Mathematics · May / June 2023 · Paper 11 · Question 7
YMA01
As printed in the paper
The question as text
7. y y = kx R2 y = 2 R1 x O x = a y = 10 – 2x Figure 2 The region R1 , shown shaded in Figure 2, is defined by the inequalities 0 y 2 y 10 – 2x y kx where k is a constant. The line x = a, where a is a constant, passes through the intersection of the lines y = 2 and y = kx Given that the area of R1 is 27 4 square units,
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[4]
0
=
10
−
2x
x
=
5
or
y
=
2,
y
=
10
−
2
x
x
=
4
Examples:
1
(
)
27
1
2
27
2 5
4
or
2 5
4
×
+
−
a
=
×
+
−
=
2
4
2
k
4
Trapezium
or
1
1
(
)
27
1
2
1
2
2
27
×
2
+
×
2 5
−
+
4
−
=
or
××
2
+
×
2 5
−
+
4
−
=
a
a
a
2
2
4
2
2
4
k
k
k
Triangle + Trapezium
or
1
(
)
1
27
1
2
2
1
27
×
2
+
2 4
−
+
××
1 2
=
o
r
××
2
+
2 4
−
+
××
1 2
=
a
a
2
2
4
2
2
4
k
k
Triangle + Rectangle + Triangle
or
1
1
(
)
27
1
1
2
27
××+
5 2
4
−
×
2
=
or
××+
5 2
4
−
×
2
=
a
2
2
4
2
2
4
k
2 Triangles
8
9
k
=
,
a
=
9
4
7(b)[2]
8
9
Two of
y
" " ,
x y
10
−
2 ,
x x
>
"
"
9
4
8
9
All three of
,
10
−
2 ,
>
y
x y
x x
9
4
Notes:
(a)
B1: Correct x intercept for
y
=
10
−
2
x
or correct x coordinate for intersection of y = 2 with
=
10
−
2
. May be seen on the diagram or in a sketch or implied by their working.
y
x
M1: Correct method for area and sets equal to 27
1
4 to form an equation in a or k. The 2
×
2
implied. May use e.g. trapezium, triangle + trapezium, triangle + rectangle + triangle or 2 triangles.
See above for some examples. Note that with a correct “5” and “4” the area is 9
−
or
9
−
a
A1: Correct value for a or k as a single value not a calculation.
A1ft: Correct values for a and k. Correct values or follow through using
2
from their first
ak =
value to the second provided the M1 is scored. Allow exact equivalents or exact ft equivalents so if
1
correct allow e.g.
=
0.8,
=
2.25 or 2 4
k
a
(b)
M1: Any two of the three correct inequalities with their k and/or their a or the letters a and k. Note
8
inequalities in y may be combined in to one so
" "
<
<
10
−
2
would count as two correct
9 x
y
x
<
or
inequalities. Inequalities must be in the correct direction but accept with
for the M. Do not
8
allow incorrect notation such as
R
" " ,
9 x R
10
−
2
x
2
2
A1: All three correct with the correct values of a and k – there is no follow though for this mark.
Note inequalities in y may be combined in to one as above.
Note that there may be an upper limit on the x inequality which is fine provided it is in an
acceptable range and consistent.
8
45
Note that
=
and y = 10 – 2x intersect when
x =
so the upper limit may be this value or
y
x
9
13
greater. In general the following are acceptable:
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