m a ths & f u r ther m a ths a l ev el s ix th f o r m p r
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SIXTH FORM PRE-COURSE WORK: MATHS & FURTHER MATHS A LEVEL
The following work is set to help with your transition from GCSE to A Level.
Please complete the following TWO booklets and return your work on the first day of Year 12.
If you have any queries, please contact the Head of Department: gareth.jones@st-edwards.co.uk
ctd..
Page 1 of 47
St Edward’s College
Maths Pre A Level Tasks
GCSE content There are two tasks to complete before you return to school. Task 1 is this booklet (approx. 3 hrs-3.5hrs) 163 marks Task 2 is an additional PDF document of UKMT problem solving surds and indies questions (approx. 1.5-2 hrs)
Name: __________________________
Comments:
Please complete both tasks in advance, showing full, detailed working for each question and bring it with you on your first day in September.
Please mark your work in a different colour using the mark scheme at the back of the booklets.
Successful completion both tasks is one of the requirements for entry onto the A Level Maths course.
On your first lesson of the course you will also be given a NON CALCULATOR baseline assessment which will also assess your suitability for the course.
Page 2 of 47
Q1.Express in the form xa
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Answer _________________________________________
(Total 3 marks)
Q2.Show that can be written in the form
where c and d are integers.
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(Total 3 marks)
Q3.Simplify
Give your answer in the form where a, b and c are integers.
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Answer _________________________________________(Total 4 marks)
Page 3 of 47
Q4.Simplify
Give your answer in the form where a and b are integers.
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Answer _________________________________________ (Total 3 marks)
Q5.A car and a lorry complete the same 240 mile journey without stopping.
The average speed of the car is x mph.
The average speed of the lorry is 12 mph slower than the car.
The lorry takes 1 hour longer than the car.
Use an algebraic method to work out the average speed of the car.
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Answer _______________________________ mph (Total 6 marks)
Page 4 of 47
Q6.Given that
Work out the value of x.
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x = __________________________________________
(Total 3 marks)
Q7.Given that 3x = 9
x+1 work out the value of x.
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x = ______________________________________ (Total 2 marks)
Q8.(a) Work out as an improper fraction.
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Answer _________________________________________
(1)
(b) Work out as a power of 2
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Answer _________________________________________ (2)
(Total 3 marks)
Page 5 of 47
Q9.(a) Work out the value of 8−2
Circle your answer.
−16 64 −64
(1)
(b) Solve 4x = 8
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x = ____________________________________________
(3)
(c) Simplify
Give your answer in the form a
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Answer _________________________________________
(3)
(Total 7 marks)
Q10.Expand and simplify (x − 4)(2x + 3y)2
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Answer _________________________________________ (Total 4 marks)
Page 6 of 47
Q11.w, x and y are three integers.
w is 2 less than x
y is 2 more than x
Prove that wy + 4 = x2
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(Total 3 marks)
Q12.(a) Solve = 4
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w = ____________________________________________
(3)
(b) Solve 4x2 − 25 < 0
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Answer _________________________________________ (3)
Page 7 of 47
(a) Solve = 5
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y = ____________________________________________
(3)
(Total 9 marks)
Q13.Expand and simplify (t + 4)³
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Answer _________________________________________
(Total 3 marks)
Q14.Solve the simultaneous equations
x + y = 4
y² = 4x + 5
Do not use trial and improvement.
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Answer _________________________________________
(Total 6 marks)
Page 8 of 47
Q15.
Solve 5x − y = 5
2y − x2 = 11
You must show your working. Do not use trial and improvement.
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Answer _________________________________________
(Total 6 marks)
Page 9 of 47
Q16.A, B and C are points on the circle x2 + y2 = 36 as shown.
A is on the y-axis.
B is on the x-axis.
M is the midpoint of AB.
COM is a straight line.
(a) Show that the coordinates of A are (0, 6)
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(b) Work out the coordinates of B.
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Answer ( ___________ , ___________ )
(1)
(c) Show that the equation of the straight line passing through C, O and M is y = x
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(d) Work out the coordinates of C. Give your answers in surd form.
