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Mark Scheme (Results) Summer 2013

GCE Core Mathematics 3 (6665/01)

Edexcel and BTEC Qualifications

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Summer 2013 Publications Code UA035670 All the material in this publication is copyright ? Pearson Education Ltd 2013

General Marking Guidance

? ?

All candidates must receive the same treatment. Examiners must mark the first candidate in exactly the same way as they mark the last. Mark schemes should be applied positively. Candidates must be rewarded for what they have shown they can do rather than penalised for omissions. Examiners should mark according to the mark scheme not according to their perception of where the grade boundaries may lie. There is no ceiling on achievement. All marks on the mark scheme should be used appropriately. All the marks on the mark scheme are designed to be awarded. Examiners should always award full marks if deserved, i.e. if the answer matches the mark scheme. Examiners should also be prepared to award zero marks if the candidate’s response is not worthy of credit according to the mark scheme. Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and exemplification may be limited. Crossed out work should be marked UNLESS the candidate has replaced it with an alternative response.

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?

?

EDEXCEL GCE MATHEMATICS General Instructions for Marking 1. The total number of marks for the paper is 75. 2. The Edexcel Mathematics mark schemes use the following types of marks: ? M marks: method marks are awarded for ‘knowing a method and attempting to apply it’, unless otherwise indicated. ? A marks: accuracy marks can only be awarded if the relevant method (M) marks have been earned. ? B marks are unconditional accuracy marks (independent of M marks) ? Marks should not be subdivided. 3. Abbreviations These are some of the traditional marking abbreviations that will appear in the mark schemes: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? bod – benefit of doubt ft – follow through the symbol will be used for correct ft cao – correct answer only cso - correct solution only. There must be no errors in this part of the question to obtain this mark isw – ignore subsequent working awrt – answers which round to SC: special case oe – or equivalent (and appropriate) dep – dependent indep – independent dp decimal places sf significant figures ? The answer is printed on the paper The second mark is dependent on gaining the first mark

4. All A marks are ‘correct answer only’ (cao.), unless shown, for example, as A1 ft to indicate that previous wrong working is to be followed through. After a misread however, the subsequent A marks affected are treated as A ft, but manifestly absurd answers should never be awarded A marks.

5. For misreading which does not alter the character of a question or materially simplify

it, deduct two from any A or B marks gained, in that part of the question affected. 6. If a candidate makes more than one attempt at any question: ? If all but one attempt is crossed out, mark the attempt which is NOT crossed out. ? If either all attempts are crossed out or none are crossed out, mark all the attempts and score the highest single attempt. 7. Ignore wrong working or incorrect statements following a correct answer. 8. In some instances, the mark distributions (e.g. M1, B1 and A1) printed on the candidate’s response may differ from the final mark scheme.?

General Principles for Core Mathematics Marking (But note that specific mark schemes may sometimes override these general principles). Method mark for solving 3 term quadratic: 1. Factorisation

( x 2 + bx + c) = ( x + p)( x + q), where pq = c , leading to x = (ax 2 + bx + c) = (mx + p)(nx + q), where pq = c and mn = a , leading to x =

2. Formula Attempt to use correct formula (with values for a, b and c). 3. Completing the square

Solving x 2 + bx + c = 0 :

b? ? ? x ± ? ± q ± c, 2? ?

2

q ≠ 0, leading to x =...

Method marks for differentiation and integration: 1. Differentiation Power of at least one term decreased by 1. ( x n → x n ?1 ) 2. Integration Power of at least one term increased by 1. ( x n → x n +1 )

Use of a formula Where a method involves using a formula that has been learnt, the advice given in recent examiners’ reports is that the formula should be quoted first. Normal marking procedure is as follows: Method mark for quoting a correct formula and attempting to use it, even if there are mistakes in the substitution of values. Where the formula is not quoted, the method mark can be gained by implication from correct working with values, but may be lost if there is any mistake in the working. Exact answers Examiners’ reports have emphasised that where, for example, an exact answer is asked for, or working with surds is clearly required, marks will normally be lost if the candidate resorts to using rounded decimals. Answers without working The rubric says that these may not gain full credit. Individual mark schemes will give details of what happens in particular cases. General policy is that if it could be done “in your head”, detailed working would not be required.

