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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

Show 49 related questions

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22M.2.SL.TZ1.8b.i:
Find $f^{- 1} (x)$ .
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.
21N.1.SL.TZ0.5b:
Find $f (4)$ .
21N.1.AHL.TZ0.2a.i:
the vertical asymptote of the graph of $f$ .
21N.1.AHL.TZ0.2a.ii:
the horizontal asymptote of the graph of $f$ .
21N.1.AHL.TZ0.2c:
Sketch the graph of $f$ on the axes below.
18N.3.AHL.TZ0.Hsrg_4b:
State with a reason whether or not $f$ and $g$ commute.
21N.1.SL.TZ0.5a:
Write down the value of $f' (4)$ .
21N.1.AHL.TZ0.2b.ii:
the $y$ -axis.
17N.1.SL.TZ0.S_5a:
Find $(g \circ f)(x)$ .
18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
19N.2.SL.TZ0.S_3b:
Let ${\text{C}}$ be a point on the graph of $h$ . The tangent to the graph of $h$ at ${\text{C}}$ is parallel to the graph of $f$ .

Find the $x$ -coordinate of ${\text{C}}$ .
19N.2.SL.TZ0.S_3a:
Find $h\left( x \right)$ .
17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
18N.3.AHL.TZ0.Hsrg_4c:
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22M.1.SL.TZ2.1b:
Find $(f \circ g) (0)$ .
EXN.1.SL.TZ0.8c:
Find an expression for $f^{- 1} (x)$ , stating its domain.
22M.1.AHL.TZ2.11c:
Given that $(h \circ g) (a) = \frac{π}{4}$ , find the value of $a$ .

Give your answer in the form $p + \frac{q}{2} \sqrt{r}$ , where $p, q, r \in ℤ^{+}$ .
22M.2.SL.TZ1.8b.ii:
Using an algebraic approach, show that the graph of $f^{- 1}$ is obtained by a reflection of the graph of $f$ in the $y$ -axis followed by a reflection in the $x$ -axis.
20N.2.SL.TZ0.S_4b.i:
Solve the equation $(f \circ g) (x) = x$ .
21M.1.SL.TZ1.1a.i:
Write down the value of $f (2)$ .
21M.1.SL.TZ1.1a.ii:
Write down the value of $(f \circ f) (2)$ .
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
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:
Given that ${\left( {g \circ f} \right)^{ - 1}}\left( a \right) = 4$ , find the value of $a$ .
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:
Show that $(f \circ g)(x) = {x^4} - 4{x^2} + 3$ .
20N.2.SL.TZ0.S_4b.ii:
Hence or otherwise, given that $g (2 a) = f^{- 1} (2 a)$ , find the value of $a$ .
EXN.1.SL.TZ0.5:
The functions $f$ and $g$ are defined for $x \in ℝ$ by $f (x) = x - 2$ and $g (x) = a x + b$ , where $a, b \in ℝ$ .

Given that $(f \circ g) (2) = - 3$ and $(g \circ f) (1) = 5$ , find the value of $a$ and the value of $b$ .
21M.2.AHL.TZ2.12d:
Find an expression for $g^{- 1} (x)$ , justifying your answer.
EXN.1.SL.TZ0.8b:
State the range of $f$ .
21M.2.SL.TZ2.5b:
Given that $(g \circ f) (x) \leq 0$ for all $x \in ℝ$ , determine the set of possible values for $c$ .
SPM.1.SL.TZ0.5a:
Show that $\left( {g \circ f} \right)\left( x \right) = 2x + 11$ .
21N.1.AHL.TZ0.2d:
The function $g$ is defined by $g (x) = \frac{a x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ and $a \in ℝ$ .

Given that $g (x) = g^{- 1} (x)$ , determine the value of $a$ .
22M.1.SL.TZ1.4a:
Find $(f \circ g) (x)$ .
21N.1.SL.TZ0.8b:
Write down an expression for $f^{- 1} (x)$ .
21N.1.AHL.TZ0.2b.i:
the $x$ -axis.
21N.1.SL.TZ0.8c:
Find the value of $f^{- 1} (\sqrt{32})$ .
21N.1.SL.TZ0.5c:
Find $h (4)$ .
16N.3.AHL.TZ0.Hsrg_2a:
(i) Sketch the graph of $y = f(x)$ and hence justify whether or not $f$ is a bijection.

(ii) Show that $A$ is a group under the binary operation of multiplication.

(iii) Give a reason why $B$ is not a group under the binary operation of multiplication.

(iv) Find an example to show that $f(a \times b) = f(a) \times f(b)$ is not satisfied for all $a,{\text{ }}b \in A$ .
20N.2.SL.TZ0.S_4a:
Find $(f \circ g) (x)$ .
16N.3.AHL.TZ0.Hsrg_2b:
(i) Sketch the graph of $y = g(x)$ and hence justify whether or not $g$ is a bijection.

(ii) Show that $g(a + b) = g(a) \times g(b)$ for all $a,{\text{ }}b \in \mathbb{R}$ .

(iii) Given that $\{ \mathbb{R},{\text{ }} + \}$ and $\{ D,{\text{ }} \times \}$ are both groups, explain whether or not they are isomorphic.
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:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .
21N.1.SL.TZ0.5d:
Hence find the equation of the tangent to the graph of $h$ at $x = 4$ .
21N.1.SL.TZ0.8a:
Show that $a = 8$ .
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