Mathematics: analysis and approaches Higher level and Standard

Mathematics: analysis and

approaches

Higher level and Standard level

Specimen papers 1, 2 and 3

First examinations in 2021

CONTENTS

Mathematics: analysis and approaches higher level

paper 1 specimen paper

M

athematics: analysis and approaches higher level

paper 1 markscheme

M

athematics: analysis and approaches higher level

paper 2 specimen paper

M

athematics: analysis and approaches higher level

paper 2 markscheme

M

athematics: analysis and approaches higher level

paper 3 specimen paper

Mathematics: analysis and approaches higher level

paper 3 markscheme

M

athematics: analysis and approaches standard level

paper 1 specimen paper

M

athematics: analysis and approaches standard level

paper 1 markscheme

M

athematics: analysis and approaches standard level

paper 2 specimen paper

M

athematics: analysis and approaches standard level

paper 2 markscheme

Candidate session number

Mathematics: analysis and approaches

Higher level

Paper 1

13 pages

Specimen paper

2 hours

16EP01

Instructions to candidates

• Write your session number in the boxes above.

• Do not open this examination paper until instructed to do so.

• You are not permitted access to any calculator for this paper.

• Section A: answer all questions. Answers must be written within the answer boxes provided.

• Section B: answer all questions in the answer booklet provided. Fill in your session number

on the front of the answer booklet, and attach it to this examination paper and your

cover sheet using the tag provided.

• Unless otherwise stated in the question, all numerical answers should be given exactly or

correct to three signicant gures.

• A clean copy of the mathematics: analysis and approaches formula booklet is required for

this paper.

• The maximum mark for this examination paper is [110 marks].

SPEC/5/MATAA/HP1/ENG/TZ0/XX

Full marks are not necessarily awarded for a correct answer with no working. Answers must be

supported by working and/or explanations. Where an answer is incorrect, some marks may be given

for a correct method, provided this is shown by written working. You are therefore advised to show all

working.

Section A

Answer all questions. Answers must be written within the answer boxes provided. Working may be

continued below the lines, if necessary.

1. [Maximum mark: 5]

Let A and B be events such that P (A) = 0.5 , P (B) = 0.4 and P (A ∪ B) = 0.6 .

Find P (A | B) .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16EP02

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 2 –

2. [Maximum mark: 5]

(a) Show that (2n - 1)

2

+ (2n + 1)

2

= 8n

2

+ 2 , where n ∈  . [2]

(b) Hence, or otherwise, prove that the sum of the squares of any two consecutive odd

integers is even. [3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Turn over

16EP03

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 3 –

3. [Maximum mark: 5]

Let

′

=

+

fx

x

()

8

21

2

. Given that f (0) = 5 , nd f (x) .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16EP04

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 4 –

4. [Maximum mark: 5]

The following diagram shows the graph of y = f (x) . The graph has a horizontal asymptote

at y = -1 . The graph crosses the x-axis at x = -1 and x = 1 , and the y-axis at y = 2 .

y

x

1

−1

−2

−2−3

−

4 2

2

3 4

3

4

y = f (x)

On the following set of axes, sketch the graph of y =

[

f (x)

]

2

+ 1 , clearly showing any

asymptotes with their equations and the coordinates of any local maxima or minima.

y

x

1

−1 0

−1

−2

−2−3

−

4 2

2

3 4

3

4

5

6

Turn over

16EP05

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 5 –

5. [Maximum mark: 5]

The functions f and g are dened such that

fx

x

()=

+

3

4

and g (x) = 8x + 5 .

(a) Show that ( g  f )(x) = 2x + 11 . [2]

(b) Given that ( g  f )

-1

(a) = 4 , nd the value of a . [3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16EP06

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 6 –

6. [Maximum mark: 8]

(a) Show that

log(cos)logcos

93

22

22xx

+= +

. [3]

(b) Hence or otherwise solve log

3

(2 sin x) = log

9

(cos 2x + 2) for

0

2

<<

x

�

. [5]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Turn over

16EP07

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 7 –

7. [Maximum mark: 7]

A continuous random variable X has the probability density function f given by

fx

xx

x

()

sin,

,

=













≤≤











� �

36 6

06

0

otherwise

.

Find P (0 ≤ X ≤ 3).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16EP08

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 8 –

8. [Maximum mark: 7]

The plane

П

has the Cartesian equation 2x + y + 2z = 3 .

The line L has the vector equation

r =−













+−













3

5

1

2

µ

p

,

µ

, p ∈  . The acute angle between

the line L and the plane

П

is 30



.

