1.
y is directly proportional to the square of x.
When x = 4, y = 25.
(a)
Find an expression for y in terms of x.
………………….……..
(3)
(b)
Calculate y when x = 2.
…………………………
(1)
(c)
Calculate x when y = 9.
…………………………
(2)
(Total 6 marks)
2.
(a)
Work out
(i)
80
……………………..
(ii)
5–2
………………………
(iii)
27

1
3
……………………….
(iv)
1
25 2
………………………..
(4)
IV
(b)
Given that x = 2k and
4
 2 c , find c in terms of k.
x
c = …………………….
(1)
(Total 5 marks)
3.
Work out
3  2 2  3 2 
8
Give your answer in its simplest form.
……………………..
(Total 3 marks)
IV
4.
Prove algebraically that the sum of the squares of any two consecutive even integers is never a
multiple of 8.
(Total 4 marks)
IV
5.
Simplify fully
(i)
(p3)3
.................................
(ii)
3q 4  2q 5
q3
.................................
(Total 3 marks)
6.
Work out
(i)
40
.................................
(ii)
4–2
.................................
3
(iii)
16 2
.................................
(Total 3 marks)
7.
The force, F, between two magnets is inversely proportional to the square of the distance, x,
between them.
When x = 3, F = 4.
(a)
Find an expression for F in terms of x.
F = ...............................
(3)
IV
(b)
Calculate F when x = 2.
.................................
(1)
(c)
Calculate x when F = 64.
.................................
(2)
(Total 6 marks)
8.
Work out
(5  3 )(5 – 3 )
22
Give your answer in its simplest form.
.................................................
(Total 3 marks)
IV
9.
(a)
Simplify
(i)
p2 × p7
…………………………
(ii)
x8  x3
…………………………
(iii)
y4  y3
y5
…………………………
(3)
(b)
Expand t(3t2 + 4)
…………………………
(2)
(Total 5 marks)
10.
Work out 5 23 – 2 34
………………………
(Total 3 marks)
IV
11.
Convert the recurring decimal 0.2 9 to a fraction.
………………………………
(Total 2 marks)
IV
12.
6 cm
4 cm
Diagram NOT
accurately drawn
A
B
Cylinder A and cylinder B are mathematically similar.
The length of cylinder A is 4 cm and the length of cylinder B is 6 cm.
The volume of cylinder A is 80 cm3.
Calculate the volume of cylinder B.
………………………… cm3
(Total 3 marks)
IV
13.
(a)
Evaluate
(i)
3–2
…………………………
1
(ii)
36 2
…………………………
2
(iii)
27 3
…………………………
(iv)
 16 
 
 81 
 34
…………………………
(5)
IV
(b)
(i)
21
Rationalise the denominator of
7
and simplify your answer.
…………………………
(ii)



Expand 5  2 3 5  2 3
Express your answer as simply as possible.
…………………………
(4)
(Total 9 marks)
14.
(a)
Simplify
k5 ÷ k2
.........................
(1)
(b)
Expand and simplify
(i)
4(x + 5) + 3(x – 7)
.........................
(ii)
(x + 3y)(x + 2y)
.........................
(4)
IV
(c)
Factorise
(p + q)2 + 5(p + q)
.........................
(1)
(d)
Simplify
(m–4)–2
.........................
(1)
(e)
Simplify 2t2 × 3r3t4
.........................
(2)
(Total 9 marks)
15.
Each side of a regular pentagon has a length of 101 mm, correct to the nearest millimetre.
(i)
Write down the least possible length of each side.
................ mm
(ii)
Write down the greatest possible length of each side.
................ mm
(Total 2 marks)
IV
16.
Mr Patel has a car.
V
1600
(2, 400)
O
t
The value of the car on January 1st 2000 was £1600
The value of the car on January 1st 2002 was £400
The sketch graph shows how the value, £V, of the car changes with time.
The equation of the sketch graph is
V = pqt
where t is the number of years after January 1st 2000.
p and q are positive constants.
(a)
Use the information on the graph to find the value of p and the value of q.
p = ........................ q = .........................
(3)
(b)
Using your values of p and q in the formula V = pqt find the value of the car on January
1st 1998.
£ .............................
(2)
(Total 5 marks)
IV
1
17.
(a)
Find the value of 16 2
........................
(1)
(b)
Given that
40  k 10 , find the value of k.
........................
(1)
( 5 + 20)
8
Diagram NOT
accurately drawn
5
2
A large rectangular piece of card is ( 5  20 ) cm long and
A small rectangle 2 cm long and
(c)
8 cm wide.
5 cm wide is cut out of the piece of card.
Express the area of the card that is left as a percentage of the area of the large rectangle.
.................................%
(4)
(Total 6 marks)
18.
IV
Rosa prepares the ingredients for pizzas.
She uses cheese, topping and dough in the ratio 2 : 3 : 5
Rose uses 70 grams of dough.
Work out the number of grams of cheese and the number of grams of topping Rosa uses.
Cheese ......................... g
Topping ....................... g
(Total 3 marks)
19.
Work out
12
1
5

