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Date	November 2021	Marks available	4	Reference code	21N.2.AHL.TZ0.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Determine	Question number	7	Adapted from	N/A

Question

A continuous random variable $X$ has a probability density function given by

$f (x) = \{\begin{matrix} arccos x & 0 \leq x \leq 1 \\ 0 & otherwise \end{matrix}$

The median of this distribution is $m$ .

Determine the value of $m$ .

[2]

Given that $P (|X - m| \leq a) = 0.3$ , determine the value of $a$ .

[4]

Markscheme

recognises that $\int_{0}^{m} arccos x d x = 0.5$ (M1)

$m arccos m - \sqrt{1 - m^{2}} - (0 - \sqrt{1}) = 0.5$

$m = 0.360034 \dots$

$m = 0.360$ A1

[2 marks]

METHOD 1

attempts to find at least one endpoint (limit) both in terms of $m$ (or their $m$ ) and $a$ (M1)

$P (m - a \leq X \leq m + a) = 0.3$

$\int_{0.360034 \dots - a}^{0.360034 \dots + a} arccos x d x = 0.3$ (A1)

Note: Award (A1) for $\int_{m - a}^{m + a} arccos x d x = 0.3$ .

${[x arccos x - \sqrt{1 - x^{2}}]}_{0.360034 \dots - a}^{0.360034 \dots + a}$

attempts to solve their equation for $a$ (M1)

Note: The above (M1) is dependent on the first (M1).

$a = 0.124861 \dots$

$a = 0.125$ A1

METHOD 2

$\int_{- a}^{a} arccos x - 0.360034 \dots d x (= 0.3)$ (M1)(A1)

Note: Only award (M1) if at least one limit has been translated correctly.

Note: Award (M1)(A1) for $\int_{- a}^{a} arccos x - m d x (= 0.3)$ .

attempts to solve their equation for $a$ (M1)

$a = 0.124861 \dots$

$a = 0.125$ A1

METHOD 3

EITHER

$\int_{- a}^{a} arccos (x + 0.360034 \dots) d x (= 0.3)$ (M1)(A1)

Note: Only award (M1) if at least one limit has been translated correctly.

Note: Award (M1)(A1) for $\int_{- a}^{a} arccos (x + m) d x (= 0.3)$ .

$\int_{2 (0.360034 \dots) - a}^{2 (0.360034 \dots) + a} arccos (x - 0.360034 \dots) d x (= 0.3)$ (M1)(A1)

Note: Only award (M1) if at least one limit has been translated correctly.

Note: Award (M1)(A1) for $\int_{2 m - a}^{2 m + a} arccos (x - m) d x (= 0.3)$ .

THEN

attempts to solve their equation for $a$ (M1)

Note: The above (M1) is dependent on the first (M1).

$a = 0.124861 \dots$

$a = 0.125$ A1

[4 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.4—Key features of graphs, intersections using technology

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