Find the possible values of p .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Turn over

16EP09

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 9 –

9. [Maximum mark: 8]

The function f is dened by f (x) = e

2x

- 6e

x

+ 5 , x ∈  , x ≤ a . The graph of y = f (x) is

shown in the following diagram.

x

y

2

0

−2

−2 2

−4

4

5

(a) Find the largest value of a such that f has an inverse function. [3]

(b) For this value of a , nd an expression for f

-1

(x) , stating its domain. [5]

(This question continues on the following page)

16EP10

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 10 –

(Question 9 continued)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Turn over

16EP11

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 11 –

Do not write solutions on this page.

Section B

Answer all questions in the answer booklet provided. Please start each question on a new page.

10. [Maximum mark: 16]

Let

fx

x

kx

()

ln

=

5

where x > 0 , k ∈ 

+

.

(a) Show that

′

=

−

fx

x

kx

()

ln

15

2

. [3]

The graph of f has exactly one maximum point P .

(b) Find the x-coordinate of P . [3]

The second derivative of f is given by

′′

=

−

fx

x

kx

()

ln25 3

3

. The graph of f has exactly one

point of inexion Q .

(c) Show that the x-coordinate of Q is

1

5

3

2

e

. [3]

The region R is enclosed by the graph of f , the x-axis, and the vertical lines through the

maximum point P and the point of inexion Q .

y

x

R

P

Q

(d) Given that the area of R is 3 , nd the value of k . [7]

16EP12

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 12 –

Do not write solutions on this page.

11. [Maximum mark: 18]

(a) Express

−+

33

i

in the form re

iθ

, where r > 0 and -π < θ ≤ π . [5]

Let the roots of the equation

z

3

33

=− + i

be u , v and w .

(b) Find u , v and w expressing your answers in the form re

iθ

, where r > 0 and -π < θ ≤ π . [5]

On an Argand diagram, u , v and w are represented by the points U , V and W respectively.

(c) Find the area of triangle UVW . [4]

(d) By considering the sum of the roots u , v and w , show that

co

scos cos

5

18

7

18

17

18

0

� � �

++ =

. [4]

12. [Maximum mark: 21]

The function f is dened by f (x) = e

sin x

.

(a) Find the rst two derivatives of f (x) and hence nd the Maclaurin series for f (x) up to

and including the x

2

term. [8]

(b) Show that the coecient of x

3

in the Maclaurin series for f (x) is zero. [4]

(c) Using the Maclaurin series for arctan x and e

3x

- 1 , nd the Maclaurin series

for arctan

(

e

3x

- 1

)

up to and including the x

3

term. [6]

(d) Hence, or otherwise, nd

lim

()

arctan

x

fx

→

−

()

0

3

1

1e

. [3]

16EP13

SPEC/5/MATAA/HP1/ENG/TZ0/XX– 13 –

Please do not write on this page.

Answers written on this page

will not be marked.

16EP14

Please do not write on this page.

Answers written on this page

will not be marked.

16EP15

Please do not write on this page.

Answers written on this page

will not be marked.

16EP16

SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

16 pages

Markscheme

Specimen paper

Mathematics:

analysis and approaches

Higher level

Paper 1

– 2 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

Instructions to Examiners

Abbreviations

M Marks awarded for attempting to use a correct Method.

A Marks awarded for an Answer or for Accuracy; often dependent on preceding M marks.

R Marks awarded for clear Reasoning.

AG Answer given in the question and so no marks are awarded.

Using the markscheme

1 General

Award marks using the annotations as noted in the markscheme eg M1, A2.

2 Method and Answer/Accuracy marks

 Do not automatically award full marks for a correct answer; all working must be checked, and

marks awarded according to the markscheme.

 It is generally not possible to award M0 followed by A1, as A mark(s) depend on the preceding

M mark(s), if any.

 Where M and A marks are noted on the same line, e.g. M1A1, this usually means M1 for an

attempt to use an appropriate method (e.g. substitution into a formula) and A1 for using the

correct values.

 Where there are two or more A marks on the same line, they may be awarded independently;

so if the first value is incorrect, but the next two are correct, award A0A1A1.

 Where the markscheme specifies M2, A2

, etc., do not split the marks, unless there is a note.

 Once a correct answer to a question or part-question is seen, ignore further correct working.

However, if further working indicates a lack of mathematical understanding do not award the final

A1. An exception to this may be in numerical answers, where a correct exact value is followed by

an incorrect decimal. However, if the incorrect decimal is carried through to a subsequent part,

and correct working shown, award FT marks as appropriate but do not award the final A1 in that

part.

