Q1.
(a)
Simplify fully
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Answer .................................................
(2)
(b)
Given that
work out the value of
Write your answer in its simplest form.
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Answer .................................................
(2)
(Total 4 marks)
Q2.
On Friday the ratio of the time Priya is sleeping to the time she is awake is 3 : 5.
She is sleeping for less time than she is awake.
(a)
Work out the number of hours that she is sleeping on Friday.
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Answer ................................................. hours
(2)
Page 1 of 117
(b)
On Saturday she sleeps for one hour more than she did on Friday.
Show that the ratio of the time she is sleeping to the time she is awake on Saturday
is 5 : 7
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(3)
(Total 5 marks)
Q3.
(a)
Simplify fully
You must show your working.
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Answer .................................................
(2)
(b)
Rationalise the denominator and simplify
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Answer .................................................
(2)
(Total 4 marks)
Page 2 of 117
Q4.
Show that
is an integer.
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(Total 2 marks)
Q5.
Here are the equations of four straight lines.
(a)
Line 1:
y=x+4
Line 2:
y = 3x
Line 3:
y = 3x + 5
Line 4:
y = –x + 5
Which two lines are parallel?
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Answer .............................. and ..............................
(1)
(b)
Which two lines intersect the y axis at the same point?
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Answer .............................. and ..............................
(1)
(Total 2 marks)
Q6.
Multiply out and simplify
(2p – 5q)(3p + q)
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Answer .................................................
(Total 3 marks)
Page 3 of 117
Q7.
The line PQ is shown on the grid.
(a)
Find the gradient of a line which is perpendicular to PQ.
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Answer .................................................
(3)
(b)
Hence find the equation of the perpendicular bisector of the line PQ.
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Answer .................................................
(2)
(Total 5 marks)
Page 4 of 117
Q8.
Solve the equation
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Answer x = .................................................
(Total 4 marks)
Q9.
(a)
Find the values of a and b such that
x 2 + 6x – 3 = (x + a)2 + b
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Answer a = ........................., b = .........................
(2)
Page 5 of 117
(b)
Hence, or otherwise, solve the equation
x 2 + 6x – 3 = 0
giving your answers in surd form.
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Answer .................................................
(3)
(Total 5 marks)
Q10.
(a)
Simplify
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Answer .................................................
(2)
Page 6 of 117
(b)
Simplify
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Answer .................................................
(3)
(Total 5 marks)
Q11.
Each term of a Fibonacci sequence is formed by adding the previous two terms.
1, 1, 2, 3, 5, 8, 13, 21, ……
A Fibonacci sequence starts a, b, a + b, …
(a)
Use algebra to show that the 6th term of this Fibonacci sequence is 3a + 5b
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(2)
Page 7 of 117
(b)
Use algebra to prove that the difference between the 9th term and 3rd term of this
sequence is four times the 6th term.
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(3)
(Total 5 marks)
Q12.
(a) Write down whether each of the following is an expression (X), an identity (I), an
equation (E) or a formula (F).
X, I, E or F
v = u + at
3n + 2n ≡ 5n
3x + 2 = 7
+ 2x – 3
(3)
Page 8 of 117
(b)
Show clearly that
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(2)
(Total 5 marks)
Q13.
The table gives the diameter, in metres, of planets in the solar system.
The diameters are given to an accuracy of 3 significant figures.
(a)
Planet
Diameter (metres)
Mercury
4.88 × 106
Venus
1.21 × 107
Earth
1.28 × 107
Mars
6.79 × 106
Jupiter
1.43 × 108
Saturn
1.21 × 108
Uranus
5.11 × 107
Neptune
4.95 × 107
Pluto
2.39 × 106
Which planet has the largest diameter?
Answer .................................................
(1)
(b)
Which planet has the smallest diameter?
Answer .................................................
(1)
Page 9 of 117
(c)
Which planet has a diameter approximately 10 times that of Venus?
Answer .................................................
(1)
(d)
Write
as an ordinary number.
Answer .................................................
(1)
(e)
What is the diameter of Pluto in kilometres?
Give your answer in standard form.
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Answer ................................................. km
(2)
(Total 6 marks)
Q14.
A is the point (2, 9)
B is the point (8, 7)
M is the midpoint of AB
C is the point (8, 18)
Not drawn accurately
Page 10 of 117
Is MC perpendicular to AB?
You must justify your answer.
Do not use graph paper to answer this question.
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(Total 4 marks)
Q15.
(a)
Show clearly that
(p + q)2 ≡ p 2 + 2pq + q2
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(1)
(b)
Hence, or otherwise, write the expression below in the form ax2 + bx + c
(2x + 3)2 + 2(2x + 3)(x – 1) + (x – 1)2
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Answer .................................................
(3)
(Total 4 marks)
Page 11 of 117
Q16.
Write each of these in the form p
, where p is an integer.
(a)
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Answer .................................................
(2)
(b)
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Answer .................................................
(2)
(c)
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Answer .................................................
(2)
(Total 6 marks)
Q17.
Evaluate
(a)
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Answer .................................................
(3)
Page 12 of 117
(b)
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Answer .................................................
(2)
(Total 5 marks)
Q18.
A is the point (1, –2).
B is the point (5, 4).
Page 13 of 117
Find the equation of the line perpendicular to AB, passing through the mid-point of AB.
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Answer .................................................
(Total 4 marks)
Q19.
Rearrange
to make x the subject.
Simplify your answer as much as possible.
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Answer .................................................
(Total 4 marks)
Q20.
(a)
Factorise
5x 2 + 20x
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Answer .................................................
(1)
Page 14 of 117
(b)
Factorise
x 2 – 49
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Answer .................................................
(1)
(c)
Factorise fully
(3x + 4)2 – (2x + 1)2
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Answer .................................................
(3)
(Total 5 marks)
Q21.
A trapezium has parallel sides of length (x + 1) cm and (x + 2) cm.
The perpendicular distance between the parallel sides is x cm.
The area of the trapezium is 10 cm2.
Not drawn accurately
Page 15 of 117
Find the value of x.
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Answer x = .............................................. cm
(Total 5 marks)
Q22.
Find the equation of the line through (0, –2) and (4, 18).
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Answer .................................................
(Total 3 marks)
Page 16 of 117
Q23.
Match each of the shaded regions to one of these inequalities.
A
y ≤ –
B
y ≤ C
y ≥ – 2x + 4
+2
+2
D
y ≥ 2x – 4
E
y ≤ 2x – 4
Region 1 ................................................
Region 2 ................................................
Region 3 ................................................
Region 4 ................................................
(Total 4 marks)
Page 17 of 117
Q24.
A shape is made from two trapezia.
Not drawn accurately
The area of this shape is given by
A=
(a + b) +
(a + h)
Rearrange the formula to make a the subject.
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Answer a = ................................................
(Total 4 marks)
Page 18 of 117
Q25.
The triangle number sequence is
1, 3, 6, 10, 15, 21, ...
The nth term of this sequence is given by
n(n + 1)
(a)
Write down an algebraic expression for the (n – 1)th term of the sequence.
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Answer .................................................
(1)
(b)
Prove that the sum of any two consecutive triangle numbers is a square number.
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(3)
(Total 4 marks)
Page 19 of 117
Q26.
Julie has a bag containing x blue marbles and y red marbles.
The ratio of blue marbles to red marbles is 2:3
She adds z blue marbles.
The ratio of blue marbles to red marbles is now 2:1
What is the ratio between x and z?
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Answer .................................................
(Total 3 marks)
Q27.
Solve these simultaneous equations
x + 3.6y = 2
x – 2.4y = 5
You must show all your working.
Do not use trial and improvement.
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Answer x = ..............................................
y = ..............................................
(Total 3 marks)
Page 20 of 117
Q28.
Make x the subject of the formula
a(x – b) = a2 + bx
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Answer ....................................................
(Total 4 marks)
Q29.
(a)
This is a page from Zoe’s exercise book.
Give a counter example to show that Zoe is wrong.
Justify your answer.
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(2)
Page 21 of 117
(b)
Prove that
(n + 5)2 – (n + 3)2 = 4(n + 4)
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(3)
(Total 5 marks)
Q30.
Some large numbers are written below.
