C4 maths mark schemes, June 2014 back to January 2010
GCE Core Mathematics C4 (6666) January 2011 1
Question Number
C4 JUNE 2014 Marks
1.
(a)
dM1
A1 cso
[5]
(b)At
see notes M1and either T:
or ,
T: or A1 cso
[2]
7
2.
(a)Either or
see notes M1
leading to A1
[2]
(b)
Either or or M1
Either or
M1
GCE Core Mathematics C4 (6666) January 2011 2
or 22.5A1
[3]
5
3.x 1 2 3 4
y 1.42857 0.90326 0.682116... 0.55556
(a) 0.68212B1 cao
[1]
(b)
Outside brackets or B1 aef
For structure of M1
anything that rounds to 2.5774 A1
[3]
(c) Overestimate
B1
and a reason such as
{top of} trapezia lie above the curve a diagram which gives reference to the extra
area concave or convex
(can be implied) bends inwards curves downwards
[1]
(d) or B1
GCE Core Mathematics C4 (6666) January 2011 3
Either or M1
with no other terms.
M1
A1 cso
Substitutes limits of 2 and 1 in u
(or 4 and 1 in x) and subtracts the correct way
round.
M1
or or
A1 oe cso
[6]
11
4. ,
M1
A1
M1 oe
or
When dependent on the previous M1
see notesdM1
or orA1 oe
GCE Core Mathematics C4 (6666) January 2011 4
[5]
5
5.
(a)M1 oe
So, Adds their expanded x (which is in terms of t) to dM
1
* Correct proofA1 *
[3]
(b)Applies to achieve
an equation containing only x’s and y’s.
M1
A1
[2]
5
6. (i)
M1
A1
A1
[3]
(ii)M1
GCE Core Mathematics C4 (6666) January 2011 5
or equivalent.
A1
{Ignore subsequent working}.
[2]
(iii)
at
or
B1 oe
Applying
M1
Integrates to give M1
A1
B1
or Use of and
in an integrated equation containing c
M1
giving A1
[7]
12
7.
(a) ,
GCE Core Mathematics C4 (6666) January 2011 6
their divided by their M1
Correct A1 oe
At ,
Some evidence of
substituting into
their
M1
applies M1
Either N: see notes M1
or
{At Q, so, } giving or or awrt
1.67
A1 cso
[6]
(b) see notes
M1
So, see notes A1
A1
Applies
M1
GCE Core Mathematics C4 (6666) January 2011 7
Dependent on the first method mark. For dM
1
A1
Dependent on
the third method
mark.
dM1
{So }
M1
A1
[9]
15
8.
(a) M1; A1
[2]
(b)
or
B1ft
[1]
(c) M1
GCE Core Mathematics C4 (6666) January 2011 8
Applies dot product formula between
their
and their
M1
Correct proof
A1 cso
[3]
(d)
or , with
either or , or
a multiple of .
M1
Correct vector equation. A1 ft
[2]
(e)
Either
or
M1
At least one set of coordinates are correct.
A1 ft
Both sets of coordinates are correct.
A1 ft
[3]
(f)M1
or
or or awrt 4.9 or equivalent
A1 oe
GCE Core Mathematics C4 (6666) January 2011 9
3 3
3023
P
82 6 3 3. 4.8989...3
h
2l114
C
3
138
B
1l
247
A
132
D
3
dM1
A1 cao
[4]
15
8. (f) Helpful Diagram!
GCE Core Mathematics C4 (6666) January 2011 10
Candidates do not need to prove this result for part (f)
GCE Core Mathematics C4 (6666) January 2011 11
Question Number
C4 JUNE 2014 (R) Marks
1. (a) B1
B1
M1
A1; A1
[5]
(b)
M1
M1
A1
[3]
8
2. (a) B1; M1
(2 dp)A1 cao
[3]
GCE Core Mathematics C4 (6666) January 2011 12
(b) Any one of
Increase the number of strips Use more trapezia Make h smaller Increase the number of x and/or y values used Shorter /smaller intervals for x More values of y. More intervals of x Increase n
B1
[1]
(c)
,
M1
A1
A1 oe
dM1
A1 oe
[5]
9
3.
