Alevel_C1C2
Alevel_C1C2
Alevel_C1C2
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8<br />
Exam-style mark scheme<br />
Integration<br />
Question Scheme Marks<br />
Number<br />
1. f(x) = x 3 + x 2 + x + c M3<br />
18 = 8 + 4 + 2 + c, therefore c = 4 B1<br />
y = x 3 + x 2 + x + 4<br />
B1<br />
y = 27 + 9 + 3 + 4 = 43<br />
B1<br />
6<br />
4 3 2<br />
3 2<br />
2. (a) ( x x x ) x<br />
x x x<br />
∫ + − − 1 d = + − − x + c<br />
B1 M4 (5)<br />
4 3 2<br />
2<br />
−<br />
− − −<br />
(b) ( x x ) x x<br />
x<br />
∫ + d = −<br />
−<br />
2<br />
+ c<br />
B2 M2 (4)<br />
⎛Or − 1<br />
−<br />
1<br />
+ c⎞<br />
⎝ x 2x<br />
2 ⎠<br />
9<br />
3. f(x) = 2x 3 - 2x 2 + c M2<br />
f(-1) = -2 - 2 + c = 6 Þ c = 10<br />
A1<br />
f(x) = 2x 3 - 2x 2 + 10<br />
B1<br />
y = f(3) = 54 - 18 + 10 = 46<br />
A1<br />
5<br />
1<br />
4. (a) t2<br />
t<br />
2<br />
t2<br />
∫ d = + c<br />
3<br />
, ⎛<br />
or 2 t t<br />
⎝<br />
⎜<br />
3<br />
3<br />
(b) t2<br />
t<br />
2<br />
t2<br />
∫ d = + c<br />
5<br />
, ⎛<br />
or 2 2<br />
t<br />
⎝<br />
⎜<br />
5<br />
3<br />
5<br />
t<br />
⎞<br />
+ c<br />
⎠<br />
⎟ M1 A1 (2)<br />
⎞<br />
+ c<br />
⎠<br />
⎟ M1 A1 (2)<br />
(c) t t t t t<br />
t t t t<br />
∫ d + ∫ d =<br />
2<br />
+<br />
2<br />
+ c<br />
3 5<br />
5. y = ax x<br />
ax<br />
∫ d = + c<br />
2<br />
2<br />
2<br />
A2 (2)<br />
M1 A1<br />
6<br />
( ) and (-2, 8) in y = 2<br />
ax<br />
+ c<br />
Substitute 1,<br />
1 2<br />
2<br />
, to find a and c:<br />
1<br />
=<br />
a<br />
+ c, then 1 = a + 2c or a = 1 – 2c …. (*)<br />
2 2<br />
M2oe<br />
8 = 2a + c ….. (**) M2oe<br />
Substitute (*) in (**):<br />
2 – 4c + c = 8, then c = -2 and a = 5 B2ft<br />
2<br />
y =<br />
5x<br />
− 2 A1<br />
2<br />
9<br />
© Oxford University Press 2008<br />
Core C1
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