Examples

Correct answer seen Further working seen Action

1.

82

5.65685...

(incorrect decimal value)

Award the final A1

(ignore the further working)

2.

1

sin 4

4

x

sin

x

Do not award the final A1

3.

log logab

log ( )ab

Do not award the final A1

– 3 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

3 Implied marks

Implied marks appear in brackets e.g. (M1), and can only be awarded if correct work is seen or if

implied in subsequent working.

 Normally the correct work is seen or implied in the next line.

 Marks without brackets can only be awarded for work that is seen.

4 Follow through marks (only applied after an error is made)

Follow through (FT) marks are awarded where an incorrect answer from one part of a question is

used correctly in subsequent part(s) or subpart(s). Usually, to award FT marks, there must be

working present and not just a final answer based on an incorrect answer to a previous part.

However, if the only marks awarded in a subpart are for the answer (i.e. there is no working

expected), then FT marks should be awarded if appropriate.

 Within a question part, once an error is made, no further A marks can be awarded for work

which uses the error, but M marks may be awarded if appropriate.

 If the question becomes much simpler because of an error then use discretion to award fewer

FT marks.

 If the error leads to an inappropriate value (e.g. probability greater than 1, use of

1r 

for the

sum of an infinite GP,

sin 1.5





, non integer value where integer required), do not award the

mark(s) for the final answer(s).

 The markscheme may use the word “their” in a description, to indicate that candidates may be

using an incorrect value.

 Exceptions to this rule will be explicitly noted on the markscheme.

 If a candidate makes an error in one part, but gets the correct answer(s) to subsequent part(s),

award marks as appropriate, unless the question says hence. It is often possible to use a

different approach in subsequent parts that does not depend on the answer to previous parts.

5 Mis-read

If a candidate incorrectly copies information from the question, this is a mis-read (MR). Apply a MR

penalty of 1 mark to that question

 If the question becomes much simpler because of the MR, then use discretion to award

fewer marks.

 If the MR leads to an inappropriate value (e.g. probability greater than

1,

sin 1.5





, non-integer

value where integer required), do not award the mark(s) for the final answer(s).

 Miscopying of candidates’ own work does not constitute a misread, it is an error.

 The MR penalty can only be applied when work is seen. For calculator questions with no

working and incorrect answers, examiners should not infer that values were read incorrectly.

– 4 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

6 Alternative methods

 Alternative methods for complete questions are indicated by METHOD 1,

METHOD 2, etc.

 Alternative solutions for part-questions are indicated by EITHER . . . OR.

7 Alternative forms

Unless the question specifies otherwise, accept equivalent forms.

 As this is an international examination, accept all alternative forms of notation.

 In the markscheme, equivalent numerical and algebraic forms will generally be written in

brackets immediately following the answer.

 In the markscheme, simplified answers, (which candidates often do not write in examinations),

will generally appear in brackets. Marks should be awarded for either the form preceding the

bracket or the form in brackets (if it is seen).

8 Accuracy of Answers

If the level of accuracy is specified in the question, a mark will be linked to giving the answer to the

required accuracy. There are two types of accuracy errors, and the final answer mark should not be

awarded if these errors occur.

 Rounding errors: only applies to final answers not to intermediate steps.

 Level of accuracy: when this is not specified in the question the general rule applies to final

answers: unless otherwise stated in the question all numerical answers must be given exactly or

correct to three significant figures.

9 Calculators

No calculator is allowed. The use of any calculator on paper 1 is malpractice, and will result in no

grade awarded. If you see work that suggests a candidate has used any calculator, please follow

the procedures for malpractice. Examples: finding an angle, given a trig ratio of 0.4235.

Candidates will sometimes use methods other than those in the markscheme. Unless the question

specifies a method, other correct methods should be marked in line with the markscheme

– 5 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

Section A

1. attempt to substitute into



PPPP

A

BABAB   

(M1)

Note: Accept use of Venn diagram or other valid method.