(a)
dM1
Simplifying gives A1 cso oe
[5]
GCE Core Mathematics C4 (6666) January 2011 13
(b) M1
So ,
M1
gives or A1 oe
ddM1
A1 cao
[5]
10
4. (a)
B1
B1 cso
M1
leading to A1
[4]
(b) B1
For their partial fraction
GCE Core Mathematics C4 (6666) January 2011 14
M1
A1ft
A1ft
dM1
So, A1 oe
[6]
10
5. (a)From question, ,
B1 oe
M1 oe
When , dM1
Hence, (cm2 s-1)A1
GCE Core Mathematics C4 (6666) January 2011 15
[4]
(b) M1; oe
When ,
Hence, (cm2 s-1)A1 cso
[2]
6
6.
(a) {B lies on B1
[1]
(b)
e.g. i: M1
So, A1
Point of intersection is B1
Finds and either
checks and is true for the third component.
substitutes into to give
and substitutes into to give
B1
[4]
(c) M1
GCE Core Mathematics C4 (6666) January 2011 16
M1
A1
[3]
(d)M1
A1
[2]
10
7.
(a) B1
M1 A1; A1
then eg either... or...
leading to with no incorrect working/statements. A1 * cso
[5]
(b) B1
So
and or
Eg. M1
GCE Core Mathematics C4 (6666) January 2011 17
So
A1
A1
[4]
(c)
(fish) (nearest 100) B1
[1]
10
8. lies on the curve,
(a)
M1
or
so A1
[2]
(b) ,
B1
B1
So,
At ,
M1;
A1
cao cso
[4]
(c) M1
GCE Core Mathematics C4 (6666) January 2011 18
gives A1
So or M1; A1
or dM1
A1
[6]
12
Question Number
C4 JUNE 2013 MARK SCHEME Marks
1. (a) , 1st Application: , 2nd Application:
, M1
A1 oe
Either
or for
M1
M1
Correct answer, with/without A1
(5)
(b)
Applies limits of 1 and 0 to an expression of the form
and and subtracts the correct way
round.
M1
GCE Core Mathematics C4 (6666) January 2011 19
csoA1 oe
(2)
[7]
2. (a)
B1
M1 A1 A1
M1
Answer is given in the question.
A1 *
(6)
(b)M1
ie: B1
so,
A1 cao
(3)
[9]
3. (a)1.154701 B1
cao
(1)
(b)
B1; M1
(4 dp)1.7787 or awrt 1.7787 A1
GCE Core Mathematics C4 (6666) January 2011 20
(3)
(c)
For .Ignore limits and .
Can be implied.
B1
M1
or equivalent
A1
A1 cao cso
(4)
[8]
4.
(a) , or At least one of or
correct.B1
Both and are correct.B1
So,
At ,
Applies their divided by
their and substitutes
into their .
M1;
Correct value for of 1
A1 cao cso
(4)
(b) M1
So, or or or equivalent.
A1 cso isw
GCE Core Mathematics C4 (6666) January 2011 21
Either or B1
(3)
(c) Range: or or B1 B1
(2)
[9]
5. (a) or or
B1
M1
A1 * cso
(3)
(b)
M1 A1
So
Integrates
to
obtain any one of or
M1
At least one term correctly followed through
A1 ft
. A1 cao
So,
Applies limits of 3 and 1 in u or 9 and 1 in x
in their integrated function and subtracts
M1
GCE Core Mathematics C4 (6666) January 2011 22
the correct way round.
A1 cso cao
(7)
[10]
6.
(a) or
B1
or
M1 A1;
M1 A1
M1
then either... or...
dddM1
A1 * leading to
(8)
(b) M1
Uses correct order of operations by moving
from
to give and
dM1
GCE Core Mathematics C4 (6666) January 2011 23
,
where
= 161 (s) (nearest second) awrt 161 A1
(3)
[11]
7.
(a)
dM1
A1 cso oe
(5)
(b) M1
A1
M1*
dM1*
A1
When , When , ddM1*
A1 cso
(7)
[12]
GCE Core Mathematics C4 (6666) January 2011 24
8. , ,
(a)
Finds the difference
between and .
Ignore labelling.
