Mark scheme - June 2007 - 6666 - Core Mathematics C4



Comparison of key skills specifications 2000/2002 with 2004 standardsX015461July 2004Issue 1

GCE Mathematics (6666/01)

June 2007

6666 Core Mathematics C4

Mark Scheme

|Question Number|Scheme | |Marks |

| |** represents a constant | | |

|1. (a) |[pic] |Takes 3 outside the bracket to give |B1 |

| | |any of [pic]or [pic]. | |

| | |See note below. | |

| | | | |

| |[pic] |Expands [pic] to give a simplified |M1; |

| |with [pic] |or an un-simplified | |

| | |[pic]; | |

| | | | |

| | |A correct simplified or an | |

| | |un-simplified [pic]expansion with | |

| | |candidate’s followed thro’ [pic] | |

| | | |A1[pic] |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] |Anything that |A1; |

| | |cancels to [pic] | |

| | |Simplified [pic] | |

| | | |A1 |

| | | | [5] |

| | | | |

| | | |5 marks |

|Question Number|Scheme | |Marks |

|Aliter | | | |

|1. |[pic] | | |

|Way 2 | | | |

| |[pic] |[pic] or[pic](See note [pic]) |B1 |

| | |Expands [pic]to give an | |

| |with [pic] |un-simplified or simplified [pic];| |

| | | | |

| | |A correct un-simplified or | |

| | |simplified [pic]expansion with | |

| | |candidate’s followed thro’[pic] | |

| | | |M1 |

| | | |A1[pic] |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] |Anything that |A1; |

| | |cancels to [pic] | |

| | |Simplified [pic] | |

| | | |A1 |

| | | | |

| | | | [5] |

| | | | |

| | | |5 marks |

Attempts using Maclaurin expansions need to be escalated up to your team leader.

If you feel the mark scheme does not apply fairly to a candidate please escalate the response up to your team leader.

|Question Number|Scheme | |Marks |

| | | | |

|2. |[pic], with substitution [pic] | | |

| | | | |

| |[pic] [pic] |[pic] or [pic] |B1 |

| | |or [pic] | |

| | | | |

| |[pic] |[pic] |M1[pic] |

| | |where k is constant | |

| | | | |

| | [pic] |[pic] |M1 |

| | | |A1 |

| | | | |

| |change limits: when x = 0 & x = 1 then u = 1 & u = 2 | | |

| | | | |

| |[pic] | | |

| | | | |

| | [pic] |Correct use of limits |depM1[pic] |

| | |u = 1 and u = 2 | |

| | | | |

| | [pic] |[pic]or [pic]or [pic] |A1 aef |

| | |Exact value only! |[6] |

| |Alternatively candidate can revert back to x … | | |

| | | | |

| |[pic] | | |

| | | | |

| | [pic] |Correct use of limits |depM1[pic] |

| | |x = 0 and x = 1 | |

| | | | |

| | [pic] |[pic]or [pic]or [pic] |A1 aef |

| | |Exact value only! | |

| | | |6 marks |

|Question Number|Scheme | |Marks |

| | | | |

|3. (a) |[pic] | | |

| | |(see note below) | |

| |[pic] |Use of ‘integration by parts’ formula in |M1 |

| | |the correct direction. | |

| | |Correct expression. | |

| | | |A1 |

| | | | |

| | [pic] |[pic] |dM1 |

| | |or [pic] with [pic] | |

| | | | |

| | [pic] |Correct expression with +c |A1 |

| | | |[4] |

| | | | |

|(b) |[pic] |Substitutes correctly |M1 |

| | |for [pic]in the | |

| | |given integral | |

| | | | |

| | [pic] | | |

| | | | |

| | [pic] |[pic] |A1;[pic] |

| | |or underlined expression | |

| | | | |

| | [pic] |Completely correct expression |A1 |

| | |with/without +c | |

| | | |[3] |

| | | | |

| | | |7 marks |

Notes:

|(b) |[pic] |This is acceptable for M1 |M1 |

| | | | |

| | [pic] | | |

| | | | |

| |[pic] |This is also |M1 |

| | |acceptable for M1 | |

|Question Number|Scheme | |Marks |

| | | | |

|Aliter |[pic] |Substitutes correctly |M1 |

|3. (b) | |for [pic]in the | |

|Way 2 | |given integral … | |

| | | | |

| |[pic] |… or | |

| | |[pic]and [pic] | |

| | | | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] |[pic] |A1[pic] |

| | |or underlined expression | |

| | | | |

| |[pic] |Completely correct expression |A1 |

| | |with/without +c | |

| | | |[3] |

| | | | |

|Aliter (b) |[pic] |Substitutes correctly |M1 |

|Way 3 | |for [pic] in[pic] | |

| | | | |

| |[pic] | | |

| | | | |

| | [pic] |[pic] |A1;[pic] |

| | |or underlined expression | |

| | | | |

| | [pic] |Completely correct expression |A1 |

| | |with/without +c | |

| | | |[3] |

| | | | |

| | | |7 marks |

|Question Number|Scheme | |Marks |

| | | | |

|4. (a) |A method of long division gives, | | |

|Way 1 | | | |

| | [pic] |[pic] |B1 |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] |Forming any one of these two identities. Can be |M1 |

| |or [pic] |implied. | |

| | | | |

| | | | |

| | | | |

| |Let [pic][pic] | | |

| | |See note below | |

| |Let [pic] [pic] |either one of [pic] or [pic] |A1 |

| | |both B and C correct |A1 |

| | | |[4] |

| | | | |

|Aliter | | | |

|4. (a) | [pic] | | |

|Way 2 | | | |

| |See below for the award of B1 |decide to award B1 here!! … |B1 |

| | |… for [pic] | |

| | | | |

| |[pic] |Forming this identity. |M1 |

| | |Can be implied. | |

| | | | |

| |Equate x2, [pic] | | |

| | | | |

| |Let [pic][pic] | | |

| | |See note below | |

| |Let [pic] [pic] |either one of [pic] or [pic] |A1 |

| | |both B and C correct |A1 |

| | | |[4] |

| | | | |

|Question Number|Scheme | |Marks |

| | | | |

|4. (b) |[pic] | | |

| | | | |

| |[pic] |Either [pic] or [pic] |M1[pic] |

| | |or either[pic] or [pic] | |

| | |[pic] |B1[pic] |

| | |[pic] |A1 |

| | |or [pic] |cso & aef |

| | |See note below. | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] |Substitutes limits of 2 and 1 |depM1[pic] |

| | |and subtracts the correct way round. (Invisible | |

| | |brackets okay.) | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] |Use of correct product (or power) |M1 |

| | |and/or quotient laws for logarithms to obtain a | |

| | |single logarithmic term for their numerical | |

| | |expression. | |

| | | | |

| |[pic] |[pic] |A1 |

| | |Or [pic] and k stated as [pic]. |[6] |

| | | | |

| | | | 10 marks |

|Question Number|Scheme | |Marks |

| | | | |

|5. (a) |If l1 and l2 intersect then: | | |

| | | | |

| |[pic] | | |

| | | | |

| | Any two of [pic] |Writes down any two of these equations correctly. |M1 |

| | | | |

| |[pic] |Solves two of the above equations to find … | |

| | |either one of [pic] or [pic] correct |A1 |

| | |both[pic] and[pic] correct |A1 |

| | | | |

| |Either [pic] |Complete method of putting their values of [pic] |B1[pic] |

| | |and [pic] into a third equation to | |

| | |show a contradiction. | |

| | | | |

| |or for example: |this type of explanation is also allowed for | |

| | |B1[pic]. | |

| |[pic] | | |

| |[pic] Lines l1 and l2 do not intersect | | |

| | | |[4] |

| | | | |

|Aliter | | | |

|5. (a) |[pic] |Uses the k component to find [pic] | |

| | |and substitutes their value of [pic] | |

| | |into either one of the i or j component. | |

|Way 2 | | | |

| |[pic] | |M1 |

| | | | |

| |[pic] |either one of the [pic]’s correct |A1 |

| | |both of the [pic]’s correct |A1 |

| | | | |

| |Either: These equations are then inconsistent |Complete method giving rise to any one of these |B1[pic] |