0.6 0.5 0.4 P( )

A

B 

(A1)



P0.3AB

(seen anywhere) A1

attempt to substitute into









P

P|

P

A

B

AB

B





(M1)

0.3

0.4







P|

A

B

3

0.75

4









A1

Total [5 marks]

2. (a) attempting to expand the LHS (M1)



22

LHS 4 4 1 4 4 1nn nn A1





2

82RHSn AG

[2 marks]

(b) METHOD 1

recognition that

21n 

and

21n 

represent two consecutive odd

integers (for all odd integers

n

) R1



22

82241nn  A1

valid reason eg divisible by

2 (2 is a factor) R1

so the sum of the squares of any two consecutive odd integers is even AG

[3 marks]

METHOD 2

recognition, eg that

n

and

2n 

represent two consecutive odd integers

(for

n

) R1



2

22

22 22nn n n  

A1

valid reason eg divisible by

2 (2 is a factor) R1

so the sum of the squares of any two consecutive odd integers

is even AG

[3 marks]

Total [5 marks]

– 6 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

3. attempt to integrate (M1)

2

d

21 4

d

u

ux x

x



2

82

dd

21

x

u

x







(A1)

EITHER



4 uC

A1

OR



2

42 1

x

C A1

THEN

correct substitution into their integrated function (must have

C

) (M1)

54 1CC  



2

42 11fx x A1

Total [5 marks]

– 7 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

4.

no

y values below 1 A1

horizontal asymptote at

2y 

with curve approaching from below as

x



A1



1,1

local minima A1



0,5 local maximum A1

smooth curve and smooth stationary points A1

Total [5 marks]

5. (a) attempt to form composition M1

correct substitution

33

85

44

xx

g











A1



211gf x x

AG

[2 marks]

(b) attempt to substitute 4 (seen anywhere) (M1)

correct equation

2411a 

(A1)

19a 

A1

[3 marks]

Total [5 marks]

– 8 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

6. (a) attempting to use the change of base rule M1

3

9

3

log (cos 2 2)

log 9

x





A1

3

1

log (cos 2 2)

2

x A1

3

log cos 2 2x AG

[3 marks]

(b)

33

log (2sin ) log cos 2 2xx

2sin cos2 2xx

M1

2

4sin cos2 2xx (or equivalent) A1

use of

2

cos 2 1 2sin

x

x (M1)

2

6sin 3x 



1

sin

2

x 

A1

π

4

x  A1

Note: Award A0 if solutions other than

π

4

x  are included.

[5 marks]

Total [8 marks]

– 9 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

7. attempting integration by parts, eg

ππ π 6π

,d d ,d sin d , cos

36 36 6 π 6

x

xx

uuxv xv

 

  

 

 

(M1)



3

0

π6 π 6 π

P0 3 cos cos d

36 π 6 π 6

xx x

X

x





 

  



 





 







(or equivalent) A1A1

Note: Award A1 for a correct

uv

and A1 for a correct dvu



.

attempting to substitute limits M1

3

0

π6 π

cos 0

36 π 6

xx













(A1)

so



3

0

1π

P0 3 sin

π6

x

X















(or equivalent) A1

1

π



A1

Total [7 marks]

8. recognition that the angle between the normal and the line is

60

(seen anywhere) R1

attempt to use the formula for the scalar product M1

2

21

12

2

cos 60

914

p

 

 



 

 







A1

2

1

2

35

p





A1

2

35 4

p

p

attempt to square both sides M1





222

95 16 7 45ppp  

5

3

7

p 

(or equivalent) A1A1

Total [7 marks]

– 10 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

9. (a) attempt to differentiate and set equal to zero M1

2

( ) 2e 6e 2e (e 3) 0

xx xx

fx



 

A1

minimum at

ln 3x 

ln 3a 

A1

[3 marks]

(b)

Note: Interchanging

x

and

y

can be done at any stage.



2

e3 4

x

y 

(M1)

e3 4

x

y 

A1

as

ln 3x 

,



ln 3 4xy

R1

so



1

ln 3 4fx x





A1

domain of

1

f



is

,4 5xx

A1

[5 marks]

Total [8 marks]

– 11 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

Section B

10. (a) attempt to use quotient rule (M1)

correct substitution into quotient rule



2

1

5ln5

5

kx k x

x

fx

kx













(or equivalent) A1



22

ln 5

,

kk x

k

kx







 A1

2

1ln5

x

kx





AG

[3 marks]

(b)



0fx





M1

2

1ln5

0

x

kx





ln 5 1

x



(A1)

e

5

x 

A1

[3 marks]

(c)



0fx





M1

3

2ln5 3

0

x

kx





3

ln 5

2

x 

A1

3

2

5ex 

A1

so the point of inflexion occurs at

3

2

1

e

5

x 

AG

[3 marks]

continued…

– 12 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

Question 10 continued

(d) attempt to integrate (M1)

d1

ln 5

d

u

ux

x



ln 5 1

d d

x

uu

kx k





(A1)