2
8
....................................
(Total 3 marks)
IV
20.
(a)
Simplify
(i)
x6
x2
..................................
(ii)
(y4)3
...................................
(2)
(b)
Expand and simplify
(t + 4)(t – 2)
...................................
(2)
(c)
Write down the integer values of x that satisfy the inequality
–2  x < 4
................................................................
(2)
(d)
Find the value of
(i)
36
–
1
2
...................................
2
(ii)
27 3
...................................
(2)
(Total 8 marks)
IV
21.
(a)
Express
6
2
in the form a b , where a and b are positive integers.
...................................
(2)
The diagram shows a right-angled isosceles triangle.
The length of each of its equal sides is
6
2
cm.
Diagram NOT
accurately drawn
6 cm
2
6 cm
2
(b)
Find the area of the triangle.
Give your answer as an integer.
............................ cm2
(2)
(Total 4 marks)
IV
22.
y
y
x
x
Graph A
Graph B
y
y
x
x
Graph C
Graph D
The graphs of y against x represent four different types of proportionality.
Write down the letter of the graph which represents the type of proportionality.
Type of proportionality
Graph letter
y is directly proportional to x
.........................
y is inversely proportional to x
.........................
y is proportional to the square of x
.........................
y is inversely proportional to the square of x
.........................
(Total 2 marks)
23.
(a)
Write down an expression, in terms of n, for the nth multiple of 5.
.............................
(1)
IV
(b)
Hence or otherwise
(i)
prove that the sum of two consecutive multiples of 5 is always an odd number,
(ii)
prove that the product of two consecutive multiples of 5 is always an even number.
(5)
(Total 6 marks)
IV
24.
Solve
2
3
5
+
= 2
x 1
x –1
x –1
x = .................................
(Total 4 marks)
25.
(a)
Expand and simplify ( x  7)( x  4)
…………………….
(2)
(b)
Expand y( y 3  2 y)
…………………….
(2)
(c)
Factorise p 2 + 6 p
…………………….
(2)
IV
(d)
Factorise completely 6 x 2  9 xy
…………………….
(2)
(Total 8 marks)
26.
(a)
Change
3
to a decimal.
11
…………………….
(1)
(b)
Prove that the recurring decimal 0.3 9 =
13
33
(3)
(Total 4 marks)
27.
d is directly proportional to the square of t.
d  80 when t  4
(a)
Express d in terms of t.
…………………….
(3)
IV
(b)
Work out the value of d when t  7
d = ………………….
(1)
(c)
Work out the positive value of t when d  45
t = ………………….
(2)
(Total 6 marks)
IV
28.
Two cylinders, P and Q, are mathematically similar.
The total surface area of cylinder P is 90 cm2.
The total surface area of cylinder Q is 810 cm2.
The length of cylinder P is 4 cm.
(a)
Work out the length of cylinder Q.
…………… cm
(3)
The volume of cylinder P is 100 cm3.
(b)
Work out the volume of cylinder Q.
Give your answer as a multiple of .
…………… cm3
(2)
(Total 5 marks)
IV
29.
(a)
Find the value of
(i)
640
……………………..
(ii)
1
64 2
…………………….
(iii)
64

2
3
…………………….
(4)
(b)
3  27  3n
Find the value of n.
n = ……………
(2)
(Total 6 marks)
30.
Estimate the value of
70.1  5.92
0.19
………………….
(Total 3 marks)
IV
31.
Simplify
(i)
32
…………..
(ii)
80
…………..
(iii)
42
1
…………..
(Total 3 marks)
32.
The engine of a new aircraft had a major inspection after 1.2 × 104 hours flying time.
The aircraft flies at an average speed of 900 km/h.
Calculate the distance travelled by the new aircraft before its engine had a major inspection.
Give your answer in standard form.
……………….… km
(Total 3 marks)
33.
Work out
(i)
80
……………………..
(ii)
5 2
………………………
(iii)
27

