m a ths & f u r ther m a ths a l ev el s ix th f o r m p r

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: [email protected]

ctd..

http://st-edwards.co.uk/

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.

of 47

Q1.Express in the form xa

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________

(Total 3 marks)

Q2.Show that can be written in the form

where c and d are integers.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(Total 3 marks)

Q3.Simplify

Give your answer in the form where a, b and c are integers.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________(Total 4 marks)

of 47

Q4.Simplify

Give your answer in the form where a and b are integers.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

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.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _______________________________ mph (Total 6 marks)

of 47

Q6.Given that

Work out the value of x.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

x = __________________________________________

(Total 3 marks)

Q7.Given that 3x = 9

x+1 work out the value of x.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

x = ______________________________________ (Total 2 marks)

Q8.(a) Work out as an improper fraction.

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(1)

(b) Work out as a power of 2

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________ (2)

(Total 3 marks)

of 47

Q9.(a) Work out the value of 8−2

Circle your answer.

−16 64 −64

(1)

(b) Solve 4x = 8

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

x = ____________________________________________

(3)

(c) Simplify

Give your answer in the form a

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(3)

(Total 7 marks)

Q10.Expand and simplify (x − 4)(2x + 3y)2

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________


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

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(Total 3 marks)

Q12.(a) Solve = 4

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

w = ____________________________________________

(3)

(b) Solve 4x2 − 25 < 0

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________ (3)

of 47

(a) Solve = 5

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

y = ____________________________________________

(3)

(Total 9 marks)

Q13.Expand and simplify (t + 4)³

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________

(Total 3 marks)

Q14.Solve the simultaneous equations

x + y = 4

y² = 4x + 5

Do not use trial and improvement.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________

(Total 6 marks)

of 47

Q15.

Solve 5x − y = 5

2y − x2 = 11

You must show your working. Do not use trial and improvement.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________

(Total 6 marks)

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)

___________________________________________________________________

_________________________________________________________________(1)

(b) Work out the coordinates of B.

___________________________________________________________________

___________________________________________________________________

Answer ( ___________ , ___________ )

(1)

(c) Show that the equation of the straight line passing through C, O and M is y = x

___________________________________________________________________

___________________________________________________________________

_________________________________________________________________(2)

(d) Work out the coordinates of C. Give your answers in surd form.

___________________________________________________________________

___________________________________________________________________

Answer ( ___________ , ___________ )(3)

(Total 7 marks)

of 47

Q17.The area of this triangle is 14 cm2

Not drawn accurately

(a) Show that 2x2 – 5x – 26 = 0

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

(3)

(b) Work out the value of x.

Give your answer to 2 significant figures.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(4)

(Total 7 marks)

Q18.

Solve x2 − x − 12 = 0

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________

(Total 3 marks)

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.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________

(Total 5 marks)

of 47

Q20.

Write in the form where a is an integer.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

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

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________


of 47

Q22.(a) Write x2 − 10x + 29 in the form (x − a)2 + b

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

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.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

c = ________________ d = ________________

(3)

(Total 5 marks)

of 47

Q23.The expression simplifies to

Work out the value of b.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

b = ______________________________________

(Total 3 marks)

Q24.The diagram shows a right-angled triangle.

Prove algebraically that x : y = 2 : 3

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(Total 6 marks)

of 47

Q25.Use the quadratic formula to solve x2 + 2x − 1 = 0

Give your answers to 2 decimal places.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

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

___________________________________________________________________

___________________________________________________________________

(1)

(d) Hence solve f(x) = g(x)

Show that a = 4

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

(4)

(Total 7 marks)

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

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

(3)

(b) The coordinates of A are (2, 7)

Work out the coordinates of B.

You must show your working.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer ( _____________ , _____________ )

(5)

(Total 8 marks)

of 47

Q28.

Work out the value of x if

Give your answer in the form of where a and b are integers.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer x = ________________________________

(Total 4 marks)

Q29.(a) Simplify fully 5x2 × 3y4 × 2x × y3

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(2)

(b) Expand and simplify (x + 7)(x – 3)

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(2)

(c) Solve (x – 8)(x + 2) = 0

___________________________________________________________________

Answer _________________________________________

(1)

(d) Factorise 8x2y + 6xy2

___________________________________________________________________

Answer _________________________________________

(2)

(Total 7 marks)

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

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

(2)

(b) 36x2 – 65x + 25 = 0

Work out the value of x.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

x = ____________________________________________

(4)

(Total 6 marks)

of 47

Q31.Solve

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________

(Total 5 marks)

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.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

Answer _________________________________________

(Total 4 marks)

of 47

Q33.(a) Factorise x2 − 9x + 20

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(2)

(b) Solve x2 − 9x + 20 = 0

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(1)

(Total 3 marks)

Q34.(a) Expand and simplify (x + 5)(x − 4)

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(2)

(b) Solve (x − 8)(x + 7) = 0

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Answer _________________________________________

(1)

(Total 3 marks)

of 47

Q35. 2m = 32 and 9p = 3m

Work out the values of m and p

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

m = ________________ p = ________________

(Total 4 marks)

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

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

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

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

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

of 47

A1

1 A1

(c) 1 seen or implied M1

A1

2 A1

[7]

Q10.


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


2x2 + 3xy − 8x − 12y

oe Allow one error

eg 2x2 + 3xy − 8x + 12y

M1

2x2 + 3xy − 8x − 12y

oe Fully correct A1

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.


(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


(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




(x =) y − 2 and (w =) y − 4

Allow (x =) w + 2 and (x =) y − 2

M1

(y)(y − 4) + 4 M1

of 47

= y2 − 4y + 4 and y2 − 4y + 4 = (y − 2)2

and (y − 2)2 = x2



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]

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

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.


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

of 47


10x − 2y = 10

Equating coefficients M1

10x − x2 = 21


oe M1

x2 − 10x + 21 = 0

Collecting terms A1

(x − 3)(x − 7) (= 0)


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


x = M1

2y − = 11


oe M1

y2 − 40y + 300 = 0

Collecting terms A1

(y − 10)(y − 30) (= 0)


M1

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


or (3, 3)

coordinates of M M1

gradient OM = 1 (and y = x)

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.


(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


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


... 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

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)

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


+ 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]

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)

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)

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

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

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

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


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

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)

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

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]

UKMT Indices & Surds Questions (Answers follow after all the questions)

2005…

2006…

2007…

2008…

2009…

2010…

2011…

2012…

2013…

2014…

2015…

2016…

UKMT Indices & Surds Answers

2005…

2006…

2007…

2008…

2009…

2010…

2011…

2012…

2013…

2014…

2015…

2016…

m a ths & f u r ther m a ths a l ev el s ix th f o r m p r

Documents