| |Or: [pic] |three explanations. | |

| |Or: Lines l1 and l2 do not intersect | | |

| | | |[4] |

| | | | |

|Question Number|Scheme | |Marks |

|Aliter | | | |

|5. (a) |If l1 and l2 intersect then: | | |

|Way 3 | | | |

| |[pic] | | |

| | | | |

| | Any two of [pic] |Writes down any two of these equations |M1 |

| | | | |

| |[pic] |either one of the [pic]’s correct |A1 |

| | |both of the [pic]’s correct |A1 |

| | | | |

| |Either: These equations are then inconsistent |Complete method giving rise to any one of these |B1[pic] |

| |Or: [pic] |three explanations. | |

| |Or: Lines l1 and l2 do not intersect | | |

| | | |[4] |

| | | | |

|Aliter | Any two of [pic] |Writes down any two of these equations |M1 |

|5. (a) | | | |

|Way 4 | | | |

| | | | |

| |[pic] |[pic] |A1 |

| | |RHS of (3) = 3 |A1 |

| | | | |

| | (3) yields [pic] |Complete method giving rise to this explanation. |B1[pic] |

| | | |[4] |

| | | | |

|Question Number|Scheme | |Marks |

| | | | |

|5. (b) |[pic] |Only one of either |B1 |

| | |[pic] or [pic] or [pic] or [pic]. | |

| | |(can be implied) | |

| | | | |

| |[pic] or [pic] |Finding the difference between their [pic] |M1[pic] |

| | |and[pic]. | |

| | |(can be implied) | |

| | | | |

| | | | |

| | |Applying the dot product formula between |M1 |

| | |“allowable” vectors. See notes below. | |

| |[pic], [pic] & [pic] is angle | | |

| | | | |

| |[pic] |Applies dot product formula between [pic] and |M1[pic] |

| | |their[pic] | |

| | |Correct expression. |A1 |

| | | | |

| |[pic] |[pic] or 0.7 or [pic] |A1 cao |

| | |but not[pic] |[6] |

| | | | |

| | | | 10 marks |

Note: If candidate use cases 2, 3, 4 and 5 they cannot gain the final three marks for this part.

Note: Candidate can only gain some/all of the final three marks if they use case 1.

Examples of awarding of marks M1M1A1 in 5.(b)

|Example |Marks |

| | |

|[pic] |M1M1A1 |

| |(Case 1) |

| | |

| | |

|[pic] |M1M0A0 |

| |(Case 2) |

| | |

| | |

|[pic] |M1M0A0 |

| |(Case 3) |

| | |

|Question Number|Scheme | |Marks |

| | | | |

|6. (a) | [pic] [pic] | | |

| | | | |

| | [pic], [pic] |Correct[pic] and [pic] |B1 |

| | | | |

| |[pic] [pic] |[pic] |M1 |

| | |[pic] | |

| | | |A1[pic] |

| | | |[3] |

| | | | |

|(b) | When [pic] [pic] (need values) |The point [pic] or[pic] |B1, B1 |

| | |These coordinates can be implied. | |

| | |([pic] is not sufficient for B1) | |

| |When [pic] m(T) =[pic] | | |

| | | | |

| |[pic] |any of the five underlined expressions or awrt 0.18 |B1 aef |

| | | | |

| | |Finding an equation of a tangent with their point and| |

| |T: [pic] |their tangent gradient or finds c by using [pic]. |M1[pic] aef |

| | | | |

| | | | |

| |T: [pic] or [pic] |Correct simplified |A1 aef cso |

| | |EXACT equation of tangent | |

| | | | |

| |or [pic] | | |

| | | | |

| |Hence T: [pic] or [pic] | | |

| | | |[5] |

| | | | |

|Question Number|Scheme | |Marks |

| | | | |

|6. (c) | [pic] [pic] | | |

|Way 1 | | | |

| | [pic] |Uses[pic] |M1 |

| | | | |

| | [pic] |Eliminates ‘t’ to write an equation involving x and |M1 |

| | |y. | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] |Rearranging and factorising with an attempt to |ddM1 |

