FOR EDEXCEL
GCE Examinations
Advanced Subsidiary
Core Mathematics C3
Paper E
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 E – Marking Guide
1.
2.
=
x 2 (2 x + 1)
( x + 2)( x − 2)
=
x2
( x + 2)( x − 3)
(a)
x−2
(2 x + 1)( x − 3)
×
M1 A2
M1 A1
sin x
cos x
LHS ≡ 2 sin x cos x −
M1
2
≡ 2sin x cos x − sin x
M1 A1
cos x
2
sin x
≡ sin x(2 cos x − 1) ≡
× cos 2x ≡ tan x cos 2x ≡ RHS
3.
tan x cos 2x = 2 cos 2x
cos 2x (tan x − 2) = 0
cos 2x = 0 or tan x = 2
2x = 90, 270 or x = 63.4
x = 45°, 63.4° (1dp), 135°
M1
A1
B1
M1 A1
(a)
f(1) = 2.30, f(1.5) = −18.5
sign change, f(x) continuous ∴ root
M1
A1
(b)
x2 + 5x − 2 sec x = 0 ⇒
2
cos x
2
x2 + 5x =
cos x =
(c)
(a)
(b)
(a)
A1
M1
1
−
1
(1 − cos x) 2
2
× sin x =
=
(ii)
= 3x2 × ln x + x3 ×
dx
=
dy
1
x
1× (3 − 2 y ) − ( y + 1) × (−2)
(3 − 2 y )2
1
5
sin x
2 1 − cos x
(11)
M1 A2
= x2(3 ln x + 1)
M1 A2
5
=
M1
A1
M1 A2
(3 − 2 y )2
(3 − 2y)2
M1 A1
(11)
3 sin θ + cos θ = R sin θ cos α + R cos θ sin α
tan α =
(c)
2
x2 + 5x
f ′(x) = 2x + 5 − 2 sec x tan x
SP: 2x + 5 − 2 sec x tan x = 0
f ′(1.05345) = 0.00046, f ′(1.05355) = −0.0022
sign change, f ′(x) continuous ∴ root ∴ x-coord of P = 1.0535 (5sf)
R cos α =
(b)
∴ g(x) =
x2 + 5x
M1 A2
dy
dx
=1÷
=
dy
dx
5.
M1
x1 = 1.3119, x2 = 1.3269, x3 = 1.3302, x4 = 1.3310 = 1.331 (3dp)
(i)
(10)
M1
x2 + 5x
2
x = arccos
4.
M1 A1
cos x
cos x
(b)
(5)
1
3
3 , R sin α = 1
, α=
π
6
maximum = 2
occurs when θ +
π
6
2 sin (θ +
3 = 0,
θ +
π
6
π
6
)+
= − π3 , −π +
∴
=
π
3
π
2
∴ R=
3 +1 = 2
3 sin θ + cos θ = 2 sin (θ +
, θ =
M1 A1
π
6
)
M1 A1
B1
M1 A1
π
3
sin (θ +
π
6
)=−
3
2
= − π3 , − 2π
3
M1
B1 M1
θ = − 5π6 , − π2
A2
(12)
 Solomon Press
C3E MARKS page 2
© Science Exam Papers
6.
(a)
f(x) ≤ 3
B1
(b)
y
y = f(x)
y = f −1(x)
O
(c)
B3
x
y = 3 − x2
x2 = 3 − y
x = ± 3− y
M1
f −1(x) =
M1 A2
3− x , x ∈ , x ≤ 3
4
3
(d)
= f( ) =
(e)
3− x =
3−x=
11
9
M1 A1
8
3− x
64
M1
(3 − x )2
(3 − x)3 = 64
3−x=4
x = −1
7.
(a)
M1
A1
10k
M1
⇒
A=
sub (2)
⇒
7 = 13e10k × e−60k
7
e−50k = 13
A1
= 0.0124 (3sf)
M1 A1
1
− 50
7
ln 13
10 × 0.01238
∴ A = 13e
(c)
13
(1)
(2)
(1)
∴ k=
(b)
18 = 5 + Ae−10k
12 = 5 + Ae−60k
⇒
⇒
t = 10, T = 18
t = 60, T = 12
(13)
e−10 k
= 13e
M1
= 14.7 (3sf)
A1
T = 5 + 14.71e−0.01238t
dT
= −0.01238 × 14.71 e−0.01238t = −0.1822e−0.01238t
dt
dT
= −0.1822e−0.01238 × 20 = −0.142
when t = 20,
dt
∴ temperature decreasing at rate of 0.142 °C per minute (3sf)
−0.01238(t − 60)
T = 5 + 14.71e
= 5 + 14.71e0.7428 − 0.01238t
= 5 + 14.71e0.7428 × e−0.01238t
= 5 + 30.9e−0.01238t,
B = 30.9 (3sf)
M1 A1
M1
A1
M1
M1
A1
(13)
Total
(75)
 Solomon Press
C3E MARKS page 3
© Science Exam Papers
Performance Record – C3 Paper E
Question
no.
Topic(s)
Marks
1
2
rational
trigonometry
expressions
5
10
3
4
5
6
numerical differentiation trigonometry functions
methods,
differentiation
11
11
12
13
7
Total
exponentials
and
logarithms,
differentiation
13
75
Student
 Solomon Press
C3E MARKS page 4
© Science Exam Papers