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Answer ( ___________ , ___________ )(3)
(Total 7 marks)
Page 10 of 47
Q17.The area of this triangle is 14 cm2
Not drawn accurately
(a) Show that 2x2 – 5x – 26 = 0
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(3)
(b) Work out the value of x.
Give your answer to 2 significant figures.
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Answer _________________________________________
(4)
(Total 7 marks)
Q18.
Solve x2 − x − 12 = 0
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Answer _________________________________________
(Total 3 marks)
Page 11 of 47
Q19. The Venn diagram shows information about a coin collection.
ξ = 120 coins in the collection
T = coins from the 20th century B = British coins
A coin is chosen at random. It is British.
Work out the probability that it is from the 20th century.
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Answer _________________________________________
(Total 5 marks)
Page 12 of 47
Q20.
Write in the form where a is an integer.
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Answer _________________________________________(Total 4 marks)
Q21.Two numbers, a and b, are combined using the operation in the following way.
a b= 2a2 − 7a − b + b2
Work out all solutions of the equation x 3 = 0
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Answer _________________________________________ (Total 4 marks)
Page 13 of 47
Q22.(a) Write x2 − 10x + 29 in the form (x − a)2 + b
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Answer _________________________________________
(2)
(b) A sketch of y = x2 + cx + d is shown.
The turning point is (3, 5)
Work out the values of c and d.
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c = ________________ d = ________________
(3)
(Total 5 marks)
Page 14 of 47
Q23.The expression simplifies to
Work out the value of b.
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b = ______________________________________
(Total 3 marks)
Q24.The diagram shows a right-angled triangle.
Prove algebraically that x : y = 2 : 3
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(Total 6 marks)
Page 15 of 47
Q25.Use the quadratic formula to solve x2 + 2x − 1 = 0
Give your answers to 2 decimal places.
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Answer ________________ and ________________ (Total 3 marks)
Q26.f(x) = 10 − x2 for all values of x.
g(x) = (x + 2a)(x + 3) for all values of x.
(a) Circle the correct value of f(−4)
26 −6 36 16 196
(1)
(b) Write down the range of f(x).
Answer _________________________________________
(1)
(c) g(0) = 24
Show that a = 4
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(d) Hence solve f(x) = g(x)
Show that a = 4
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(4)
(Total 7 marks)
Page 16 of 47
Q27.The line y = 3x + p and the circle x2 + y2 = 53 intersect at points A and B.
p is a positive integer.
(a) Show that the x-coordinates of points A and B satisfy the equation
10x2 + 6px + p2 − 53 = 0
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(3)
(b) The coordinates of A are (2, 7)
Work out the coordinates of B.
You must show your working.
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Answer ( _____________ , _____________ )
(5)
(Total 8 marks)
Page 17 of 47
Q28.
Work out the value of x if
Give your answer in the form of where a and b are integers.
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Answer x = ________________________________
(Total 4 marks)
Q29.(a) Simplify fully 5x2 × 3y4 × 2x × y3
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Answer _________________________________________
(2)
(b) Expand and simplify (x + 7)(x – 3)
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Answer _________________________________________
(2)
(c) Solve (x – 8)(x + 2) = 0
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Answer _________________________________________
(1)
(d) Factorise 8x2y + 6xy2
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Answer _________________________________________
(2)
(Total 7 marks)
Page 18 of 47
Q30.The square and the rectangle have the same area.
All lengths are in centimetres.
Not drawn accurately
(a) Show that 36x2 – 65x + 25 = 0
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(2)
(b) 36x2 – 65x + 25 = 0
Work out the value of x.
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x = ____________________________________________
(4)
(Total 6 marks)
Page 19 of 47
Q31.Solve
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Answer _________________________________________
(Total 5 marks)
Page 20 of 47
Q32.
Here is a sketch of y = x2 + bx + c
The curve intersects
the x-axis at (5, 0) and point P
the y-axis at (0, −10)
Work out the x-coordinate of the turning point of the graph.