Question Number

Scheme

Marks

1

3 x2 ? 2 x + 7 x2 (+0 x) ? 4 3 x 4 ? 2 x 3 ? 5 x2 + (0 x) ? 4 3 x 4 + 0 x 3 ? 12 x2 ? 2 x 3 + 7 x2 + 0 x ?2 x 3 + 0 x 2 + 8 x 7 x2 ? 8 x ? 4 7 x2 + 0 x ? 28 ? 8 x + 24

By Division

a=3

3 x ? 2 x....... x (+0 x) ? 4 3 x 4 ? 2 x 3 ? 5 x2 + (0 x) ? 4

Long division as far as

2 2

B1

3 x 4 + 0 x 3 ? 12 x2 ? 2 x + ................ ?2 x 3 + .................

Two of b = ?2 All four of b = ?2

3

M1

c = 7 d = ?8 e = 24 c = 7 d = ?8 e = 24

A1 A1

(4 marks)

B1 M1

Notes for Question 1 Stating a = 3 . This can also be scored by the coefficient of x 2 ?in 3 x2 ? 2 x + 7 ?

Using long division by x 2 ? 4 and getting as far as the ‘x’ term. The coefficients need not be correct. Award if you see the whole number part as ...x 2 + ...x following some working. You may also see this in a table/ grid. Long division by ( x + 2) will not score anything until ( x ? 2) has been divided into the new quotient. It is

A1 A1

very unlikely to score full marks and the mark scheme can be applied. Achieving two of b = ?2 c = 7 d = ?8 e = 24 . The answers may be embedded within the division sum and can be implied. Achieving all of b = ?2 c = 7 d = ?8 and e = 24

Accept a correct long division for 3 out of the 4 marks scoring B1M1A1A0 Need to see a=…, b=…, or the values embedded in the rhs for all 4 marks

Question Number Alt 1 By Multiplicat ion

Scheme

Marks

* 3 x 4 ? 2 x 3 ? 5 x2 ? 4 ≡ (ax2 + bx + c)( x2 ? 4) + dx + e Compares the x 4 terms a = 3 Compares coefficients to obtain a numerical value of one further constant B1 M1

?2 = b, ? 5 = ?4a + c ? c = ..,

Two of b = ?2 All four of b = ?2

c = 7 d = ?8 e = 24 c = 7 d = ?8 e = 24

A1 A1

(4 marks)

B1 M1

Notes for Question 2 Stating a = 3 . This can also be scored for writing 3 x 4 = ax 4

Multiply out expression given to get *. Condone slips only on signs of either expression. Then compare the coefficient of any term (other than x 4 ) to obtain a numerical value of one further constant. In reality this means a valid attempt at either b or c The method may be implied by a correct additional constant to a.

A1 A1

Achieving two of b = ?2 c = 7 d = ?8 e = 24 Achieving all of b = ?2 c = 7 d = ?8 and e = 24

? ? ? ? ? ? ? ? ?

Question Number

Scheme

Marks

2(i)

ln graph crossing x axis at (1,0) B1 and asymptote at x=0

Shape including cusp B1ft

2(ii)

Touches or crosses the x axis at (1,0) B1ft Asymptote given as x=0 B1

Shape B1 Crosses at (5, 0) B1ft

2(iii)

Asymptote given as x=4 B1

(7 marks)

Notes for Question 2

(i) B1 Correct shape, correct position and passing through (1, 0). Graph must ‘start’ to the rhs of the y - axis in quadrant 4 with a gradient that is large. The gradient then decreases as it moves through (1, 0) into quadrant 1. There must not be an obvious maximum point but condone ‘slips’. Condone the point marked (0,1) on the correct axis. See practice and qualification for clarification. Do not with hold this mark if (x=0) the asymptote is incorrect or not given.

(ii) B1ft

Correct shape including the cusp wholly contained in quadrant 1. The shape to the rhs of the cusp should have a decreasing gradient and must not have an obvious maximum.. The shape to the lhs of the cusp should not bend backwards past (1,0) Tolerate a ‘linear’ looking section here but not one with incorrect curvature (See examples sheet (ii) number 3. For further clarification see practice and qualification items. Follow through on an incorrect sketch in part (i) as long as it was above and below the x axis.

