IB Maths AA HL Sl 2 7 Quadratic Equations and Inequalities Questions

Practise IB Maths AA HL 2.7 by solving parameterised quadratic equations and inequalities using discriminants and graph conditions.

Exam points

Solve quadratic equations and inequalities by factoring, formula, completing the square or graphing.
Interpret solution sets and parameter conditions using complete mark-scheme evidence.

[Maximum number: 8]

The functions f and g are defined by

\begin{aligned} & f(x)=\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}, x \in \mathbb{R} \\ & g(x)=\frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}, x \in \mathbb{R} \end{aligned}

(a)

Let h(x)=n f(x)+g(x) where $n \in \mathbb{R}, n>1$ .

[ 8 ]

(i)

By forming a quadratic equation in $\mathrm{e}^{x}$ , solve the equation h(x)=k, where $k \in \mathbb{R}^{+}$ .

[ 5 ]

$\quad(n+1) \mathrm{e}^{2 x}-2 k \mathrm{e}^{x}+(n-1)=0$ $\mathrm{e}^{x}=\frac{2 k \pm \sqrt{4 k^{2}-4\left(n^{2}-1\right)}}{2(n+1)}$ $x=\ln \left(\frac{k \pm \sqrt{k^{2}-n^{2}+1}}{n+1}\right)$

(ii)

Hence or otherwise show that the equation h(x)=k has two real solutions provided that $k>\sqrt{n^{2}-1}$ and $k \in \mathbb{R}^{+}$ .

[ 3 ]

for two real solutions, we require $k>\sqrt{k^{2}-n^{2}+1}$
and we also require $k^{2}-n^{2}+1>0$ $k^{2}>n^{2}-1$ $\Rightarrow k>\sqrt{n^{2}-1}\left(k \in \mathbb{R}^{+}\right)$

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IB Maths AA HL Sl 2 7 Quadratic Equations and Inequalities Questions question 1

Question (a)

Question (i)

Question (ii)