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Answer _________________________________________
(Total 4 marks)
Page 21 of 47
Q33.(a) Factorise x2 − 9x + 20
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Answer _________________________________________
(2)
(b) Solve x2 − 9x + 20 = 0
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Answer _________________________________________
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(Total 3 marks)
Q34.(a) Expand and simplify (x + 5)(x − 4)
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Answer _________________________________________
(2)
(b) Solve (x − 8)(x + 7) = 0
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Answer _________________________________________
(1)
(Total 3 marks)
Page 22 of 47
Q35. 2m = 32 and 9p = 3m
Work out the values of m and p
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m = ________________ p = ________________
(Total 4 marks)
Page 23 of 47
Mark schemes
Q1.
or a = –
B3
[3]
Q2.Alternative method 1
or × M1
or
or M1dep
A1
Alternative method 2
−
or −
oe common denominator
eg − M1
Page 24 of 47
− or
oe common denominator and common surd in numerator
− or M1dep
A1
Q3.
oe
stated or in correct place in expression or implied by multiplier of 2 or 4
B1
oe
stated or in correct place in expression or implied by multiplier of 2 or 4
B1
oe rationalisation of their denominator
M1
oe in the form where A is a positive integer
Page 25 of 47
A1
Q4.
oe eg
B2
or
and
or
B1
or or B3
Q5.
or
x − 12 = or x = M1
and
x − 12 = and x = M1dep
− = 1
− 12 = M1dep
240 x − 240( x − 12) = x ( x − 12)
240(t + 1) − 12 t(t + 1) = 240t
or 2880 = x 2 − 12 x
or t2 + t − 20 = 0
or x2 − 12x − 2880 = 0 oe
M1 dep
Page 26 of 47
(x + 48)(x − 60)
(t + 5)(t − 4)
or correct use of formula
or correct use of formula
or (x − 6 )2 −36 −2880 = 0
M1
60
SC2 for 60 from trial and error A1
[6]
Q6.
3x – (x – 5)
Condone omission of brackets M1
2x + 5 = 17
M1
6
SC2 11 A1
Alternative 1
23x = 217 × 2x – 5
M1
3x = 12 + x
M1
6
SC2 11 A1
Alternative 2
Substitutes a value for x and evaluates correctly as a power of 2.
M1
Substitutes a different value for x and evaluates correctly as a power of 2 which is
closer to 17. M1
6
Page 27 of 47
SC2 11 A1
[3]
Q7.
(x =) 2(x + 1) or 2x + 1
or x (= x + 1)
oe May be seen as an index is (32)x + 1
or 91/2x
M1
–2
Correct answer is 2 marks even if working nonsense or wrong.
A1
[2]
Q8.
(a)
oe improper fraction
B1
(b)
oe M1
A1
Additional Guidance
[3]
Q9.
(a) B1
(b) 22
or or
or M1
4
Page 28 of 47
A1
1 A1
(c) 1 seen or implied M1
A1
2 A1
[7]
Q10.
Alternative method 1
4x2 + 6xy + 6xy + 9y
2
oe Allow one error
Implied by 4x2 + 12xy + … or … + 12xy + 9y
2
M1
4x2 + 6xy + 6xy + 9y
2 or 4x2 + 12xy + 9y
2
oe Fully correct A1
4x3 + 6x
2y + 6x
2y + 9xy
2
or 4x3 + 12x
2y + 9xy
2
or − 16x2 − 24xy − 24xy − 36y
2
or − 16x2 − 48xy − 36y
2
oe
ft correct multiplication of their expansion by x or by −4 if
their expansion for first M1 has at least 3 terms after simplification
M1dep
4x3 + 12x
2y + 9xy
2 − 16x2 − 48xy − 36y
2
ft M1A0M1 if their first expansion has at least 3 terms after simplification
A1ft
Alternative method 2
2x2 + 3xy − 8x − 12y
oe Allow one error
eg 2x2 + 3xy − 8x + 12y
M1
2x2 + 3xy − 8x − 12y
oe Fully correct A1
Page 29 of 47
4x3 + 6x
2y − 16x
2 − 24xy or (+) 6x2y + 9xy
2 − 24xy −36y2
oe ft correct multiplication of their expansion by 2x or by 3y if
their expansion for first M1 has at least 3 terms after simplification
M1dep
4x3 + 12x
2y + 9xy
2 − 16x2 − 48xy − 36y
2
ft M1A0M1 if their first expansion has at least 3 terms after simplification
A1ft
Q11.
