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Date	November Example questions	Marks available	3	Reference code	EXN.2.SL.TZ0.2
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	2	Adapted from	N/A

Question

The following diagram shows a circle with centre $O$ and radius $3$ .

Points $A$ , $P$ and $B$ lie on the circumference of the circle.

Chord $[AB]$ has length $L$ and $A \hat{O} B = θ$ radians.

Show that arc $APB$ has length $6 π - 3 θ$ .

[2]

a.

Show that $L = \sqrt{18 - 18 \cos θ}$ .

[2]

b.

Arc $APB$ is twice the length of chord $[AB]$ .

Find the value of $θ$ .

[3]

c.

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

EITHER

uses the arc length formula (M1)

arc length is $3 (2 π - θ)$ A1

OR

length of arc $AB$ is $3 θ$ A1

the sum of the lengths of arc $AB$ and arc $APB$ is $6 π$ A1

THEN

so arc $APB$ has length $6 π - 3 θ$ AG

[2 marks]

a.

uses the cosine rule (M1)

$L^{2} = 3^{2} + 3^{2} - 2 (3) (3) \cos θ$ A1

so $L = \sqrt{18 - 18 \cos θ}$ AG

[2 marks]

b.

$6 π - 3 θ = 2 \sqrt{18 - 18 \cos θ}$ A1

attempts to solve for $θ$ (M1)

$θ = 2.49$ A1

[3 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

Syllabus sections

Topic 2—Functions » SL 2.5—Composite functions, identity, finding inverse

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18N.2.AHL.TZ0.H_9c.i:
sketch the graph of $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
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21N.1.AHL.TZ0.2a.i:
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21N.1.AHL.TZ0.2a.ii:
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19N.2.SL.TZ0.S_3b:
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19N.2.SL.TZ0.S_3a:
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22M.2.SL.TZ1.8b.ii:
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20N.2.SL.TZ0.S_4b.i:
Solve the equation $(f \circ g) (x) = x$ .
21M.1.SL.TZ1.1a.i:
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21M.1.SL.TZ1.1a.ii:
Write down the value of $(f \circ f) (2)$ .
18N.2.AHL.TZ0.H_9d:
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18N.3.AHL.TZ0.Hsrg_4a.i:
Find $\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .
SPM.1.SL.TZ0.5b:
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18N.3.AHL.TZ0.Hsrg_4a.ii:
Find $\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .
17M.2.SL.TZ2.S_6a:
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20N.2.SL.TZ0.S_4b.ii:
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EXN.1.SL.TZ0.5:
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EXN.1.SL.TZ0.8b:
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21N.1.AHL.TZ0.2d:
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21N.1.SL.TZ0.8b:
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21N.1.AHL.TZ0.2b.i:
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21N.1.SL.TZ0.8c:
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16N.3.AHL.TZ0.Hsrg_2a:
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(iii) Give a reason why $B$ is not a group under the binary operation of multiplication.

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20N.2.SL.TZ0.S_4a:
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16N.3.AHL.TZ0.Hsrg_2b:
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EXN.1.SL.TZ0.8d:
Find the coordinates of the point(s) where the graphs of $y = f (x)$ and $y = f^{- 1} (x)$ intersect.
18N.2.AHL.TZ0.H_9b:
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21N.1.SL.TZ0.5d:
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21N.1.SL.TZ0.8a:
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21N.1.SL.TZ0.8d.ii:
Find the value of $p$ and the value of $q$ .
21N.1.SL.TZ0.8d.i:
Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

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Topic 2—Functions