| | |make[pic]the subject. | |

| | | | |

| |[pic] |[pic] |A1 |

| | | |[4] |

|Aliter | | | |

|6. (c) | [pic] |Uses[pic] |M1 |

|Way 2 | | | |

| | [pic] |Uses [pic] |M1 implied |

| | | | |

| |Hence, [pic] |Eliminates ‘t’ to write an equation involving x and |ddM1 |

| | |y. | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

| | | | |

|Question Number|Scheme | |Marks |

|Aliter | | | |

|6. (c) | [pic] [pic] | | |

|Way 3 | | | |

| | [pic] |Uses[pic] |M1 |

| | | | |

| | [pic] |Uses [pic] |M1 |

| | | | |

| | [pic] | | |

| | | | |

| |Hence, [pic] |Eliminates ‘t’ to write an equation involving x and |ddM1 |

| | |y. | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | | |

| | | |[4] |

|Aliter | | | |

|6. (c) |[pic] |Uses[pic] |M1 |

|Way 4 | | | |

| | [pic] |Uses [pic] |M1 |

| | | | |

| | [pic] |then uses [pic] |ddM1 |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

| | | | |

|Question Number|Scheme | |Marks |

|Aliter | | | |

|6. (c) | [pic] [pic] | | |

|Way 5 | | | |

| | [pic] | | |

| | | | |

| | [pic] |Draws a right-angled triangle and places |M1 |

| | |both[pic]and 1 on the triangle | |

| | | | |

| | | | |

| | |Uses Pythagoras to deduce the hypotenuse | |

| | | | |

| | | | |

| | | |M1 |

| | | | |

| |Hence, [pic] |Eliminates ‘t’ to write an equation involving x and |ddM1 |

| | |y. | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

| | | | |

| | | | 12 marks |

|Question Number|Scheme | |Marks |

| | | | |

|7. (a) | |x |0 |

| |Enter marks into ePEN in the correct order. |0.446 or awrt 0.44600 |B1 |

| | |awrt 0.64359 |B1 |

| | |awrt 0.81742 |B1 |

| | |0 can be implied | | | |[3] |

| | | | | | | |

|(b) |[pic] |Outside brackets |B1 |

|Way 1 | |[pic] or [pic] | |

| | |For structure of trapezium rule[pic]; | |

| | |Correct expression | |

| | |inside brackets which all must be | |

| | |multiplied by[pic]. | |

| | | |M1[pic] |

| | | |A1[pic] |

| | | | |

| |[pic](4dp) |for seeing 0.4726 |A1 cao |

| | | |[4] |

| | | | |

| | | | |

|Aliter | |[pic] and a divisor of 2 on all terms |B1 |

|(b) |[pic] |inside brackets. | |

|Way 2 | |One of first and last ordinates, two of | |

| | |the middle ordinates inside brackets | |

| |which is equivalent to: |ignoring the 2. | |

| | |Correct expression inside brackets if | |

| |[pic] |[pic] was to be factorised out. | |

| | | |M1[pic] |

| | | |A1[pic] |

| | | | |

| |[pic] | 0.4726 |A1 cao |

| | | |[4] |

| | | | |

|Question Number|Scheme | |Marks |

| | | | |

|7. (c) |Volume [pic] |[pic] or [pic] |M1 |

| | |Can be implied. | |

| | |Ignore limits and[pic] | |

| | | | |

| | [pic] or [pic] |[pic] |A1 |

| | |or [pic] | |

| | | | |

| | [pic] |The correct use of limits on a function |dM1 |

| |or |other than tan x; ie | |

| |[pic] |[pic] ‘minus’ [pic]. [pic] may be | |

| | |implied. Ignore[pic] | |

| | | | |

| | [pic] | | |

| |or | | |

| |[pic] | | |

| | | | |

| |[pic] or [pic] or [pic] or [pic] or [pic] |[pic]or[pic] |A1 aef |

| | |or [pic] or [pic] | |

| | |or [pic] | |

| | |must be exact. |[4] |

| | | | |

| | | | 11 marks |

Beware: In part (c) the factor of [pic]is not needed for the first three marks.