M1
Correct difference. A1
M1
A1 cso
(4)
(b) M1
So, caoA1 cao
It follows that, or B1 ft
{Note that }
or
and
Uses a correct method in order to find both possible
sets of coordinates of B.M1
GCE Core Mathematics C4 (6666) January 2011 25
Both coordinates are correct.
A1 cao
(5)
[9]
Question Number C4 JUNE 2013 (R) MARK SCHEME
Marks
1. At least one of “A”
or “C” are correct.
B1
Breaks up their partial
fraction correctly into
three terms and
both and .
B1 cso
1.
2. Writes down a correct
identity and attempts to
find the value of either one “A” or “ B”
or “C”.
M13.
Either
leading to
Correct value for “B” which is found using
a correct identity and
follows from their partial
fraction decompositio
n.
A1 cso
GCE Core Mathematics C4 (6666) January 2011 26
So,
[4]
2.
(ignore)
B1 oe
Differentiates implicitly to include
either
.
M1*
B1
A1
4. Substitutes into
their differentiated equation or expression.
dM1*
5.
dM1*
Uses to
achieve A1 cso
[7]
3.
6.
or
Either M1
Either A1
8. Correct substitution
(Ignore integral sign and A1
An attempt to divide each term by u.
dM1
GCE Core Mathematics C4 (6666) January 2011 27
ddM1
A1 ft
Applies limits of 5 and 3 in u
or 4 and 0 in x in their integrated function and
subtracts the correct way round.
M1
A1
cao cso
[8]
4. (a)Power
of
M1
9.
or B1
10.
M1 A1
A1; A1
(6)
(b)
Writes down or
uses B1
When M1
GCE Core Mathematics C4 (6666) January 2011 28
So, 19.2201
csoA1 cao
(3)
[9]
5. (a) 6.248046798... = 6.248 (3dp) 6.248 or awrt 6.248 B1
(1)
(b)B1; M1
(2 dp) 49.37 or awrt 49.37 A1
(3)
(c)
M1
A1
B1
A1
Substitutes limits of 8 and 0 into an
integrated function of the form of either
or
and subtracts the
correct way round.
dM1
GCE Core Mathematics C4 (6666) January 2011 29
A1
(6)
(d)
Difference
1.46 or awrt 1.46 B1
(1)
[11]
6.
, ,
(a)
A is on l, so
B1
Substitutes their value of into M1
A1 cao
(3)
(b)
Finds the difference
between and .
Ignore labelling.
M1
M1; A1 ft
GCE Core Mathematics C4 (6666) January 2011 30
ddM1;
A1 cso cao
(5)
(c) M1
So, A1 cao
(2)
(d) M1;A1 cao
(2)
[12]
7.
(a) ,
At least one of
or correct.
B1
Both and are correct.
B1
At ,
Applies their
divided by their
M1;
A1 cao cso
(4)
(b) M1
A1 * cso
or B1
GCE Core Mathematics C4 (6666) January 2011 31
(3)
(c)
For
or Ignore limits and .
Can be implied.
B1
Either or
oe
M1
oeA1
Substitutes limits of 125 and 27 into an integrated function
and subtracts the correct way round.
dM1
or or A1
(5)
[12]
8. where M is a constant
(a)
is the rate of increase of the mass of waste products.
M is the total mass of unburned fuel and waste fuel
(or the initial mass of unburned fuel)
Any one correct explanation.
B1
Both explanations are correct.
B1
(2)
(b) or B1
GCE Core Mathematics C4 (6666) January 2011 32
or M1 A1
M1
then either... or...
ddM1
A1 * cso leading to or
oe
(6)
(c)
M1
So A1
dM1
A1 cso
(4)
[12]
GCE Core Mathematics C4 (6666) January 2011 33
Question Number
C4 JANUARY 2013 MARK SCHEME Marks
1.or
B1
see notesM1
A1
See notes
below!
A1;
A1
[5]
5
2. (a)
,
In the form
M1
simplified or
un-simplified.
A1
simplified
or un-simplified.
GCE Core Mathematics C4 (6666) January 2011 34
dM1
Correct answer,
with/without A1
[5]
(b)
Applies limits of 2
and 1 to their part
(a) answer and
subtracts the correct
way round.
