Mathematics · Oct / Nov 2022 · Paper 21 · Question 3
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As printed in the paper
The question as text
3. A sequence a1, a2, a3, … is defined by nπ an = cos2 3 Find the exact values of
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
3(a)[3]
1
a =
1
4
(ii)
1
a =
2
4
(iii)
a =
1
3
50 2
[
]
+
49 ( 1275)
=
oe
2
50
nπ
=
1
cos
2
34
×
+
16 1
×
3
4
n
=
1
49
2599
1275
+
=
2
2
(a)
B1: 1
4 or 0.25
B1: 1
4 or 0.25
B1: 1 (which has clearly not come from a rounded degrees decimal answer)
Note that use of degrees gives
a
=
0.9996659868...,
a
=
0.9986643935...,
a
=
0.9969965583...
1
2
3
and scores no marks.
(b)
M1: Correct attempt to find the sum of 1
+
2
+ +
3
...
+
50
50 2
50 1 50
[
]
[
]
A1:
+
49
or e.g.
+
or 1275
2
2
Award for any correct numerical expression or for 1275. May be implied or may be seen as
part of a complete calculation. A correct answer only of 1275 implies both of the first 2 marks.
50
nπ
1
1
1
M1: Correct attempt to find
cos
2
e.g. by
34
×
+
16 1
× or
17
×
+
17
×
+
16 1
× or
3
4
4
4
n
=
1
16 (
×
1
+ +
1
1)
+ + or
1
1
17 (
×
1
+ +
1
1) 1
−
3(b)[4]
50 2
[
]
+
49 ( 1275)
=
oe
2
50
nπ
=
1
cos
2
34
×
+
16 1
×
3
4
n
=
1
49
2599
1275
+
=
2
2
(a)
B1: 1
4 or 0.25
B1: 1
4 or 0.25
B1: 1 (which has clearly not come from a rounded degrees decimal answer)
Note that use of degrees gives
a
=
0.9996659868...,
a
=
0.9986643935...,
a
=
0.9969965583...
1
2
3
and scores no marks.
(b)
M1: Correct attempt to find the sum of 1
+
2
+ +
3
...
+
50
50 2
50 1 50
[
]
[
]
A1:
+
49
or e.g.
+
or 1275
2
2
Award for any correct numerical expression or for 1275. May be implied or may be seen as
part of a complete calculation. A correct answer only of 1275 implies both of the first 2 marks.
50
nπ
1
1
1
M1: Correct attempt to find
cos
2
e.g. by
34
×
+
16 1
× or
17
×
+
17
×
+
16 1
× or
3
4
4
4
n
=
1
16 (
×
1
+ +
1
1)
+ + or
1
1
17 (
×
1
+ +
1
1) 1
−
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