Mathematics · May / June 2024 · Paper 21 · Question 7
YMA01
As printed in the paper
The question as text
7. The circle C1 has equation 2 + y 2 + 8x – 10y = 29 x
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[3]
x
y
x
y
Attempts (
+
4
)
2
+
(
−
5
)
2
...
=
...
x
y
Correct centre (
)
−
4, 5
Exact radius 70
Attempts distance (or distance 2) from (
''
−
4, 5 ''
)
to (
'' 5,
−
8 ''
)
7(b)[3]
to (
)
2
2
Full method. E.g. compares
9
+
13
70
+
52
States that
distance between centres=
250
=
15.81
sum of radii = 70+
52
=
15.58
Concludes as sum of radii < distance between centres, result proven *
(a)
Attempts to complete the square for both variables. Look for (
) (
M1:
±
4
2
,
±
x
y
or may be implied by (
)
by inspection from the equation.
±
4,
±
5
(
)
A1:
−
4, 5
A1:
70 which may be scored following (
)
(
)
x
±
4
2
+
y
±
5
2
=
70
(b)
Attempts distance (or distance 2) from (
)
to (
)
M1:
''
−
4, 5 ''
'' 5,
−
8 ''
(condone sign errors)
to their (
)
2
2
dM1: Full method. E.g. compares their
9
+
13
'' 70 ''
+
52
A1*: Requires correct calculation(s), correct reason and minimal conclusion. Must be
evidence to support the inequality – e.g. decimal values – not just the inequality
stated. May use the difference between sum of radii and distance between centres.
(b) Alt
2
2
2
2
M1:
Attempts to solve
x
+
y
+
8
x
−
10
y
=
29
and
x
+
y
−
10
x
+
16
y
= −
37
forming a
linear equation (need not be simplified for this mark)
FYI: Linear equation is 18
x
−
26
y
=
66
o.e. if the calculations are correct
dM1: Attempts to solves their 18
−
26
=
66
simultaneously with either circle equation
x
y
reaching a 3 term quadratic for either x or y.
A1*: Requires
correct equationscorrect ''solution'' stating no roots and concluding no points of intersection andminimal conclusion
9
−
33
x
2
2
2
E.g. Substitute
y
=
into
x
+
y
+
8
x
−
10
y
=
29
gives
250
x
−
412
x
13
2
and
''
−
4
''
= −
308256
<
0
so no solutions
b
ac
FYI the equation in y (if correct) is
2
250
y
+
984
y
+
1116
=
0
sin
sin
x
x
States or uses
tan
x
=
5sin
x
×
+
13
=
cos
x
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