1 million = 106
1 billion = 109
1 trillion = 1012
(a)
How many millions are there in one trillion?
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Answer ....................................................
(1)
(b)
Write 8 billion in standard form.
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Answer ....................................................
(1)
(c)
Work out 8 billion multiplied by 3 trillion.
Give your answer in standard form.
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Answer ....................................................
(2)
(Total 4 marks)
Page 22 of 117
Q31.
Annie, Bert and Charu are investigating the number sequence
21, 40, 65, 96, 133, ...
(a)
Annie has found the following pattern.
1st term
1 × 2 + 32 + 2 × 5 =
21
2nd term
2 × 3 + 42 + 3 × 6 =
40
3rd term
3 × 4 + 52 + 4 × 7 =
65
4th term
4 × 5 + 62 + 5 × 8 =
96
5th term
5 × 6 + 72 + 6 × 9 = 133
Complete the nth term for Annie’s pattern.
nth term
n × (n + 1) + ........................ + ........................ × ........................
(2)
(b)
Bert has found this formula for the nth term
(3n + 1)(n + 3) + 5
Charu has found this formula for the nth term
(2n + 3)2 – (n + 1)2
Prove that these two formulae are equivalent.
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(3)
(Total 5 marks)
Page 23 of 117
Q32.
(a)
Find the equation of the line AB.
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Answer .................................................
(3)
(b)
Give the y-coordinate of the point on the line with an x-coordinate of 6.
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Answer .................................................
(2)
(c)
Write down the gradient of a line perpendicular to AB.
Answer .................................................
(1)
(Total 6 marks)
Page 24 of 117
Q33.
(a)
Find the value of
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Answer .................................................
(1)
(b)
Find the value of 8x 0
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Answer .................................................
(1)
(Total 2 marks)
Q34.
Solve the simultaneous equations
y=x+2
y = 3x 2
You must show your working.
Do not use trial and improvement.
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Answer .....................................................................................................................
(Total 5 marks)
Page 25 of 117
Q35.
(a)
n is a positive integer.
(i)
Explain why n(n + 1) must be an even number.
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(1)
(ii)
Explain why 2n + 1 must be an odd number.
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(1)
(b)
Expand and simplify (2n + 1)2
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Answer .................................................
(2)
(c)
Prove that the square of any odd number is always 1 more than a multiple of 8.
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(3)
(Total 7 marks)
Q36.
(a)
(i)
Factorise x 2 – 10x + 25
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Answer .................................................
(2)
Page 26 of 117
(ii)
Hence, or otherwise, solve the equation
(y – 3)2 – 10(y – 3) + 25 = 0
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Answer y = .................................................
(2)
(b)
Simplify
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Answer .................................................
(3)
(Total 7 marks)
Q37.
(a)
Factorise 2n2+ 5n + 3
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Answer .................................................
(2)
Page 27 of 117
(b)
Hence, or otherwise, write 253 as the product of two prime factors.
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Answer .................................................
(1)
(Total 3 marks)
Q38.
The diagram shows the graph of an equation of the form y = x 2+ bx + c
Find the values of b and c.
You must show your method.
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Answer b = ................. , c = ...................
(Total 3 marks)
Page 28 of 117
Q39.
Find the equation of the straight line passing through the point (0, 5) which is perpendicular
to the line
y=
x+3
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Answer .................................................
(Total 2 marks)
Q40.
Solve the simultaneous equations
4x + 3y = 14
2x + y = 5
You must show your working.
Do not use trial and improvement.
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Answer x = .................... , y = ......................
(Total 3 marks)
Page 29 of 117
Q41.
A special packet of breakfast cereal contains 20% more than a normal packet. The special
packet contains 600 g of cereal. How much cereal does the normal packet contain?
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Answer ................................................. g
(Total 3 marks)
Q42.
Make x the subject of the formula
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Answer x = .......................................
(Total 4 marks)
Page 30 of 117
Q43.
Solve the equation
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(Total 5 marks)
Q44.
Two gas supply companies have different ways of charging for the gas they supply.
Alpha gasCO
Fixed Charge
Price per kilowatt hour of gas
£9.60
First 5 kilowatt hours free then
£1.30 for every kilowatt hour over 5.
Beta gasCO
Fixed Charge
Price per kilowatt hour of gas
No fixed charge
£1.50 for every kilowatt hour.
Page 31 of 117
Find the number of kilowatt hours after which Alpha gasCo becomes cheaper than Beta gasCo.
You might want to use some graph paper.
You must show your method clearly.
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Answer ................................... kilowatt hours
(Total 4 marks)
Q45.
Simplify fully
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Answer ............................................
(Total 4 marks)
Page 32 of 117
Q46.
(a)
(i)
Evaluate
13z0
Answer ............................................
(1)
(ii)
Evaluate
(13z)0
Answer ............................................
(1)
(b)
If 3x =
, find the value of x.
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Answer x = ............................................
(2)
(c)
If 4y =
, find the value of y.
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Answer y = ............................................
(2)
(Total 6 marks)
Page 33 of 117
Q47.
On the grid below, indicate clearly the region defined by the three inequalities
x ≥ 1
y ≥ x – 1
x + y ≤ 7
Mark the region with an R.
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(Total 3 marks)
Page 34 of 117
Q48.
Simplify
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Answer .................................................
(Total 4 marks)
Q49.
Solve the equation
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Answer .................................................
(Total 5 marks)
Page 35 of 117
Q50.
The region R is shown shaded below.
Write down three inequalities which together describe the shaded region.
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Answer .................................................
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(Total 3 marks)
Q51.
Make x the subject of the formula
w = x2 + y
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Answer x = .................................................
(Total 2 marks)
Page 36 of 117
Q52.
Find the values of a and b such that
x 2 – 10x + 18 = (x – a)2 + b
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Answer a = ..........................., b = ...........................
(Total 2 marks)
Q53.
The line l on the graph passes through the points A (0, 3) and B (–4, 11).
Page 37 of 117
(a)
Calculate the gradient of the line l.
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Answer ......................................................
(2)
(b)
Write down the equation of the line l.
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Answer ......................................................
(1)
(c)
Write down the equation of the line which also passes through the point (0, 3) but is
perpendicular to line l.
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Answer ......................................................
(2)
(Total 5 marks)
Q54.
(a)
Write down the value of 110
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Answer ....................................................
(1)
(b)
Find the value of 8
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Answer ....................................................
(2)
Page 38 of 117
(c)
Simplify 6–2 × 1440.5
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Answer ......................................................
(3)
(Total 6 marks)
Q55.
(a)
Factorise 7x + 14
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Answer .................................................
(1)
(b)
Expand and simplify 4(m + 3) + 3(2m – 5)
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Answer .................................................
(2)
(c)
Solve the simultaneous equations:
2x + 3y = 9
3x + 2y = 1
You must show all your working.
Do not use trial and improvement.
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Answer x = ......................., y = .........................
(4)
Page 39 of 117
(d)
Factorise x 2 + 6x – 16
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Answer .................................................
(2)
(Total 9 marks)
Q56.
A sketch of the line 2y – x = 4 is shown.
The line crosses the axes at A and B.
(a)
Calculate the coordinates of A and B.
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Answer A(..........., .............), B (............., ............)
(2)
(b)
Calculate the gradient of the line AB.
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Answer .................................................
(2)
(Total 4 marks)
Page 40 of 117
Q57.
James invests £700 for 2 years at 10% per year compound interest.
How much interest does he earn?
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Answer £ .................................................
(Total 2 marks)
Q58.
(a)
Write down the next odd number after 2n + 1
Answer ............................................
(1)
(b)
Show that (2n + 1)2 ≡ 4n2 + 4n + 1
.........................................................................................................................
.........................................................................................................................
(1)
(c)
Prove that the difference between the squares of consecutive odd numbers is
a multiple of 8
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(3)
(Total 5 marks)
Page 41 of 117
Q59.
The diagram shows the cross-section of a river bed, ABCDE.
BD is the surface of the river.
C is the midpoint of the river.
A and E are points on the river bank that are 1 metre above the river surface.
Not drawn accurately
Taking the origin at B, 1 metre as 1 unit on both
axes and the river surface as the x-axis, the
curve ABCDE can be represented by the
equation
(a)
Show that the depth of the river at the mid-point C is 2 metres.
