IB Maths AA HL 1.6 Deductive proof and generalisation Question Bank

Practise IB Maths AA HL 1.6 by proving general identities and extending results to parameterised or inequality statements.

[Maximum number: 2]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

Show that $x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A}$ .

METHOD 1

\begin{aligned} & x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}+A\right)-2 A x}{x^{2}+A}\left(\equiv \frac{x^{3}+A x-2 A x}{x^{2}+A}\right) \\ & \equiv \frac{x^{3}-A x}{x^{2}+A}\left(\equiv \frac{x\left(x^{2}+A-2 A\right)}{x^{2}+A}\right) \\ & x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A} \end{aligned}

Note: Award no marks for numerical approaches.

METHOD 2

EITHER
attempt to do long division on $\frac{x\left(x^{2}-A\right)}{x^{2}+A}$ or $\frac{x^{3}-A x}{x^{2}+A}$
quotient of x and remainder of -2 A x
$x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A}$

OR
$\frac{x^{3}}{x^{2}+A}-\frac{A x}{x^{2}+A} \equiv \frac{x^{3}+A x}{x^{2}+A}-\frac{A x+A x}{x^{2}+A}$ $\equiv \frac{x\left(x^{2}+A\right)}{\left(x^{2}+A\right)}-\frac{2 A x}{\left(x^{2}+A\right)}$ $\equiv x-\frac{2 A x}{x^{2}+A} \equiv \frac{x\left(x^{2}-A\right)}{x^{2}+A}$

Note: Award no marks for numerical approaches.

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SL 1.6—Deductive proof question 1