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Date November 2016 Marks available 4 Reference code 16N.2.hl.TZ0.5
Level HL only Paper 2 Time zone TZ0
Command term Solve Question number 5 Adapted from N/A

Question

Consider the function f defined by f(x)=3xarccos(x) where 1.

Sketch the graph of f indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.

[3]
a.

State the range of f.

[2]
b.

Solve the inequality \left| {3x\arccos (x)} \right| > 1.

[4]
c.

Markscheme

N16/5/MATHL/HP2/ENG/TZ0/05.a/M

correct shape passing through the origin and correct domain     A1

 

Note: Endpoint coordinates are not required. The domain can be indicated by - 1 and 1 marked on the axis.

(0.652,{\text{ }}1.68)    A1

two correct intercepts (coordinates not required)     A1

 

Note: A graph passing through the origin is sufficient for (0,{\text{ }}0).

 

[3 marks]

a.

[-9.42,{\text{ }}1.68]{\text{ }}({\text{or }} - 3\pi ,{\text{ }}1.68])    A1A1

 

Note: Award A1A0 for open or semi-open intervals with correct endpoints. Award A1A0 for closed intervals with one correct endpoint.

 

[2 marks]

b.

attempting to solve either \left| {3x\arccos (x)} \right| > 1 (or equivalent) or \left| {3x\arccos (x)} \right| = 1 (or equivalent) (eg. graphically)     (M1)

N16/5/MATHL/HP2/ENG/TZ0/05.c/M

x =  - 0.189,{\text{ }}0.254,{\text{ }}0.937    (A1)

- 1 \leqslant x <  - 0.189{\text{ or }}0.254 < x < 0.937    A1A1

 

Note: Award A0 for x <  - 0.189.

 

[4 marks]

c.

Examiners report

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

Syllabus sections

Topic 3 - Core: Circular functions and trigonometry » 3.5 » The inverse functions x \mapsto \arcsin x , x \mapsto \arccos x , x \mapsto \arctan x ; their domains and ranges; their graphs.

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