Mathematics · May / June 2021 · Paper 31 · Question 6
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As printed in the paper
The question as text
6. Given that k is a positive constant,
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6(a)[4]
k
Shape
k
O
k
Correct intercepts.
−
2
2
(ii)
Shape
k
3
Correct intercepts.
O
k
6
k
k
Attempts to solve either
2
2
or
2
2
k
+
x
= −
x
+
k
−
x
=
x
−
3
3
1
1
Achieves either
or
oe
x
= −
k
x
=
k
6
3
k
k
Attempts to solve both
and
k
+
2
x
= −
2
x
+
k
−
2
x
=
2
x
−
3
3
1
1
Achieves either
and
x
= −
k
x
=
k
6
3
(a)
y
=
k
−
2
x
.
M1: For the shape of
Score for an upside down V with maximum on the yaxis with the branches
at least reaching the xaxis.
k
A1: Intercepts at k on the yaxis and
on the xaxis. Allow as values or coordinates and allow coordinates
±
2
the wrong way round (e.g. (k, 0) for (0, k)) as long as they are in the correct places. The graph must cross
the axes at these points.
k
M1: For the shape of
y
=
2
x
−
.
Score for a V shape in quadrants 1 and 2 with a minimum on the xaxis and
3
at least reaches the yaxis.
k on the yaxis and 6
k on the xaxis. Allow as values or coordinates and allow coordinates
A1: Intercepts at 3
the wrong way round (e.g. (k/3, 0) for (0, k/3)) as long as they are in the correct places. The graph must
cross the yaxis and touch the xaxis.
(b)
M1: Attempts to solve one correct equation not involving moduli (or equivalent equations)
2
k
A1: One correct solution. Allow unsimplified forms such as
−
12
A1: Two correct solutions (simplified) and no other solutions given.
Some may square in (b) e.g.
k
2
4
kx
k
2
8
kx
8
k
2
k
(
)
2
2
x
−
=
k
−
2
x
⇒
4
x
2
−
+
=
k
2
−
4
kx
+
4
k
2
⇒
=
⇒
x
=
3
3
9
3
9
3
Score M1 for squaring to obtain at least 3 terms on both sides and solving for x and A1 for
Candidates are unlikely to find the other root using this approach.
6(b)[4]
k
k
Attempts to solve either
2
2
or
2
2
k
+
x
= −
x
+
k
−
x
=
x
−
3
3
1
1
Achieves either
or
oe
x
= −
k
x
=
k
6
3
k
k
Attempts to solve both
and
k
+
2
x
= −
2
x
+
k
−
2
x
=
2
x
−
3
3
1
1
Achieves either
and
x
= −
k
x
=
k
6
3
(a)
y
=
k
−
2
x
.
M1: For the shape of
Score for an upside down V with maximum on the yaxis with the branches
at least reaching the xaxis.
k
A1: Intercepts at k on the yaxis and
on the xaxis. Allow as values or coordinates and allow coordinates
±
2
the wrong way round (e.g. (k, 0) for (0, k)) as long as they are in the correct places. The graph must cross
the axes at these points.
k
M1: For the shape of
y
=
2
x
−
.
Score for a V shape in quadrants 1 and 2 with a minimum on the xaxis and
3
at least reaches the yaxis.
k on the yaxis and 6
k on the xaxis. Allow as values or coordinates and allow coordinates
A1: Intercepts at 3
the wrong way round (e.g. (k/3, 0) for (0, k/3)) as long as they are in the correct places. The graph must
cross the yaxis and touch the xaxis.
(b)
M1: Attempts to solve one correct equation not involving moduli (or equivalent equations)
2
k
A1: One correct solution. Allow unsimplified forms such as
−
12
A1: Two correct solutions (simplified) and no other solutions given.
Some may square in (b) e.g.
k
2
4
kx
k
2
8
kx
8
k
2
k
(
)
2
2
x
−
=
k
−
2
x
⇒
4
x
2
−
+
=
k
2
−
4
kx
+
4
k
2
⇒
=
⇒
x
=
3
3
9
3
9
3
Score M1 for squaring to obtain at least 3 terms on both sides and solving for x and A1 for
Candidates are unlikely to find the other root using this approach.
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