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IV
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) 3
1
27
……………………….
(iv) 2
1
25
……………………….. (4)
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(b) Given that x = 2k and
c
x2
4 , find c in terms of k.
c = ……………………. (1)
(Total 5 marks)
3. Work out
8
23223
Give your answer in its simplest form.
…………………….. (Total 3 marks)
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4. Prove algebraically that the sum of the squares of any two consecutive even integers is never a
multiple of 8.
(Total 4 marks)
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5. Simplify fully
(i) (p3)3
.................................
(ii) 3
54 23
q
................................. (Total 3 marks)
6. Work out
(i) 40
.................................
(ii) 4–2
.................................
(iii) 2
3
16
................................. (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)
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(b) Calculate F when x = 2.
................................. (1)
(c) Calculate x when F = 64.
................................. (2)
(Total 6 marks)
8. Work out
22
)3–5)(35(
Give your answer in its simplest form.
................................................. (Total 3 marks)
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9. (a) Simplify
(i) p2 × p
7
…………………………
(ii) x8 x
3
…………………………
(iii) 5
34
y
yy
………………………… (3)
(b) Expand t(3t2 + 4)
………………………… (2)
(Total 5 marks)
10. Work out 532 – 2
43
……………………… (Total 3 marks)
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11. Convert the recurring decimal 92.0 to a fraction.
……………………………… (Total 2 marks)
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12.
Diagram accurately drawn
NOT
A B
4 cm6 cm
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)
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13. (a) Evaluate
(i) 3–2
…………………………
(ii) 2
1
36
…………………………
(iii) 3
2
27
…………………………
(iv)
4
3
81
16
………………………… (5)
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(b) (i) Rationalise the denominator of 7
21 and simplify your answer.
…………………………
(ii) Expand 325325
Express your answer as simply as possible.
………………………… (4)
(Total 9 marks)
14. (a) Simplify k5 ÷ k
2
......................... (1)
(b) Expand and simplify
(i) 4(x + 5) + 3(x – 7)
.........................
(ii) (x + 3y)(x + 2y)
......................... (4)
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(c) Factorise (p + q)
2 + 5(p + q)
......................... (1)
(d) Simplify (m–4
)–2
......................... (1)
(e) Simplify 2t2 × 3r
3t4
......................... (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)
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16. Mr Patel has a car.
1600
(2, 400)
O t
V
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)
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17. (a) Find the value of 2
1
16
........................ (1)
(b) Given that 1040 k , find the value of k.
........................ (1)
Diagram accurately drawn
NOT
8
2
5
( 5 + 20)
A large rectangular piece of card is ( 205 ) cm long and 8 cm wide.
A small rectangle 2 cm long and 5 cm wide is cut out of the piece of card.
(c) Express the area of the card that is left as a percentage of the area of the large rectangle.
.................................% (4)
(Total 6 marks)
18. Rosa prepares the ingredients for pizzas.
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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 122
1
8
5
.................................... (Total 3 marks)
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20. (a) Simplify
(i) 2
6
x
x
..................................
(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 2
1–
...................................
(ii) 27 3
2
................................... (2)
(Total 8 marks)
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21. (a) Express 2
6 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 2
6 cm.
6
6
2
2
cm
cm
Diagram accurately drawn
NOT
(b) Find the area of the triangle.
Give your answer as an integer.
............................ cm2
(2)
(Total 4 marks)
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22.
x
x
x
x
y
y
y
y
Graph A
Graph C
Graph B
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)
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(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)
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24. Solve 1
2
x +
1–
3
x =
1–
52x
x = ................................. (Total 4 marks)
25. (a) Expand and simplify )4)(7( xx
……………………. (2)
(b) Expand )2( 3 yyy
……………………. (2)
(c) Factorise p 2
+ 6 p
……………………. (2)
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(d) Factorise completely xyx 96 2
……………………. (2)
(Total 8 marks)
26. (a) Change 11
3 to a decimal.