B1ft The curve touches or crosses the x axis at (1, 0). Allow for the curve passing through a point marked ‘1’ on the x axis. Condone the point marked on the correct axis as (0, 1) Follow through on an incorrect intersection in part (i). B1 Award for the asymptote to the curve given/ marked as x = 0. Do not allow for it given/ marked as ‘the y axis’. There must be a graph for this mark to be awarded, and there must be an asymptote on the graph at x = 0. Accept if x=0 is drawn separately to the y axis.

(iii) B1 Correct shape. The gradient should always be negative and becoming less steep. It must be approximately infinite at the lh end and not have an obvious minimum. The lh end must not bend ‘forwards’ to make a C shape. The position is not important for this mark. See practice and qualification for clarification. The graph crosses (or touches) the x axis at (5, 0). Allow for the curve passing through a point marked ‘5’ on the x axis. Condone the point marked on the correct axis as (0, 5) Follow through on an incorrect intersection in part (i). Allow for ( (i ) + 4,0 )

B1ft

B1 The asymptote is given/ marked as x = 4. There must be a graph for this to be awarded and there must be an asymptote on the graph (in the correct place to the rhs of the y axis).

If the graphs are not labelled as (i), (ii) and (iii) mark them in the order that they are given.

?

Examples?of?graphs?in?number?2? Part?(i)? Condoned????????????????????????????????????????????????????????????????????Not?condoned????????????????????????????????????????????????Condone? ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????Y?axis?? ? ? ? ??????????????????(0,1)????????????????????????????????????????????????????????1???????????????????????????????????????????????????????????????????????????????????1? ? ? Part?(ii)? ????????????1,1,1????????????????????????????????????????????????????????????????0,1,1,?????????????????????????????????????????????????????????????????0,1,0,? ? ?

x=0

x=0

x=0

????????????????1???????????????????????????????????????????????????????????????????????1??????????????????????????????????????????????????????????????????????????1?????? ? Example?of?follow?through?in?part?(ii)?and?(iii)? ???(i)???B0? ? ? ? ? ?????????????0????????????????????????????????????????????????????????????????????0??????????????????????????????????????????????????????????????????????????????????????????4? ? ? ? ? ? ? ? (ii)?B1ftB1ftB0???????????????????????????????????????(iii)???????????B0B1ftB0? ?

Question Number 3(a) ? ? ?

Scheme

Marks

2cos x cos 50 ? 2sin x sin 50 = sin x cos 40 + cos x sin 40

sin x (cos 40 + 2 sin 50) = cos x (2 cos 50 ? sin 40) ÷ cos x ? tan x (cos 40 + 2sin 50) = 2 cos 50 ? sin 40

M1

M1 A1

tan x =

2 cos 50 ? sin 40 , cos 40 + 2sin 50

(or numerical answer awrt 0.28) cao

? (b) ?

o States or uses cos50 = sin40 and cos40 =sin50 and so tan x o = 1 3 tan 40 *

A1 *

(4)

Deduces

tan 2θ = 1 3 tan 40

M1 so

2θ = 15.6

θ = awrt 7.8(1) One answer

A1 M1

Also 2θ = 195.6, 375.6 , 555.6 ? θ = ..

?

θ = awrt 7.8 , 97.8, 187.8, 277.8

All 4 answers

A1

(4)

? ? Alt 1 3(a) ? ? ???

[8 marks ]

2cos x cos 50 ? 2sin x sin 50 = sin x cos 40 + cos x sin 40 2cos x sin 40 ? 2sin x cos 40 = sin x cos 40 + cos x sin 40 ÷ cos x ? 2sin 40 ? 2 tan x cos 40 = tan x cos 40 + sin 40

sin 40 1 ( or numerical answer awrt 0.28) , ? tan x = tan 40 3cos 40 3

2cos( x + 50) = sin( x + 40) ? 2sin(40 ? x ) = sin( x + 40)

M1

M1

?

tan x =

Alt 2 3(a)

A1,A1

? ?

