Mathematics · Oct / Nov 2024 · Paper 21 · Question 8
YMA01
As printed in the paper
The question as text
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. C1 y Q C2 R P x O Figure 1 Figure 1 shows a sketch of part of the curve C1 with equation y = x 2 + 3 x > 0 and part of the curve C2 with equation 9 y x > 0 13 x 2 The curves C1 and C2 intersect at the points P and Q as shown in Figure 1.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8(a)[4]
9
x
2
+
3
=
13
−
x
4
+
3
x
2
=
13
x
2
−
9
x
2
4
−
10
2
+
9
=
0
x
x
(
)(
)
4
−
10
2
+
9
=
0
2
−
1
2
−
9
=
0
2
=
...
x
x
x
x
x
=
...
x
=
x
=
1,
x
3
(a)
Note we are now marking part (a) as M1A1M1A1 not M1M1A1A1
M1:
Equates the 2 curves and multiplies through by x2 to obtain a quadratic equation in x2
A1(M1 on EPEN): Correct 3TQ in x2 with terms collected and all on one side.
Note that the “= 0” may be implied by their attempt to solve.
M1(A1 on EPEN): Solves a 3TQ in x2 by a suitable method e.g. factorising, completing the
square or quadratic formula and takes the square root to obtain at least one value for
x. They may use a calculator, but the solution must be via x2.
E.g.
4
−
10
2
+
9
=
0
2
=
1, 9
=
1, 3
is acceptable for this mark as is
x
x
x
x
(
)(
)
x
4
−
10
x
2
+
9
=
0
x
2
−
1
x
2
−
9
=
0
x
=
1, 3
They cannot go from e.g.
4
−
10
2
+
9
=
0
to
x =
1, 3
x
x
(
)(
)(
)(
)
Note
4
−
10
2
+ =
9
0
−
1
+
1
−
3
+
3
=
0
=
1, 3
scores M0
x
x
x
x
x
x
x
A1:
Both correct and no other values. Condone any confusion with which is P and which
is Q and condone e.g. P = 1, Q = 3
Examples:
9
x
2
+
3
=
13
−
x
4
+
3
x
2
=
13
x
2
−
9
x
2
4
−
10
2
+
9
=
0
=
1, 3
x
x
x
Scores M1A1M0A0
9
2
+
3
=
13
−
4
+
3
2
=
13
2
−
9
x
x
x
x
2
x
x
4
−
10
x
2
+
9
=
0
x
2
=
1, 9
x
=
1, 9
Scores M1A1M1A0 (BOD)
Attempts to solve
4
10
2
9
0
by identifying roots/factors and then using e.g. long
x
−
x
+
=
division should be sent to review.
Part (a) may also be done without obtaining a quartic:
9
9
9
y
=
x
2
+
3,
y
=
13
−
x
2
+
3
=
13
−
x
2
−
10
+
=
0
x
2
x
2
x
2
9
1
x
−
x
−
=
0
x
x
9
1
x
−
=
0
x
2
=
9
x
=
3
or
x
−
=
0
x
2
=
1
x
=
1
x
x
α
β
M1: Solves simultaneously and attempts to factorise to
x
−
x
−
x
x
A1: Correct factorisation
M1: Attempts to solve via x2
A1: x = 1 and x = 3. Both correct and no other values. Condone any confusion with which is P and
which is Q and condone e.g. P = 1, Q = 3
Part (a) may also be done via y e.g.:
9
9
2
3,
13
13
2
16
48
0
y
=
x
+
y
=
−
y
=
−
y
−
y
+
=
x
2
y
−
3
y
2
−
16
y
+
48
=
0
y
=
4, 12
=
4
=
1,
=
12
=
3
y
x
y
x
M1: Solves simultaneously to obtain a 3TQ in y
A1: Correct 3TQ in y
M1: Solves their 3TQ in y and uses at least one value of y to find a value for x
A1: x = 1 and x = 3. Both correct and no other values. Condone any confusion with which is P and
which is Q and condone e.g. P = 1, Q = 3
8(b)[4]
9
9
3
(
)
✓
x
(
)
13
−
−
x
2
+
3
d
x
=
13
x
+
−
−
3
x
+
c
2
3
x
x
or
9
9
3
(
)
x
(
)
(
)
13
−
d
x
=
13
x
+
+
c
,
x
2
+
3 d
x
=
+
3
x
+
c
x
2
x
3
3
9
3
9
3
3
1
x
( )
10
x
+
−
=
10 3
+
−
−
10
+
9
−
=
...
3
3
3
3
x
1
or
3
9
3
3
1
✓
x
(
)
13
x
+
−
+
3
x
=
39
+ −
3
13
+
9
−
9
+ −
9
+
3
=
...
3
3
x
1
1
16
=
3
Note that this question has the calculator warning so attempts that do not use algebraic
integration score no marks.
(b)
M1:
Evidence of integration
for at least one term for either curve or for the
x
n
→
x +
n
1
difference between the 2 curves e.g. it is not for integrating
4
10
2
9
x
−
x
+
A1:
Correct integration for both curves either as a difference (either way round) or
9
x
3
9
x
3
separately. E.g.
13
3
(
)
or
13x
+
and
+
3
±
x
+
−
−
x
+
c
x
3
x
3
x
Allow simplified or unsimplified but indices must be processed.
dM1: Depends on the first method mark.
Substitutes their positive values from part (a) and subtracts if attempted as a
difference or substitutes their values into the separate integrations, subtracts and then
subtracts the 2 results either way round to obtain a value.
Condone poor bracketing as long as the intention is clear.
May be implied by the correct final answer following correct integration.
If the integration and/or limits are incorrect and the substitution is not shown
explicitly then you may need to check.
16
or exact equivalent. 16
16
A1:
scores A0 unless it gets corrected to
−
+
3
3
3
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