IBDP Past year exam questions -Quadratics

1.  [M18/P1/TZ1]

Solve  ${\left(\ln x\right)}^2–\left(\ln 2\right)\left(\ln x\right)<2{\left(\ln 2\right)}^2$ .   [6 marks]

2.  [M18/P2/TZ2] 

Given  $g\left(t\right)={\left(\frac{1+t}{1–t}\right)}^2$ . Let  $\alpha , \beta$ be the roots of  $g\left(t\right)=k$ , where  $0<k<1$ .

(i)   Find  $\alpha$ and  $\beta$ in terms of  $k$ .

(ii)   Show that  $\alpha +\beta <–2$ .   [7 marks]

3.  [N17/P2/TZ0] 

In the quadratic equation $7x^2–8x+p=0$ , $\left(p\in \mathrm{\mathbb{Q}}\right)$ , one root is three times the other root. Find the value of $p$ .