……………………. (1)
(b) Prove that the recurring decimal 93.0 = 33
13
(3)
(Total 4 marks)
27. d is directly proportional to the square of t.
80d when 4t
(a) Express d in terms of t.
……………………. (3)
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(b) Work out the value of d when 7t
d = …………………. (1)
(c) Work out the positive value of t when 45d
t = …………………. (2)
(Total 6 marks)
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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)
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29. (a) Find the value of
(i) 064
……………………..
(ii) 2
1
64
…………………….
(iii) 3
2
64
……………………. (4)
(b) n3273
Find the value of n.
n = …………… (2)
(Total 6 marks)
30. Estimate the value of 19.0
92.51.70
…………………. (Total 3 marks)
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31. Simplify
(i) 32
…………..
(ii) 80 …………..
(iii) 2
1
4 ………….. (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) 25
………………………
(iii) 3
1
27
……………………….
(iv) 2
1
25
……………………….. (Total 4 marks)
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34. (a) Express 72.0 as a fraction in its simplest form.
…………………………… (3)
x is an integer such that 1 x 9
(b) Prove that 99
0.0x
x
(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)
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36. Work out (4 × 103) ÷ (8 × 10
5)
Give your answer in standard form.
…………………………… (Total 2 marks)
37. Find the value of
(i) 21
36
……………………………
(ii) 32
…………………………… (Total 2 marks)
38. (a) Simplify
x5 ÷ x
2
…………………………… (1)
(b) Simplify
2w4y × 3w
3y
2
…………………………… (2)
(Total 3 marks)
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39. (a) Work out 18
7 × 5
3
1
.................... (2)
(b) Work out 32
1 ÷ 2
5
4
.................... (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)
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41. Work out 232 × 3
2
1
Give your answer as a mixed number in its simplest form.
……………………… (Total 3 marks)
42. Convert the recurring decimal 310.0 to a fraction.
………………………….. (Total 3 marks)
43. (a) Write down the value of 36 21
……………… (1)
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(b) 4n 2
3
= 3
1–
8
Find the value of n.
n = ………..…… (3)
(Total 4 marks)
44. (a) 75 × 7
6 = 7
3 × 7
k
Find the value of k.
k = …………… (2)
(b) Simplify 32
73
3
15
ba
ba
………..…… (2)
(Total 4 marks)
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45. (i) Convert the recurring decimal 63.0 to a fraction.
……………………
(ii) Convert the recurring decimal 2. 631 to a mixed number.
Give your answer in its simplest form.
…………………… (Total 5 marks)
46. Find the value of 10
75
5
55
……………….. (Total 2 marks)
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47. Find the value of 4
3
16 × 2
1
)04.0(
…..…..……………… (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)
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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)
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x and y are different prime numbers.
(b) (i) Write down the Highest Common Factor (HCF) of the two expressions
x2y xy
2
....................................
(ii) Write down the Lowest Common Multiple (LCM) of the two expressions
x2y xy
2
.................................... (3)
(Total 4 marks)
51. Simplify
(a)
34 × 3
6 .............................
(1)
(b)
10
5
3
3 .............................
(1)
(Total 2 marks)
52. Evaluate 3
2
8
.................................. (Total 2 marks)
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53. Write 140 as the product of its prime factors.
.............................................................................. (Total 2 marks)
.................................. (Total 2 marks)
54. (i) Write 638 000 in standard form.
.....................................................
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(ii) Write 5.03 × 10
–2 as an ordinary number.
..................................................... (Total 2 marks)
55. Simplify
(i) a6 × a
3
..............................
(ii) 2
8
c
c
..............................
(iii) (e4)5
.............................. (Total 3 marks)
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56. Express the recurring decimal 60.2 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
7
85
2
22
.............................. (2)
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(b) Write down the value of 6
0
.............................. (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 13
2 + 2
4
3
Give your answer as a fraction in its simplest form.
…………… (Total 3 marks)