M1
or
equivalent
.
A1
[2]
7
3. Method 1: Using one identity
their constant term B1
Forming a correct
identity.B1
Either
or
Attempts to find the
value of either one of
their B or their C from
their identity.
M1
Correct values for
their B and their C,
which are found using
a correct identity.
A1
GCE Core Mathematics C4 (6666) January 2011 35
[4]
Method 2: Long Division
their constant term B1
So,
Forming a correct
identity.B1
Either
or
Attempts to find the
value of either one of
their B or their C from
their identity.
M1
Correct values for
their B and their C,
which are found usingA1
So, [4]
4
4. (a)1.0981 B1
cao
[1]
(b)B1;
M1
(3 dp)
2.843 or awrt 2.843 A1
GCE Core Mathematics C4 (6666) January 2011 36
[3]
(c) or
B1
M1
A1
Expands to give a “four
term” cubic in u.
Eg: M1
An attempt to divide at least three terms in their
cubic by u. See notes.M1
A1
=
Applies limits of 3 and 2 in u or 4 and 1 in x and subtracts
either way round.
M1
Correct exact answer
or equivalent.A1
[8]
12
5. Working parametrically:
GCE Core Mathematics C4 (6666) January 2011 37
(a)Applies to obtain a
value for t.M1
When , Correct value for y.A1
[2]
(b)
Applies to obtain a
value for t.
(Must be seen in part (b)).
M1
When ,
A1
[2]
(c) and either
or
B1
Attempts their divided by
their
M1
At A, so
Applies and M1
or
or equivalent.
M1 A1 oe cso
[5]
(d)Complete substitution for both
andM1
B1
GCE Core Mathematics C4 (6666) January 2011 38
Either
or
or
M1*
A1
Depends on the previous
method mark.
Substitutes their changed limits
in t and subtracts either way
round.
dM1*
or equivalent.A1
[6]
15
6. (a), seen or implied.
M1
At least one correct value of x. (See notes).
A1
Both A1 cso
[3]
(b)
V
For .Ignore limits and
B1
See notes.
M1
GCE Core Mathematics C4 (6666) January 2011 39
Attempts to give any two of or
.
M1
Correct integration. A1
Applying limits the
correct way round. Ignore
M1
Two term exact answer.
A1
[6]
9
7. (a)
Any two equations.
(Allow one slip).M1
Eg: or An attempt to eliminate one of the parameters.
M1
Leading to Either A1
or M1 A1
[5]
(b)
,
Realisation that the dot product is required
between and
M1
GCE Core Mathematics C4 (6666) January 2011 40
.
Correct equation. A1
awrt 69.1 A1
[3]
(c)
,
M1 A1
M1
leading to A1
Position vector
M1 A1
[6]
14
8. (a) or
B1
or See notes. M1 A1
Correct completion to
.
GCE Core Mathematics C4 (6666) January 2011 41
A1 *
[4]
(b) ; See notes. M1; A1
Substitutes into an equation
of the form
or equivalent.
