12 questions · 100 marks · ≈ 2 hours
A = 233257
B = 2345211
Find the highest common factor (HCF) of A and B.
A = 233257
B = 2345211
Find the lowest common multiple (LCM) of A and B.
A = 233257
B = 2345211
Find the least number that A must be multiplied by to give a square number.
C =
A = 233257
B = 2345211
Work out the value of C, giving your answer as a number in standard form.
Question 1 continued
The points with coordinates (2, 1), (5, 1), (3, 3) and (6, 3) are the vertices of
parallelogram A.
On the grid, draw and label parallelogram A.
2
Parallelogram A is transformed to parallelogram B under the translation
−
5
The points with coordinates (2, 1), (5, 1), (3, 3) and (6, 3) are the vertices of
parallelogram A.
On the grid, draw and label parallelogram B.
Parallelogram B is transformed to parallelogram C by a reflection in the line with
equation x = 2
The points with coordinates (2, 1), (5, 1), (3, 3) and (6, 3) are the vertices of
parallelogram A.
On the grid, draw and label parallelogram C.
Parallelogram C is transformed to parallelogram D by a rotation through 90°
anticlockwise about the point with coordinates (–5, 0)
The points with coordinates (2, 1), (5, 1), (3, 3) and (6, 3) are the vertices of
parallelogram A.
On the grid, draw and label parallelogram D.
Question 2 continued
y
x
Turn over for a spare grid if you need to redraw your parallelograms.
Question 2 continued
Question 2 continued
Only use this grid if you need to redraw your parallelograms.
y
x
O
B
25°
C
A
15 cm
Figure 1
In Figure 1, A and B are two points on a circle with centre O. The line AC is the tangent
to the circle at A and OBC is a straight line.
AC = 15 cm and ÐOCA = 25°
Calculate the area, in cm² to 3 significant figures, of the circle.
The region inside triangle OAC but outside the circle is shown shaded in Figure 1
O
B
25°
C
A
15 cm
Figure 1
In Figure 1, A and B are two points on a circle with centre O. The line AC is the tangent
to the circle at A and OBC is a straight line.
AC = 15 cm and ÐOCA = 25°
Calculate the area, in cm² to 3 significant figures, of this shaded region.
Question 3 continued
In a race, Akaash ran a distance of (8x2 - 105) metres in (6x + 1) seconds.
His average speed for the race was x m/s.
Show that x satisfies the equation 2x2 - x - 105 = 0
In a race, Akaash ran a distance of (8x2 - 105) metres in (6x + 1) seconds.
His average speed for the race was x m/s.
Hence calculate the distance, in metres, that Akaash ran in the race.
Show clear algebraic working.
Question 4 continued
When Joe goes to school in winter, the probability that he wears a coat is 5
8
If he wears a coat, the probability that he wears a scarf is 1
4
If he does not wear a coat, the probability that he wears a scarf is 1
6
Complete the probability tree diagram.
Scarf
Coat
No scarf
Scarf
No coat
No scarf
On a day Joe goes to school in winter, calculate the probability that
When Joe goes to school in winter, the probability that he wears a coat is 5
8
If he wears a coat, the probability that he wears a scarf is 1
4
If he does not wear a coat, the probability that he wears a scarf is 1
6
he is not wearing a coat and is not wearing a scarf,
When Joe goes to school in winter, the probability that he wears a coat is 5
8
If he wears a coat, the probability that he wears a scarf is 1
4
If he does not wear a coat, the probability that he wears a scarf is 1
6
he is wearing a coat or he is wearing a scarf but he is not wearing both a coat and a scarf.
On a day Joe goes to school in winter, if he is wearing a coat and a scarf then the
probability that he is also wearing a hat is 3
5
When Joe goes to school in winter, the probability that he wears a coat is 5
8
If he wears a coat, the probability that he wears a scarf is 1
4
If he does not wear a coat, the probability that he wears a scarf is 1
6
Calculate the probability, that on a day Joe goes to school in winter, he is not wearing
all three of a coat, a scarf and a hat.
Question 5 continued
f and g are two functions such that
f : x ↦ x2 + x - 6 where x ≥ −1
2
x
−
g : x ↦ 2
24
x
−
3
2
State the value of x that must be excluded from any domain of g
f and g are two functions such that
f : x ↦ x2 + x - 6 where x ≥ −1
2
x
−
g : x ↦ 2
24
x
−
3
2
Find the value of a and the value of b such that
f (x) = (x + a)2 + b
f and g are two functions such that
f : x ↦ x2 + x - 6 where x ≥ −1
2
x
−
g : x ↦ 2
24
x
−
3
2
Hence write down the range of f
f and g are two functions such that
f : x ↦ x2 + x - 6 where x ≥ −1
2
x
−
g : x ↦ 2
24
x
−
3
2
Find the value of c for which g–1(0) = c
Given that f (x) = g (x)
f and g are two functions such that
f : x ↦ x2 + x - 6 where x ≥ −1
2
x
−
g : x ↦ 2
24
x
−
3
2
show that 2x3 - x2 - 13x - 6 = 0
Show clear algebraic working.
f and g are two functions such that
f : x ↦ x2 + x - 6 where x ≥ −1
2
x
−
g : x ↦ 2
24
x
−
3
2
Use the factor theorem to show that (2x + 1) is a factor of 2x3 - x2 - 13x - 6
The curve with equation y = f (x) intersects the curve with equation y = g (x) at the point A
with coordinates ( p, q), where p > −1
2
f and g are two functions such that
f : x ↦ x2 + x - 6 where x ≥ −1
2
x
−
g : x ↦ 2
24
x
−
3
2
Find the coordinates of A.
