Find the value of k.

Find ${\text{E}}(X)$.

Find the median of X.

$k\int_0^{\frac{\pi }{2}} {\sin x{\text{d}}x = 1}$     M1

$k[ - \cos x]_0^{\frac{\pi }{2}} = 1$

k = 1     A1

[2 marks]

${\text{E}}(X) = \int_0^{\frac{\pi }{2}} {x\sin x{\text{d}}x}$     M1

integration by parts     M1

$[ - x\cos x]_0^{\frac{\pi }{2}} + \int_0^{\frac{\pi }{2}} {\cos x{\text{d}}x}$     A1A1

= 1     A1

[5 marks]

$\int_0^M {\sin x{\text{d}}x}  = \frac{1}{2}$     M1

$[ - \cos x]_0^M = \frac{1}{2}$     A1

$\cos M = \frac{1}{2}$

$M = \frac{\pi }{3}$     A1

Note: accept $\arccos \frac{1}{2}$

[3 marks]

Most candidates scored maximum marks on this question. A few candidates found k = –1.

Most candidates scored maximum marks on this question. A few candidates found k = –1.

Most candidates scored maximum marks on this question. A few candidates found k = –1.