M1
Correct algebra to ,
where k is a positive value.M1
awrt 77 A1
[5]
9
Question Number C4 JUNE 2012 MARK SCHEME Marks
1. (a) B1 M1 any two constants correct A1 Coefficients of all three constants correct A1 (4)
(b) (i) GCE Core Mathematics C4 (6666) January 2011 42
M1 A1ft A1ft
(ii) M1
M1
A1 (6) [10]
2. (a) csoB1 (1)
(b) M1
At x = 8, A1 (2)(c) B1
M1
At x = 8 A1 (3)
GCE Core Mathematics C4 (6666) January 2011 43
[6]Question Number Scheme Marks
3. (a) M1 , , 2 or equivalent B1 M1; A1ft
or A1
A1 (6)
(b) B1ft (1)
(c) M1 A1 (2)
[9]
4. Can be implied. Ignore integral signs B1
=GCE Core Mathematics C4 (6666) January 2011 44
M1 A1
M1Leading to or equivalent
A1 (5) [5]
GCE Core Mathematics C4 (6666) January 2011 45
Question Number Scheme Marks
5. (a) Differentiating implicitly to obtain and/or M1
A1
or equivalent B1
M1
A1 (5)
(b) M1 Using or or M1Leading to
or M1 or or A1 A1
GCE Core Mathematics C4 (6666) January 2011 46
Substituting either of their values into to obtain a value of the other variable. M1
both A1 (7) [12]
Question Number Scheme Marks
6. (a) B1
M1 A1
M1
A1 (5)
(b) When can be implied B1
M1
M1GCE Core Mathematics C4 (6666) January 2011 47
A1 (4)
(c) M1 or equivalent
M1 A1 (3) [12]
Question Number Scheme Marks
7. (a) x 1 2 3 4y ln2 2ln8
0.6931 1.9605 3.1034 4.1589M1
B1 M1 7.49 cao
A1 (4)(b) M1 A1
GCE Core Mathematics C4 (6666) January 2011 48
M1 A1 (4)
(c) M1 Using or implying M1
A1 (3) [11]
Question Number Scheme Marks
8. (a) M1 A1 (2)
(b)
M1 A1ft (2)
(c) M1 A1
M1
GCE Core Mathematics C4 (6666) January 2011 49
Leading to A1
Position vector of P is M1 A1 (6) [10]
Question Number
C4 JANUARY 2012 MARK SCHEME Marks
1. (a)
not necessari
ly required.
At dM1 A1 cso
[5]
(b)So, m(N) = M1
N: M1
N: A1
[3]
(8
marks)
2. (a) M1 A1
GCE Core Mathematics C4 (6666) January 2011 50
A1
[3]
(b) M1 A1
A1 isw
Ignore subseque
nt working
[3]
(6
marks)
3. (a)or
B1
M1 A1ft
A1; A1
[5]
(b)Can be implied by
later work even in part (c).
M1
x terms:
giving, A1
[2]
(c) terms:
M1
GCE Core Mathematics C4 (6666) January 2011 51
So, or or A1
[2]
(9
marks
)
4.
Volume Use of . B1
M1
A1
Substitutes limits of 2 and 0
and subtracts the correct way round.
dM1
So Volume or
A1 oe isw
[5]
(5
marks
)
5. (a) ,
B1 B1
So,
B1
oe
(3)
(b)M1 oe
M1
GCE Core Mathematics C4 (6666) January 2011 52
A1 A1 A1
(5)
(8
mark
s)
6. (a)0.73508 B1
cao (1)
(b)B1 M1
(4 dp)
awrt 1.1504A1 (3)
(c) B1
B1
M1
dM1
AG
A1 cso (5)
(d)
Applying limits
and either way round.
M1
or
or awrt
A1
GCE Core Mathematics C4 (6666) January 2011 53
14
4371
d
l
C
109
D
BA
ˆBAD
awrt 0.077
or awrt 6.3(%)
A1 cso (3)
(12
mark
s)
Question Number
Scheme Marks
7.
(a) M1; A1
[2]
(b)
or
M1 A1ft
[2]
Let d be the shortest distance from C to l.
(c) M1
Applies dot product formula
between
their
M1
GCE Core Mathematics C4 (6666) January 2011 54
and their
Correct followed through expression
or equation.A1
awrt 109
A1 cso AG
[4]
(d) M1
So, A1
[2]
(e)M1; dM1 A1
[3]
(f) or M1
awrt 3.54 A1
[2]
(15
marks)
Question Number
Scheme Marks
8. (a) Can be implied. M1
Either one. A1
giving
A1 cao, aef
GCE Core Mathematics C4 (6666) January 2011 55
[3]
(b) B1
M1*
A1ft
dM1*
eg:
Using any of the subtraction (or
addition) laws for logarithms
CORRECTLY
dM1*
eg: or eg: Eliminate ln’s correctly.
dM1*
gives
Make P the subject.
dM1*
or
etc.
A1
[8]
(c) . So population cannot exceed 5000.