EITHER

2

u

k



A1

so

3

2

1

d

2

u

uu

kk











A1

OR



2

ln 5

2

x

k



A1

so



3

2

1

e

2

5

e

5

ln 5

d

2

x

kx k











A1

THEN

19

1

24k









5

8k



A1

setting their expression for area equal to

3 M1

5

3

8k



5

24

k 

A1

[7 marks]

Total [16 marks]

– 13 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

11. (a) attempt to find modulus (M1)





23 12r 

A1

attempt to find argument in the correct quadrant (M1)

3

πarctan

3





 





A1

5π

6



A1

5πi 5πi

66

33i12e 23e



  





[5 marks]

(b) attempt to find a root using de Moivre’s theorem M1

15πi

618

12 e

A1

attempt to find further two roots by adding and subtracting

2π

3

to

the argument M1

17πi

618

12 e



A1

117πi

618

12 e A1

Note: Ignore labels for

,uv

and

w

at this stage.

[5 marks]

continued…

– 14 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

Question 11 continued

(c) METHOD 1

attempting to find the total area of (congruent) triangles

UOV, VOW

and

UOW

M1

11

66

12π

Area 3 12 12 sin

23















A1A1

Note: Award A1 for

11

66

12 12







and A1 for

2π

sin

3

.

1

3

33

12

4









(or equivalent) A1

[4 marks]

METHOD 2

22

11 11

2

66 66

2π

UV 12 12 2 12 12 cos

3

 



 

 

(or equivalent) A1

1

6

UV 3 12









(or equivalent) A1

attempting to find the area of

UVW

using

1

Area UV VW sin

2



  

for example M1

11

66

1π

Area 3 12 3 12 sin

23



 





1

3

33

12

4









(or equivalent) A1

[4 marks]

(d)

0uvw 

R1

1

6

7π 7π 5π 5π 17π 17π

12 cos isin cos i sin cos isin 0

18 18 18 18 18 18



 

     

 



 



A1

consideration of real parts M1

1

6

7π 5π 17π

12 cos cos cos 0

18 18 18



  





7π 7π

cos cos

18 18









explicitly stated A1

5π 7π 17π

cos cos cos 0

18 18 18

 

AG

[4 marks]

Total [18 marks]

– 15 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

12. (a) attempting to use the chain rule to find the first derivative M1









sin

cos e

x

fx x



 A1

attempting to use the product rule to find the second derivative M1



sin 2

ecos sin

x

f

xxx





(or equivalent) A1

attempting to find



0f

,



0f



and





0f



M1





01f 

;

  

sin 0

0cos0e 1f





;



sin 0 2

0e cos0sin01f





A1

substitution into the Maclaurin formula

2

( ) (0) (0) (0) ...

2!

x

fx f xf f



  

M1

so the Maclaurin series for



f

x

up to and including the

2

x

term is

2

1

2

x

x

A1

[8 marks]

(b) METHOD 1

attempting to differentiate

()

f

x



M1

  



 

sin 2 sin

cos e cos sin cos e 2sin 1

xx

fx x x x x x





(or equivalent) A2

substituting

0x 

into their



f

x



M1

    

01101010f





so the coefficient of

3

x

in the Maclaurin series for



f

x is zero AG

METHOD 2

substituting

sin

x

into the Maclaurin series for e

x

(M1)

23

sin

sin sin

e 1 sin ...

2! 3!

x

xx

x   

substituting Maclaurin series for

sin

x

M1

23

33

3

sin

... ...

3! 3!

e 1 ... ...

3! 2! 3!

x

xx

x



 







     





A1

coefficient of

3

x

is

11

0

3! 3!

 A1

so the coefficient of

3

x

in the Maclaurin series for



f

x

is zero AG

[4 marks]

continued…

– 16 – SPEC/5/MATAA/HP1/ENG/TZ0/XX/M

Question 12 continued

(c) substituting

3

x

into the Maclaurin series for e

x

M1

 

23

3

33

e 1 3 ...

2! 3!

x

xx

x   

A1

substituting





3

e1

x

 into the Maclaurin series for

arctan

x

M1





35

33

e1 e1

arctan e 1 e 1 ...

35

xx



   

 

3

23

33

3

2! 3!

33

3 ...

2! 3! 3

xx

x

xx

x











   





A1

selecting correct terms from above M1

  

23 3

33 3

3

2! 3! 3

x

xx

x



  





23

99

3

22

x

x 

A1

[6 marks]

(d) METHOD 1

substitution of their series M1

2

0

...

2

lim

9

3 ...

2

x





A1

0

1...

