A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
1 Scheme Marks AOs
and Progress
descriptor

States that: M1 2.2a 5th

Decompose
algebraic
Equates the various terms. M1* 2.2a fractions into
partial fractions −
Equating the coefficients of x: two linear factors.
Equating constant terms:

Multiplies both of the equations in an effort to equate one of the M1* 1.1b
two variables.

Finds A = 8 A1 1.1b

Find B = −2 A1 1.1b

(5 marks)

Notes
Alternative method

Uses the substitution method, having first obtained this equation:

Substitutes to obtain B = 27 (M1)

Substitutes to obtain A = 43.2 (M1)

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
2 Scheme Marks AOs
and Progress
descriptor

Begins the proof by assuming the opposite is true. B1 3.1 7th

‘Assumption: there do exist integers a and b such that Complete proofs
’ using proof by
contradiction.
M1 2.2a
Understands that
‘As both 25 and 15 are multiples of 5, divide both sides by 5 to

leave ’

Understands that if a and b are integers, then 5a is an integer, M1 1.1b

3b is an integer and 5a + 3b is also an integer.

B1 2.4
Recognises that this contradicts the statement that ,

as is not an integer. Therefore there do not exist integers a and

b such that ’

(4 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
3 Scheme Marks AOs
and Progress
descriptor

(a) M1 1.1b 6th

Finds and Differentiate
simple functions
Writes −2sin 2t = − 4sin t cos t M1 2.2a defined
parametrically
A1 1.1b including
application to
Calculates
tangents and
normals.

(3)

(b) A1 ft 1.1b 6th

Evaluates at Differentiate
simple functions
defined
parametrically
including
application to
M1 ft 1.1b tangents and
normals.
Understands that the gradient of the tangent is , and then the
gradient of the normal is −2.

M1 ft 1.1b
Finds the values of x and y at

and

M1 ft 2.2a
Attempts to substitute values into

For example, is seen.

Shows logical progression to simplify algebra, arriving at: A1 2.4

or

(5)

(8 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme
(b) Award ft marks for a correct answer using an incorrect answer from part a.

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
4 Scheme Marks AOs
and Progress
descriptor

M1 2.2a 6th
States that Integrate using
trigonometric
M1 2.2a identities.

Makes an attempt to find

Writing or writing ln (sin x) constitutes an

attempt.

A1 1.1b
States a fully correct answer

(3 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
5 Scheme Marks AOs
and Progress
descriptor
M1 2.2a 5th
Demonstrates an attempt to find the vectors , and
Find the
A1 1.1b magnitude of a
Finds , and vector in 3
dimensions.
M1 2.2a
Demonstrates an attempt to find , and
A1 1.1b
Finds

Finds

Finds
M1 2.2a
States or implies in a right-angled triangle
B1 2.1
States that

(6 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
6 Scheme Marks AOs
and Progress
descriptor

(a) M1 2.2a 5th

States or implies that
Find composite
M1 2.2a functions.
States or implies that

M1 1.1b
Makes an attempt to solve . For example,
or is seen.

States that . Must show all steps and a logical A1 1.1b

progression.

(4)

(b) M1* 2.2a 5th

Find the domain
B1* 3.2b and range of
States that as there are no real solutions to the
composite
equation.
functions.

(2)

(6 marks)

Notes
(b) Alternative Method

M1: Uses the method of completing the square to show that or

B1: Concludes that this equation will have no real solutions.

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
7 Scheme Marks AOs
and Progress
descriptor

Begins the proof by assuming the opposite is true. B1 3.1 7th

‘Assumption: there is a finite amount of prime numbers.’ Complete proofs
using proof by
Considers what having a finite amount of prime numbers means M1 2.2a contradiction.
by making an attempt to list them:

Let all the prime numbers exist be

Consider a new number that is one greater than the product of M1 1.1b
all the existing prime numbers:

Let

Understands the implication of this new number is that division M1 1.1b

by any of the existing prime numbers will leave a remainder of
1. So none of the existing prime numbers is a factor of N.

Concludes that either N is prime or N has a prime factor that is B1 2.4

not currently listed.

Recognises that either way this leads to a contradiction, and B1 2.4

therefore there is an infinite number of prime numbers.

(6 marks)

Notes

If N is prime, it is a new prime number separate to the finite list of prime numbers, .
If N is divisible by a previously unknown prime number, that prime number is also separate to the finite list of
prime numbers.

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
8 Scheme Marks AOs
and Progress
descriptor

Attempts to write a differential equation. M1 3.1a 7th

Construct simple
differential
For example, or is seen.
equations.
A1 3.1a
States

(2 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
9 Scheme Marks AOs
and Progress
descriptor

(a) Recognises that it is a geometric series with a first term M1 3.1a 6th
and common ratio Use geometric
sequences and
Attempts to use the sum of a geometric series. For example, M1* 2.2a series in context.

or is seen.

A1 1.1b
Finds
(3)

(b) M1 3.1a 5th

Use arithmetic
States or
sequences and
M1 1.1b series in context.
Begins to simplify. or
Applies law of logarithms correctly M1 2.2a
or

A1 1.1b

States
(4)

(c) Uses the sum of an arithmetic series to state M1 3.1a 5th

Use arithmetic
sequences and
Solves for d. d = £11.21 series in context.
A1 1.1b

(2)
(9 marks)
Notes
M1
Award mark if attempt to calculate the amount of money after 1, 2, 3,….,8 and 9 months is seen.

