CAIE A-Level Mathematics 6.3 Continuous Random Variables Question Bank

$\int_{0}^{\sqrt{2}}\left(a x-x^{3}\right) \mathrm{d} x=1$

M1

Attempted integration of f(x) and equated to 1.

$$\left[a \frac{x^{2}}{2}-\frac{x^{4}}{4}\right]_{0}^{\sqrt{2}}=1$$$a-\frac{4}{4}=1$

A1

Correct integration and substitute correct limits.

a=2

A1

AG Convincingly obtained and no errors seen.

$\int_{0}^{m}\left(2 x-x^{3}\right) d x=$

M1

Attempt integrate f(x) with limits 0 to m (or m to $\sqrt{2}$ ) and equate to $\frac{1}{2}$.

$m^{2}-\frac{m^{4}}{4}=\frac{1}{2}$

A1

For correct quartic in any form.

$m^{4}-4 m^{2}+2=0 \Rightarrow m^{2}=\frac{4 \pm \sqrt{16-8}}{2} \quad[=2 \pm \sqrt{2}]$

M1

For solving their three term quartic to find $m^{2}$.

$m=\sqrt{2-\sqrt{2}}$ or 0.765(3 sf)

A1

$\int_{0}^{\sqrt{2}}\left(2 x^{2}-x^{4}\right) \mathrm{d} x$

M1

Attempt to integrate x f(x). Ignore limits.

$\left[\frac{2 x^{3}}{3}-\frac{x^{5}}{5}\right]_{0}^{\sqrt{2}}$

A1

Correct integration and correct limits.

$\left[=\frac{4 \sqrt{2}}{3}-\frac{4 \sqrt{2}}{5}\right]=\frac{8}{15} \sqrt{2}$

A1

OE For single exact term.