2

lim

9

3...

2

x









1

3



A1

METHOD 2

use of l’Hôpital’s rule M1







sin

3

0

2

3

cos e

lim

3e

1e 1

x





(or equivalent) A1

1

3



A1

[3 marks]

Total [21 marks]

12EP01

Candidate session number

Mathematics: analysis and approaches

Higher level

Paper 2

12 pages

Specimen

2 hours

Instructions to candidates

• Write your session number in the boxes above.

• Do not open this examination paper until instructed to do so.

• A graphic display calculator is required for this paper.

• Section A: answer all questions. Answers must be written within the answer boxes provided.

• Section B: answer all questions in the answer booklet provided. Fill in your session number

on the front of the answer booklet, and attach it to this examination paper and your

cover sheet using the tag provided.

• Unless otherwise stated in the question, all numerical answers should be given exactly or

correct to three signicant gures.

• A clean copy of the mathematics: analysis and approaches formula booklet is required for

this paper.

• The maximum mark for this examination paper is [110 marks].

SPEC/5/MATAA/HP2/ENG/TZ0/XX

Full marks are not necessarily awarded for a correct answer with no working. Answers must be

supported by working and/or explanations. Solutions found from a graphic display calculator should be

supported by suitable working. For example, if graphs are used to nd a solution, you should sketch

these as part of your answer. Where an answer is incorrect, some marks may be given for a correct

method, provided this is shown by written working. You are therefore advised to show all working.

Section A

Answer all questions. Answers must be written within the answer boxes provided. Working may be

continued below the lines, if necessary.

1. [Maximum mark: 6]

The following diagram shows part of a circle with centre O and radius 4 cm .

O

B

A

θ

4 cm

5 cm

Chord AB has a length of 5 cm and AÔB = θ .

(a) Find the value of θ , giving your answer in radians. [3]

(b) Find the area of the shaded region. [3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP02

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 2 –

2. [Maximum mark: 6]

On 1st January 2020, Laurie invests $P in an account that pays a nominal annual interest

rate of 5.5 % , compounded quarterly.

The amount of money in Laurie’s account at the end of each year follows a geometric

sequence with common ratio, r .

(a) Find the value of r , giving your answer to four signicant gures. [3]

Laurie makes no further deposits to or withdrawals from the account.

(b) Find the year in which the amount of money in Laurie’s account will become double the

amount she invested. [3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP03

Turn over

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 3 –

3. [Maximum mark: 6]

A six-sided biased die is weighted in such a way that the probability of obtaining a “six” is

7

10

.

The die is tossed ve times. Find the probability of obtaining

(a) at most three “sixes”. [3]

(b) the third “six” on the fth toss. [3]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP04

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 4 –

4. [Maximum mark: 7]

The following table below shows the marks scored by seven students on two dierent

mathematics tests.

Test 1 (x)

15 23 25 30 34 34 40

Test 2 ( y)

20 26 27 32 35 37 35

Let L

1

be the regression line of x on y . The equation of the line L

1

can be written in the

form x =ay + b .

(a) Find the value of a and the value of b . [2]

Let L

2

be the regression line of y on x . The lines L

1

and L

2

pass through the same point

with coordinates ( p , q) .

(b) Find the value of p and the value of q . [3]

(c) Jennifer was absent for the rst test but scored 29 marks on the second test. Use an

appropriate regression equation to estimate Jennifer’s mark on the rst test. [2]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP05

Turn over

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 5 –

5. [Maximum mark: 7]

The displacement, in centimetres, of a particle from an origin, O, at time t seconds, is given

by s (t) = t

2

cos t+2t sin t , 0 ≤ t ≤ 5 .

(a) Find the maximum distance of the particle from O. [3]

(b) Find the acceleration of the particle at the instant it rst changes direction. [4]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP06

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 6 –

6. [Maximum mark: 6]

In a city, the number of passengers, X , who ride in a taxi has the following probability distribution.

x

1 2 3 4 5

P (X = x)

0.60 0.30 0.03 0.05 0.02

After the opening of a new highway that charges a toll, a taxi company introduces a charge

for passengers who use the highway. The charge is $ 2.40 per taxi plus $ 1.20 per passenger.

Let T represent the amount, in dollars, that is charged by the taxi company per ride.

(a) Find E (T ) . [4]

(b) Given that Var (X ) = 0.8419 , nd Var (T ) . [2]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP07

Turn over

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 7 –

7. [Maximum mark: 5]

Two ships, A and B , are observed from an origin O . Relative to O , their position vectors at

time t hours after midday are given by

r

A

=













+













4

3

5

8

t

r

B

=

−













+













7

3

0

12

t

where distances are measured in kilometres.