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(3)
Page 42 of 117
(b)
Show that the distance AE is approximately 12.25 metres.
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(5)
(Total 8 marks)
Q60.
The function denoted by INT(x) is defined as the integer part of the positive number x.
For example,
(a)
INT(2.7) = 2,
INT(0.68) = 0,
INT (5.432) = 5
Write down the value of INT (9.44)
.........................................................................................................................
Answer ................................................
(1)
(b)
Write down a value of x for which INT(x) = x
.........................................................................................................................
Answer ................................................
(1)
Page 43 of 117
(c)
y is a positive number less than 0.5
Explain why
INT(2y) = INT(y)
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(2)
(Total 4 marks)
Q61.
(a)
Show that (p + q)2 ≡ p2 + 2pq + q2
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.........................................................................................................................
.........................................................................................................................
(1)
(b)
p and q are two numbers.
The sum of p and q is 10
The product of p and q is 18
Work out the value of p2 + q2
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Answer ...............................................
(3)
(Total 4 marks)
Page 44 of 117
Q62.
Andy and Ben both have a collection of marbles.
If Andy gives Ben 15 marbles, they will have an equal number.
If Ben gives Andy 15 marbles, Andy will have four times as many as Ben.
How many marbles does Andy have?
You must show your working.
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Answer ..............................................
(Total 4 marks)
Q63.
Solve the equation
(2x – 3)2 = (x – 1)(x + 1)
Give your solutions to 2 decimal places.
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Answer ................................................
(Total 5 marks)
Page 45 of 117
Q64.
(a)
Simplify
(5y 5)2
.........................................................................................................................
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.........................................................................................................................
Answer .................................................
(2)
(b)
Expand and simplify
(x – 3)(x + 5)
.........................................................................................................................
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.........................................................................................................................
Answer .................................................
(2)
(c)
Rearrange the formula
w = 3(2x + y) – 5
to make x the subject.
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.........................................................................................................................
Answer x = ..........................................
(3)
(Total 7 marks)
Page 46 of 117
Q65.
The diagram shows a sequence of triangle patterns using shaded and unshaded triangles.
The number of unshaded triangles in Pattern n is
The number of shaded triangles in Pattern n is
Work out the difference between the number of unshaded triangles and shaded triangles in
Pattern 100.
You must show your working.
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Answer ................................................
(Total 3 marks)
Q66.
(a)
Expand
3(x + 5)
.........................................................................................................................
Answer ................................................
(1)
Page 47 of 117
(b)
Factorise
6a – 9
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.........................................................................................................................
Answer ................................................
(1)
(Total 2 marks)
Q67.
(a)
Solve the equation
2(4y – 1) = 18
.........................................................................................................................
.........................................................................................................................
.........................................................................................................................
Answer y = ............................................
(3)
(b)
The diagram shows a rectangle and a square.
The rectangle has length
3x – 5 and width x + 1
The rectangle and the square have the same perimeter.
Not drawn accurately
Work out the length of the side of the square.
Give your answer as an expression in terms of x in its simplest form.
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Answer ................................................
(4)
(Total 7 marks)
Page 48 of 117
Q68.
(a)
A straight line has gradient 2 and passes through the point (0, 8)
Write down the equation of the line.
Answer ..................................................
(1)
(b)
Write down the gradient of a line that is perpendicular to the line
y = 4x
Answer ..................................................
(1)
(c)
Write down the equation of a line that is perpendicular to the line
y = 4x
Answer ..................................................
(1)
(Total 3 marks)
Q69.
(a)
Expand and simplify
2x 2(x + 6) + 3x(x – 5)
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Answer ..................................................
(3)
(b)
Factorise fully
3mh2 – 15m 2h
.........................................................................................................................
.........................................................................................................................
Answer ..................................................
(2)
(c)
Simplify fully
4rs2 × 5r3s 4
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.........................................................................................................................
Answer ..................................................
(2)
Page 49 of 117
(d)
Solve
9x 2 + 29x – 28 = 0
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Answer ..................................................
(3)
(Total 10 marks)
Q70.
(a)
Work out the Highest Common Factor (HCF) of 42 and 98.
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........................................................................................................................
Answer ................................................
(2)
(b)
Write
+
in the form
where a and b are integers.
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........................................................................................................................
Answer ................................................
(2)
(Total 4 marks)
Q71.
(a)
Work out
........................................................................................................................
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........................................................................................................................
Answer .................................................
(3)
Page 50 of 117
(b)
Write down the value of
(i)
Answer .................................................
(1)
(ii)
Answer .................................................
(1)
(Total 5 marks)
Q72.
Make x the subject of the formula
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Answer ............................................
(Total 5 marks)
Q73.
(a)
Show that
(x – 2)(x + 2) ≡ x 2 – 4
........................................................................................................................
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........................................................................................................................
(1)
Page 51 of 117
(b)
Hence, simplify
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Answer .............................................
(1)
(Total 2 marks)
Q74.
Prove that
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(Total 4 marks)
Q75.
(a)
Simplify
(4y 3)2
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.........................................................................................................................
Answer ............................................
(1)
Page 52 of 117
(b)
Expand and simplify
(a + 2b)(3a – 4b)
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Answer ............................................
(3)
(Total 4 marks)
Q76.
Solve the simultaneous equations
x + 3y = 11
2x – y = 1
You must show your working.
Do not use trial and improvement.
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Answer x = ............. , y = ...............
(Total 3 marks)
Q77.
(a)
Multiply out
x(x + 7)
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Answer ............................................
(1)
Page 53 of 117
(b)
Find both solutions of the quadratic equation
x(x + 7) = 8
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Answer ............................................
(3)
(Total 4 marks)
Q78.
(a) Three friends share a pizza.
One eighth of it falls on the floor and is thrown away.
The remainder is shared equally.
What fraction of the whole pizza does each person get?
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........................................................................................................................
Answer ................................................
(3)
(b)
The pizza costs £6.40
This is 20% less than the original price.
What was the original price of the pizza?
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Answer £ ..............................................
(3)
(Total 6 marks)
Page 54 of 117
Q79.
(a)
(i)
Factorise
2
x –x–2
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...............................................................................................................
Answer ..............................................
(2)
(ii)
2
Hence, solve x – x – 2 = 0
...............................................................................................................
Answer ..............................................
(1)
(b)
Simplify
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Answer ..............................................
(3)
(Total 6 marks)
Page 55 of 117
Q80.
Work out the values of a and b.
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Answer a = ..................., b = ..................
(Total 3 marks)
Q81.
You are given the identity
2
2
x – 12x + 10 ≡ (x + a) + b
Work out the values of a and b.
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..................................................................................................................................
Answer a = .......................... b = ..........................
(Total 2 marks)
Q82.
x and y are integers.
x<4
y<3
x+y>4
Page 56 of 117
Use a graphical method to show that there is only one possible pair of values of x and y.
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(Total 4 marks)
Q83.
(a)
Tom finds the value of
Sam finds the value of
(2n – 1)(n + 1)
(2n – 1)(n + 1)
when n = 1
when n = 2
Work out the difference between Tom’s value and Sam’s value.
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Answer .......................................................................
(3)
Page 57 of 117
(b)
Expand and simplify
(2n – 1)(n + 1)
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Answer .......................................................................
(2)
(Total 5 marks)
Q84.
Simplify fully
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Answer .......................................................................
(Total 4 marks)
Q85.
Here are the equations of two straight lines.
y = 7x – 6
y = mx + c
(a)
Write down the value of m if the lines are perpendicular.
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Answer .......................................................................
(1)
Page 58 of 117
(b)
Write down the values of m and c if the lines are reflections of each other in the y axis.
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Answer m = ...............................................................
c = ...............................................................
(2)
(Total 3 marks)
Q86.
Solve the simultaneous equations
2x + 5y = 16
4x + 3y = 11
You must show your working.
Do not use trial and improvement.
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Answer .......................................................................
(Total 3 marks)
Q87.
(a)
Expand and simplify
2
(x + 4)
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Answer .......................................................................
(2)
Page 59 of 117
(b)
Hence or otherwise, show that
2
(x + 4) – 4(x + 4) ≡ x (x + 4)
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(2)
(Total 4 marks)
Q88.
