Mathematics · Oct / Nov 2022 · Paper 11 · Question 2
YMA01
As printed in the paper
The question as text
2. Given that (x – 5)(2x + 1)(x + 3) ≡ ax3 + bx2 –32x – 15 where a and b are constants,
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
2(a)[2]
−
2
−
17
x−
2
all on one line or exact simplified equivalent. Allow
1x . Withhold the
4 x
x
final mark if they attempt to multiply all the terms by 4 for example or a +c appears.
into their d
y
M1
Attempts to substitute
x =
2
which must be a changed function to find the
d
x
gradient of the tangent at P
13
using their d
y
dM1
It is for the method of finding a line passing through
2, 2
2
at
x =
d
x
eg it cannot be the gradient of the normal.
13
21
Score for sight of
−
= −
"
"(
−
2)
with both coordinates substituted in correctly
y
x
2
4
or if they use the form y
=
mx
+
c
they must proceed as far as c = … It is dependent on the
previous method mark.
A1
21
x
+
4
y
−
68
=
0
or any equivalent multiple where the coefficients are integers and all terms
are on one side of the equation. If they state values which contradict the equation then the
equation takes precedence.
2(a)
a =
2
b = −
3
(b)
2
3
3
2
32
15
2
5
3
3
32
1
1
x
−
x
−
x
−
−
Any two term of
d
=
−
−
−
3
d
x
x
2
x
2
x
2
x
2
x
5
5
5
5
x
x
n
→
x +
n
1
4
7
6
5
64
3
1
6
x
2
−
x
2
−
x
2
−
x
2
+
c
35
25
15
(a)
If there is a contradiction between their stated values and the embedded values in a cubic
expression then the embedded values take precedence.
B1
a =
2
which may be embedded
B1
b = −which may be embedded
3
(b)
M1
Attempts to write as a sum of terms. Award for any term with a correct index from correct
working. The index does not need to be processed.
A1
Any two correct unsimplified or simplified terms of the expression
2
5
3
3
32
1
1
5
3
1
1
−
−
−
−
−
3
0.4
−
0.6
−
6.4
−
3
x
2
x
2
x
2
x
2
oe eg
x
2
x
2
x
2
x
2
(indices must be processed)
5
5
5
x +
1
M1
Increases the power of any of their noninteger terms by 1. (
x
n
→
n
) This mark cannot be
awarded for just increasing the power by 1 on a numerator or denominator term. The index
does not need to be processed.
A1
Any two terms correct unsimplified or simplified (see below). May appear in a list or on
separate lines.
7
5
3
1
4
6
64
A1
−
−
−
6
+
all on one line (including the constant c and all simplified)
x
2
x
2
x
2
x
2
c
35
25
15
Allow exact equivalents but not rounded decimals so 4
35 must be written as a fraction.
4
7
5
3
1
4
Accept
35 x
2
−
0.24
x
2
−
4.26
x
2
−
6
x
2
+
c
. Also allow terms written as eg
3
x Withhold
35 x
this mark if the final answer has the integral sign and dx around it or any other spurious notation.
2(b)[5]
into their d
y
M1
Attempts to substitute
x =
2
which must be a changed function to find the
d
x
gradient of the tangent at P
13
using their d
y
dM1
It is for the method of finding a line passing through
2, 2
2
at
x =
d
x
eg it cannot be the gradient of the normal.
13
21
Score for sight of
−
= −
"
"(
−
2)
with both coordinates substituted in correctly
y
x
2
4
or if they use the form y
=
mx
+
c
they must proceed as far as c = … It is dependent on the
previous method mark.
A1
21
x
+
4
y
−
68
=
0
or any equivalent multiple where the coefficients are integers and all terms
are on one side of the equation. If they state values which contradict the equation then the
equation takes precedence.
2(a)
a =
2
b = −
3
(b)
2
3
3
2
32
15
2
5
3
3
32
1
1
x
−
x
−
x
−
−
Any two term of
d
=
−
−
−
3
d
x
x
2
x
2
x
2
x
2
x
5
5
5
5
x
x
n
→
x +
n
1
4
7
6
5
64
3
1
6
x
2
−
x
2
−
x
2
−
x
2
+
c
35
25
15
(a)
If there is a contradiction between their stated values and the embedded values in a cubic
expression then the embedded values take precedence.
B1
a =
2
which may be embedded
B1
b = −which may be embedded
3
(b)
M1
Attempts to write as a sum of terms. Award for any term with a correct index from correct
working. The index does not need to be processed.
A1
Any two correct unsimplified or simplified terms of the expression
2
5
3
3
32
1
1
5
3
1
1
−
−
−
−
−
3
0.4
−
0.6
−
6.4
−
3
x
2
x
2
x
2
x
2
oe eg
x
2
x
2
x
2
x
2
(indices must be processed)
5
5
5
x +
1
M1
Increases the power of any of their noninteger terms by 1. (
x
n
→
n
) This mark cannot be
awarded for just increasing the power by 1 on a numerator or denominator term. The index
does not need to be processed.
A1
Any two terms correct unsimplified or simplified (see below). May appear in a list or on
separate lines.
7
5
3
1
4
6
64
A1
−
−
−
6
+
all on one line (including the constant c and all simplified)
x
2
x
2
x
2
x
2
c
35
25
15
Allow exact equivalents but not rounded decimals so 4
35 must be written as a fraction.
4
7
5
3
1
4
Accept
35 x
2
−
0.24
x
2
−
4.26
x
2
−
6
x
2
+
c
. Also allow terms written as eg
3
x Withhold
35 x
this mark if the final answer has the integral sign and dx around it or any other spurious notation.
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