2cos x sin 40 ? 2sin x cos 40 = sin x cos 40 + cos x sin 40 ÷ cos x ? 2sin 40 ? 2 tan x cos 40 = tan x cos 40 + sin 40

sin 40 1 ( or numerical answer awrt 0.28 ) , ? tan x = tan 40 3cos 40 3

M1 M1

?

tan x =

A1,A1

Notes for Question 3

(a) M1 Expand both expressions using cos( x + 50) = cos x cos 50 ? sin x sin 50 and

sin( x + 40) = sin x cos 40 + cos x sin 40 . Condone a missing bracket on the lhs.

The terms of the expansions must be correct as these are given identities. You may condone a sign error on one of the expressions. Allow if written separately and not in a connected equation. M1 Divide by cos x to reach an equation in tan x . Below is an example of M1M1 with incorrect sign on left hand side

2cos x cos50 + 2sin x sin 50 = sin x cos 40 + cos x sin 40 ? 2cos50 + 2 tan x sin 50 = tan x cos 40 + sin 40

This is independent of the first mark.

A1

2 cos 50 ? sin 40 cos 40 + 2sin 50 Accept for this mark tan x = awrt 0.28... as long as M1M1 has been achieved. tan x =

States or uses cos50=sin40 and cos40=sin50 leading to showing

A1*

tan x =

2 cos 50 ? sin 40 sin 40 1 = = tan 40 cos 40 + 2 sin 50 3cos 40 3

This is a given answer and all steps above must be shown. The line above is acceptable. Do not allow from tan x = awrt 0.28... (b) M1 A1 For linking part (a) with (b). Award for writing tan 2θ = 1 3 tan 40 Solves to find one solution of θ which is usually (awrt) 7.8

M1 Uses the correct method to find at least another value of θ . It must be a full method but can be implied by any correct answer. Accept θ = A1

180 + theirα 360 + theirα 540 + theirα , (or ) , (or ) 2 2 2

Obtains all four answers awrt 1dp. θ = 7.8, 97.8, 187.8, 277.8 . Ignore any extra solutions outside the range. Withhold this mark for extras inside the range. Condone a different variable. Accept x= 7.8, 97.8, 187.8, 277.8

Answers fully given in radians, loses the first A mark. Acceptable answers in rads are awrt 0.136, 1.71, 3.28, 4.85 Mixed units can only score the first M 1

Question Number 4(a) ? ?

Scheme

Marks

f ′ ( x) = 50 x 2 e 2 x + 50 xe 2 x ?????????????????oe.

Puts f ′ ( x ) = 0 to give x = -1 and x = 0 or one coordinate Obtains (0,-16) and (-1, 25e-2 -16) CSO

M1A1 dM1A1 A1

(5)

(b)? ? (c) ?

Puts 25 x 2 e 2 x ? 16 = 0 ? x 2 =

16 ?2 x 4 e ? x = ± e? x 25 5

B1*

(1)

Subs x0 = 0.5 into x =

4 ?x e ? x1 = awrt 0.485 5

M1A1 A1

(3)

? x2 = awrt 0.492, x3 = awrt 0.489

?????????????? ????????????????????????????? α = 0.49 ?? ?? f (0.485) = ?0.487, f (0.495) = (+)0.485 , sign change and deduction

(d)

B1 B1

(2) (11 marks)

Notes for Question 4

No marks can be scored in part (a) unless you see differentiation as required by the question. (a)

M1

Uses vu '+ uv ' . If the rule is quoted it must be correct. It can be implied by their u = .., v = ..., u ' = ..., v ' = ... followed by their vu '+ uv ' If the rule is not quoted nor implied only accept answers of the form Ax 2 e2 x + Bxe2 x

A1

?

f ′ ( x) = 50 x 2 e2 x + 50 xe2 x .??

????Allow un simplified forms such as? f ′ ( x) = 25 x 2 × 2e 2 x + 50 x × e2 x

dM1 A1

Sets f ′ ( x ) = 0 , factorises out/ or cancels the e 2 x leading to at least one solution of x This is dependent upon the first M1 being scored. Both x = -1 and x = 0 or one complete coordinate . Accept

(0,-16) and (-1, 25e-2 -16) or

( ?1, awrt ? 12.6 )

A1 CSO. Obtains both solutions from differentiation. Coordinates can be given in any way.

x = - 1, 0

y=

25 ? 16, ? 16 or linked together by coordinate pairs (0,-16) and (-1, 25e-2 -16) but 2 e

the ‘pairs’ must be correct and exact.

