CAIE A-Level Mathematics A2 4.2.3 Complex Loci Questions

For attempt at integration

M1*

The power of t must increase by 1 with a change of coefficient in the same term.

Use of s=v t scores M0.

$\frac{1}{2+1} k t^{2+1}-\frac{4}{2} t^{1+1}+3 t\left[=\frac{1}{3} k t^{3}-2 t^{2}+3 t\right][+c]$
Marking guidance:

Allow unsimplified.

$\frac{1}{3} k \times 2^{3}-2 \times 2^{2}+3 \times 2[-0]=6$
Use of limits 0 and 2 with 6 to form an equation in k only
(without c but allow with +c-c ).

k=3

$2 \times 3 t-4$

Or at min value $t=\frac{-b}{2 a}=\frac{4}{2 \times 3}$
For attempt at differentiation. Must have expression of the form
a t+b with $a \neq 3$, unless their $k=\frac{3}{2}$. Allow 2 k t-4.

$[2 \times 3 t-4=0 \Rightarrow] t=\frac{2}{3}$

A1FT

OE

FT their $k t=\frac{2}{\text { their } k}$. Allow without working.

$v\left[=3 \times\left(\frac{2}{3}\right)^{2}-4 \times \frac{2}{3}+3\right]=\frac{5}{3} \mathrm{~ms}^{-1}$
OE

Marking guidance:

Allow 1.67 or better for v.

Alternative Method for Question 4(b): Using completing the square

Attempt at completing the square
Must have $\left(t-\frac{2}{3}\right)^{2}$ OE, or $\left(t-\frac{2}{\text { their } k}\right)^{2}$.

$3\left(t-\frac{2}{3}\right)^{2}-\frac{4}{3}+3$

(A1FT)

FT their $k k\left(t-\frac{2}{k}\right)^{2}-\frac{4}{k}+3$.

$v=\frac{5}{3} \mathrm{~m} \mathrm{~s}^{-1}$
OE

Allow 1.67 or better.