x + 1
Find g
The three functions, f, g and h, are defined as
f : x ↦ 4x3 + 4x2 − 5x − 3
g : x ↦ x
−
7
x
−
3
2
h : x ↦ 2x + 1
Express the inverse function g−1 in the form g−1(x) = …
The three functions, f, g and h, are defined as
f : x ↦ 4x3 + 4x2 − 5x − 3
g : x ↦ x
−
7
x
−
3
2
h : x ↦ 2x + 1
Solve the equation g(x) = h(x)
The three functions, f, g and h, are defined as
f : x ↦ 4x3 + 4x2 − 5x − 3
g : x ↦ x
−
7
x
−
3
2
h : x ↦ 2x + 1
Use the factor theorem to show that (2x + 3) is a factor of f (x)
The three functions, f, g and h, are defined as
f : x ↦ 4x3 + 4x2 − 5x − 3
g : x ↦ x
−
7
x
−
3
2
h : x ↦ 2x + 1
Hence solve the equation f (x) = 0
Show clear algebraic working.
Question 11 continued
Question 11 continued
Question 11 continued