21N.2.AHL.TZ0.7

recognises that $\underset{0}{\overset{m}{\int }}\text{arccos} x dx=0.5$                     (M1)

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

$m=0.360034\dots$

$m=0.360$                     A1

[2 marks]

METHOD 1

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

$\text{P}\left(m-a\le X\le m+a\right)=0.3$                   

$\underset{0.360034\dots -a}{\overset{0.360034\dots +a}{\int }}\text{arccos} x dx=0.3$                     (A1)

Note: Award (A1) for $\underset{m-a}{\overset{m+a}{\int }}\text{arccos} x dx=0.3$.

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

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

$\underset{-a}{\overset{a}{\int }}\text{arccos\hspace{0.05em}}\overline{)x-0.360034\dots \overline{) dx \left(=0.3\right)}}$                     (M1)(A1)

 

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

Note: Award (M1)(A1) for $\underset{-a}{\overset{a}{\int }}\text{arccos\hspace{0.05em}}\overline{)x-m\overline{) dx \left(=0.3\right)}}$.

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

$a=0.124861\dots$

$a=0.125$                       A1

 

METHOD 3

EITHER 

$\underset{-a}{\overset{a}{\int }}\text{arccos\hspace{0.05em}}\left(x+0.360034\dots \right) dx \left(=0.3\right)$                     (M1)(A1)

 

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

Note: Award (M1)(A1) for $\underset{-a}{\overset{a}{\int }}\text{arccos\hspace{0.05em}}\left(x+m\right) dx \left(=0.3\right)$.

OR

$\underset{2\left(0.360034\dots \right)-a}{\overset{2\left(0.360034\dots \right)+a}{\int }}\text{arccos\hspace{0.05em}}\left(x-0.360034\dots \right) dx \left(=0.3\right)$                     (M1)(A1)

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

Note: Award (M1)(A1) for $\underset{2m-a}{\overset{2m+a}{\int }}\text{arccos\hspace{0.05em}}\left(x-m\right) dx \left(=0.3\right)$.

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]

[N/A]

[N/A]