1
3
……………………….
(iv)
1
25 2
………………………..
(Total 4 marks)
IV
34.
(a)
Express 0.2 7 as a fraction in its simplest form.
……………………………
(3)
x is an integer such that 1  x  9
(b)
Prove that 0.0 x 
x
99
(2)
(Total 5 marks)
35.
The length of a path is 14 m correct to the nearest metre.
(i)
Write down the minimum possible length of the path.
……………………………m
(ii)
Write down the maximum possible length of the path.
……………………………m
(Total 2 marks)
IV
36.
Work out (4 × 103) ÷ (8 × 105)
Give your answer in standard form.
……………………………
(Total 2 marks)
37.
Find the value of
1
(i)
36 2
……………………………
(ii)
32
……………………………
(Total 2 marks)
38.
(a)
Simplify
x5 ÷ x2
……………………………
(1)
(b)
Simplify
2w4y × 3w3y2
……………………………
(2)
(Total 3 marks)
IV
39.
(a)
Work out 1
7
1
×5
8
3
....................
(2)
(b)
Work out 3
1
4
÷2
2
5
....................
(2)
(Total 4 marks)
40.
A field is in the shape of a rectangle.
The length of the field is 340 m, to the nearest metre.
The width of the field is 117 m, to the nearest metre.
Calculate the upper bound for the perimeter of the field.
.............................................. m
(Total 2 marks)
IV
41.
Work out
1
22 ×3
3
2
Give your answer as a mixed number in its simplest form.
………………………
(Total 3 marks)
42.
Convert the recurring decimal 0.0 13 to a fraction.
…………………………..
(Total 3 marks)
1
43.
(a)
Write down the value of 36 2
………………
(1)
IV
3
(b)
4n 2 = 8
–
1
3
Find the value of n.
n = ………..……
(3)
(Total 4 marks)
44.
(a)
75 × 76 = 73 × 7k
Find the value of k.
k = ……………
(2)
(b)
Simplify
15a 3 b 7
3a 2 b 3
………..……
(2)
(Total 4 marks)
IV
45.
(i)
Convert the recurring decimal 0.3 6 to a fraction.
……………………
(ii)
Convert the recurring decimal 2. 13 6 to a mixed number.
Give your answer in its simplest form.
……………………
(Total 5 marks)
46.
Find the value of
55  5 7
510
………………..
(Total 2 marks)
IV
3
47.
Find the value of 16 4 × (0.04)

1
2
…..…..………………
(Total 3 marks)
48.
Using the information that
73 × 154 = 11 242
write down the value of
(i)
7.3 × 1.54
............................
(ii)
112 420 ÷ 0.73
.............................
(Total 2 marks)
IV
49.
Alex and Ben were given a total of £240
They shared the money in the ratio 5 : 7
Work out how much money Ben received.
£ .............................
(Total 2 marks)
50.
p is a prime number not equal to 7
(a)
Write down the Highest Common Factor (HCF) of
49p
and
7p2
....................................
(1)
IV
x and y are different prime numbers.
(b)
(i)
Write down the Highest Common Factor (HCF) of the two expressions
x2 y
xy2
....................................
(ii)
Write down the Lowest Common Multiple (LCM) of the two expressions
x2 y
xy2
....................................
(3)
(Total 4 marks)
51.
Simplify
(a)
34 × 36
.............................
(1)
(b)
35
310
.............................
(1)
(Total 2 marks)
52.
Evaluate
8

2
3
..................................
(Total 2 marks)
IV
53.
Write 140 as the product of its prime factors.
..............................................................................
(Total 2 marks)
..................................
(Total 2 marks)
54.
(i)
Write 638 000 in standard form.
.....................................................
IV
(ii)
Write 5.03 × 10–2 as an ordinary number.
.....................................................
(Total 2 marks)
55.
Simplify
(i)
a6 × a3
..............................
(ii)
c8
c2
..............................
(iii)
(e4)5
..............................
(Total 3 marks)
IV
56.
Express the recurring decimal 2.06 as a fraction.
Write your answer in its simplest form.
....................................
(Total 3 marks)
57.
Jerry measures a piece of wood as 60 cm correct to the nearest centimetre.
(i)
Write down the minimum possible length of the piece of wood.
............................. cm
(ii)
Write down the maximum possible length of the piece of wood.
............................. cm
(Total 2 marks)
58.
(a)
Work out the value of
2 5  28
27
..............................
(2)
IV
(b)
Write down the value of 60
..............................
(1)
(Total 3 marks)
59.
p is inversely proportional to m.
p = 48 when m = 9
Calculate the value of p when m = 12
..................................
(Total 2 marks)
60.
Work out the value of 1
2
3
+2
3
4
Give your answer as a fraction in its simplest form.
……………
(Total 3 marks)
IV