Find the value of $A$ to three decimal places.

Find the mode of $X$.

Find the value ${\text{P}}(X \le 3|X \ge 2)$.

$A\int_1^5 {\sin (\ln x){\text{d}}x = 1}$     (M1)

$A = 0.323{\text{ (3 dp only)}}$     A1

[2 marks]

either a graphical approach or $f'(x) = \frac{{\cos (\ln x)}}{x} = 0$     (M1)

$x = 4.81\;\;\;\left( { = {{\text{e}}^{\frac{\pi }{2}}}} \right)$     A1

Note:     Do not award A1FT for a candidate working in degrees.

[2 marks]

${\text{P}}(X \le 3|X \ge 2) = \frac{{{\text{P}}(2 \le X \le 3)}}{{{\text{P}}(X \ge 2)}}\;\;\;\left( { = \frac{{\int_2^3 {\sin \left( {\ln (x)} \right){\text{d}}x} }}{{\int_2^5 {\sin \left( {\ln (x)} \right){\text{d}}x} }}} \right)$     (M1)

$= 0.288$     A1

Note:     Do not award A1FT for a candidate working in degrees.

[2 marks]

Total [6 marks]