 ```FOR OCR
GCE Examinations
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 using a valid 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.
C3 Paper F – Marking Guide
1.
3
= [ 29 (3x − 2) 2 ] 62
=
2.
3.
2
9
(i)
(64 − 8) =
M1 A1
12 94
= −3(2 x − 7)
M1 A1
− 32
×2=−
6
M1 A1
3
(2 x − 7) 2
(ii)
= 2x × e−x + x2 × (−e−x) = xe−x(2 − x)
M1 A2
(i)
LHS ≡
M1 A1
2 (cos x cos 45 − sin x sin 45) + 2(cos x cos 30 + sin x sin 30)
≡
2(
1
2
cos x −
≡ cos x − sin x +
(ii)
let x = 75,
(4)
1
2
sin x) + 2(
3
2
cos x +
3 cos x + sin x ≡ (1 +
1
2
sin x)
(5)
M1
3 ) cos x ≡ RHS
A1
2 cos 120° + 2 cos 45° = (1 + 3 ) cos 75°
2 ( − 12 ) + 2( 1 ) = (1 + 3 ) cos 75°
M1
2
1
2
cos 75° =
4.
2 = (1 +
2
2(1 + 3)
3 ) cos 75°
1
2+ 6
=
A1
(i)
f(1) = 2.30, f(1.5) = −18.5
sign change, f(x) continuous ∴ root
(ii)
x2 + 5x − 2 sec x = 0 ⇒ x2 + 5x =
cos x =
5.
(i)
2
cos x
2
x2 + 5x
2
−1
(ii)
x2 + 5x
M1
M1
) ∴ xn + 1 = cos−1 (
6.
)
A1
M1 A1
A1
= f(2) = 2 + ln 4
M1 A1
f ′(x) =
×3=
3
3x − 2
(i)
A1
M1 A1
3x − 2 = ey − 2
f −1(x) = 13 (2 + ex − 2)
y = 2 + ln (3x − 2),
x = 13 (2 + ey − 2),
(8)
M1
x = 1, y = 2, grad = 3
y − 2 = 3(x − 1)
[ y = 3x − 1 ]
(iii)
2
xn 2 + 5 xn
x0 = 1.25, x1 = 1.31191, x2 = 1.32686, x3 = 1.33024,
x4 = 1.33100, x5 = 1.33116, x6 = 1.33120
∴ α = 1.331 (3dp)
1
3x − 2
(7)
M1
A1
x = cos (
(iii)
M1
M1
A1
(8)
y
y = 3x + 5a
B3
y = x − a
(0, 5a)
B2
O
( − 53 a, 0)
(−a, 0)
(ii)
−x − a = 3x + 5a
(a, 0)
(0, −a)
x
⇒ x = − 32 a
−x − a = −(3x + 5a) ⇒ x = −2a,
M1 A1
x = −2a, − 32 a
 Solomon Press
C3F MARKS page 2
M1 A1
(9)
7.
(i)
=
4
∫2
1x
(2x − e 2 ) dx
1x
= [x2 − 2e 2 ] 42
= (16 − 2e2) − (4 − 2e) = 12 + 2e − 2e2
(ii)
V = π∫
4
M1 A1
M1 A1
1x
(2x − e 2 )2 dx
2
M1
2
x
1x
2
3
4
2
(2x − e ) 1.6428 2.3053 0.3733
I ≈ 13 × 1 × [1.6428 + 0.3733 + 2(2.3053)] = 3.7458
∴ V ≈ 3.7458π = 11.8 (3sf)
8.
(i)
M1
M1 A1
A1
(9)
y
−1
y = cos (2x)
y = sin−1 x
O
(ii)
(iii)
x
b = sin−1 a
⇒ a = sin b
−1
b = cos 2a ⇒ 2a = cos b
∴ 2 sin b = cos b
sin b
cos b
=
1
2
tan b =
1
2
tan2 b =
2
cos b =
cos b = ±
1
2
A1
1
4
=
5
4
M1
4
5
2
5
A1
cos b = ±
1
5
M1
1
5
from diagram, a > 0 ∴ a =
9.
M1
M1
1
4
sec2 b = 1 +
a=
B3
=
1
5
5
A1
(i)
f(x) > −2
B1
(ii)
x = 0, y = e − 2 ∴ P (0, e − 2)
y = 0, 0 = e3x + 1 − 2
3x + 1 = ln 2
x = 13 (ln 2 − 1) ∴ Q ( 13 (ln 2 − 1), 0)
B1
M1
A1
(iii)
f ′(x) = 3e3x + 1
∴ y − (e − 2) = 3e(x − 0)
y = 3ex + e − 2
M1
A1
M1
A1
(iv)
tangent at Q: y − 0 = 6(x −
1
3
(ln 2 − 1))
y = 6x − 2 ln 2 + 2
intersect: 3ex + e − 2 = 6x − 2 ln 2 + 2
x(3e − 6) = 4 − e − 2 ln 2
x=
4 − e − 2 ln 2
3e − 6
= −0.0485 (3sf)
(10)
B1
M1
M1
A1
(12)
Total
(72)
 Solomon Press
C3F MARKS page 3
Performance Record – C3 Paper F
Question no.
Topic(s)
1
2
3
4
5
6
7
8
9
Total
integration differentiation trigonometry numerical functions functions integration, trigonometry functions,
methods
Simpson’s
exponentials
rule
and
logarithms
Marks
4
5
7
8
8
Student
 Solomon Press
C3F MARKS page 4
9
9
10
12
72
``` # AS Entrance Examination Sample Paper Mathematics Instructions   