B1
[1]
(12
marks)
GCE Core Mathematics C4 (6666) January 2011 56
Question Number C4 JUNE 2011 MARK SCHEME Marks
1. B1
M1
Any two of A, B, C A1
terms All three correct A1 (4) [4]
2. M1 , or
B1
n not a natural number, M1
ft their A1 ft
A1GCE Core Mathematics C4 (6666) January 2011 57
A1 (6) [6]
3. (a) or equivalent M1 A1
At , M1 A1 (4)
(b) or M1
At , awrt 0.031A1 (2) [6]
Question Number Scheme Marks
4. (a) , awrt , B1 B1 (2)GCE Core Mathematics C4 (6666) January 2011 58
(b) B1 M1 Accept 1.3 A1 (3)
(c) B1 B1 M1Hence cso
A1 (4)
(d) M1 A1
M1 A1
GCE Core Mathematics C4 (6666) January 2011 59
= M1 A1 (6)
[15]
Question Number Scheme Marks
5. … B1
… M1 A1At , M1leading to Accept A1
At M1
A1 (7) [7]
Question Number Scheme Marks
GCE Core Mathematics C4 (6666) January 2011 60
6. (a) i: j: Any two equations M1 leading to , M1 A1
or M1 A1
k: , B1 (6) (As LHS = RHS, lines intersect)
(b) M1 A1 Acute angle is awrt 69.1 A1 (3)
(c) B1 (1)
(d) Let d be shortest distance from B to
M1
= awrt 7.5 A1
GCE Core Mathematics C4 (6666) January 2011 61
M1 awrt 6.99 A1 (4)
[14]
Question Number Scheme Marks
7. (a) or M1 awrt 1.05
A1 (2)
(b) , M1 A1At P, Can be implied A1 Using , M1 For normal M1 At Q, leading to 1.0625 A1 (6)
GCE Core Mathematics C4 (6666) January 2011 62
(c) M1 A1
A1 M1 A1
M1 A1 (7)
[15]
Question Number Scheme Marks
(a) M1 A1 (2)
(b) B1
M1
GCE Core Mathematics C4 (6666) January 2011 63
Using M1leading to A1 M1
or equivalent A1 (6) [8]
C4 MARK SCHEME JANUARY 2011Questio
n Number
Scheme Marks
1. M1 A1 A1
M1
M1 A1
[6]
2. M1 A1
GCE Core Mathematics C4 (6666) January 2011 64
At M1
M1 A1
[5]
Question
NumberScheme Marks
3.
(a) M1 A1
A1 (3)
(b)
ft constants M1 A1ft A1ft
(3)
(c) M1
M1 A1
depends on first two Ms in (c) M1 dep
GCE Core Mathematics C4 (6666) January 2011 65
Using depends on first two Ms in (c) M1 dep
A1 (6)
[12]
Question
NumberScheme Marks
4.
(a) M1 A1 (2)
(b) M1 A1ft (2)
or (c)
or B1
M1
Leading to M1 A1
(4)
(d) M1
accept A1
GCE Core Mathematics C4 (6666) January 2011 66
awrt 6.8 (2)
[10]
Question
NumberScheme Marks
5.
(a) B1
M1 A1
M1 A1
(5)
(b)
Coefficient of x; M1
Coefficient of ; A1 either M1 A1
GCE Core Mathematics C4 (6666) January 2011 67
correct Leading to M1 A1 (5)
(c) Coefficient of is M1 A1ft
cao A1 (3)
[13]
GCE Core Mathematics C4 (6666) January 2011 68
Question
NumberScheme Marks
6.
(a) , M1 A1
Using , at M1 A1
M1 A1 (6)
(b) B1
M1 A1 (3)
(c) M1
M1
M1 A1
M1
A1
(6)
[15]
Alternative to (c) using parameters
M1
GCE Core Mathematics C4 (6666) January 2011 69
M1
M1 A1
The limits are and M1
A1
(6)
Question
NumberScheme Marks
7.