Alternative method 1
(w =) x − 2 and (y =) x + 2
Allow (x =) w + 2 and (x =) y − 2
M1
(x − 2)(x + 2) + 4 or wy = (x − 2)(x + 2) and wy = x2 − 4
M1
= x2 − 4 + 4 and x2 − 4 + 4 = x2
All steps must be seen
SC1 correct numerical example with all steps shown A1
Alternative method 2
(x =) w + 2 and (y =) w + 4
Allow (x=) w + 2 and (x =) y − 2
M1
(w)(w + 4) + 4
M1
= w2 + 4w + 4 and w2 + 4w + 4 = (w + 2)2
and (w + 2)2 = x2
All steps must be seen
SC1 correct numerical example with all steps shown A1
Alternative method 3
(x =) y − 2 and (w =) y − 4
Allow (x =) w + 2 and (x =) y − 2
M1
(y)(y − 4) + 4 M1
Page 30 of 47
= y2 − 4y + 4 and y2 − 4y + 4 = (y − 2)2
and (y − 2)2 = x2
All steps must be seen
SC1 correct numerical example with all steps shown A1
Q12.
(a) 2w − 3 = 24
M1
2w = 24 + 3 or 2w = 27
M1dep
13.5
oe A1
(b) x2 − < 0 or 4x
2 < 25 or (2x − 5)(2x + 5) < 0
M1
x2 <
or 2.5 or −2.5 seen M1dep
−2.5 < x < 2.5
oe A1
(c) 1 = 5(y − 6)
or 1 = 5y − 30
M1
1 + 30 = 5y
or 31 = 5y
or = y − 6
M1dep
oe A1
[9]
Page 31 of 47
Q13.
(t + 4)(t2 + 4t + 4t + 16)
oe Must be correct M1
t3 + 4t
2 + 4t2 + 16t + 4t
2 + 16t + 16t + 64
ft From their (t + 4)(t2 + 4t + 4t + 16)
oe Must have at least 4 terms correct
M2 t3 + 3t2(4) + 3t (4)2 + 43 oe
M1
t3 + 12t
2 + 48t + 64
A1
[3]
Q14.
(4 − x)2 = 4x + 5
M1
16 − 4x − 4x + x2 = 4x + 5
Allow one error but must be a quadratic in x
M1dep
x2 − 12x + 11 (= 0)
oe Must be 3 terms A1
(x − 11)(x − 1) (= 0)
or
(x − 6)2 − 36 + 11 = 0 oe M1
x = 11 and x = 1
Must have M3 to ft
x = 11 and y = −7 or x = 1 and y = 3
A1ft
x = 11 and y = −7 and
x = 1 and y = 3
A1
Alternative method y
2 = 4(4 − y) + 5 M1
y2 = 16 − 4y + 5
Allow one error but must be a quadratic in y
M1dep
y2 + 4y − 21 (= 0)
oe Must be 3 terms
Page 32 of 47
A1
(y + 7)(y − 3) (= 0)
or
(y + 2)2 − 4 − 21 = 0 oe
M1
y = −7 and y = 3
Must have M3 to ft
x = 11 and y = −7 or
x = 1 and y = 3
A1ft
x = 11 and y = −7 and
x = 1 and y = 3
A1
[6]
Q15.
