FOR EDEXCEL
GCE Examinations
Advanced Subsidiary
Core Mathematics C3
Paper F
MARKING GUIDE
This guide is intended to be as helpful as possible to teachers by providing
concise solutions and indicating how marks could be awarded. There are
obviously alternative methods that would also gain full marks.
Method marks (M) are awarded for knowing and using a method.
Accuracy marks (A) can only be awarded when a correct method has been used.
(B) marks are independent of method marks.
Written by Shaun Armstrong
 Solomon Press
These sheets may be copied for use solely by the purchaser’s institute.
© Science Exam Papers
C3 Paper F – Marking Guide
1.
2.
3.
3
sin θ
= −8 cos θ
M1
3 = −8 sin θ cos θ = −4 sin 2θ
sin 2θ = − 34
M1
A1
2θ = 180 + 48.590, 360 − 48.590 = 228.590, 311.410
θ = 114.3, 155.7 (1dp)
M1
A2
(a)
g(x) = (x + a)2 − a2 + 2
∴ g(x) ≥ 2 − a2
M1 A1
A1
(b)
gf(3) = g(1 − 3a) = (1 − 3a)2 + 2a(1 − 3a) + 2
∴ 1 − 6a + 9a2 + 2a − 6a2 + 2 = 7, 3a2 − 4a − 4 = 0
(3a + 2)(a − 2) = 0
a = − 23 , 2
M1
A1
M1
A1
(a)
3x + 1 = e2
x = 13 (e2 − 1)
M1
M1 A1
(b)
consider ln (3x2 + 5x + 3) ≥ 0
⇒
3x2 + 5x + 3 ≥ 1
3x2 + 5x + 2 ≥ 0
(3x + 2)(x + 1) ≥ 0
−1 − 23
M1
A1
∴ if (e.g.) x = − 34 , ln (3x2 + 5x + 3) = ln
∴ if x = − 34 , ln (3x2 + 5x + 3) < 0
4.
(a)
dx
=1×
dy
1 − 2y + y ×
1 − 2y −
=
y
1− 2 y
1
−
1
(1 − 2 y ) 2
2
(1 − 2 y ) − y
1− 2y
=
15
16
= −0.0645...
M1
∴ statement is false
A1
× (−2)
=
1− 3y
1− 2 y
M1
M1 A1
y = −1, x = − 3 , grad =
∴ y+1=
1
4
3 (x +
1
4
B1
3)
M1
3 x − 4y − 1 = 0
(a)
3
3x+3
4y + 4 =
5.
[ p = −4, q = −1 ]
A1
(8)
y
( π6 , 3)
( 7π
, 1)
6
O
x
x=
(b)
(8)
M1 A1
dy
dx
1− 2 y
=1÷
=
1− 3y
dy
dx
(b)
(7)
M1
2
3
x ≤ −1 or x ≥ −
(6)
2 + sec (x −
x−
π
6
=π−
x=
5π
6
3π
2
,
2π
3
x=
π
6
) = 0,
π
3
,π+
sec (x −
π
3
=
2π
3
,
M2 A3
5π
3
π
6
) = −2,
cos (x −
4π
3
π
6
) = − 12
M1
B1 M1
A2
(10)
 Solomon Press
C3F MARKS page 2
© Science Exam Papers
6.
(a)
y
(−3, 6)
(3, 6)
(0, 4)
O
(b)
x
y
B3
(c)
y
(−3, 3)
(0, 8)
(0, 2)
O ( 32 , 0) x
M2 A2
( − 92 , 2)
O
7.
(a)
(b)
f(x) = 1 +
y=
x
4x
2x − 5
15
(2 x − 5)( x − 1)
−
=
2 x 2 − 7 x + 5 + 4 x( x − 1) − 15
(2 x − 5)( x − 1)
=
6 x 2 − 11x − 10
(2 x − 5)( x − 1)
3x + 2
,
x −1
M2 A2
=
B1
M1 A1
(3x + 2)(2 x − 5)
(2 x − 5)( x − 1)
=
3x + 2
x −1
M1 A1
y(x − 1) = 3x + 2
M1
x(y − 3) = y + 2
M1
y+2
y−3
x=
x+2
x−3
3( x − 1) + 5
=
x −1
∴ f −1(x) =
f(x) =
A1
3+
5
x −1
M1
x < 1 ∴ f(x) < 3 ∴ domain of f −1(x) is x ∈
8.
, x<3
A1
−1
(c)
f(x) = 2 ⇒ x = f (2) = −4
(a)
dy
= 2x −
dx
1
(4 + ln
2
x)
x = 1, y = −1, grad =
∴ y+1=
7
4
− 12
×
M1 A1
1
x
= 2x −
1
2 x 4 + ln x
7
4
A1
(x − 1)
SP:
2x −
1
2 x 4 + ln x
let f(x) = 2x −
M1
A1
=0
1
2 x 4 + ln x
M1
, f(0.3) = −0.40, f(0.4) = 0.088
M1
sign change, f(x) continuous ∴ root
(c)
2x −
1
2 x 4 + ln x
=0
⇒
A1
2x =
1
2 x 4 + ln x
x2 =
1
4 4 + ln x
=
1
(4 + ln
4
x)
x=
(d)
(12)
M1 A1
4y + 4 = 7x − 7
7x − 4y = 11
(b)
(11)
1
(4 + ln
4
− 12
=
x)
− 12
1
(4 + ln
2
M1
x)
x1 = 0.38151, x2 = 0.37877, x3 = 0.37900, x4 = 0.37898 (5dp)
− 14
A1
M1 A2
(13)
Total
(75)
 Solomon Press
C3F MARKS page 3
© Science Exam Papers
Performance Record – C3 Paper F
Question
no.
Topic(s)
1
2
3
4
5
6
7
8
Total
rational differentiation,
trigonometry functions exponentials differentiation trigonometry functions
numerical
expressions,
and
methods
functions
logarithms,
proof
Marks
6
7
8
8
10
11
12
13
75
Student
 Solomon Press
C3F MARKS page 4
© Science Exam Papers