© Pearson Education Ltd 2018 10

 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
10 Scheme Marks AOs
and Progress
descriptor

M1 2.2a 6th
Selects as the appropriate trigonometric
identity. Integrate using
trigonometric
M1 1.1b identities.
Manipulates the identity to the question:

M1 1.1b
States that

Makes an attempt to integrate the expression, x and sin x are M1 1.1b

seen.

A1 1.1b

Correctly states

(5 marks)

Notes
Student does not need to state ‘+C’ to be awarded the third method mark. Must be stated in the final answer.

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
11 Scheme Marks AOs
and Progress
descriptor

(a) Writes tanx and secx in terms of sinx and cosx. For example, M1 2.1 5th
Understand the
functions sec,
cosec and cot.

M1 1.1b
Manipulates the expression to find

A1 1.1b
Simplifies to find

(3)

(b) B1 2.2a 6th

States that or
Use the functions
M1 1.1b sec, cosec and cot
to solve simple
Writes that or trigonometric
problems.
A1 1.1b
Finds

(3)

(6 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
12 Scheme Marks AOs
and Progress
descriptor

(a) M1 1.1b 8th

Rearranges to obtain Use parametric
equations in
M1 1.1b modelling in a
variety of
Substitutes into
contexts.

For example, is seen.

A1 1.1b
Finds

(3)

(b) Deduces that the width of the arch can be found by substituting M1 3.4 8th
into Use parametric
equations in
Finds x = 0 and x = 160 and deduces the width of the arch is A1 3.2a modelling in a
160 m. variety of
contexts.

(2)

(c) Deduces that the greatest height occurs when M1 3.4 8th
Use parametric
equations in
modelling in a
Deduces that the height is 100 m. A1 3.2a variety of
contexts.

(2)

(7 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
13 Scheme Marks AOs
and Progress
descriptor

Makes an attempt to set up a long division. M1 2.2a 5th

Divide
polynomials by
For example: is seen. linear expressions
with a remainder.
Award 1 accuracy mark for each of the following: A4 1.1b

seen, 2x seen, −21 seen.

For the final accuracy mark either D = 138 or or the

remainder is 138 must be seen.

(5 marks)

Notes

This question can be solved by first writing and then solving for
A, B, C and D. Award 1 mark for the setting up the problem as described. Then award 1 mark for each correct
coefficient found. For example:
Equating the coefficients of x3: A = 1
Equating the coefficients of x2: 6 + B = 8, so B = 2
Equating the coefficients of x: 12 + C = −9, so C = −21
Equating the constant terms: −126 + D = 12, so D = 138.

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
14 Scheme Marks AOs
and Progress
descriptor

M1 3.1a 8th
Recognises the need to use the chain rule to find Construct
differential
equations in a
For example is seen. range of contexts.

M1 2.2a
Finds and

Makes an attempt to substitute known values. For example, M1 1.1b

A1 1.1b
Simplifies and states

(4 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
Scheme Marks AOs
and Progress
descriptor

15 M1 2.2a 6th
Recognises the need to write
Integrate using
Selects the correct trigonometric identity to write M1 2.2a trigonometric
identities.
. Could also write

M1 1.1b
Makes an attempt to find

A1 1.1b
Correctly states answer

(4 marks)

Notes

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
16 Scheme Marks AOs
and Progress
descriptor

(a) Finds and M1 3.1a 7th

Use numerical
Change of sign and continuous function in the interval A1 2.4 methods to solve
problems in
root context.

(2)

(b) Makes an attempt to differentiate h(t) M1 2.2a 7th

Use numerical
A1 1.1b
methods to solve
Correctly finds problems in
context.

Finds and M1 1.1b

M1 1.1b
Attempts to find

A1 1.1b
Finds

(5)

(c) Demonstrates an understanding that x = 19.3705 and M1 2.2a 7th

x = 19.3715 are the two values to be calculated.
Use numerical
methods to solve
Finds and M1 1.1b
problems in
context.
Change of sign and continuous function in the interval A1 2.4

root

(3)

(10 marks)

Notes
(a) Minimum required is that answer states there is a sign change in the interval and that this implies a root in
the given interval.

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 A level Pure Maths: Practice Set 7 mark scheme

Pearson
Progression Step
17 Scheme Marks AOs
and Progress
descriptor

(a) Demonstrates an attempt to find the vectors , and M1 2.2a 6th

Solve geometric
A1 1.1b problems using
Finds , and
vectors in 3
dimensions.
M1 2.2a
Demonstrates an attempt to find , and

A1 1.1b
Finds

Finds

Finds

Demonstrates an understanding of the need to use the Law of M1 ft 2.2a

Cosines. Either (or variation) is seen,
or attempt to substitute into formula is made

Makes an attempt to simplify the above equation. For example, M1 ft 1.1b

is seen.

Shows a logical progression to state B1 2.4

(7)

(b) States or implies that is isosceles. M1 2.2a 6th

Solve geometric
Makes an attempt to find the missing angles M1 1.1b problems using
vectors in 3
dimensions.

States . Accept awrt 56.8° A1 1.1b

(3)

(10 marks)

Notes
(b) Award ft marks for a correct answer to part a using their incorrect answer from earlier in part a.

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