Find the minimum distance between the two ships.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP08

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 8 –

8. [Maximum mark: 7]

The complex numbers w and z satisfy the equations

w

z

= 2i

z

∗

- 3w = 5 + 5i .

Find w and z in the form a + bi where a , b ∈  .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP09

Turn over

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 9 –

9. [Maximum mark: 5]

Consider the graphs of

y

x

=

−

2

3

and y = m (x + 3) , m ∈  .

Find the set of values for m such that the two graphs have no intersection points.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12EP10

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 10 –

Do not write solutions on this page.

Section B

Answer all questions in the answer booklet provided. Please start each question on a new page.

10. [Maximum mark: 15]

The length, X mm , of a certain species of seashell is normally distributed with mean 25 and

variance, σ

2

.

The probability that X is less than 24.15 is 0.1446 .

(a) Find P (24.15 < X < 25) . [2]

(b) (i) Find σ , the standard deviation of X .

(ii) Hence, nd the probability that a seashell selected at random has a length

greater than 26 mm . [5]

A random sample of 10 seashells is collected on a beach. Let Y represent the number of

seashells with lengths greater than 26 mm .

(c) Find E (Y ) . [3]

(d) Find the probability that exactly three of these seashells have a length greater

than 26 mm . [2]

A seashell selected at random has a length less than 26 mm .

(e) Find the probability that its length is between 24.15 mm and 25 mm . [3]

12EP11

Turn over

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 11 –

Do not write solutions on this page.

11. [Maximum mark: 21]

A large tank initially contains pure water. Water containing salt begins to ow into the tank

The solution is kept uniform by stirring and leaves the tank through an outlet at its base.

Let x grams represent the amount of salt in the tank and let t minutes represent the time

since the salt water began owing into the tank.

The rate of change of the amount of salt in the tank,

d

x

t

, is described by the dierential

equation

d

e

x

t

x

t

=−

+

−

10

1

4

.

(a) Show that t + 1 is an integrating factor for this dierential equation. [2]

(b) Hence, by solving this dierential equation, show that

xt

t

()

=

−+

+

−

200 40 5

1

4

e

. [8]

(c) Sketch the graph of x versus t for 0 ≤ t ≤ 60 and hence nd the maximum amount of

salt in the tank and the value of t at which this occurs. [5]

(d) Find the value of t at which the amount of salt in the tank is decreasing most rapidly. [2]

The rate of change of the amount of salt leaving the tank is equal to

x

t + 1

.

(e) Find the amount of salt that left the tank during the rst 60 minutes. [4]

12. [Maximum mark: 19]

(a) Show that

cot

tan

2

1

2

θ

=

−

. [1]

(b) Verify that x = tan θ and x =- cot θ satisfy the equation x

2

+ (2 cot 2θ) x - 1= 0 . [7]

(c) Hence, or otherwise, show that the exact value of

ta

n

�

12

23

= −

. [5]

(d) Using the results from parts (b) and (c) nd the exact value of

tanc

ot

� �

24 24

−

.

Give your answer in the form

ab+ 3

where a , b ∈  . [6]

12EP12

SPEC/5/MATAA/HP2/ENG/TZ0/XX– 12 –

SPEC/5/MATAA/HP2/ENG/TZ0/XX/M

16 pages

Markscheme

Specimen paper

Mathematics:

analysis and approaches

Higher level

Paper 2

– 2 – SPEC/5/MATAA/HP2/ENG/TZ0/XX/M

Instructions to Examiners

Abbreviations

M Marks awarded for attempting to use a correct Method.

A Marks awarded for an Answer or for Accuracy; often dependent on preceding M marks.

R Marks awarded for clear Reasoning.

AG Answer given in the question and so no marks are awarded.

Using the markscheme

1 General

Award marks using the annotations as noted in the markscheme eg M1, A2.

2 Method and Answer/Accuracy marks

 Do not automatically award full marks for a correct answer; all working must be checked, and

marks awarded according to the markscheme.

 It is generally not possible to award M0 followed by A1, as A mark(s) depend on the preceding

M mark(s), if any.

 Where M and A marks are noted on the same line, e.g. M1A1, this usually means M1 for an

attempt to use an appropriate method (e.g. substitution into a formula) and A1 for using the

correct values.

 Where there are two or more A marks on the same line, they may be awarded independently;

so if the first value is incorrect, but the next two are correct, award A0A1A1.