Alternative method 1
y = 5x − 5
M1
2(5x − 5) − x2 = 11 or 10x − 10 − x2 = 11
Eliminating a variable
oe M1
x2 − 10x + 21 = 0
Collecting terms A1
(x − 3)(x − 7) (= 0)
Correct and accurate method to solve their 3-term quadratic equation
M1
x = 3 and x = 7
or x = 3 and y = 10
or x = 7 and y = 30
A1
x = 3, y = 10 and x = 7, y = 30 A1
Page 33 of 47
Alternative method 2
10x − 2y = 10
Equating coefficients M1
10x − x2 = 21
Eliminating a variable
oe M1
x2 − 10x + 21 = 0
Collecting terms A1
(x − 3)(x − 7) (= 0)
Correct and accurate method to solve their 3-term quadratic equation
M1
x = 3 and x = 7
or x = 3 and y = 10 or x = 7 and y = 30
A1
x = 3, y = 10 and x = 7, y = 30
A1
Alternative method 3
x = M1
2y − = 11
Eliminating a variable
oe M1
y2 − 40y + 300 = 0
Collecting terms A1
(y − 10)(y − 30) (= 0)
Correct and accurate method to solve their 3-term quadratic equation
M1
Page 34 of 47
y = 10 and y = 30
or x = 3 and y = 10
or x = 7 and y = 30
A1
x = 3, y = 10 and x = 7, y = 30
A1
[6]
Q16.
(a) (02 +) 62 = 36
or (OA =) radius = 6
or
oe B1
Additional Guidance
0 + 36 = 36 B0
(b) (6, 0) B1
(c) Alternative method 1
or
gradient AB M1
gradient OM × gradient AB = −1
and
gradient OM = 1 (and y = x)
must see correct working for M1 A1
Alternative method 2
or (3, 3)
coordinates of M M1
gradient OM = 1 (and y = x)
Page 35 of 47
or (0, 0) and (3, 3) (and y = x)
must see correct working for M1 A1
(d) x2 + x2 = 36 or 2x2 = 36
or y2 + y2 = 36 or 2y2= 36
or (−)6 cos 45° or (−)6 sin 45°
oe equation M1
or
or or M1
or
or
oe surd form A1
[7]
Q18.
Alternative method 1
(x + a)(x + b)
where ab = ±12 or a + b = −1 M1
(x − 4)(x + 3)
A1
4 and −3
SC1
4 or −3 with no or one incorrect answer A1
Alternative method 2
Page 36 of 47
oe
allow one sign error M1
oe
fully correct M1
4 and −3
SC1
4 or −3 with no or one incorrect answer
Alternative method 3
... M1
oe equation A1
4 and −3
SC1
4 or −3 with no or one incorrect answer
Q19.
x(x − 15) + x + x − 2 + 32 = 120
M1
x2 − 13x − 90 = 0
A1
(x − 18)(x + 5) = 0
oe
(x + a)(x + b)
where ab = –90 and a + b = –13
M1
their 18 + their 18 − 2 or 34 M1
oe
Page 37 of 47
SC2 for A1
[5]
Q20.
Use of or
or = or =
eg or
eg or or M1
One term simplified
ie
or A1
Two terms simplified
ie
or A1
or a = 21
A1
[4]
Q21.
2x2 − 7x − 3 + 32
M1
2x2 − 7x + 6
A1
(2x + a)(x + b) (= 0)
ab = ± their 6
Must be a quadratic in 2x2
Substitution in quadratic formula (if used) must be correct for M1 eg for 2x
2 − 7x + 6 (= 0)
Page 38 of 47
M1
1.5 and 2
oe
SC3 for 2x2 − 7x + 3 (= 0)
leading to answers of 0.5 and 3 A1
[4]
Q22.
(a) (x − 5)2 or 2a = 10 or a = 5 or a2 + b = 29
M1
(x − 5)2 + 4 or a = 5 and b = 4
A1
(b) Alternative method 1
(x − 3)2 + 5 M1
x2 − 3x − 3x + 9 + 5 or x2 − 6x + 14
Correct expansion of their (x + m)2 + n
M1
c = − 6 and d = 14
A1
Alternative method 2
+ d − M1
= −3 and d − = 5
Equates coefficients for their (x + a)2 + b
M1
hc = − 6 and d = 14 A1
Additional Guidance 9 + 3c + d = 5
M0
[5]
Page 39 of 47
Q23.
