Question 11:
--- 11(a) ---
11(a) | f(x)
Use correct quotient rule NB the question asks for so need complete form | M1 | Or correct product rule.
4e2x(e2x −3ex +2)−2e2x(2e2x −3ex)
Obtain correct derivative in any form, e.g.
( )2
e2x −3ex +2 | A1
Equate their derivative to zero | *M1 | Can be implied by numerator equated to zero for
( )
quotient rule. 8=6ex
Solve for x to obtain x = lna | DM1 | a positive.
4
Obtain x =ln and y = –16
3 | A1 | No errors seen.
8
Accept equivalent exact forms, e.g. x=ln .
6
Question | Answer | Marks | Guidance
11(a) | Alternative Method for Question 11(a)
f(x)
Complete method to express in partial fractions | M1 | q r
As far as p+ + with values for p,
ex −2 ex −1
 8 2 
q and r 2+ − .
 ex −2 ex −1
( )
Allow in u u=ex .
sex tex
Differentiate to obtain f(x)= +
( )2 ( )2
ex −2 ex −1 | *M1 | Note: the question requires f’(x) so if they have
substituted for ex, they will also need chain rule.
−8ex 2ex
Obtain f(x)= +
( )2 ( )2
ex −2 ex −1 | A1 | From correct work.
Equate derivative to zero and solve for x to obtain x = lna | DM1 | Must follow correctly to give a positive value of
a.
4
Obtain x =ln and y = –16
3 | A1 | No errors seen.
8
Accept x=ln ,or equivalent.
6
5
Question | Answer | Marks | Guidance
--- 11(b) ---
11(b) | du
State or imply =ex
dx | B1
2u
Obtain  du or equivalent
u2 −3u+2
2  8 2
Or  +  − du
1 u u ( u−2 ) u ( u−1 ) | B1 | Correct expression in u.
Condone missing du or missing integral but not
both.
Allow FT if using their partial fractions from
(a).
A B
State or imply partial fractions of the form +
u−1 u−2
C D E 2u−3 3 2u−3 F G
Or + + Or + = + +
1 2
u u−2 u−1 u2 −3u+2 u2 −3u+2 u2 −3u+2 u−2 u−1 | B1 FT | Complete reduction to partial fractions.
Correct form for their integrand.
Use a correct method for finding a constant | M1 | Available if they have incorrect form.
−2 4
Obtain correct +
u−1 u−2
2u−3 3 3
Or + −
2
u2 −3u+2 u−2 u−1 | A1
Integrate to obtain aln(u – 1) +bln(u – 2) or equivalent | *M1 | M0 if they have additional terms that do not
cancel out.
Obtain correct –2ln(u – 1) + 4ln(u – 2) or equivalent | A1FT | FT values of their partial fraction coefficients.
Correctly use limits u = 5 and 3 in an expression of the form aln(u – 1) + bln(u – 2)
or x = ln5 and ln3 in an expression of the form aln(ex – 1) + bln(ex – 2) | DM1
81
Obtain ln
4 | A1 | Accept ln20.25.
9