Mathematics · Oct / Nov 2023 · Paper 21 · Question 8
YMA01
As printed in the paper
The question as text
8. In a large theatre there are n rows of seats, where n is a constant. The number of seats in the first row is a, where a is a constant. In each subsequent row there are 4 more seats than in the previous row so that • in the 2nd row there are (a + 4) seats • in the 3rd row there are (a + 8) seats • the number of seats in each row form an arithmetic sequence Given that the total number of seats in the first 10 rows is 360
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8(a)[2]
Uses
=
360
2
×+ ×
9 4
=
360
S
a
10
2
2
+
36
=
72
=
18
a
a
n
(
)
8(b)[2]
Uses
=
2146
2 ''18''
×
+
−
1
×
4
=
2146
S
n
2
n
16
+
2
=
2146
n
n
2
+
8
−
1073
=
0
n
n
*
8(c)[3]
(i) States 29
(ii) Attempts ''18'' (''29'' 1) 4
+
−
×
=
130
(a)
10 2
M1: Uses
=
360
with a correct formula to set up a linear equation in ''a''. E.g.
×+ ×
9 4
S
a
10
2
Alternatively uses the sum of all 10 terms
(
) (
) (
) (
) (
) (
) (
) (
) (
)
a
+
a
+
4
+
a
+
8
+
a
+
12
+
a
+
16
+
a
+
20
+
a
+
24
+
a
+
28
+
a
+
32
+
a
+
36
=
360
A1:
a =
18
following a correct equation
(b)
n
(
)
M1: Attempts to use
=
2146
2 ''18''
×
+
−
1
×
4
=
2146
following through on their value of ''a''
S
n
2
n
2
+
8
−
1073
=
0
A1*: Proceeds to
n
n
showing sufficient working, which in most circumstances should be
regarded as one correct simplified intermediate line. (See main scheme)
(c)(i)
B1: States 29.
The 37
−
if written down must be deleted/crossed out, or the 29 chosen by being ''ringed'' or underlined
(c)(ii)
(
)1
+
−
×
4
M1: Attempts
a
n
with their values of a and n. E.g. ''18'' (''29'' 1) 4
+
−
×
29
Alternatively uses
2146
''18''
+
=
...
l
l
2
=
130
l =
130
A1: Maximum number of seats
which may be labelled
This alone scores both marks (but only if they had a = 18 and n =29 earlier)
2
2
=
−
9
+
13
y
x
x
3
d
y
4
9
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