(a)
Work out
.........................................................................................................................
.........................................................................................................................
Answer .................................................
(3)
(b)
Write
as an improper fraction.
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Answer .................................................
(2)
(Total 5 marks)
Q89.
For each calculation circle the answer that is correct and is in standard form.
(a)
5
7
(3 × 10 ) × (4 × 10 )
Answer
12 × 10
12
1.2 × 10
36
35
12 × 10
13
1.2 × 10
(1)
Page 60 of 117
(b)
–8
–2
(4 × 10 ) ÷ (8 × 10 )
Answer
–6
0.5 × 10
5 × 10
4
–7
5 × 10
5 × 10
–5
(1)
(Total 2 marks)
Q90.
Evaluate
–2
x 10
Give your answer as a fraction in its simplest form.
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Answer ................................................
(Total 3 marks)
Q91.
The value of a vintage car rises from £36 000 to £63 000.
Work out the percentage increase in the price of the car.
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Answer ............................................%
(Total 3 marks)
Page 61 of 117
Q92.
Work out the Highest Common Factor (HCF) of 63 and 105.
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Answer ................................................
(Total 2 marks)
Q93.
Solve
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Answer ..................................................
(Total 6 marks)
Page 62 of 117
Q94.
The difference between the squares of two consecutive even numbers is twice the sum of
the numbers.
For example
2
2
8 – 6 = 28
2 × (8 + 6) = 28
Prove this result algebraically.
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(Total 4 marks)
Q95.
(a)
Solve the inequality
3x + 7 > x + 8
.........................................................................................................................
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.........................................................................................................................
Answer .................................................
(2)
(b)
Make a the subject of the formula
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Answer .................................................
(2)
(Total 4 marks)
Page 63 of 117
(a)
Q96.
Simplify
(9 + √7)(9 + √7)
Give your answer in the form
a + b√7
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Answer ...................................................
(2)
(b)
Prove that
.........................................................................................................................
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(4)
(Total 6 marks)
Q97.
Two families go to a pantomime.
The Khan family of two adults and three children pay £69.
The Lewis family of three adults and five children pay £109.
Work out the cost of an adult ticket and the cost of a child ticket.
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Answer
Adult ticket £ ..........................
Child ticket £ ..........................
(Total 5 marks)
Page 64 of 117
Q98.
(a)
3
Factorise fully
12x – 8xyz
.........................................................................................................................
Answer .................................................
(2)
(b)
Factorise
2
x + 3x + 2
.........................................................................................................................
Answer .................................................
(2)
(c)
Simplify
.........................................................................................................................
Answer .................................................
(1)
(d)
Factorise fully
2
10x – 40y
2
.........................................................................................................................
.........................................................................................................................
Answer .................................................
(3)
(Total 8 marks)
Q99.
Greg thinks of a positive whole number smaller than 15.
He subtracts 4 from the number and then doubles his result.
He subtracts 4 from the new number and then doubles this result.
He repeats this process several times.
He stops when he gets an answer of 40.
(a)
Show that when Greg starts with 12 he gets an answer of 40.
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.........................................................................................................................
(2)
Page 65 of 117
(b)
Find another number that Greg could have started with to get an answer of 40.
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Answer .................................................
(2)
(Total 4 marks)
Q100.
(a)
A sequence starts
2
7
17
.......
The rule for finding the next term in this sequence is to multiply the previous term by 2 and
then add on 3
Work out the next term.
.........................................................................................................................
.........................................................................................................................
Answer ..................................................
(1)
(b)
The rule for finding the next term in a different sequence is to multiply the previous term by
2 and then add on a, where a is an integer.
The first term is 8 and the fourth term is 127
8
........
.........
127
Work out the value of a.
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Answer a = ..........................................
(4)
(Total 5 marks)
Page 66 of 117
Q101.
(a)
Simplify
3
m ×m
5
Answer ..................................................
(1)
(b)
Simplify
Answer ..................................................
(1)
(c)
Simplify fully
.........................................................................................................................
.........................................................................................................................
Answer ..................................................
(2)
(Total 4 marks)
Q102.
(a)
A magazine contains adverts, photographs and features.
of the pages are adverts and
are photographs.
What fraction are features?
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.........................................................................................................................
Answer ..................................................
(3)
(b)
There are 24 photographs in the magazine.
The ratio of sports photographs to other photographs is 5 : 3
How many sports photographs are there in the magazine?
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.........................................................................................................................
Answer ..................................................
(2)
(Total 5 marks)
Page 67 of 117
(a)
M1.
√16 – √4 (= 4 – 2)
or
or
√2(2√2 – √2)
= √2(√2)
both steps needed
or
Both steps needed
M1
2
A1
(b)
or
or
or
Do not allow for
B1
oe
B1
[4]
M2.
(a)
Condone
3 unsupported is M0
M1
9
Do not allow
(of a day)
SC1 Answer 15 or 9 and 15
A1
Page 68 of 117
(b)
(their 9 + 1) : 24 – (their 9 + 1)
10 and 14 seen
M1
10:14
Must be integers
A1 ft
5:7
Must have seen previous ratio
A1
[5]
(a)
M3.
5√3 or 3√3
M1
8√3 A1
(b)
M1
3√7 A1
[4]
5√2 (– √2 = 4√2)
If attempts to square the bracket
√2500 ± √50√2 ± √50√2 ± √4 M1
M4.
B1
32
32 A1
B1
[2]
Page 69 of 117
M5.
(a)
2 and 3
oe
B1
(b)
3 and 4
oe
B1
[2]
M6.
6p2 + 2pq – 15pq – 5q2
For 3 correct terms
M1
6p2 + 2pq – 15pq – 5q2
Fully correct
A1
6p2 – 13pq – 5q2
From 4 terms
Do not ignore fw
B1 ft
[3]
M7.
(a)
Gradient of PQ =
M1
Perp. grad. =
(=
oe)
Drawing method:
Perpendicular line drawn and attempt at finding its gradient M2
M1 dep
oe
A1
Page 70 of 117
(b)
y = (their –
)x+c
M1
y=–
x+
oe
Accept 1.4 to 1.6 for
from graph
A1
[5]
M8.
3(3x + 1) –2 (2x + 5)
Could have 6 as denominator here
Condone lack of brackets
M1
9x + 3 – 4x – 10
A1
(their 5x – 7) = 6
M1 dep
x = 2.6
or
A1
[4]
M9.
(a)
(a =) 3
B1
(b =) –12
Allow 12 if –12 given in working
B1
Page 71 of 117
(b)
(x + 3)2 = 12
or (x =)
Using their values from (a)
Substitution into formula (allow 1 error)
M1
x + 3 = √12
or (x =)
Using their values from (a)
M1 dep
(x =)
±√12 – 3
or
A1
[5]
M10.
(a)
3(x + 5) or 3x + 15
B1 for 3
B1 for x + 5
B1 for
B2
(b)
(x – 3)(x + 3)
M1
x(x + 3)
M1
Do not ignore further working
A1
[5]
Page 72 of 117
M11.
(a)
4th term = a + 2b
or (a = 1 and b = 1 and)
3(1) + 5(1)
oe
Accept 5th term = 2a + 3b (oe) for M1 if 4th
term not seen.
M1
6th term
Must see 4th and 5th terms
A1
(b)
Continuing sequence to 9th term
= 3a + 5b, 5a + 8b, 8a + 13b, 13a + 21b
Must come from continuing sequence and not
from 4 × 6th – 3rd
M1
Allow subtraction to be ‘assumed’. Condone
missing bracket if answer correct
A1
Either way round, expansion or factorisation
A1
[5]
M12.
Allow embedded solutions, but if contradicted M marks only
(a)
F, I, E, X
–1eeoo
B3
(b)
Must have 4 terms
Condone 1 sign error
M1
Must show cancellation, either by ‘crossing out’
or stating ab – ab = 0
A1
[5]
Page 73 of 117
M13.
(a)
Jupiter
B1
(b)
Pluto
B1
(c)
Saturn
B1
(d)
4 880 000
B1
(e)
or 2 390 oe
B1
A1
[6]
M14.