Beware: In part (b) a candidate can also add up individual trapezia in this way:

[pic]

|Question Number|Scheme | |Marks |

| | | | |

|8. (a) | [pic] and [pic] | | |

| | | | |

| |[pic] |Separates the variables with [pic] and |M1 |

| | |[pic]on either side with integral signs | |

| | |not necessary. | |

| | | | |

| |[pic] |Must see [pic] and [pic]; |A1 |

| | |Correct equation with/without + c. | |

| | | | |

| |When [pic] |Use of boundary condition (1) to attempt |M1 |

| |[pic] |to find the constant of integration. | |

| | | | |

| |[pic] [pic] | | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

| | | | |

|(b) |[pic] [pic] |Substitutes[pic]into an expression |M1 |

| | |involving P | |

| | | | |

| |[pic] or [pic] |Eliminates [pic] and takes |M1 |

| |…or [pic] or [pic] |ln of both sides | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] or [pic] (to nearest minute) |awrt [pic] or [pic] |A1 |

| | | |[3] |

| | | | |

|Question Number|Scheme | |Marks |

| | | | |

|8. (c) | [pic] and [pic] | | |

| | | | |

| |[pic] |Separates the variables with [pic] and |M1 |

| | |[pic]on either side with integral signs | |

| | |not necessary. | |

| | | | |

| |[pic] |Must see [pic] and [pic]; |A1 |

| | |Correct equation with/without + c. | |

| | | | |

| |When [pic] |Use of boundary condition (1) to attempt |M1 |

| |[pic] |to find the constant of integration. | |

| | | | |

| |[pic] [pic] | | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

| | | | |

|(d) |[pic] [pic] | | |

| | | | |

| |[pic] |Eliminates [pic] and makes [pic] or |M1 |

| |…or … [pic] |[pic]the subject by taking ln’s | |

| | | | |

| |[pic] |Then rearranges |dM1 |

| | |to make t the subject. | |

| | |(must use sin-1) | |

| |[pic] | | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] or [pic] (to nearest minute) |awrt [pic] or [pic] |A1 |

| | | |[3] |

| | | | |

| | | | 14 marks |

|Question Number|Scheme | |Marks |

| | | | |

| | [pic] and [pic] | | |

| | | | |

|Aliter |[pic] |Separates the variables with [pic] and |M1 |

|8. (a) | |[pic]on either side with integral signs | |

|Way 2 | |not necessary. | |

| | | | |

| |[pic] |Must see [pic] and [pic]; |A1 |

| | |Correct equation with/without + c. | |

| | | | |

| |When [pic] |Use of boundary condition (1) to attempt |M1 |

| |[pic] |to find the constant of integration. | |

| | | | |

| |[pic] [pic] | | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

| | | | |

|Aliter |[pic] |Separates the variables with [pic] and |M1 |

|8. (a) | |[pic]on either side with integral signs | |

|Way 3 | |not necessary. | |

| | | | |

| |[pic] |Must see [pic] and [pic]; |A1 |

| | |Correct equation with/without + c. | |

| | | | |

| |When [pic] |Use of boundary condition (1) to attempt |M1 |

| |[pic] |to find the constant of integration. | |

| | | | |

| |[pic] [pic] | | |

| |[pic] | | |

| |[pic] | | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

|Question Number|Scheme | |Marks |

| | | | |

| | [pic] and [pic] | | |

| | | | |

|Aliter |[pic] |Separates the variables with [pic] and |M1 |

|8. (c) | |[pic]on either side with integral signs | |

|Way 2 | |not necessary. | |

| | | | |

| |[pic] |Must see [pic] and [pic]; |A1 |

| | |Correct equation with/without + c. | |

| | | | |

| |When [pic] |Use of boundary condition (1) to attempt |M1 |

| |[pic] |to find the constant of integration. | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] | | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