Notes for Question 4 Continued

(b) B1 This is a show that question and all elements must be seen Candidates must 1) State that f(x)=0 or writes 25 x 2 e 2 x ? 16 = 0 or 25 x 2 e 2 x = 16 2) Show at least one intermediate (correct) line with either 16 16 ?2 x e oe x 2 or x the subject. Eg x 2 = e ?2 x , x = 25 25 or square rooting 25 x 2e 2 x = 16 ? 5 xe x = ±4 or factorising by DOTS to give (5 xe x + 4)(5 xe x ? 4) = 0 3) Show the given answer x = ±

4 ?x e . 5

Condone the minus sign just appearing on the final line. A ‘reverse’ proof is acceptable as long as there is a statement that f(x)=0 (c) M1 Substitutes x0 = 0.5 into x =

A1 A1

4 ?x e ? x1 = .... 5 4 This can be implied by x1 = e ?0.5 , or awrt 0.49 5 x1 = awrt 0.485 3dp. Mark as the first value given. Don’t be concerned by the subscript.

x2 = awrt 0.492, x3 = awrt 0.489 3dp. Mark as the second and third values given.

States α = 0.49 Justifies by either calculating correctly f (0.485) and f (0.495) to awrt 1sf or 1dp,

(d)

B1 B1

f (0.485) = ?0.5, f (0.495) = (+) 0.5 rounded f (0.485) = ?0.4, f (0.495) = (+) 0.4 truncated

giving a reason – accept change of sign, >0 <0 or f (0.485) × f (0.495) < 0 and giving a minimal conclusion. Eg. Accept hence root or α = 0.49 A smaller interval containing the root may be used, eg f (0.49) and f (0.495). Root = 0.49007

or by stating that the iteration is oscillating or by calculating by continued iteration to at least the value of x4= awrt 0.491 and stating (or seeing each value round to) 0.49

Question Number 5(a)

Scheme

Marks

dx = 2 × 3sec 3 y sec 3 y tan 3 y = ( 6sec 2 3 y tan 3 y ) dy

Uses

? 6sin 3 y ? ? oe ? cos3 3 y ? ?

M1A1 M1

(2)

(b)

dy 1 = to obtain d x dx

dy

dy 1 = 2 dx 6sec 3 y tan 3 y

tan 2 3 y = sec 2 3 y ? 1 = x ? 1 Uses sec 2 3 y = x and tan 2 3 y = sec 2 3 y ? 1 = x ? 1 to get dy 1 = dx 6 x( x ? 1) 1 2

?1

B1

dy dx or in just x. dx dy

CSO

M1

?

A1*

(4)

(c)

d 2 y 0 ? [6( x ? 1) 2 + 3 x( x ? 1) 2 ] ????????????????????????? 2 = dx 36 x 2 ( x ? 1)

????????????????????????

1

M1A1

? ? ? Alt 1 to 5(a) Alt 2 to 5 (a) Alt 1

d2y 6 ? 9x = 3 2 dx 36 x 2 ( x ? 1) 2

=

2 ? 3x 12 x 2 ( x ? 1) 2

3

dM1A1

(4) (10 marks)

x = (cos3 y )?2 ?

dx = ?2(cos3 y )?3 × ?3sin 3 y dy

M1A1

x = sec3 y × sec3 y ?

dx = sec3 y × 3sec3 y tan 3 y + sec3 y × 3sec3 y tan 3 y dy

M1A1

????????????????????? To 5 (c) ? ? ? ??????????????????

d 2 y 1 ?1 1 ?3 ?1 = 6 [ x (? 2 )( x ? 1) 2 + (?1) x ?2 ( x ? 1) 2 ] 2 dx

?2 1 2 =1 6 x ( x ? 1) [ x ( ? 2 ) + ( ?1)( x ? 1)] ?3 ?3

M1A1

dM1 oe A1

(4)

1 ?????????????????????????????? = 12 x ?2 ( x ? 1) 2 [2 ? 3 x]

Notes for Question 5

(a) M1 Uses the chain rule to get A sec3 y sec 3 y tan 3 y = A sec2 3 y tan 3 y . There is no need to get the lhs of the expression. Alternatively could use the chain rule on (cos3 y ) ?2 ? A(cos3 y ) ?3 sin 3 y or the quotient rule on A1

(

)

1 ± A cos3 y sin 3 y ? 2 (cos3 y ) (cos3 y )4

dx = 2 × 3sec3 y sec3 y tan 3 y or equivalent.?There is no need to simplify the rhs but dy

both sides must be correct.