 Where the markscheme specifies M2, A2

, etc., do not split the marks, unless there is a note.

 Once a correct answer to a question or part-question is seen, ignore further correct working.

However, if further working indicates a lack of mathematical understanding do not award the final

A1. An exception to this may be in numerical answers, where a correct exact value is followed by

an incorrect decimal. However, if the incorrect decimal is carried through to a subsequent part,

and correct working shown, award FT marks as appropriate but do not award the final A1 in that

part.

Examples

Correct answer seen Further working seen Action

1.

82

5.65685...

(incorrect decimal value)

Award the final A1

(ignore the further working)

2.

1

sin 4

4

x

sin

x

Do not award the final A1

3.

log logab

log ( )ab

Do not award the final A1

– 3 – SPEC/5/MATAA/HP2/ENG/TZ0/XX/M

3 Implied marks

Implied marks appear in brackets e.g. (M1), and can only be awarded if correct work is seen or if

implied in subsequent working.

 Normally the correct work is seen or implied in the next line.

 Marks without brackets can only be awarded for work that is seen.

4 Follow through marks (only applied after an error is made)

Follow through (FT) marks are awarded where an incorrect answer from one part of a question is

used correctly in subsequent part(s) or subpart(s). Usually, to award FT marks, there must be

working present and not just a final answer based on an incorrect answer to a previous part.

However, if the only marks awarded in a subpart are for the answer (i.e. there is no working

expected), then FT marks should be awarded if appropriate.

 Within a question part, once an error is made, no further A marks can be awarded for work

which uses the error, but M marks may be awarded if appropriate.

 If the question becomes much simpler because of an error then use discretion to award fewer

FT marks.

 If the error leads to an inappropriate value (e.g. probability greater than 1, use of

1r  for the

sum of an infinite GP,

sin 1.5





, non integer value where integer required), do not award the

mark(s) for the final answer(s).

 The markscheme may use the word “their” in a description, to indicate that candidates may be

using an incorrect value.

 Exceptions to this rule will be explicitly noted on the markscheme.

 If a candidate makes an error in one part, but gets the correct answer(s) to subsequent part(s),

award marks as appropriate, unless the question says hence. It is often possible to use a

different approach in subsequent parts that does not depend on the answer to previous parts.

5 Mis-read

If a candidate incorrectly copies information from the question, this is a mis-read (MR). Apply a MR

penalty of 1 mark to that question

 If the question becomes much simpler because of the MR, then use discretion to award

fewer marks.

 If the MR leads to an inappropriate value (e.g. probability greater than

1,

sin 1.5





, non-integer

value where integer required), do not award the mark(s) for the final answer(s).

 Miscopying of candidates’ own work does not constitute a misread, it is an error.

 The MR penalty can only be applied when work is seen. For calculator questions with no

working and incorrect answers, examiners should not infer that values were read incorrectly.

– 4 – SPEC/5/MATAA/HP2/ENG/TZ0/XX/M

6 Alternative methods

 Alternative methods for complete questions are indicated by METHOD 1,

METHOD 2, etc.

 Alternative solutions for part-questions are indicated by EITHER . . . OR.

7 Alternative forms

Unless the question specifies otherwise, accept equivalent forms.

 As this is an international examination, accept all alternative forms of notation.

 In the markscheme, equivalent numerical and algebraic forms will generally be written in

brackets immediately following the answer.

 In the markscheme, simplified answers, (which candidates often do not write in examinations),

will generally appear in brackets. Marks should be awarded for either the form preceding the

bracket or the form in brackets (if it is seen).

8 Accuracy of Answers

If the level of accuracy is specified in the question, a mark will be linked to giving the answer to the

required accuracy. There are two types of accuracy errors, and the final answer mark should not be

awarded if these errors occur.

 Rounding errors: only applies to final answers not to intermediate steps.

 Level of accuracy: when this is not specified in the question the general rule applies to final

answers: unless otherwise stated in the question all numerical answers must be given exactly or

correct to three significant figures.

9 Calculators

A GDC is required for paper 2, but calculators with symbolic manipulation features/ CAS functionality

are not allowed.

Calculator notation

The subject guide says:

Students must always use correct mathematical notation, not calculator notation.

Do not accept final answers written using calculator notation. However, do not penalize the use of

calculator notation in the working.

Candidates will sometimes use methods other than those in the markscheme. Unless the question

specifies a method, other correct methods should be marked in line with the markscheme

– 5 – SPEC/5/MATAA/HP2/ENG/TZ0/XX/M

Section A