(x − 3)(x + 3)
Substitutes any value for x into both expressions but not x = 0
M1
(x − 3)(x + 5)
Sets up a correct equation in b
M1dep
(b =) 2 or x 2 + 2x − 15
A1
[3]
Q24.
(8x − y)2 = (6x)2 + (x + y)2
oe
Allow (8x − y) (8x − y) and (x + y) (x + y)
Condone 6x2
M1
Expands (8x − y)2 to 4 terms with 3 correct from 64x
2 − 8xy − 8xy + y2
oe
If going straight to 3 terms must be
64x2 − 16xy + ky
2 (k ≠ 0) or
ax2 − 16xy + y2 (a ≠ 0)
M1
Expands (x + y)² to 4 terms with 3 correct from x
2 + xy + xy + y2
oe
If going straight to 3 terms must be
x2 + 2xy + ay
2 (a ≠ 0) or
bx2 + 2xy + y2 (b ≠ 0)
M1
27x2 − 18xy (= 0) or 27x
2 = 18xy
or better e.g.1 9x
2 − 6xy (= 0) e.g.2 3x
2 = 2xy
64x − 16y = 36x + x + 2y
or equivalent linear equation
e.g.1 64x − 16y − 36x = x + 2y
e.g.2 64x − 16y − x − 2y = 36x
A1
Any correct factorisation of their px
2 + qxy or correct division of their px
2 = qxy by a multiple of x
(p and q non zero) e.g.1 9x (3x − 2y) (= 0)
Page 40 of 47
e.g.2 3x (9x − 6y) (= 0) e.g.3 27x = 18y
e.g.4 9x = 6y
Correct collection and correct simplification of terms for their linear equation in x and y
e.g. 27x = 18y
To gain this mark there must have been some expansion of brackets seen
M1
3x = 2y or or
or or or
or
Must see M1 M1 M1 A1
Do not allow if a contradictory statement is also seen M1
[6]
Q25.
Allow one error M1
or
or −1 ±
Fully correct A1
0.41 and −2.41
SC2 for 0.41 or − 2.41 A1
[3]
Q26.
(a) −6 B1
(b) f(x) ≤ 10 or 10 ≥ f(x)
Condone y ≤ 10 or 10 ≥y
B1
(c) 6a = 24 (so a = 4)
Page 41 of 47
B1 for 2a × 3 = 24
B1 for 24 = (0 + 8)(0 + 3)
8 × 3 = 24 ...on its own ... is B0 B1
(d) 10 − x² = (x + 8)(x + 3)
or 10 − x² = x² + 2ax + 3x + 6a
oe M1
2x² + 11x + 14 (= 0)
oe allow one error M1dep
(2 x + c)(x + d (= 0)
cd = 14 or c + 2d = 11
ft from their quadratic (factorising or correct substitution in quadratic formula)
M1dep
−3.5 and −2
oe A1
[7]
Q27.
(a) x2 + (3x + p)2 = 53
oe M1
9x2+ 3xp + 3xp + p2
or 9x2 + 6xp + p2
Expands (3x + p)2 correctly
M1
x2 + (3x + p)2 = 53
and x2 + 9x2 + 3xp + 3xp + p2 = 53
and 10x2 + 6px + p2 − 53 = 0
or
x2 + (3x + p)2 = 53
and x2 + 9x2 + 6xp + p2 = 53
and 10x2 + 6px + p2 − 53 = 0
A1
(b) 7 = 3 × 2 + p or 7 = 6 + p
or p = 1
Page 42 of 47
oe
Substitutes x = 2 into given equation
10(2)2 + 6p(2) + p2 − 53 = 0
or p2 + 12p − 13 = 0
or (p − 1)(p + 13)
or p = 1 (and p = −13)
M1
10x2 + 6x + 1 − 53 (= 0)
or 10x2 + 6x − 52 (= 0)
or 5x2 + 3x − 26 (= 0)
oe equation
Substitutes their p into given equation
M1dep
(5x + 13)(x − 2)
or
or
oe
Correct factorisation of their 3-term quadratic
or correct substitution in formula for their 3-term quadratic
or correct completion of square to expression for x
M1
(x =) −2.6
oe A1
(−2.6, −6.8)
oe A1
Q28.