(b) M1 Uses

dy 1 dy dx = to get an expression for . Follow through on their dx dy d x dx

dy

Allow slips on the coefficient but not trig expression. B1 Writes tan 2 3 y = sec 2 3 y ? 1 or an equivalent such as tan 3 y = sec 2 3 y ? 1 and

uses x = sec 2 3 y to obtain either tan 2 3 y = x ? 1 or tan 3 y = ( x ? 1) 2 All elements must be present.

x

1

Accept

3y

1

x ?1

cos3 y =

1 ? tan 3 y = x ? 1 x x ?1 x

If the differential was in terms of sin 3 y,cos3 y it is awarded for sin 3 y =

M1

Uses sec 2 3 y = x and tan 2 3 y = sec 2 3 y ? 1 = x ? 1 or equivalent to get

just x. Allow slips on the signs in tan 2 3 y = sec 2 3 y ? 1 .

dy in dx

It may be implied- see below

A1* CSO. This is a given solution and you must be convinced that all steps are shown.

Note that the two method marks may occur the other way around Eg.

1 dx dy = 6sec2 3 y tan 3 y = 6 x( x ? 1) 2 ? = dy dx

1 6 x( x ? 1) 2

1

Scores the 2nd method Scores the 1st method The above solution will score M1, B0, M1, A0

Notes for Question 5 Continued

Example 1- Scores 0 marks in part (b)

dx dy 1 1 1 = 6sec 2 3 y tan 3 y ? = = = 1 2 dy dx 6sec 3 x tan 3 x 6sec 2 3 x sec 2 3 x ? 1 6 x( x ? 1) 2

Example 2- Scores M1B1M1A0

dx dy 1 1 1 ? = 2sec 2 3 y tan 3 y ? = = = 1 2 dy dx 2sec 3 y tan 3 y 2sec 2 3 y sec 2 3 y ? 1 2 x( x ? 1) 2

(c) Using Quotient and Product Rules M1

vu '? uv ' with u = 1and v = 6 x( x ? 1) 2 and achieving Uses the quotient rule v2

1

u ' = 0 and v ' = A( x ? 1) 2 + Bx( x ? 1)

?

1

?

1 2

.?

If the formulae are quoted, both must be correct. If they are not quoted nor implied by their working allow expressions of the form

?

? d 2 y ? 0 ? [ A( x ? 1) 2 + Bx( x ? 1)? 2 ] ? 2?= 1 2 ? ? ? dx ? 2 ? 6 x( x ? 1) ? ? ?

1 1

? d 2 y ? 0 ? A( x ? 1) 2 ± Bx( x ? 1) ? 2 or ? 2 ? = Cx 2 ( x ? 1) ? dx ?

1 1

A1

d 2 y 0 ? [6( x ? 1) 2 + 3 x( x ? 1) 2 ] = Correct un simplified expression dx 2 36 x 2 ( x ? 1)

1

1

?1

oe

dM1

Multiply numerator and denominator by ( x ? 1) 2 producing a linear numerator which is then simplified by collecting like terms. Alternatively take out a common factor of ( x ? 1)

? 1 2

from the numerator and collect like terms from the

linear expression This is dependent upon the 1st M1 being scored. A1 Correct simplified expression

d 2y 2 ? 3x = 3 oe 2 dx 12 x 2 ( x ? 1) 2

Notes for Question 5 Continued

(c) Using Product and Chain Rules

1 ? dy 1 ?1 Writes = = Ax ( x ? 1) 2 and uses the product rule with u or v = Ax ?1 and 1 dx 6 x( x ? 1) 2

M1

v or u = ( x ? 1) 2 . If any rule is quoted it must be correct.

If the rules are not quoted nor implied then award if you see an expression of the form

3 2 1

?

1

( x ? 1)

A1

?

