Cambridge Assessment International Education Cambridge Ordinary Level

[Pages:10]Cambridge Assessment International Education Cambridge Ordinary Level

ADDITIONAL MATHEMATICS Paper 2 MARK SCHEME Maximum Mark: 80

4037/22 October/November 2019

Published

This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners' meeting before marking began, which would have considered the acceptability of alternative answers.

Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers.

Cambridge International will not enter into discussions about these mark schemes.

Cambridge International is publishing the mark schemes for the October/November 2019 series for most Cambridge IGCSETM, Cambridge International A and AS Level components and some Cambridge O Level components.

? UCLES 2019

This document consists of 10 printed pages.

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4037/22

Cambridge O Level ? Mark Scheme PUBLISHED

October/November 2019

Generic Marking Principles

These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles.

GENERIC MARKING PRINCIPLE 1:

Marks must be awarded in line with:

? the specific content of the mark scheme or the generic level descriptors for the question ? the specific skills defined in the mark scheme or in the generic level descriptors for the question ? the standard of response required by a candidate as exemplified by the standardisation scripts.

GENERIC MARKING PRINCIPLE 2:

Marks awarded are always whole marks (not half marks, or other fractions).

GENERIC MARKING PRINCIPLE 3:

Marks must be awarded positively:

? marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate

? marks are awarded when candidates clearly demonstrate what they know and can do ? marks are not deducted for errors ? marks are not deducted for omissions ? answers should only be judged on the quality of spelling, punctuation and grammar when these

features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous.

GENERIC MARKING PRINCIPLE 4:

Rules must be applied consistently e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors.

GENERIC MARKING PRINCIPLE 5:

Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen).

GENERIC MARKING PRINCIPLE 6:

Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

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Cambridge O Level ? Mark Scheme PUBLISHED

October/November 2019

MARK SCHEME NOTES

The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes.

Types of mark

M Method marks, awarded for a valid method applied to the problem.

A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied.

B Mark for a correct result or statement independent of Method marks.

When a part of a question has two or more `method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation `dep' is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme.

Abbreviations

awrt cao dep FT isw nfww oe rot SC soi

answers which round to correct answer only dependent follow through after error ignore subsequent working not from wrong working or equivalent rounded or truncated Special Case seen or implied

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4037/22

Question Answer 1 A

Cambridge O Level ? Mark Scheme PUBLISHED

Marks B1

B

October/November 2019

Guidance

B1

B 1

2

dy dx

=

6

cos

3x

-3sin 3x

d2 y dx2

=

-18sin

3x

-

9 cos 3x

Insert and collect like terms

3(i) 3(ii) 3(iii)

k = ?15 14P5 or 14 ? 13 ? 12 ? 11 ? 10 240 240 3P1 ? 5P2 ? 6P2 or 3 ? (5 ? 4) ? (6 ? 5)

= 1800 6P2 ? 8P3 or (6 ? 5) ? (8 ? 7 ? 6)

= 10 080

B1

B1

B1

FT Correct derivative of their

dy dx

M1

Must insert for y, their

dy dx

and

d2 y dx 2

correctly resulting in 6 terms.

A1 Allow ?15sin3x seen nfww

M1

A1 cao

M1 Two of the three elements multiplied by ...

A1

M1 One element multiplied by ... Clear intention to multiply

A1

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4037/22

Cambridge O Level ? Mark Scheme PUBLISHED

October/November 2019

Question 4

Answer kx + 3 = x2 + 5x + 12 x2 + (5 ? k)x + 9(= 0) Use discriminant of their quadratic. (5 ? k)2 ? 36 oe k = ?1 and 11 ?1 < k < 11

OR 2x + 5 ~ k

y = (2x + 5) x + 3

2x2 + 5x + 3 = x2 + 5x +12 x2 = 9 x = ? 3 k = 11 or k = ?1 ?1 < k < 11

5(i)

dy dx

=

(

-2k

x +1)3

( x +1)3

Gradient of normal = 2k

or Gradient of tangent = -3

8 2k

=1 3

or

2 k 8

= -3

k = 12

5(ii)

x

=

2

dy dx

=-8 9

or their

-2k 27

y = 4 or their k

3

9

Marks

Guidance

M1 Equate and attempt to simplify to all terms on one side.

M1 dep

A1 Unsimplified

A1 Both boundary values A1 Must be in terms of .

M1 Connect gradients of line and curve M1 Eliminate and y.