(a) awrt B1
awrt or B1
(2)
(b) B1 M1 A1ft
0.542 or 0.543 A1 (4)
(c) B1
M1
A1
GCE Core Mathematics C4 (6666) January 2011 70
M1 A1
, B1
M1
A1
(8)
[14]
Question Number C4 JUNE 2010 MARK SCHEME Marks
1. (a) accept awrt 4 d.p.B1 B1 (2)
(b)(i) B1 for
B1 M1 cao A1
(ii) B1 for B1 M1 cao A1 (6)
[8]GCE Core Mathematics C4 (6666) January 2011 71
2. B1 M1 A1 ft sign error A1ft or equivalent with u M1 cso A1 (6)
[6]
3. B1 M1 A1= A1Substituting M1 Accept exact equivalents
M1 A1 (7) [7]
GCE Core Mathematics C4 (6666) January 2011 72
Question Number Scheme Marks
4. (a) B1 B1
or equivalentM1 A1 (4)
(b) At , , B1
M1 A1
M1 M1 A1 (6)
[10]
5. (a) B1 M1 A1 A1 (4)
(b) M1 B1
GCE Core Mathematics C4 (6666) January 2011 73
B1
M1 ft their A1 ft stated or implied
A1 A1 (7) [11]
Question Number Scheme Marks
6. (a) M1 M1 cso
A1 (3)
(b) M1 A1 A1 M1 A1
M1
GCE Core Mathematics C4 (6666) January 2011 74
A1 (7) [10]
7. (a) j components M1 A1 Leading to accept vector forms A1 (3)
(b) Choosing correct directions or finding and M1
use of scalar productM1 A1
awrt A1 (4)
(c)
M1 A1 A1 , M1 A1 (5)
GCE Core Mathematics C4 (6666) January 2011 75
awrt 34 [12]
Question Number
Scheme Marks
8. (a) M1 A1GCE Core Mathematics C4 (6666) January 2011 76
B1 M1 Leading to cso
A1 (5)
(b) separating variables M1 M1 A1 When , M1 When awrt 10.4
M1 A1 (6)
[11]
GCE Core Mathematics C4 (6666) January 2011 77
Question
NumberC4 JANUARY 2010 MARK SCHEME Marks
Q1 (a) + … M1 A1
… A1; A1 (4)
(b) M1
csoA1 (2)
(c) M1
M1
cao A1 (3)
[9]
Q2 (a) 1.386, 2.291 awrt 1.386, 2.291 B1 B1 (2)
(b) B1
M1
ft their A1ft
GCE Core Mathematics C4 (6666) January 2011 78
(a) cao A1
(4)
(c)(i) M1 A1
M1 A1
(ii) M1
seen or implied M1
A1 (7)
[13]
GCE Core Mathematics C4 (6666) January 2011 79
Question
NumberScheme Marks
Q3 (a) M1 A1
Accept , A1 (3)
(b) At , M1
A1
awrt 0.349A1 (3)
(c) At , M1
M1
Leading to A1 (3)
[9]
GCE Core Mathematics C4 (6666) January 2011 80
A X
Y1l
2ld4 26
Question
Number
Scheme Marks
GCE Core Mathematics C4 (6666) January 2011 81
Q4 (a) A: Accept vector forms B1 (1)
(b) M1 A1
awrt 0.73A1 (3)
(c) X: Accept vector forms B1 (1)
(d) Either orderM1
caoA1 (2)
(e) M1
Do not penalise if consistent A1 (2)
incorrect signs in (d)(f) Use of correct right angled triangle M1
GCE Core Mathematics C4 (6666) January 2011 82
M1
awrt 27.9A1 (3)
[12]
GCE Core Mathematics C4 (6666) January 2011 83
Question
NumberScheme Marks
Q5 (a) M1
A1 (2)
(b) Integral signs not necessary B1
M1
ft their A1ft
, 3
28
23=9+61 n 1+C M1
A1
A1
(6)
[8]
GCE Core Mathematics C4 (6666) January 2011 84
Question
NumberScheme Marks
Q6 B1
B1
When M1
M1
awrt 0.299 A1
[5]
GCE Core Mathematics C4 (6666) January 2011 85
Question
NumberScheme Marks
Q7 (a) Any one correct value B1
At , Method for finding one value of x M1
At , At A, ; at B, Both A1
(3)
(b) Seen or implied B1
M1 A1
A1
M1
A1 (6)
[9]
GCE Core Mathematics C4 (6666) January 2011 86
Q8 (a) B1
M1
Use of M1
M1
M1
M1
A1 (7)
(b) M1
integral in (a) M1
their answer to A1ft (3)
GCE Core Mathematics C4 (6666) January 2011 87
part (a) [10]
GCE Core Mathematics C4 (6666) January 2011 88