Question 6 continued
Question 6 continued
Question 6 continued
2h cm
4 cm
Figure 2
Figure 2 shows a right circular cylinder cut from a solid sphere of radius 4 cm. The height of
the cylinder is 2h cm.
The volume of the cylinder is V cm³
Show that V = 32h - 2h3
2h cm
4 cm
Figure 2
Figure 2 shows a right circular cylinder cut from a solid sphere of radius 4 cm. The height of
the cylinder is 2h cm.
The volume of the cylinder is V cm³
Using calculus, find the stationary value, to 3 significant figures, of V.
Show your working clearly.
2
r h
=
Volume of cylinder
Question 7 continued
Part of the curve with equation V = 32h - 2h3 is drawn on the grid.
V
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
O
h
0.5
1
1.5
2
2.5
3
3.5
4
2h cm
4 cm
Figure 2
Figure 2 shows a right circular cylinder cut from a solid sphere of radius 4 cm. The height of
the cylinder is 2h cm.
The volume of the cylinder is V cm³
Find an estimate, to one decimal place, of each of the two values of h for which the
volume of the cylinder is 100 cm³
Question 7 continued
Question 7 continued
x metres
C
D
A
B
y metres
E
F
Figure 3
Figure 3 shows the plan of a floor ABCDEF in which all the corners are right angles and
CD = x metres AF = y metres
The length of DE is twice the length of CD.
The length of FE is twice the length of AF.
The perimeter of the floor plan is 68 metres.
Find and simplify an equation in terms of x and y for this information.
The area of the floor plan is 248 m²
x metres
C
D
Diagram NOT
accurately drawn
A
B
y metres
E
F
Figure 3
Figure 3 shows the plan of a floor ABCDEF in which all the corners are right angles and
CD = x metres AF = y metres
The length of DE is twice the length of CD.
The length of FE is twice the length of AF.
The perimeter of the floor plan is 68 metres.
Show that 2y 2 + 2x2 - xy = 248
x metres
C
D
Diagram NOT
accurately drawn
A
B
y metres
E
F
Figure 3
Figure 3 shows the plan of a floor ABCDEF in which all the corners are right angles and
CD = x metres AF = y metres
The length of DE is twice the length of CD.
The length of FE is twice the length of AF.
The perimeter of the floor plan is 68 metres.
Find the possible lengths of AB.
Show clear algebraic working.
Question 8 continued
Question 8 continued
Question 8 continued
−
−
2
2
1
2
The transformation with matrix A is equivalent to the transformation with matrix B
followed by the transformation with matrix C, where C is a 2 × 2 matrix.
Find matrix C.
a
b
d
b
−
1
The inverse of matrix
is
c
d
c
−
ad
bc
a
−
Question 9 continued
Triangle ABC has sides AB = 10 cm, BC = 7 cm and CA = 6 cm.
Calculate the area, in cm² to 3 significant figures, of triangle ABC.
C
7 cm
6 cm
A
B
10 cm
Figure 4
Figure 4 shows triangle ABC and a circle inside the triangle so that each side of the
triangle is a tangent to the circle.
Triangle ABC has sides AB = 10 cm, BC = 7 cm and CA = 6 cm.
Calculate the radius, in cm to 2 significant figures, of the circle.
a
b
c
bc
A
2
2
2
=
−
Cosine rule:
2
cos
1
ab
C
=
Area of triangle
sin
2
Question 10 continued
Question 10 continued
Question 10 continued
Find the range of values of x for which both
2x + 3(4 - 3x) < 8x and 6x - 5 ≤ (2x - 3)2
Show clear algebraic working.
Question 11 continued
A company produces bird food made with sunflower seeds, corn and millet.
The weights of sunflower seeds, corn and millet in each bag of the bird food are in the ratios
sunflower seeds : corn : millet = 8 : 3 : 1
The total weight of the bird food in each bag is 30 kg.
Calculate the weight, in kg, of corn in each bag of the bird food.
Tom buys bags of the bird food from the company and sells them in his shop.
He sells each bag for $15.04
He makes a profit of 17.5% on each bag he sells.
A company produces bird food made with sunflower seeds, corn and millet.
The weights of sunflower seeds, corn and millet in each bag of the bird food are in the ratios
sunflower seeds : corn : millet = 8 : 3 : 1
The total weight of the bird food in each bag is 30 kg.
Calculate, in $, how much Tom pays the company for each bag of the bird food he buys.
The bags are filled with the bird food at a rate of 530 grams per second.
A company produces bird food made with sunflower seeds, corn and millet.
The weights of sunflower seeds, corn and millet in each bag of the bird food are in the ratios
sunflower seeds : corn : millet = 8 : 3 : 1
The total weight of the bird food in each bag is 30 kg.
Calculate the number of bags of the bird food that are filled completely in 6 hours.
Sunflower seeds cost £150.75 for 30 kg from farmer A.
Sunflower seeds cost $162.35 for 25 kg from farmer B.
Using an exchange rate of £1 = $1.315
A company produces bird food made with sunflower seeds, corn and millet.
The weights of sunflower seeds, corn and millet in each bag of the bird food are in the ratios
sunflower seeds : corn : millet = 8 : 3 : 1
The total weight of the bird food in each bag is 30 kg.
find which of the two farmers, A or B, sells sunflower seeds at the cheaper cost per kilogram.
You must give a reason for your answer.
Question 12 continued
Question 12 continued