Mid point (5, 8)
B1
Gradient AB
Accept any indication
eg, 6 across, 2 down
B1
Attempt to find gradient MC or ‘stepping’ from M to C
M1 for using ‘Their gradient’
M1
Valid conclusion with justification.
eg, No because gradient MC not 3
Accept any indication
eg, (5, 8) plus (3, 9) = (8, 17), mm’ ≠ – 1
A1
Page 74 of 117
Alt
Mid point (5, 8)
B1
Use of Pythagoras
M1
Three correct lengths
A1
Correct conclusion at least 2 correct values
A1
[4]
M15.
(a)
Convincing algebra
Must see
or
box method and
B1
(b)
Allow one sign or coefficient error
For middle term accept
or
M1
A1
ft if M1 awarded and no further errors
A1 ft
[4]
M16.
(a)
√300
oe eg, √(2 × 3) × √(2 × 25) or √(2 × 2 × 3 × 25)
√ (correct product of factors which includes ‘3’)
M1
10√3 SC1 for 5√12 or 2√75
A1
Page 75 of 117
(b)
4√3 or 5√3 seen
M1
9√3 A1
(c)
Attempt to rationalise
ie, Multiply num. and denom. By √3
oe eg,
scores M1
M1
6√3 A1
[6]
M17.
(a)
(±)6
B1
B1
(±)1.5
oe
B1
(b)
oe eg, 0.01
B1 for 100 or
or
B2
[5]
Page 76 of 117
M18.
Attempt to find gradient of perpendicular line
Must be negative reciprocal of their gradient for AB
M1
(Gradient =) –
oe
eg –0.66, –0.67
A1
Use of midpoint (3, 1)
Must be used either on the diagram with an attempt
at a perpendicular or in y = mx + c to find c.
M1
y=–
x +3
ft their gradient if first M1 awarded
Accept equivalents eg 3y + 2x = 9
A1 ft
[4]
M19.
y(3x – 4) = xy + 2
y × 3x – 4 = xy + 2 is M0 unless recovered
M1
3xy – 4y = xy + 2
A1
2xy = 4y + 2
dep 3xy – xy = 4y + 2 Allow one ‘sign’ error
M1
oe Do not award if x = not written
SC x =
B2
A1
Page 77 of 117
Alt.
y(3x – 4) = xy + 2
y × 3x – 4 = xy + 2 is M0 unless recovered
M1
3x – 4 = x +
3x – 4 =
A1
2x = 4 +
3x – x = 4 +
Allow one ‘sign’ error
M1 dep
x=2+
oe Deduct mark if x = not written
SC x =
B2
A1
[4]
M20.
(a)
5x (x + 4)
B1
(b)
(x + 7)(x – 7)
B1
(c)
M1 for expanding and collecting to general quad form,
allow one error but expansions must have x 2 term,
x term and constant term.
Allow misuse of minus.
eg 9x2 + 24x + 16 – 4x2 + 4x + 1
Difference of two squares
((3x + 4) – (2x + 1)) × ((3x + 4) + (2x + 1))
M1
5x 2 + 20x + 15
A1 for either (x + 3) or (5x + 5) if difference of 2 squares used.
A1
5 (x + 3)(x + 1)
Accept
(x + 3)(5x + 5) or (5x + 15)(x + 1)
A1
[5]
Page 78 of 117
M21.
(Area =)
x (x + 1 + x + 2)
oe
(x + 1) +
× x × (1)
M1
2x 2 + 3x – 20 = 0
oe
eg
x 2 + 1.5x – 10 = 0
A1
(2x – 5)(x + 4) = 0
M1 for an attempt at using an algebraic method such as
factorising, formula (allow one error) or completing the square
(allow one error) to solve the quadratic
eg for (2x + a)(x + b) where ab = ± 20
A1 for a completely correct method
M1dep
A1
x = 2.5
Do not award last A1 if a negative value given as possible answer
eg if –4 given
2.5 seen with no or incomplete work SC2
2.5 after first M1, A1 give 5/5
A1
[5]
M22.
Identifying –2 as constant term in equation y = mx + c
B1
Gradient =
Attempt to find gradient
M1
y = 5x – 2
oe
A1
[3]
Page 79 of 117
M23.
1 → D
1 → y ≥ 2x – 4
B1
2 → C
2 → y ≥ –2x + 4
B1
3 → E
3 → y ≤ 2x – 4
B1
4 → A
4 → y ≤ –
x+2
B1
[4]
M24.
2A = ah + bh + ab + bh
Accept A= ah/2 + bh/2 + ab/2 + bh/2
Allow one error
NB 4A = ah + bh + ab + bh is one error.
M1
2A – 2bh = ah + ab
A – bh = ah/2 + ab/2
A1
2A – 2bh = a(h + b)
For factorising
DM1
or equivalent ft if both Ms awarded.
oe e.g.
A1 ft
[4]
M25.
(a)
oe e.g.
B1
Page 80 of 117
(b)
oe e.g
ft their a
M1
n2 + 2n + 1
A1
n2
(n + 1)2
A1
[4]
M26.
6:3 or numerical values in the
ratios 2:3 and 6:3
(x + z) : y = 2: 1
3x = 2y
M1
Finding ‘z’ e.g. 4 or appropriate numerical value
x + z = 2y
If both correct. Accept x + z = 2y
A1
1: 2
oe Accept words e.g. z is twice x.
A1
[3]
Page 81 of 117
M27.
trial and improvement is 0
1st-2nd
6y = – 3 allow 1 error eg, 12y = – 3 6y = 3
2 – 3.6y = 5 + 2.4y allow 1 error or
2.4equation(l) + 3.6equation(2)
M1
y = – 0.5 or x = 3.8
A1
y = – 0.5 and x = 3.8
Must have both.
Allow reversed if both seen correct in working
ft if Ml awarded
A1 ft
[3]
M28.
ax – ab = a2 + bx
Allow ax + ab =
M1
ax – bx = a2 + ab
A1
x(a – b) = a2 + ab
For factorising
NB sc x(a – b) = a2 + b Allow Ml and Al if
DM1
oe, e.g.
Follow through on factorisation if DM1 awarded.
Do not award if x = not shown,
fw such as cancelling a’s do not award last Al.
A1 ft
[4]
Page 82 of 117
M29.
(a)
Continuation at least once more
e.g. 53 – 43 = 61,
63 – 53 = 91 (allow this to be prime if stated)
Correctly evaluated.
M1
Justification that the answer is not prime.
e.g. 91 = 7 × 13.
83 – 73 = 169 = 13 × 13
Must show the factors.
NB 13 – 03 = 1 (1 not prime) Ml, A1
A1
(b)
n2 + 5n + 5n + 25 – (n2 + 3n + 3n + 9)
Ml for expanding and subtracting (allow 1
arithmetical error). Condone ‘invisible bracket’
M1
n2 + 10n + 25 – n2 – 6n – 9
Al for all terms collected and correct signs
or clear evidence of subtraction.
A1
4n + 16 = 4(n + 4)
Factorisation must be shown. Expanding is AO.
A1
[5]
M30.
(a)
106
oe
B1
(b)
8 × 109
B1
(c)
2.4 × 1022
B1 2.4 or 22 as power or
24000000000000000000000
oe
e.g. 24 × 1021
B2
[4]
Page 83 of 117
M31.
(a)
(n + 2)2, (n + 1)(n + 4)
–1 eeoo
B2
(b)
Expand Bert 3n2 + 10n + 8
Allow one error but not 3n2 + 10n + 3
M1
Expand Charu 4n2 + 12n + 9 – (n2 + 2n + 1)
4n2 + 6n + 6n + 9 – n2 + n + n + 1
M1
Convincing algebra that these are equivalent.
Allow dealing correctly with – (n2 + 2n + 1) as minimum.
e.g. 4n2 + 12n + 9 – n2 – 2n – 1 is M1 A1
A1
[5]
M32.
(a)
Intercept = 9
i.e. identifying that 9 is the constant
term in the equation.
B1
Gradient = –
=–3
Any attempt at gradient for M1.
i.e ±9/ ±3
M1
y = – 3x + 9
Accept equivalent forms.
NB y = 3x + 9 is B1, Ml, A0
A1
(b)
Substitute x = 6 into their equation
Or recognise that y-step from 0 to 3
is the same as 3 to 6. eg sight of 9.
