Mathematics · May / June 2022 · Paper 11 · Question 5
YMA01
As printed in the paper
The question as text
5. The curve C has equation y = f(x) Given that • f(x) is a quadratic expression • the maximum turning point on C has coordinates (−2, 12) • C cuts the negative x-axis at −5
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
5(a)[4]
or (
a x
−
1)(
x
+
5)
2
4
2
12
or
y
=
a
x
+
bx
+
c
a
−
b
+
c
=
and 25
a
−
5
b
+
c
=
0
0
12
a
( 5 2)
2
a
..
.
E.g.
=
−
−
+
=
or
12
=
a
( 2 1)( 2
−−
−
+
5)
a
=
..
.
or
2 ( 2)
a
−
+
b
=
0,25
a
−
5
b
+
c
=
0,4
a
−
2
b
+
c
=
1
2
a
=
...,
b
=
.. ,
.
c
=
..
.
4
4 (
4
16
20
12
−
3 x
(
+
2)
2
or
−
3 x
−
1)(
x
+
5)
or
−
x
2
−
x
+
oe
3
3
3
Gradient of 2l is 5
− o.e.
5(b)[3]
4
5
Equation of 2l is
y
= −
"
"(
x
+
5)
4
5
25
y
= −
x
−
4
4
"
5
25
"
4
"
4
16
20
"
5(c)[2]
For two of
y
x
;
y
x y
;
x
2
x
−
−
−
−
+
4
4
5
3
3
3
Or with strict inequalities
"
5
25
"
4
"
4
16
20
"
For all three of
y
−
x
−
;
y
x y
;
−
x
2
−
x
+
4
4
5
3
3
3
Or with strict inequalities
Notes:
(a) Just the expression for f (x) is required but you may see y = ... or f(x) =.. . which is fine
M1: For knowing a quadratic form and using two correct pieces of information. This can be scored by
12
±
a x
(
±
2)
2
using the maximum value occurs at (−2,12) to state the form
use symmetry, usually with roots of −5 and 1 to state the form (
a x
±
1)(
x
±
5)
and substituting in two points, usually (
)
y
ax
2
bx
c
using the general form
=
+
+
−
2,12
and
(
)
. An alternative is using just one point and setting d
y
−
5,0
x =
0 at
x
= −−
2
4
a
+
b
=
d
Condone for each of these: (1) slips (e.g incorrect squaring) or (2) the use of a = ± 1
A1: For a correct equation/expression set up with the correct unknown constants included. It is for getting
in a position where only one more equation or constant, usually the value of a, needs to be found. If a is
set = ± 1 then this mark cannot be awarded
Usually scored for one of
using the maximum of (−2,12) to state the form ( )
y
=
12
±
a x
(
+
2)
2
using the symmetry to deduce roots, usually −5 and 1 to state the form ( )
y
=
a x
(
−
1)(
x
+
and substituting in two points, usually (
)
using the general form ( )
y
=
ax
2
+
bx
+
c
−
2,12
(
)
−
5,0
to form two different equations in a, b and c. Award for example for 4
a
−
2
b
+
c
25
a
−
5
b
+
c
=
0
dM1: For a full method of finding f( )
x in any allowable form.
with usually (
)
y
=
12
±
a x
(
+
2)
2
−
5,0
using the form
to find a
with usually(
)
using the form
y
=
a x
(
−
1)(
x
+
5)
−
2,12
to find a
using the general form
y
=
ax
2
+
bx
+
c
scored for forming three equations and solving to find values
for a, b and c. E.g Uses (
)
and (
)
−
2,12
−
5,0
in y = f(x) and then uses 2
ax
+
b
=
0
at x = − 2
A1: Correct equation or expression for f( )x or y. Accept in any equivalent form
4
(
)
f( )
x
=
12
−
(
x
+
2)
2
3
4 (
(
)
y
= −
x
−
1)(
x
+
5)
3
4
16
20
()
2
y
= −
x
−
x
+
3
3
3
(b)
M1: Applies perpendicular condition to find gradient of 2l
Look for 5
−, 1.25
−
5
or it may be implied by an equation for 2l of
−
y
=
x
+
...
4
4
y
"
m
"(
x
M1: Uses their changed gradient with (−5,0) to find the equation of the line. Look for
=
n
If they use the form y
=
mx
+
c
with (−5,0) they must proceed as far as c = ...
5
25
A1: Correct equation
y
= −
x
−
o.e. Allow exact equivalents but must be in the form y
=
mx
+
4
4
(c)
M1: For identifying any two of the three restrictions given in the scheme. FT on their answers to (a) and (b)
where (a) is a quadratic and (b) is linear. Allow for this mark only the inequality for the quadratic to be
( )
y
ax
2
+
bx
+
c
or
y
f
x
if that is what they identify as the curve. They may combine two of these
4
4
16
20
as one statement, e.g.
x
y
x
2
x
, which is fine (and scores the M1). Use of R
−
−
+
5
3
3
3
instead of y is M0 A0
A1ft: For all three of the restrictions given in the scheme, follow through their answers to (a) and (b) Accept
with strict inequalities, but should be consistent in all inequalities.
2
xy
−
3
x
2
=
50
;
y
−
x
3
+
6
x
=
0
Make a free account to save your answers and see your weak topics and ask the AI why a wrong answer is wrong.