Mathematics · Oct / Nov 2023 · Paper 31 · Question 7
YMA01
As printed in the paper
The question as text
7. y O x Figure 1 Figure 1 shows a sketch of the curve C with equation y = f (x) where 2 2 x f ( ) x e 2 x 2 3
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[2]
f(0)
=
(0
−
3)
2
=
9
0
f( )
x
9
7(b)[4]
f '( )
x
= −
2 e
x
−
x
2
(2
x
2
−
3)
2
+
e
−
x
2
×
8 (2
x
x
2
−
3)
(
)
(
)
2
2
=
2 (2
x
x
2
−
3)e
−
x
−
(2
x
2
−
3)
+
4
=
2 e
x
−
x
(2
x
2
−
3) 7
−
2
x
2
7(c)[4]
3 7
7
7
7
2
7
−
−
x =
2
,
f
=
e
2
×
2
−
3
=
16e
2
2
2
2
2
7
−<
16e
2
k
<
9
Notes
(a) Work for part (a) must be seen in part (a) not recovered in part (c).
M1:
Substitutes
x =
0
and proceeds to find a value for y. Implied by sight of 9.
A1:
0
f( )
x
9
Accept with y or just f, but not with x. Accept interval, [0, 9], or set notation.
(b)
M1:
Attempts the product rule and chain rule achieving
±
P
x
e
−
x
2
(2
x
2
±
3)
2
+
e
−
x
2
×
Qx
(2
x
P, Q > 0. Alternatively, attempts the quotient rule on (
)
2
2
x −
2
3
achieving
2
e
x
(
)
(
)
Px
2
x
2
±
3 e
x
2
−
Qx
e
x
2
2
x
2
±
3
2
P, Q > 0
(
)
2
e
2
x
A1:
2
2
oe (need not be simplified)
−
2 e
x
−
x
(2
x
2
−
3)
2
+
e
−
x
×
8 (2
x
x
2
−
3)
dM1: Must have scored previous M. Attempts to takes out a factor of
2 e
x
−
x
2
(2
x
2
3)
to obtain a
−
factor (
)
±
C
±
Dx
2
which may be unsimplified. Allow if e.g. the x or
− is dropped when
2
e
x
taking out the factor.
A1:
Achieves
2 e
x
−
x
2
(2
x
2
−
3)(7
−
2
x
2
)
with no errors seen.
(c) Note : allow all the marks in (c) from answers which were a constant multiple out in (b) or
missing the factor x, which lead to the correct answers here.
M1:
Attempts to find a y value (which may be zero) for at least one of the valid nonzero roots for
3
A
f’(x)=0. Allow for an attempt at either
or f
leading to a value, where A and B
f
±
±
2
B
A
are their values from (b) with AB > 0. Note e.g. f
attempted is M0.
B
7
− Accept awrt 0.483 for this mark following the award of M.
A1:
For obtaining
16e
2
dM1: Attempts the inside region, allowing
, using their y intercept from (a) and their positive y
value, less than their intercept, from an attempt at the at the y value of the maxima. Allow a
positive y value from an attempt at any nonzero root of f’(x) as such an attempt. It is dependent
on the previous method mark.
16 ,9
k ∈
7
A1:
or in any equivalent form e.g. interval notation
Allow y instead of k
−<
16e
2
k
<
9
72
e
Ignore references to “k = 0”
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