M1 can be implied by answer only.
M1
–9
A1
Page 84 of 117
(c)
ft on their gradient in (a), Allow an
'embedded' answer in an equation,
e.g. y =
x+9
B1 ft
[6]
M33.
(a)
4
B1
(b)
8
B1
[2]
Page 85 of 117
3x 2 = x + 2
M34.
y = 3(y – 2)2
M1
3x 2 – x – 2 = 0
3y2 – 13y + 12 = 0
A1
(3x + 2)(x – 1) = 0
or (x –
)2 = ±√(
) or ±
x=
(3y – 4)(y – 3) = 0 (Reverse A1 s below)
Must be for factorising a quadratic.
x (or y) terms must have product equal to square term and
number terms must have a product equal to ± constant term.
If completing the square or formula used must be to at least the
stage shown for Method mark. or (y –
)2 = ±√(
) or ±
y=
M1
x = 1 and –
Need both
A1
y = 3 and
Must match appropriate values of y with x
Must use y = x + 2, or x = y – 2. Answers without any working is
B1, otherwise answers must be supported by an algebraic method.
Graphical method is M0.
Special case: x = 1, y = 3 without working B1. (Can be guessed).
NB only award this if no other marks awarded.
A1 ft
[5]
M35.
(a)
(i)
Even × odd, so even product
or equivalent
B1
(ii)
2 × n always even, so 2n + 1 is odd
or equivalent
B1
(b)
4n2 + 2n + 2n + 1
3 or 4 terms correct
M1
4n2 + 4n + 1
Must simplify
A1
Page 86 of 117
(c)
Odd2 – 1 = (2n + 1)2 –1
= 4n2 + 4n
= 4n(n + 1)
Must factorise
B1
= 4 × even
Deduce ‘even’ connection
B1
= multiple of 8
Concluding statement
B1
[7]
M36.
(a)
(i)
(x – 5)(x – 5) or (x – 5)2
B1 for incorrect signs
B2
(ii)
Indicating replacement of x by y – 3
Might just see (y – 3 – 5)2 or (y – 8)2
Re-starting ?... must get as far as
y 2 –16y + 64 or (y – 8)2 to score M1
M1
y=8
A1
(b)
(x – 3)(x + 3)
M1
x(x + 3)
M1
A1
[7]
Page 87 of 117
M37.
(a)
(2n ± 3)(n ± 1) or (2n ± 1)(n ± 3)
M1
(2n + 3)(n + 1)
A1
(b)
23 × 11
Must see both factors
B1
[3]
M38.
Either,
(x + 3)(x – 5) = x 2 – 2x – 15
Expansion not necessary for M1
M1
b = –2
A1
c = –15
A1
Note starting with (x – 3)(x + 5) could give c = –15 and will score
M1, A0, A1
OR, substituting coordinates (–3,0) and (5,0) into equation to get:
0 = 9 –3b + c and
0 = 5 + 5b + c
correct substitution which might be unsimplified
eg. 0 = (–3)2 – 3b + c and 0 = 52 + 5b + c
M1
Solving to give b = –2
A1
c = –15
A1
[3]
M39.
Sight of –1
or –1.5 or –3/2
accept –1 / (
) or –1 / 0.66 ... for M1 only
M1
y=–
x+5
oe eg. 2y = –3x + 10
A1
[2]
Page 88 of 117
M40.
4x + 3y = 14 4x + 3y = 14
4x + 2y = 10 6x + 3y = 15
allow error in one term
M1
y=4
2x = 1
correct elimination from their equations
M1
x=
and y = 4
oe
SC correct answers with no working or using T & I
A1
[3]
120% → 600
1.2
M41.
B1
600 ÷ 120 × 100
600 ÷ 1.2
M1
500
A1
[3]
M42.
y(x – 3) = 3x + 4
M1 for cross-multiplying and expanding bracket
M1
yx – 3y = 3x + 4
A1 correct expansion
A1
yx – 3x = 3y + 4
M1
x(y – 3) = 3y + 4
M1 for clollecting terms and factorising
A1
x = (3y + 4)/(y – 3)
A1 correct factorisation and division
[4]
Page 89 of 117
M43.
(x – 2) + 5x(x + 1) = 3(x + 1)(x – 2)
Allow 1 error
M1
5x 2 + 6x – 2 = 3x 2 – 3x – 6
A1
2x 2 + 9x + 4 = 0
M1
(2x + 1)(x + 4) = 0
A1
x = –1/2, –4
A1
[5]
M44.
9.60 + (x – 5) × 1.30
Alt: M1 for graph of Alpha parcels
M1
= 1.50x
M1 for graph of Beta
M1
3.10 = 0.20x
A1 accuracy
A1
x = 15.5
A1 answer. Accept 16 but not 15.
T&I gets M1 iff taken as far as 15.
A1 for both schemes at 15
A1 for both schemes at 16
A1 conclusion
A1
[4]
Page 90 of 117
M45.
Numerator = (x + 4)(x – 4)
B1
Denominator = (3x 2)(x 4)
M1
= (3x – 2)(x + 4)
A1
2
or 3x + 12x – 2x – 8
or 3x2 – 2x + 12x – 8
Answer = (x – 4)/(3x – 2)
A1
[4]
M46.
(a)
(i)
13
B1
(ii)
1
B1
(b)
3x = 3–3
M1
x = –3
M1 for writing 1/27 as a power of 3, correctly
allow embedded answer
A1
(c)
4y = 41½
M1
y = 1½
M1 for 4y = 8
allow embedded answer
A1
[6]
M47.
Correct region indicated
Award marks dependent upon number of lines drawn correctly and
extent of shading
B3
[3]
Page 91 of 117
M48.
(5x ± a)(x ± b)
M1 for attempt to factorise. Must have (5x ± a)(x ± b)
where ab = ± 3, a, b must be integers.
M1
(5x – 1)(x + 3)
A1
(x – 3)(x + 3)
B1
(5x – 1)(x – 3)
Answer seen and further work then deduct last B1.
B1
[4]
M49.
LHS x(x – 1) – 2(x + 1)
Give M1 for x2 – 3x + 2 if first line seen
Allow invisible bracket if recovered.
M1
LHS = x 2 – 3x – 2
Terms need not be collected. e.g.x2 – x – 2x – 2
A1
(x – 1)(x + 1)(= x 2 – 1)
On RHS or as denominator.
x 2 – 1 can be written as x2 – x + x – 1
M1
Their (x 2 – 3x – 2) = their (x 2 – 1)
Dependent on first 2 M1’s
DM1
–
(= 0.33(3...))
Do not follow through.
NB ‘cancelling’ x2 on top and bottom of
Gives correct answer. Give M1, A1, M1. M0, A0.
A1
[5]
Page 92 of 117
M50.
y ≥ 0
Accept y > 0, or 0 ≤ y ≤ 3,
B1
x ≤ 6
Accept x < 6 or 0 ≤ x ≤ 6,
B1
y ≤ ×
Accept y <
y=<
x or x ≥ 2y or equivalent.
x
Any order.
B1
Special case: All three equations given (no inequalities) B1
Special case: All three inequalities the wrong way around B2.
[3]
M51.
x 2 = w – y.
Or equivalent -x 2 = y – w
B1
x = √(w – y)
Accept ± √(w – y) and – √(w – y)
B1
[2]
M52.
a=5
from expansion x2 – 2ax + a2 and comparing coeffs.
or simply spotting that a = 5
B1
b=–7
ft. from their a using a2 + b = 18
ie. b = 18 – a2
or by inspection
B1 ft
[2]
Page 93 of 117
M53.
(a) Any correct attempt at
(y-step) ÷ (x-step)
Might be marked on diagram
M1
–2
A1
(b)
y = -2x + 3
ft. their gradient
B1
(c)
Gradient =
Attempt at gradient of perpendicular line, ft. from their gradient in
part (a) using (m1 × m2 = –1) as long as there is no contradiction
between parts (a) and (b)
M1
y = ½x + 3
or equivalent
A1 ft
[5]
M54.
(a)
1
B1
(b)
(3√8)2 or 3√(82)
Cube root and square attempted
M1
4
A1
(c)
62 = 1/62 = or 1/36
M1
1440.5 = √144 or 12
M1
or 0.33 ...
