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Date	November 2016	Marks available	8	Reference code	16N.3.AHL.TZ0.Hsrg_2
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Sketch, Show that, and Explain	Question number	Hsrg_2	Adapted from	N/A

Question

Let $A$ be the set $\{ x|x \in \mathbb{R},{\text{ }}x \ne 0\}$ . Let $B$ be the set $\{ x|x \in ] - 1,{\text{ }} + 1[,{\text{ }}x \ne 0\}$ .

A function $f:A \to B$ is defined by $f(x) = \frac{2}{\pi }\arctan (x)$ .

Let $D$ be the set $\{ x|x \in \mathbb{R},{\text{ }}x > 0\}$ .

A function $g:\mathbb{R} \to D$ is defined by $g(x) = {{\text{e}}^x}$ .

(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 $a,{\text{ }}b \in A$ .

[13]

a.

(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.

[8]

b.

Markscheme

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

(i) N16/5/MATHL/HP3/ENG/TZ0/SG/M/02.a.i A1

Notes: Award A1 for general shape, labelled asymptotes, and showing that $x \ne 0$ .

graph shows that it is injective since it is increasing or by the horizontal line test R1

graph shows that it is surjective by the horizontal line test R1

Note: Allow any convincing reasoning.

so $f$ is a bijection A1

(ii) closed since non-zero real times non-zero real equals non-zero real A1R1

we know multiplication is associative R1

identity is 1 A1

inverse of $x$ is $\frac{1}{x}(x \ne 0)$ A1

hence it is a group AG

(iii) $B$ does not have an identity A2

hence it is not a group AG

(iv) $f(1 \times 1) = f(1) = \frac{1}{2}$ whereas $f(1) \times f(1) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ is one counterexample A2

hence statement is not satisfied AG

[13 marks]

a.

N16/5/MATHL/HP3/ENG/TZ0/SG/M/02.b

award A1 for general shape going through (0, 1) and with domain $\mathbb{R}$ A1

graph shows that it is injective since it is increasing or by the horizontal line test and graph shows that it is surjective by the horizontal line test R1

Note: Allow any convincing reasoning.

so $g$ is a bijection A1

(ii) $g(a + b) = {{\text{e}}^{a + b}}$ and $g(a) \times g(b) = {{\text{e}}^a} \times {{\text{e}}^b} = {{\text{e}}^{a + b}}$ M1A1

hence $g(a + b) = g(a) \times g(b)$ AG

(iii) since $g$ is a bijection and the homomorphism rule is obeyed R1R1

the two groups are isomorphic A1

[8 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

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

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