Mathematics · May / June 2022 · Paper 11 · Question 9

p

(ii)

−

(iii) 3

−

p

Correct shape, same

height starting at O,

scaling may be

incorrect.

Two repeats of the sinx

graph each side

α

For

x

=

2

180

°−

α

Attempt at second root E.g.

x

=

2

α

x

=

9

0

°−

2

Notes:

(a)

(i) B1: For 2p. Condone p + p. Award when 2p is the y coordinate of a coordinate pair. E.g (

180

−α

. Award when − p is the y coordinate of a coordinate pair. E.g (

)

−

p

α−

180,

−

p

(ii) B1: For

. Award when 3 − p is the y coordinate of a coordinate pair. E.g (

(iii) B1: For 3

−

p

180

+ α

, 3

−

p

(b)

M1: Same shape, starting at O, and same height as original graph, but the scaling may be incorrect.

Mark the intention. For example their graph may be slightly taller or shorter than the given curve.

One complete sine curve on just one side of the yaxis is sufficient for the method mark.

Also allow this mark when candidates use Diagram 1 and mark the x intercepts as below.

If they ignore the given Diagram and use their own axes you must be convinced their sketch for

y

=

sin 2

x

is the same eight / amplitude as

y

=

sin

x

but a different scaling / frequency. So you must be able

to see some numbers marked on the axes or graph

A1: Correct sketch. Look for intersections at (

)

and (

)

±

360 ,0

°

±

180 ,0

°

in addition to the above

(c)

α

x

=

given as one solution.

B1:

2

M1: A second solution which may not be in the given range. Any of 180

°−

α

α

°+ α

, 90

°−

, 180

2

2

α

α

αare examples of using the symmetry to find a second solution.

−

9

0

°−

,

−

180

°

+

, 36

0

°+

2

2

2

α

or 180

°−

α

A1:

x

=

9

0

°−

as second solution with no additional solutions within the given range.

2

2

d

y

6

2

5