CAIE IGCSE Additional Math 4 equations inequalities and graphs

$k>\frac{49}{4}$ oe

B1

FT on $\mid$ their $\left.-\frac{49}{4} \right\rvert\,$ excluding k>12

x * 3
B1
* can be any inequality sign or =.
-3x+4<x+2
or
3x-4>-x-2
M1
$x * \frac{1}{2}$
A1
* can be any inequality sign or =.
$\frac{1}{2}<x<3$
A1
Do not accept $\frac{1}{2}<x,\ x<3$ unless "and" is between the inequalities.
Alternative method:
$9x^2-24x+16 * x^2+4x+4$
(M1) Square both sides; * can be any inequality sign or =.
(2x-1)(x-3) * 0
(M1) * can be any inequality sign or =.
Critical values $\frac{1}{2},\ 3$
(A1)
$\frac{1}{2}<x<3$
(A1)

Solves or factorises $x^{\frac{1}{3}}-x^{\frac{1}{6}}-2=0$ oe:
$$\left(x^{\frac{1}{6}}+1\right)\left(x^{\frac{1}{6}}-2\right)$$

or when substituting $y=x^{\frac{1}{3}}$ factorises to
$$\left(y^{\frac{1}{2}}+1\right)\left(y^{\frac{1}{2}}-2\right)$$

M1

A substitution may be used
$$\left(x^{\frac{1}{3}}\right)^{6}-\left(x^{\frac{1}{6}}\right)^{6}=2^{6} \text { scores } 0 \text { marks }$$$\left(x^{\frac{1}{3}}\right)-\left(x^{\frac{1}{6}}\right)=x^{\log _{x} 2}$ then $\frac{1}{3}-\frac{1}{6}=\log _{x} 2$ scores 0 marks

$\left[x^{\frac{1}{6}}=-1\right], \quad x^{\frac{1}{6}}=2$

A1

x=64[x=1]

A1

Rejects $x^{\frac{1}{6}}=-1$ or x=1 ignored at some stage,
soi

B1