Allow 12/36 or equivalent
1/36 × 12 is not fully simplified
... A0
A1
[6]
M55.
(a)
7(x + 2)
allow one error
B1
Page 94 of 117
(b)
4m + 12 + 6m – 15
M1
10m – 3
allow 10m + –3
A1
(c)
6x + 9y = 27
and
or
6x + 4y = 2
4x + 6y = 18
and
9x + 6y = 3
5y = 25
or
5x = –15
y=5
or
x=–3
x=–3
and
y=5
M1
M1 dep
A1
A1
SC1 correct answer with no working or using T&I
(d)
(x + 8)(x – 2)
B1 (x ± 8)(x ± 2)
B2
[9]
M56.
(a)
A(–4, 0)
or 2y = 4 and – x = 4 seen
B1
B(0,2)
SC1 reversed answers
B1
(b)
(their difference in y’s) ÷ (± their difference in x’s)
or attempt to rearrange 2y – x = 4 to y = 0.5x + 2
and 2y = 4 + x or y – 0.5x = 2 seen
ft condone A(0, ?) and/or B(?, 0) from (a)
M1
0.5 or
ft for 0 < gradient < 1 only
A1 ft
[4]
Page 95 of 117
700 × 1.12 – 700
or 700 × 0.1 or 70 or 700 × 1.1 or 770
or 700 × 1.12
or 847 or 140
M57.
M1
147(.00)
A1
[2]
M58.
(a)
2n + 3
oe 2n + 1 + 2
B1
(b)
4n2 + 2n + 2n + 1
B1
(c)
(2n + 3)2 – (2n + 1)2
oe vice versa
M1
4n2 + 12n + 9 – (4n2 + 4n + 1)
Allow lack of brackets
A1
8n + 8 (= 8(n + 1))
Accept 8n + 8 as sufficient justification
– 8n – 8 OK
A1
[5]
M59.
(a)
2x(x – 10) = 0
M1
River ( x = ) 10 ( m wide)
A1
When x = 5,
y = (50 – 100) ÷ 25 = –2
Working must be seen
A1
Page 96 of 117
(b)
2x 2 – 20x = 25
M1
2x 2 – 20x – 25 = 0
M1
Use quadratic formula, completing the square to solve equation
Allow one error but not wrong formula
M1dep
x = –1.123 or 11.123
A1
AE = 1.123 + 11.123 ≈ 12.25 m
NB error with – b gives 1.123 and – 11.123
Allow last A1
A1
[8]
M60.
(a)
9
B1
(b)
Any integer value (accept 0)
B1
(c)
If y <
then INT (y) = 0 and
2y < 1 so INT (2y) = 0
E1 For a partial explanation, eg 1 numerical example
E2
[4]
M61.
(a)
Evidence of 4 terms:
2
p + pq + pq (qp) + q2
B1
(b)
p2 + q2 = (p + q)2 –2pq
100 = p2 + q2 + 2 × 18 oe
M1
(p2 + q2 =) 102 – 2 × 18
A1
64
A1
[4]
Page 97 of 117
M62.
a – 15 = b + 15
T&I using two numbers with a difference of 30
a – b = ± 30
a and b have a difference of 30
M1
4(b – 15) = a + 15
Trial and improvement:
Checking ratio of 2 numbers used
M1
4(a – 30 – 15) = a + 15
Dependent on both Ms
4(b – 15) – (b + 15) = 30 ⇒ b = 35
M1dep
(a =) 65
A1
[4]
M63.
4x 2 – 12x + 9
or 4x2 – 6x – 6x + 9
3 correct terms out of 4 seen
B1
x2 – 1
or x2 – x + x – 1
3 correct terms out of 4 seen
B1
(their) 3x 2 – 12x + 10 (= 0)
ft (their) 4x2 – 12x + 9 and (their) x2 – 1
oe
M1
ft (their) 3x2 – 12x + 10
Must be in the form ax2 + bx + c (= 0)
M1
2.82 and 1.18
A1
[5]
Page 98 of 117
M64.
(a)
25y
10
B1 ?y 10 or 25y ?
B2
(b)
x 2 + 2x – 15
B1 x2 – 3x + 5x – 15
3 correct terms out of 4 terms seen
B2
(c)
(w =) 6x + 3y – 5
or w + 5 = 3(2x + y)
M1
(6x) = w – 3y + 5
(w + 5)/3 = 2x + y
M1
(x) = (w – 3y + 5)/6
oe
eg, (x) = [(w + 5)/3 – y]/2
A1ft
[7]
M65.
100 × (100 + 1) ÷ 2 (or 5050)
or
100 × (100 – 1) ÷ 2 (or 4950)
1
3–1
6–3
10 – 6 …
M1
(their) 5050 – (their) 4950
Pattern 1 → 1
Pattern 2 → 2 …
Pattern n → n
n
M1 dep
100
A1
[3]
Page 99 of 117
M66.
(a)
3x + 15
Allow 3 × x + 15
B1
(b)
3(2a – 3)
3 × (2a – 3)
B1
[2]
M67.
(a)
8y – 2 (= 18)
(4y – 1 =) 18 ÷ 2
M1
8y = 18 + 2
4y = 9 + 1
M1
2.5
oe
A1
(b)
3x – 5 + 3x – 5 + x + 1 + x + 1
oe
Allow one error
M1
8x – 8
oe
A1
(their) (8x – 8) ÷ 4
oe
Condone missing brackets
M1
2x – 2
oe
SC2 x – 1 from 3x – 5 + x + 1 or 4x – 4
A1 ft
[7]
M68.
(a)
y = 2x + 8
B1
(b)
–
oe –0.25
B1
Page 100 of 117
(c)
y=–
(+ c) or x = –4y (+ c)
B1 ft
[3]
M69.
(a)
2x 3 + 12x 2 + 3x 2 – 15x
Allow 1 error
M1
2x 3 + 12x 2 + 3x 2 – 15x
Fully correct
A1
2x 3 + 15x 2 – 15x
Ignore factorising after final answer
ft From 4 terms where simplification is possible
A0 For fw eg, incorrect attempt to collect
terms
A1 ft
(b)
3mh(h – 5m)
B1 For partial factorisation with two factors
3m(h2 – 5mh)
3h(mh – 5m2)
mh(3h – 15m)
B2
(c)
20r 4s
6
B1 For 2 terms correct
B2
(d)
Attempt to factorise
or attempt to use formula
(allow one error)
(ax + b)(cx + d) where ac = 9 and bd = ± 28
M1
(9x – 7)(x + 4)
or formula fully correct
A1
7 / 9 and –4
Accept 0.77 ... or 0.78
A1
[10]
Page 101 of 117
M70.
(a) (42 =) 2 (×) 3 (×) 7 or
(98 =) 2 (×) 7 (×) 7
Accept on factor trees / repeated division / list or similar
Condone redundant (× 1)
M1
14
A1
(b)
√99 = 3√11 or √44 = 2√11
M1
5√11
or a = 5, b = 11
A1
[4]
M71.
(a)
At least one correct
M1
ft Their attempts at improper fractions
M1
oe eg,
A1
Alt
1+3+
oe
M1
(4 +)
oe
M1
oe eg,
A1
Page 102 of 117
(b)
(i)
1
B1
(ii)
5 or –5
Either of 5 or –5 scores the B1
B1
[5]
M72.
y(x – 4) = 2x + 3
M1
xy – 4y = 2x + 3
M1dep
xy – 2x = 4y + 3
Allow one error
M1
x(y – 2) = 4y + 3
M1
(x =)
oe
A1
[5]
M73.
(a)
x 2 – 2x + 2x – 4
B1
(b)
x+2
B1
[2]
M74.
One fraction correct
or n(n + 6) or (n + 5)(n – 4)
M1
n2 + 6n and n2 + 5n – 4n – 20
n2+ 6n or n2 + n – 20
Condone – n2 + 5n – 4n – 20
M1dep
Page 103 of 117
n2 + 6n – n2 – 5n + 4n + 20
n2 + 6n – n2 – n + 20
A1
5n + 20 and
Answer given
Must see all working for final mark
Must see use of denominators for final mark
A1
[4]
M75.