| | | | |

|Question Number|Scheme | |Marks |

| | | | |

| | [pic] and [pic] | | |

| | | | |

|Aliter |[pic] |Separates the variables with [pic] and |M1 |

|8. (c) | |[pic]on either side with integral signs | |

|Way 3 | |not necessary. | |

| | | | |

| |[pic] |Must see [pic] and [pic]; |A1 |

| | |Correct equation with/without + c. | |

| | | | |

| |When [pic] |Use of boundary condition (1) to attempt |M1 |

| |[pic] |to find the constant of integration. | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] | | |

| | | | |

| |[pic] | | |

| |[pic] | | |

| | | | |

| |Hence, [pic] |[pic] |A1 |

| | | |[4] |

| | | | |

Note: dM1 denotes a method mark which is dependent upon the award of the previous method mark.

ddM1 denotes a method mark which is dependent upon the award of the previous two method marks.

depM1[pic] denotes a method mark which is dependent upon the award of [pic].

ft denotes “follow through”

cao denotes “correct answer only”

aef denotes “any equivalent form”

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If you see this integration applied anywhere in a candidate’s working then you can award M1, A1

To award this M1 mark, the candidate must use the appropriate law(s) of logarithms for their ln terms to give a one single logarithmic term. Any error in applying the laws of logarithms would then earn M0.

Note: The x and y coordinates must be the right way round.

A candidate who incorrectly differentiates [pic]to give [pic] or [pic]is then able to fluke the correct answer in part (b). Such candidates can potentially get: (a) B0M1A1[pic] (b) B1B1B1M1A0 cso.

Note: cso means “correct solution only”.

Note: part (a) not fully correct implies candidate can achieve a maximum of 4 out of 5 marks in part (b).

t

[pic]

[pic]

1

Candidates can score this mark if there is a complete method for finding the dot product between their vectors in the following cases:

Case 2: [pic]

and [pic]

[pic]

Case 3: [pic]

and [pic]

[pic]

Case 4: their ft [pic]

and [pic]

[pic]

Case 1: their ft [pic]

and [pic]

[pic]

Case 5: their [pic]

and their [pic]

[pic]

Some candidates may find rational values for B and C. They may combine the denominator of their B or C with (2x +1) or (2x – 1). Hence:

Either [pic] or [pic] is okay for M1.

Candidates are not allowed to fluke [pic] for A1. Hence cso. If they do fluke this, however, they can gain the final A1 mark for this part of the question.

Note: This is not a dependent method mark.

[pic], gains B0M1A1A0

In (a) for [pic] writing 0.4459959… then 0.45600 gains B1 for awrt 0.44600 even though 0.45600 is incorrect.

In (b) you can follow though a candidate’s values from part (a) to award M1 ft, A1 ft

Special Case: If you see the constant [pic]in a candidate’s final binomial expression, then you can award B1

Special Case: If you see the constant [pic]in a candidate’s final binomial expression, then you can award B1

Note: You would award: B1M1A0 for

[pic]

because [pic] is not consistent.

If a candidate states one of either B or C correctly then the method mark M1 can be implied.

If a candidate gives the correct exact answer and then writes 1.088779…, then such a candidate can be awarded A1 (aef). The subsequent working would then be ignored. (isw)

There are other acceptable answers for A1, eg: [pic]

NB: Use your calculator to check

eg. 0.240449…

[pic] is an acceptable response for the final accuracy A1 mark.

[pic] is an acceptable response for the final accuracy A1 mark.

[pic] is an acceptable response for the final accuracy A1 mark.

There are so many ways that a candidate can proceed with part (c). If a candidate produces a correct solution then please award all four marks. If they use a method commensurate with the five ways as detailed on the mark scheme then award the marks appropriately. If you are unsure of how to apply the scheme please escalate your response up to your team leader.

[pic] written down without the first M1 mark given scores all four marks in part (a).

[pic] written down without the first M1 mark given scores all four marks in part (c).

[pic] written down without the first M1 mark given scores all four marks in part (a).

[pic] written down without the first M1 mark given scores all four marks in part (c).

Mark Scheme (Results)

Summer 2007

Edexcel Limited. Registered in England and Wales No. 4496750

Registered Office: One90 High Holborn, London WC1V 7BH

GCE

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