M1
or better M1
oe eg or 2x2 = 484
their 22 must be an integer
Dependent on the first M1
Page 43 of 47
M1 dep
A1
[4]
Q29.
(a) 30x3y
7
B1 for two correct terms B2
(b) x2 – 3x + 7x – 21
Allow one error M1
x2 + 4x – 21
A1
(c) 8 and –2
or x = 8 and x = –2
Any order B1
(d) 2xy (4x + 3y)
B1 for a correct partial factorisation
x (8xy + 6y2)
y (8x2 + 6xy)
2 (4x2y + 3xy2)
2x (4xy + 3y2)
2y (4x2 + 3xy)
xy (8x + 6y)
B2
[7]
Q30.
(a) (6x – 5)2 = 5x
oe allow invisible brackets
ie 6x – 5 × 6x – 5 = 5x
M1
36x2 – 30x – 30x + 25 = 5x
oe A1
(b) Alternative method 1
(ax ± c)(bx ± d)
ab = 36 and cd = 25 but not (6x – 5)(6x – 5)
M1
Page 44 of 47
(4x – 5)(9x – 5)
A1
and seen
oe eg 1.25 and (0.55 minimum)
ft on (4x ± 5)(9x ± 5) only
A1ft
given as answer and shown to give a negative length
Strand (ii)
oe ft their values, evaluated correctly from their factorisation, for x if a valid conclusion reached
Q1ft
Alternative method 2
Allow 1 error, but not wrong formula, eg + instead of ±, 2 instead of 2a or only dividing root by 2a.
M1
oe A1
and seen
ft on –65 only for –b giving – and
(oe) A1ft
given as answer and shown to give a negative length
Strand (ii)
oe ft their values for x if a valid conclusion reached Q1ft
6(x + 3) or (–)2(x – 2) or 6x + 18 or 2x – 4 or –2x + 4 or (x – 2)(x + 3)
M1
6x + 18 – 2x + 4
Page 45 of 47
or 4x + 22 or x2 – 2x + 3x – 6 or x2 + x – 6
allow three correct terms after expansion ignore RHS and denominator
allow three correct terms after expansion as denominator or
RHS M1
x2 – 3x – 28 = 0
A1
(x – 7)(x + 4) (= 0)
correct method to solve their quadratic equation by
correct substitution into the quadratic formula
or correct completion of the square
or correct factorisation M1
(x =) 7 and (x =) – 4
SC2 (x =) 7 or (x =) – 4
A1
Additional Guidance
Correct substitution into quadratic formula
[5]
Q32.
0 = 52 + 5b + c
or –10 = 02 + b(0) + c
or c = –10
oe M1
b = –3 or x2 – 3x + c
or (y =) x2 – 3x – 10
oe
(x – 5)(x + k) and –5k = –10
M1dep
(x – 5)(x + 2)
Page 46 of 47
or 2x – 3 = 0 or x-coordinate of P = –2
or two symmetrical coordinates
e.g. (1, –12) and (2, –12)
oe
Correctly factorises the 3-term quadratic expression or correctly substitutes into quadratic formula for the 3-term quadratic
Dep on M1 M1 M1dep
oe
Accept (1.5, –12.25) A1
[4]
Q33.
(a) (x − 4)(x − 5)
B1 for (x − a)(x − b) where ab = 20
or a + b = −9
B2
(b) 4 and 5
ft their part (a) provided two brackets B1ft
[3]
Q34.
(a) x2− 4x + 5x − 20
Allow one error M1
x2 + x − 20
A1
(b) 8 and −7 B1
[3]
Q35.
m = 5
B1
(32)p = 3m or 32p = 3m
or (32)p = 3their 5 or 32p = 3their 5
Page 47 of 47
or 35 = 243 or 3their 5
or 3their 5 correctly evaluated
9p or 9p = 3their 5
or 9p = 243 or 32p = 243
oe M1
2p = m or 2p = their 5 or 9p =
oe M1
p = 2.5
oe
ft for values of m and p where p= A1ft
[4]
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