Mathematics · May / June 2024 · Paper 31 · Question 4
YMA01
As printed in the paper
The question as text
4. f (x) = 8 sin x cos x + 4 cos 2 x – 3
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
4(a)[3]
States or uses sin 2
=
2sin
cos
or
cos2
= ±
2cos
2
±
1
x
x
x
x
x
Uses sin 2
and
2
±in f (x)
x
=
2sin
x
cos
x
cos2
x
= ±
2cos
x
1
(
)
f( )
x
=
8sin cos
x
x
+
4cos
2
x
−
3
=
4sin 2
x
+
2cos2
x
−
1
2
2
2
4(b)[3]
R
=
a
+
b
R
=
20
or 2 5
(
)
b
tan
"awrt 0.464"
α
=
α=
=
a
(
)
(
)
f( )
x
=
2 5 sin 2
x
+
0.464
−
1
(i) Maximum value = "2 5
−
1"
4(c)[3]
5
π
(ii) Solves
2
+
α=
=
...
x
x
2
(
) awrt 3.69
(or (
) awrt 3.70
x =
x =
)
(a)
For the method marks condone working in mixed variables provided the intention is clear
M1: Either states one of the correct identities sin 2
=
2sin
cos
or
cos2
=
2cos
2
−
1
x
x
x
x
x
or uses sin 2
or
cos2
2cos
2
±(may be implied by use of e.g.
1
x
=
2sin
x
cos
x
x
= ±
x
cos
2
x
= ±±
1 sin
2
x
and
cos2
x
= ±±
1 2sin
2
x
or
cos2
x
= ±
cos
2
x
±
sin
2
x
)
Can be implied by either of
a =
4
= ±
2
= −
5 or
−
1
b
and
c
dM1: Uses sin 2
=
2sin
cos
and
cos2
= ±
2cos
2
± (which may be implied as above for the first
1
x
x
x
x
x
method mark) in f (x) to produce an expression of the required form sin 2
a
x
+
b
cos2
Condone slips when substituting in.
Can be implied by
a =
4
and
b
= ±
2
and
c
= −
5 or
−
1
A1:
4sin 2
+
2cos2
− Correct answer scores all three marks. Must be in terms of x.
1
x
x
(b)
Full marks can only be scored provided correct values for a, b and c are found in (a).
2
2
2
B1ft: Finds the exact value of R using
R
=
a
+
b
(or may use trigonometry using their value for α)
Can be implied by a correct exact value.
Follow through on their a or b but the correct values should give
R =
20
or 2 5
M1:
Uses a ''correct '' method to find αusing their a and b. Accept for example tan
α
= ±
a
tan
α
= ±
α
=
...
(You may need to check this on your calculator if only the angle is seen)
b
b
a
May also be seen using e.g. sin
α
= ±
α
=
...
or cos
α
= ±
α
=
...
Allow the angle to
"
"
"
"
R
R
be found in degrees for this mark.
α=
"awrt 26.6 "
° (1dp)
A1cso: (
)
(
)
− o.e. e.g. (
)
(
)
f( )
x
=
2 5 sin 2
x
+
awrt 0.464
1
f( )
x
=
20 sin 2
x
+
0.464
− (may be
1
awarded if seen in (c)) Can only be scored provided correct values for a, b and c are found in (a).
(c) (i)
B1ft: "2 5
−
1"
o.e. e.g. " 20
−
1"
but follow through on their R
+
as long as R has been correctly
c
found (ie awarded in (b)). Ft on their c which may be different in (a) and (b) but cannot be 0
Allow to be a decimal if R was given as a decimal in (b) at any point. isw following a correct
answer.
(c)(ii)
5
π
M1:
Solves
2
+
α=
=
...
using their αfound in (b) (note that this is still the equation which
x
x
2
will need to be solved if they differentiate f(x) first and set equal to 0). You may need to check this
on your calculator.
Allow this mark to be scored even if there are additional equations formed (and possibly solved).
They must be consistent in their use of degrees or radians in an equation that they are solving.
Make a free account to save your answers and see your weak topics and ask the AI why a wrong answer is wrong.