Mathematics · Oct / Nov 2025 · Paper 11 · Question 8
YMA01
As printed in the paper
The question as text
8. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A curve has equation y = f(x), x > 0 Given that 8 • x x k where k is a constant f ( ) 2 , x 2 • the equation of the tangent to the curve at x = 2 is y x 5 3 2
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8(a)[2]
(
)
8
f
′
2
=
5
2 2
+
+
k
=
5
k
=
...
2
1 2 2
−
(
)
8(b)[2]
=
=
−
=
x
2
y
5 2
3 2
2 2
1
(
)
−
= −
−
So
y
"2 2"
x
2
5
or
1
1
= −
+
= −
+
=
y
x
c
"2 2"
2
c
c
...
5
5
1
11 2
= −
+
y
x
5
5
Notes
Mark all parts together.
(a)
( )
x =
2
′
M1: Substitutes
into
f
x
, sets = 5 and solves for k.
Condone slips in evaluating (
)
f
′
2
as long as the intention is clear.
k = −
−
A1:
Correct exact value
1 2 2
oe e.g. 1
8
(b)
1
M1: Correct attempt at the equation of the normal using a gradient of
−
,
x =
2
and y from
5
x =
=
−
substituting
2
into
y
5
x
3 2
If they use y = mx + c then they must reach as far as c = …
(
)
1
A1:
Correct equation in any form e.g.
y
−
2 2
= −
x
−
2
or 5
y
+
x
=
11 2
or
y
= −
5
Apply isw as soon as a correct equation is seen.
8(c)[4]
8
(
)
8
(
)
′
2
f ( )
x
=
2
x
+
+
k
f( )
x
=
x
−
+
kx
+
c
2
x
x
(
)
(
)
(
)
8
2 2
=
2
−
+
1 2 2
−
2
+
c
c
=
... 2
+
5 2
2, 2 2
2
(
)
(
)
8
=
2
−
+
−
+ +
f( )
x
x
1 2 2
x
2
5 2
x
Notes
Mark all parts together.
(c)
M1: Integrates with two terms having a correct index. Allow indices unprocessed.
(
)
8
...
(
)
(
It is for 2 from:
2
x
→
...
x
2
or
x
1 1
+
,
→
or ...
x
−
1
or ...
x
−+
2 1
,
k
→
...
x
or ...
x
x
2
x
A1ft: Correct integration in any form with indices processed.
(
)
8
(
)
2
f( )x
=
x
−
+
kx
+
c
but follow through on their k which may be a decimal and allow the
x
x = is not required and “+ c” is not required.
letter k. "f ( )
"
Brackets are required around their 1 2 2
−
but they may be implied by their attempt to find “c”
so you may need to check their calculation for c.
x =
x =
dM1: Uses
2
and their attempt at y using
2
in
y
=
5
x
−
3 2
in their integrated function and
finds the value of c.
Correct simplified exact expression(
)
(
)
8
2
f( )
x
=
x
−
+
1 2 2
−
x
+ +
2
5 2
A1:
or e.g.
x
(
)
8
2
=
−
+
−
+ +
. "f ( )
x = is not required.
"
f( )
x
x
x
2 2
x
2
5 2
x
Brackets are required around their 1 2 2
−
if written as in the first example above.
2
2
2
2
x but allow 8 for 2 2 .
Do not allow
1x or e.g.
2 x for
Apply isw once a correct function is seen.
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