A1 A1 A1 B1 oe Unsimplified

M1 Gradient of normal = - 1

gradient of tangent

M1

1

Equate gradient of normal to at

3

x = 1

or equate gradient of tangent to ?3 at x =1

A1

B1 FT

B1 FT

y

-

4 3

x-2

=-8 9

or

y

=-8x+ 9

28 9

B1 isw

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Cambridge O Level ? Mark Scheme PUBLISHED

October/November 2019

Question Answer

6(i)

sin x

cos x +

1 cos x

+ 1

1 cos x

+1

sin x cos x

1

sin x + cos

x

+

1

+ cos sin x

x

sin2 x +1+ 2cos x + cos2 x

(1+ cos x)sin x

2(1+ cos x) (1+ cos x)sin x

All correct AG

OR

tan2 x + (sec x + 1)2 tan x (sec x +1)

=

2sec2 x + 2sec x

tan x(sec x +1)

2sec x tan x

2 cos

x

?

cos sin

x x

All correct AG

6(ii)

3sin2x + sinx ? 2 = 0 oe

(3sinx ? 2)(sinx + 1) = 0

41.8? awrt

138.2? awrt

Marks

Guidance

M1

Use

tan x =

sin x cos x

and

sec x =

1 cos x

throughout

M1 dep Multiply by cos

M1 dep Add their fractions correctly and expand (1 + cosx)2 correctly

M1 dep Use sin2x + cos2x = 1 and take out a factor of 2.

A1 Do not award if brackets missing at any point or missing more than twice or misplaced. Do not credit mixed variables.

M1 Add fractions

M1 dep Expand brackets correctly and use 1 + tan2x = sec2x

M1 dep Cancel sec + 1

M1 dep

Use

tan

x

=

sin cos

x x

and

sec

x

=

1 cos

x

oe

A1 Do not award if brackets missing at any point or missing more than twice or misplaced. Do not credit mixed variables.

B1 Obtain three term quadratic.

M1 Solve three term quadratic

A1

A1 Mark final answers This mark is not awarded if there are more solutions in the range.

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Cambridge O Level ? Mark Scheme PUBLISHED

October/November 2019

Question Answer

7(a)

2 ? 4 ? p = 40 p = 5

(x ? 2)(x ? 4)(x ? p) = 0

a = ?11

b = 38

OR

2 ? 4 ? p = 40 p = 5

Obtain equations 4a + 2b = 32 16a + 4b = ?24

and attempt to solve

a = ?11 b = 38

7(b)

Find x = ?1

(x + 1)(x2 ? 6x ? 40) (= 0) or (x + 4)(x2 ? 9x ? 10)(= 0) or (x ? 10)(x2 + 5x + 4)(= 0)

(x + 1)(x + 4)(x ? 10) (= 0)

x = ?1, ?4, 10

OR

Uses factor theorem to find a root (?1)3 ? 5(?12) ? 46(?1) ? 40 or ?1 ? 5 + 46 ? 40 = 0 x = ?1

Uses factor theorem to attempt to find further roots

(?4)3 ? 5(?4)2 ? 46(?4) ? 40 or ? 64 ? 80 + 184 ? 40 = 0 x = ?4

(10)3 ? 5(10)2 ? 46(10) ? 40 or 1000 ? 500 ? 460 ? 40 = 0 x = 10

Marks

Guidance

B1 May be obtained later.

M1 Factorise cubic

A1 Expand and identify

A1

B1 May be obtained later. M1

A1 A1 M1 Trial value/s and finds a root or shows

that (x + 1) or (x + 4) or (x ? 10) divides into x3 ? 5x2 ? 46x ? 40. A1 Factorise to give linear and quadratic factor

M1 Solve the quadratic to give 2 roots A1

M1 This may be awarded for x = ?4 or x = 10.

M1 At least two more trials.

A1

A1

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Cambridge O Level ? Mark Scheme PUBLISHED

October/November 2019

Question Answer

8(i)

52 + 122 = 13

v A

= - 5 i - 6j or 2

1 (-5i -12j)

2

8(ii)

vB = 1212 + (-9)2

15

8(iii)

rA

=

20

-7

+

t

-2.5 -6

or

rA = (20 - 2.5t )i + (-7 - 6t ) j

rB

=

-67

11

+

t

12 -9

or

rB = (-67 +12t )i + (11- 9t ) j

8(iv) 20 ? 2.5t = ?67 + 12t or ?7 ? 6t = 11 ? 9t

t = 6

r

=

5 -43

only

or r = 5i ? 43j

9(i)

Midpoint (1, 2)

Gradient of AB= - 3 4

Gradient of PM

=

-1 their gradient of

AB

=

4 3

Equation PM

y x

- -

2 1

=

4 3

y= 4x+2 33

9(ii)

s= 4r+ 2

33

Marks M1 A1

Guidance

M1 Use Pythagoras

A1 Do not allow ? 15. Mark final answer. B1 FT on their vA only if of the form

k(?5i ? 12j) where k 1 or 0.

B1

M1 Equate or coordinates. Must have two terms in both coordinates.

A1 nfww Ignore other value of t. A1 A0 if further value of found.

B1 May be seen on diagram B1

M1 Use m1 ? m2 = ?1

M1 dep Attempt to find equation of line with their midpoint and their gradient of PM. If y = mx + c used c must be found.

A1

B1 FT

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