Mathematics · May / June 2023 · Paper 31 · Question 4
YMA01
As printed in the paper
The question as text
4. The function f is defined by 2 – 5 x 0 x ∈ f(x) = 2x
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
4(a)[1]
y
=
f( )
x
Curve starting on negative xaxis and passing
through positive yaxis, in quadrants 1 and 2 only.
−
1
y
=
f
( )
x
Shape and position correct.
4(b)[2]
x
+
5
x
+
5
or (
)
2
2
2
5
2
2
5
2 2
2
5
5
x
−
=
x
or
x
−
=
or
x
=
x
−
−
=
x
4(c)[3]
2
2
Full attempt to solve
2
x
2
−
x
−
5
=
0
x
=
...
exact
1
+
41
x
=
4
Notes
(a) Mark the question as a whole if (c) answered as (b) allow the marks.
)
B1
Correct range. Accept
y
−
5,
f( )
x
−
5,
f
∈−
5,
∞
or correct formal set notation but not just
x
−
5.
(b) Note: if a sketch is redrawn score for the sketch of the inverse only.
M1
For a curve starting on the negative x axis and passing through the positive y axis, in quadrants 1
and 2 only.
A1
Correct shape (curvature) and position. Must be increasing (not bending back on itself) with
decreasing gradient, though be tolerant with pen slips at the end. Do not penalise incorrect
intercepts.
(c)
B1
Sets up a correct equation for the solution, as shown in scheme or equivalents. Should be an
equation but allow “=0” implied if there is an attempt to solve. Just
2
x
2
−
x
−
5
is B0 with no
further working.
M1
Full attempt to solve a correct equation leading to exact answers.Attempts via
f( )
x
will lead a quartic (
)
8
x
4
−
4
0
x
2
−
x
+
45
=
0
if co
r
re
c
t
but will likely not lead to exact answers.
Note exact answers following a quadratic is fine, but method should be shown for a quartic.
Decimal answer only is M0.
1
+
41
A1
x
=
ONLY.
4
Some examples of curves for question 4(b).
M1A0: Meets conditions
M1A1bod: Curve slips
M1A1bod: Gradient is
for M1 but gradient not
downward at the end due to slip
decreasing on the whole.
decreasing.
of pen. This is a borderline case.
M0A0: Does not start on
M1A1bod: Position and curvature
M1A0: Meets condition for M1
negative xaxis. Curvature
correct – ignore possible y = x
but curvature out of tolerance,
would be unacceptable.
line, mark the curve.
gradient increases after
intersection.
M1A0: Curve is clearly going
downward on the righthand
side.
States
x =
2
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