× Bx ?1 ± C ( x ? 1) 2 × x ?2

?

d 2 y 1 ?1 1 ?3 ?1 = 6 [ x (? 2 )( x ? 1) 2 + (?1) x ?2 ( x ? 1) 2 ] 2 dx

?2 Factorises out / uses a common denominator of x ( x ? 1) 2 producing a linear factor/numerator which ?3

dM1

must be simplified by collecting like terms. Need a single fraction. A1 Correct simplified expression

d 2y ?3 1 ?2 == 12 x ( x ? 1) 2 [2 ? 3 x] oe ? 2 dx

(c) Using Quotient and Chain rules Rules M1 Uses the quotient rule

? 3 2 ? vu '? uv ' with u = ( x ? 1) 2 and v = 6 x and achieving 2 v 1

u ' = A( x ? 1) and v ' = B .?

? If the formulae is quoted, it must be correct. If it is not quoted nor implied by their working allow an expression of the form

?

? d 2 y ? Cx( x ? 1) ? 2 ? D( x ? 1) ? 2 ? 2?= Ex 2 ? dx ?

3 1

A1

? ? 1 6 x × ? ( x ? 1) 2 ? ( x ? 1) 2 × 6 2 d y 2 = Correct un simplified expression 2 dx 2 (6x)

3

1

dM1

Multiply numerator and denominator by ( x ? 1) 2 producing a linear numerator which is then simplified by collecting like terms. Alternatively take out a common factor of ( x ? 1)

? 3 2

3

from the numerator and collect like terms from the

linear expression This is dependent upon the 1st M1 being scored. A1

d 2y 2 ? 3x d 2 y ( 2 ? 3 x ) x ( x ? 1) Correct simplified expression = oe = 3 dx 2 12 x 2 ( x ? 1) 2 dx 2 12

?2

?3 2

Notes for Question 5 Continued

(c) M1

Using just the chain rule

Writes

?1 dy 1 1 = = = (36 x3 ? 36 x 2 ) 2 and proceeds by the chain rule to 1 1 dx 6 x( x ? 1) 2 (36 x3 ? 36 x 2 ) 2

A(36 x3 ? 36 x 2 ) 2 ( Bx 2 ? Cx) .

M1?

?3

Would automatically follow under this method if the first M has been scored??

? ? ?

Question Number

Scheme

Marks

6(a) ln(4 ? 2 x )(9 ? 3 x), = ln( x + 1) 2

So 36 ? 30 x + 6 x 2 = x 2 + 2 x + 1 and 5 x 2 ? 32 x + 35 = 0 Solve M1, M1 A1 M1A1

5 x 2 ? 32 x + 35 = 0 to give x =

7 oe ( Ignore the solution x = 5 ) 5

(5) (b)

Take loge’s to give ln 2 +lne

x 3x +1

= ln10

M1 M1 dM1

x ln 2+(3x +1)lne = ln10

?????????????????????????????????????????????? x (ln 2+3lne) = ln10 ? ln e ? x =..

and uses lne = 1

?1 + ln10 ???????????????????????????????????????????????????????????????? x = 3 + ln 2

M1 A1 (5)

Note that the 4th M mark may occur on line 2 (10 marks) Notes for Question 6

(a) M1 Uses addition law on lhs of equation. Accept slips on the signs. If one of the terms is taken over to the rhs it would be for the subtraction law. M1 A1 Uses power rule for logs write the 2 ln( x + 1) term as ln( x + 1) 2 . Condone invisible brackets Undoes the logs to obtain the 3TQ =0. 5 x 2 ? 32 x + 35 = 0 . Accept equivalences. The equals zero may be implied by a subsequent solution of the equation. M1 Solves a quadratic by any allowable method. The quadratic cannot be a version of (4 ? 2 x )(9 ? 3 x ) = 0 however. Deduces x = 1.4 or equivalent. Accept both x=1.4 and x=5. Candidates do not have to eliminate x = 5 . You may ignore any other solution as long as it is not in the range ?1 < x < 2 . Extra solutions in the range scores A0.

