Mathematics · Oct / Nov 2022 · Paper 31 · Question 3
YMA01
As printed in the paper
The question as text
3. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. y A O (2, 0) x Figure 1 Figure 1 shows a sketch of part of the curve with equation y = f(x) where f(x) = (x − 2) 2 e 3x x ∈ The curve has a maximum turning point at A and a minimum turning point at (2, 0)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
3(a)[5]
(
)
2
f ( )
′
x
=
2(
x
−
2) e
×
3
x
+
x
−
2
×
3
e
3
x
(
)
f ( )
′
x
0
e
3
x
(
x
2
)
2
3
(
x
2
)
0
x
..
.
=
−
+
−
=
=
4
x
=
3
4
4 4
y =
4e
so is
A
,
e
4
9
3 9
4
3(b)[2]
k >
0
or
k „
their
e
4
or with … or < respectively.
9
4
0
<
k
„
the
ir
e
4
9
Notes:
(a)
(
)
2
M1: Differentiates using the product rule reaching the form
α
(
x
−
2) e
×
3
x
+
x
−
2
×
β
e
3
x
A1: Correct derivative, need not be simplified.
M1: Sets their derivative equal to zero and attempts to cancel or factorise out the exponential term and the
term (or multiple thereof) and solve for x. Alternatively, they may expand to a 3 term quadratic
(
x −
2)
(
)
2
3x
−
10
x
+
8
and attempt to solve this via correct method. Allow recovery on slips in the exponent for this
mark (and the A’s if a correct quadratic expression is solved).
A1: Correct x value.
A1: Correct simplified y value. Allow if seen in part (b). Isw, after a correct answer, but the follow through in
(b) will not apply on an incorrectly simplified answer if this mark has been awarded.
(b)
M1: Identifies one correct boundary for k (the direction of inequality must be correct). Accept in terms of y
or f(x) for this mark.
A1ft: Correct range for k following their positive y coordinate in (a). Must have correct inequalities. Accept
awrt 24.3 for the upper bound. Allow as separate inequalities, or set or interval notation.
y
=
log (2
x
+
1)
10
y
=
2
x
+
1
x
=
...
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