Mathematics · Oct / Nov 2020 · Paper 41 · Question 5
YMA01
As printed in the paper
The question as text
5. y R O 2 4 x Figure 3 ln x ∫
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
5(a)[3]
d
x
=
x
−
2
ln
x x
d
= −
x
−
1
ln
x
+
x
−
1
×
d
x
x
2
x
= −
x
−
1
ln
x
−
x
−
1
(
+
c
)
3
+
2
x
+
ln
x
5(b)[4]
=
3
x
−
2
+
2
x
−
1
+
x
−
2
ln
x
oe
x
2
4
3
ln
x
1
Area =
−
+
2ln ''
x
−
−
''
x
x
x
2
3
ln 4
1
3
ln 2
1
= −
+
2ln 4
−
−
−−
+
2ln 2
−
−
=1
2ln 2
+
4
4
4
2
2
2
(a)
∫
∫
1
M1: Attempts by parts to reach a form
x
−
2
ln
x x
d
ax
−
1
ln
x
b
x
−
1
d
x
where ,
= ±
±
×
a b ≠
x
If a formula is stated it must be correct.
dM1: Integrates again to reach a form
±
ax
−
1
ln
x
±
bx
−
1
ln
x
1
A1:
−
x
−
1
ln
x
−
x
−
1
(
+
c
)
o.e such as
−
−
+
c
x
x
(
)
Condone the omission of the constant of integration. Condone
−
x
−
1
ln
x
+ −
x
−
1
(b)
3
+
2
x
+
ln
x
3
2
ln x
B1:
=
3
x
−
2
+
2
x
−
1
+
x
−
2
ln
x
or
+
+
.
x
2
x
2
x
x
2
The 2
∫
2
x
may be implied by later work. That is if they state
d
2ln
or
ln x
2
x
=
x
x
x
2
4
3
ln
x
1
M1: Area =
2ln ''
x
''
following through on their part (a)
−
+
−
−
x
x
x
2
p
ln
x
s
4
M1: For substituting 2 and 4 into an expression of the form
+
q
ln
x
+
r
+
and attempts to
x
x
x
2
simplify by using one log law correctly. (Note that the two terms in 1
x may be combined before substitution)
A1: Exact answer: Likely to be 1
or 1
+
2ln 2
+
ln 4
........................................................................................................................................................................
For those who don't read the instruction in (b) to use (a) and do via parts score
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