FOR EDEXCEL
GCE Examinations
Advanced Subsidiary
Core Mathematics C3
Paper H
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.
C3 Paper H – Marking Guide
1.
(a)
(b)
= f(2) = −2
gf(x) = g(2 − x2) =
6 − 3x 2
∴
5(sec2 2θ
5 sec2 2θ
(5 sec 2θ
sec 2θ =
=
3 − 2 x2
x2 =
2.
M1 A1
9
4
1
2
2
3(2 − x )
2(2 − x 2 ) − 1
6 − 3x
=
2
M1 A1
3 − 2 x2
2(6 − 3x2) = 3 − 2x2
,
x = ± 32
,
M1 A1
− 1) − 13 sec 2θ = 1
− 13 sec 2θ − 6 = 0
+ 2)(sec 2θ − 3) = 0
− 52 or 3
M1
M1
A1
cos 2θ = − 52 (no solutions) or
1
3
2θ = 70.529, 360 − 70.529, 360 + 70.529, 720 − 70.529
= 70.529, 289.471, 430.529, 649.471
θ = 35.3°, 144.7°, 215.3°, 324.7° (1dp)
3.
(2 x − 3)( x + 3)
(2 x − 3)( x − 2)
(a)
=
(b)
ln (2x2 + 3x − 9) − ln (2x2 − 7x + 6) = 2,
ln
x+3
x−2
= 2,
2
B1 M1
A2
x+3
x−2
=
x+3
x−2
4.
(a)
2 x2 + 3x − 9
ln
2x2 − 7 x + 6
=2
= e2
M1
A1
e2 − 1
( 3π
, 5)
2
B3
x
( π2 , −1)
(
3π
2
⇒ −1 = a + b
, −5) ⇒ −5 = a − b
adding,
B1
−6 = 2a ∴ a = −3, b = 2
−3 + 2 cosec x = 0, cosec x =
x = 0.73, π − 0.7297,
5.
3
2
,
sin x =
M1 A1
2
3
x = 0.73, 2.41 (2dp)
⇒ 18 000 = 2000e3k, e3k = 9
k = 13 ln 9 = 0.732 (3sf)
M1
A2
(a)
t = 3, N = 18 000
(b)
4000 = 2000e0.7324t
1
t = 0.7324
ln 2 = 0.9464 hours
M1
M1 A1
∴ doubles in 57 minutes (nearest minute)
A1
(c)
(7)
y
O
(c)
M1
A1
2e 2 + 3
( π2 , 1)
(b)
(7)
M1 A2
x + 3 = e (x − 2)
3 + 2e2 = x(e2 − 1)
x=
(6)
N = 2000e0.7324t,
when t = 3,
dN
= 0.7324 × 2000e0.7324t = 1465e0.7324t
dt
M1
M1 A1
M1 A1
dN
= 13 200 ∴ increasing at rate of 13 200 per hour (3sf) A1
dt
 Solomon Press
C3H MARKS page 2
(9)
(10)
6.
(a)
d
d
(sec x) =
[(cos x)−1]
dx
dx
= −(cos x)−2 × (−sin x)
=
sin x
cos 2 x
=
1
cos x
×
M1 A1
sin x
cos x
= sec x tan x
(b)
(c)
7.
A1
dy
= 2e2x × sec x + e2x × sec x tan x = e2x sec x (2 + tan x)
dx
x = 0, y = 1, grad = 2
∴ y = 2x + 1
e2x sec x (2 + tan x) = 0
tan x = −2
x = −1.11 (2dp)
SP:
M1 A1
M1
A1
M1
M1
A1
(11)
(a)
f(x) = (x − 1)2 − 1 + 5 = (x − 1)2 + 4
M1 A1
(b)
f(x) ≥ 4
B1
(c)
y = (x − 1)2 + 4
(x − 1)2 = y − 4
x − 1 = ± y−4
M1
x=1±
y−4
−1
x−4
f (x) = 1 +
(d)
(e)
M1 A1
translation by 4 units in negative x direction
translation by 1 unit in negative y direction (either first)
dy
−1
= 12 ( x − 4) 2
dx
x = 8, y = 3, grad =
(a)
(b)
(c)
1
4
A1
[ y = 35 − 4x ]
dy
= −e2x−2 + ex
dx
M1 A1
M1
A1
e2, grad =
M1
x = 2, y =
y=
3
2
3
4
3
2
2
e =
3
4
(12)
M1 A1
SP: −e2x−2 + ex = 0
let f(x) = −e2x−2 + ex
f(1.3) = −0.70, f(1.4) = 0.29
sign change, f(x) continuous ∴ root
∴ y−
(d)
B2
M1
∴ grad of normal = −4
∴ y − 3 = −4(x − 8)
8.
M1
3
4
e2
2
e (x − 2)
M1
M1 A1
2
ex
∴ x = 0 ⇒ y = 0 so passes through origin
A1
x1 = −1.125589, x2 = −1.125803, x3 = −1.125804 (7sf)
∴ x-coordinate of B = −1.1258 (5sf)
M1 A2
A1
(13)
Total
(75)
 Solomon Press
C3H MARKS page 3
Performance Record – C3 Paper H
Question
no.
Topic(s)
Marks
1
2
functions trigonometry
6
7
3
4
5
7
8
Total
rational
functions, exponentials differentiation functions, differentiation,
expressions, trigonometry
and
differentiation numerical
exponentials
logarithms,
methods
and
differentiation
logarithms
7
9
10
Student
 Solomon Press
C3H MARKS page 4
6
11
12
13
75