A1

Notes for Question 6 Continued

(b) M1 Takes logs of both sides and splits LHS using addition law. If one of the terms is taken to the other side it can be awarded for taking logs of both sides and using the subtraction law. M1 Taking both powers down using power rule. It is not wholly dependent upon the first M1 but logs of both sides must have been taken. Below is an example of M0M1

ln 2 x × lne3x +1 = ln10 ? x ln 2 × (3x + 1) ln e = ln10

dM1 This is dependent upon both previous two M’s being scored. It can be awarded for a full method to solve their linear equation in x. The terms in x must be collected on one side of the equation and factorised. You may condone slips in signs for this mark but the process must be correct and leading to x = … M1 Uses ln e = 1. This could appear in line 2, but it must be part of their equation and not just a statement. Another example where it could be awarded is e3 x+1 =

10 ? 3x + 1 = ... 2x

A1

Obtains answer x =

?1 + ln10 ? ln10 ? 1 ? ? log e 10 ? 1 ? =? ? oe . DO NOT ISW HERE ?=? 3 + ln 2 ? 3 + ln 2 ? ? 3 + log e 2 ?

Note 1: If the candidate takes log10 ’s of both sides can score M1M1dM1M0A0 for 3 out of 5. Answer = x =

? log e + log10 ? ? log e + 1 ? =? ? 3log e + log 2 ? 3log e + log 2 ? ?1 + log10 without reference to natural logs then award M4 but with hold 3 + log 2

Note 2: If the candidate writes x =

the last A1 mark, scoring 4 out of 5.

Question Number

Scheme

Marks

Alt 1 to 6(b) Writes lhs in e’s

2 x e3 x+1 = 10 ? e x ln 2e3 x +1 = 10

1st M1 2nd M1, 4th M1 dM1

?????????????????????????????????????????????????????????????????? ? e x ln 2+ 3 x+1 = 10, x ln 2 + 3 x + 1 = ln10 ?????????????????????????????????????????????????????????????????????????? x (ln 2+3) = ln10 ? 1 ? x =.. ?????????????????????????????????????????????????????????????????????????????????????????????? x =

?1 + ln10 3 + ln 2 Notes for Question 6 Alt 1

A1

(5)

M1 M1 dM1

Writes the lhs of the expression in e’s. Seeing 2 x = e x ln 2 in their equation is sufficient Uses the addition law on the lhs to produce a single exponential Takes ln’s of both sides to produce and attempt to solve a linear equation in x? You may condone slips in signs for this mark but the process must be correct leading to x= .. Uses ln e = 1. This could appear in line 2

M1

Question Number

Scheme ??????????? 0-f ( x)-10 ?

Marks

7(a)

B1

(1)

(b)

?

ff(0) = f(5), = 3?

B1,B1

(2)

(c)

?????????????????????????????????? y =

4 + 3x ? y (5 ? x) = 4 + 3 x ? 5? x

M1

??????????????????????????????????????????????????????? ? 5 y ? 4 = xy + 3 x ? ?????????????????????????????????????????????????????? ? 5 y ? 4 = x( y + 3) ? x =

5y ? 4 ? y+3

5x ? 4 ? 3+ x

dM1

??????????????????????????????????????????????????????????????????????????????????????????? g ?1 ( x) = ?

A1

(3)

(d)

???????????????????????????????????? gf ( x) = 16 ? f ( x) = g ?1 (16) = 4 ???oe? ????????????????????????????????????????????????????????????? f ( x ) = 4 ? x = 6 ? ???????????????????????????????????????????????????????????? f ( x ) = 4 ? 5 ? 2.5 x = 4 ? x = 0.4 oe ?

M1A1 B1 M1A1

(5)

(11 marks) Alt 1 to 7(d) 4 + 3(ax + b) = 16 ?? 5 ? (ax + b)

?????????????????? gf ( x) = 16 ?

M1 A1 B1 M1 A1

(5)

?????????????????????????????????????????????? ax + b = x ? 2 or 5 ? 2.5 x ??????????????????????? ????????????????????????????????????????????????????????????????????????? ? x = 6 ? ??????????????????????????????????????????????????????

4 + 3(5 ? 2.5 x) = 16 ? x = ... ? 5 ? (5 ? 2.5 x)

???????????????????????????????????????????????????????????????????????? ? x = 0.4 ??????oe ? ?

Notes for Question 7

(a) B1 Correct range. Allow 0-f ( x)-10 , 0-f-10 , 0-y-10 , 0-range-10 , [ 0,10] Allow f ( x).0 and f ( x)-10

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