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Date May 2019 Marks available 6 Reference code 19M.1.AHL.TZ2.H_9
Level Additional Higher Level Paper Paper 1 (without calculator) Time zone Time zone 2
Command term Determine Question number H_9 Adapted from N/A

Question

Consider the functions f and g defined on the domain  0 < x < 2 π by  f ( x ) = 3 cos 2 x and  g ( x ) = 4 11 cos x .

The following diagram shows the graphs of  y = f ( x ) and  y = g ( x )

Find the x -coordinates of the points of intersection of the two graphs.

[6]
a.

Find the exact area of the shaded region, giving your answer in the form  p π + q 3 , where p q Q .

[5]
b.

At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the y -coordinate of A on the graph of g .

[6]
c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

3 cos 2 x = 4 11 cos x

attempt to form a quadratic in  cos x      M1

3 ( 2 co s 2 x 1 ) = 4 11 cos x      A1

( 6 co s 2 x + 11 cos x 7 = 0 )

valid attempt to solve their quadratic     M1

( 3 cos x + 7 ) ( 2 cos x 1 ) = 0

cos x = 1 2      A1

x = π 3 , 5 π 3      A1A1

Note: Ignore any “extra” solutions.

[6 marks]

a.

consider (±)  π 3 5 π 3 ( 4 11 cos x 3 cos 2 x ) d x      M1

= ( ± ) [ 4 x 11 sin x 3 2 sin 2 x ] π 3 5 π 3      A1

Note: Ignore lack of or incorrect limits at this stage.

attempt to substitute their limits into their integral     M1

= 20 π 3 11 sin 5 π 3 3 2 sin 10 π 3 ( 4 π 3 11 sin π 3 3 2 sin 2 π 3 )

= 16 π 3 + 11 3 2 + 3 3 4 + 11 3 2 + 3 3 4

= 16 π 3 + 25 3 2      A1A1

[5 marks]

b.

attempt to differentiate both functions and equate     M1

6 sin 2 x = 11 sin x      A1

attempt to solve for x      M1

11 sin x + 12 sin x cos x = 0

sin x ( 11 + 12 cos x ) = 0

cos x = 11 12 ( or sin x = 0 )      A1

y = 4 11 ( 11 12 )      M1

y = 169 12 ( = 14 1 12 )      A1

[6 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.7—Circular functions: graphs, composites, transformations
Show 89 related questions
Topic 5 —Calculus » SL 5.11—Definite integrals, areas under curve onto x-axis and areas between curves
Topic 3— Geometry and trigonometry
Topic 5 —Calculus

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