Mathematics · Oct / Nov 2025 · Paper 31 · Question 7
YMA01
As printed in the paper
The question as text
7. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A curve C has equation 2 x k x y 2 x e ( 3 2 ) where k is a constant. Given that C has two distinct turning points, find the range of possible values of k. (7)
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Show the answer and mark scheme
d
y
(
)
=
2
+
−
+
2
+
−
×
+
−
2e
x
(3
k
2)
x
2 e
x
x
(3
k
2)
x
2
x
(3
k
2)
d
x
2
x
2
+
(3
k
−
2)
x
+ =
1
0
(3
k
−
2)
2
−××
4 2 1
(
)
2
−
−
>
9
k
12
k
4
0
12
±
( 12)
−
2
−××−
4 9 ( 4)
−
−
=
=
=
9
k
2
12
k
4
0
k
k
...
2 9
×
−
+
2
2 2
2
2 2
k
<
or
k
>
3
3
(
)
(
d
y
x
2
+
(3
k
−
2)
x
x
2
+
(3
k
−
2)
x
M1:
Attempts the product rule achieving
=
A
e
+
e
d
x
where f(x) is a quadratic function of x, condone missing/misplaced
brackets for the M.
2 (3
2ex
+
k
−
2)
x
2 (3
(
)
A1:
One of
or
2 e
x
x
+
k
−
2)
x
×
2
x
+
(3
k
−
2)
(unsimplified)
A1:
Correct unsimplified derivative
dM1: Sets their derivative equal to zero (may be implied), collects terms and
attempts
on a 3 term quadratic – must be extracted not just part
b
2
−
4
ac
of the quadratic formula. It is dependent on the previous method mark.
A1:
or equivalent quadratic expression or multiple thereof.
Need not be simplified.
M1:
Must have attempted
and reached a quadratic in k. Attempts to
solve their quadratic in k (including via a calculator) and attempts the
outside region for their critical values (accept with endpoints included.).
Accept with decimals answer correct to 3 s.f. for this mark.
2
−
2 2
2
+
2 2
A1:
<
or
>
or exact equivalent. Allow with “and” instead of
k
k
3
3
“or” when given as two ranges, but use of set notation must use union, not
intersection. E.g.
Alternative method via implicit differentiation scores the same way with
M1A1A1 for correct differentiation. M1 for a correct form of derivative
statement up to constant/sign errors.
NB: method must be seen for the dM mark, if critical values appear from a
derivative statement with no method shown (by calculator) then it is dM0. The
discriminant must be considered. But the last mark is not dependent, so may
be scored for selecting the correct “outside” region for their cvs.
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