(a)
16y 6
B1
(b)
3a2 – 4ab + 6ab – 8b2
3 out of 4 terms correct
M1
3a2 – 4ab + 6ab – 8b2
A1
3a2 + 2ab – 8b2
A1
[4]
M76.
2x + 6y = 22
6x – 3y = 3
M1
7y = 21
7x = 14
A1
y = 3 and x = 2
A1
[3]
Page 104 of 117
M77.
x 2 + 7x
(a)
B1
(b)
x 2 + 7x – 8 (= 0)
M1
(x + 8)(x – 1)
(x + a)(x + b) where ab = ± 8
Use of formula, allow one error
M1
–8 and 1
A1
[4]
M78.
(a)
1–
oe
M1
Their
oe
M1 dep
oe
A1
(b)
Sight of 80% or 0.8 or 1.25 or
B1
× 10
oe
M1
8(.00)
A1
[6]
Page 105 of 117
M79.
(a)
(i)
(x ± a)(x ± b)
ab = ± 2
M1
(x + 1)(x – 2)
A1
(ii)
(x = ) –1 and 2
ft Their brackets if M1 awarded
B1ft
(b)
ft Their factorisation if M1 awarded in (a)
M1
Numerator 3x + 2 + x – 2
ft Their factorisation
A1ft
or
No ft
A1
[6]
M80.
a+b=7
M1
ab = 10
M1
a = 2, b = 5
a = 5, b = 2
B1
[3]
M81.
(a =) –6
B1
(b =) –26
B1
[2]
Page 106 of 117
M82.
x = 4, y = 3 and x + y = 4 drawn
B1 One line correct
B2
x = 3 and y = 2 identified
eg, as circled or marked point on graph
(no lines or regions necessary)
ft x = 2 and y = 3 for lines x = 3 and y = 4 interchanged
B1ft Correct region identified from (their) three lines
Correct region from two correct lines
or (Their) x = 3 and y = 2 from (their) two lines
B2ft
[4]
M83.
(a) (2 × 1 – 1) × (1 + 1)
or
(2 × 2 – 1) × (2 + 1)
or 2 × 12 + 1 – 1
or 2 × 22 + 2 – 1
M1
2 or 9
A1
(±) 7
A1
(b)
2n2 + n – 1
B1 2n2 + 2n – n – 1 any three out of four terms correct
B2
[5]
M84.
6x(3x – 2)
or 2x(9x – 6) or
B1
(2)(3x + 2)(3x – 2)
(6x + 4)(3x – 2) or (3x + 2)(6x – 4)
B1
Page 107 of 117
Partial simplification
or
or
M1
A1
[4]
M85.
(a)
–0.14(…)
B1
(b)
–7
B1
–6
B1
[3]
M86.
4x + 10y = 32
(4x + 3y = 11)
6x + 15y = 48
20x + 15y = 55
oe
Allow one error
M1
7y = 21 or 14x = 7
oe
A1 ft
y = 3 and x =
SC1 for no working or T&I
A1
[3]
M87.
(a)
x 2 + 4x + 4x + 16
At least three terms correct
M1
x 2 + 8x + 16
A1
Page 108 of 117
(b)
– 4x – 16 or (x + 4) (x + 4 – 4)
M1
x 2 + 4x = x(x + 4) or (x + 4) x
A1
[4]
M88.
(a)
72 and / or –72
B1 Sight of 2 (and / or –2) also
B1 Sight of 36 or
(but not –36 or –
)
B3
(b)
B1 Sight of 1
or
SC1
B2
[5]
M89.
(a)
Circles 1.2 × 1013
B1
(b)
Circles 5 × 10–7
B1
[2]
M90.
5
or –5 or ±5
B1
oe
B1
SC2 0.05 or equivalent fraction
B1
[3]
Page 109 of 117
M91.
M1
× 100
M2 complete and correct build up method. If
any numerical errors calculations must be
shown to give M2.
M1dep
75
A1
[3]
M92.
Prime factorisation of either number correct (any form)
63 = 32 × 7
105 = 3 × 5 × 7
M1
21
A1
Alt
List factors of at least one number correctly (excl 1 and itself)
63 = (1,) 3, 7, 9, 21 (, 63)
105 = (1,) 3, 5, 7, 15, 21, 35 (, 105)
M1
21
A1
[2]
M93.
Sight of 10x or –3(2x – 1) or 3x(2x – 1)
M1
–6x + 3 or 6x 2 – 3x
M1 dep
6x 2 – 7x – 3 (= 0)
A1
Page 110 of 117
(2x – 3)(3x + 1) (= 0)
M1
x = 1.5 or –
A1
Full answer with stages clearly shown
Strand (ii)
Q1
[6]
M94.
(2n + 2)2 – (2n)2
M1
4n2 + 8n + 4 – 4n2)
M1 dep
8n + 4
A1
8n + 4 = 2(2n + 2 + 2n)
or
2(2n + 2 + 2n) = 8n + 4
A1
Alternate method
Let n be even
(n + 2)2 – n2
M1
n2 + 4n + 4 – n2
M1 dep
4n + 4
A1
2(n + n + 2) = 2(2n + 2) = 4n + 4
or
4n + 4 = 2(2n + 2) = 2(n + n + 2)
A1
[4]
M95.
(a)
3x – x > 8 – 7
M1
x>
oe
A1
Page 111 of 117
(b)
a + 3 = b2
M1
a = b2 – 3
A1
[4]
M96.
(a)
81 + 9
+9
+
or better
4 terms and any 3 correct
M1
88 + 18
a = 88 b = 18
A1
(b)
M1
A1
=2+2
M1
= 2(1 +
)
Strand (ii)
Correct answer with a logical argument showing key steps
Q1
Page 112 of 117
Alternate method 1
M1
A1
=2+2
M1
= 2(1 +
)
Strand (ii)
Correct answer with a logical argument showing key steps
Q1
Alternate method 2
+6=2
(1 +
)
M1
=2
+2×3
A1
=
+6
M1
12 +
Note: This is not a full proof
Q0
[6]
M97.
2a + 3c = 69
3a + 5c = 109
B1 one equation correct
Any letters may be used but need to be consistent for B2
B2
× 1st by 3 or 5
× 2nd by 2 or 3
oe (to obtain consistent coefficients)
M1
Page 113 of 117
Two equations (max one error) and subtraction
eg, 6a + 9c = 207
6a + 10c = 218 and subtraction
M1 dep
Adult (a =) 18 Child (c =) 11
A1
[5]
M98.
(a)
4x(3x 2 – 2yz)
B1 one correct factor
eg, 4(3x3 – 2xyz) or x(12x2 – 8yz)
B2
(b)
(x ± 1)(x ± 2)
M1
(x + 1)(x + 2)
A1
(c)
B1
(d)
10(x 2 – 4y 2)
M1
10(x + 2y)(x – 2y)
A1 for both ± 2y
or 10(x + 4y)(x – y)
A2
[8]
M99.
(a)
(12 – 4) × 2 (= 16)
oe
M1
(16 – 4) × 2 (= 24) and
(24 – 4) × 2 (= 40)
oe
A1
Page 114 of 117
(b)
12 ÷ 2 + 4
or (40,) 24, 16, 12, 10(, 9)
M1
10 or 9
A1
[4]
M100.
(a)
37
B1
(b)
16 + a
(127 – a) ÷ 2
B1
2 × their (16 + a) + a
32 + 3a, 2(16 + a) + a
M1
2 × their (32 + 3a) + a = 127
oe 64 + 7a = 127
M1
(a =) 9
A1
Alternate method
Evidence of multiplying 8 by 2 and adding any number
Evidence of subtracting a number from 127 and dividing by 2
M1
Evidence of multiplying their answer by 2 and adding the same number
Evidence of subtracting the same number from their answer
and dividing by 2
M1
Refined attempt
M1
(a =) 9
A1
[5]
M101.
(a)
m8
B1
(b)
m –2
or
B1
Page 115 of 117
(c)
B1
seen or implied by cancelling common factors
B2
[4]
M102.
(a)
Either
oe
or
M1
oe
A1
A1 ft
(b)
× 24
oe
M1
15
A1
[5]
Page 116 of 117
Page 117 of 117