Mathematics · Oct / Nov 2021 · Paper 41 · Question 5
YMA01
As printed in the paper
The question as text
5. y C O x Figure 1 Figure 1 shows a sketch of the curve C with parametric equations π t 4 π x = 5 + 2 tan t y = 8 sec2 t −3
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Show the answer and mark scheme
5(a)[4]
=
=
=
8tan
t
d
x
d
x
2sec
2
t
d
t
x
=
3,tan
t
= −⇒ Gradient = −8
1
At
(
)
2
x
−
5
y
5(b)[3]
1
+
tan
2
t
=
sec
2
t
⇒+
1
=
Attempts to use
4
8
(
)
2
y
=
2
x
−
5
+
8
8
„
f
„
32
5(c)[2]
(a)
d
y
d
y
. Condone incorrect attempts on d
y
t and d
x
d
t
=
M1: Attempts to use the rule
t
d
x
d
x
d
d
d
t
d
y
16sec
2
t
tan
t
A1: A correct expression for
=
or unsimplified equivalent. Note that there may be many
d
x
2sec
2
t
different versions of this including ones that have used double angle formulae. Look carefully
dM1: Dependent upon the previous M. It is for
t = −into their d
y
either substituting tan
1
x =
g( )
t
to find the gradient
d
t = −π into their d
y
or substituting
x =
g( )
t
to find the gradient. Condone
4
d
t =
awrt
−
0.785
(
)
A1: CSO Gradient = − 8. This cannot be awarded from differentiation of
y
=
2
x
−
5
(b)
M1: Attempts to use
± ±
1
tan
2
t
= ±
sec
2
t
with tant being replaced by an expression in x and
replaced by an expression in y
(
)
2
x
−
5
y
A1: A correct unsimplified equation
1
+
=
o.e.
4
8
y
2
(
x
5
)
2
8
f( )
x
2
(
x
5
)
2
8
A1:
=
−
+
.
=
−
+
is also fine
d
y
(
)
x =
4
x
−
5
NB 1: It is possible to use part (a), find
, and then integrate using a point such as (3, 16) to find
d
''c''
NB2: It is possible to use points to generate f (x).
2
For the M1 you should expect to see
f ( )
x
=
ax
+
bx
+
c
, or equivalent, and the use 3 points to set up
3 simultaneous equations in a, b and c which must be then solved. At least one of three points used must be
5, 8 , 3,16 (
correct. Examples of possible points that can be used are (
) (
)
)
5
−
2 3, 32
2
For the A1 you need to see a correct equation, e.g.
y
=
2
x
−
20
x
+
58
(
)
y
=
2
x
−
5
2
+
8
The final A1 will be
.
If you see a solution worthy of merit and you cannot see how to award the marks then send to review.
(c)
M1: One correct end found, condoning strict inequalities.
Look for the lower value of the range to be 8 (or their c) or the higher value to be 32
or equivalent such as [
]
A1: 8
„
f
„
32
8,32
d
u
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