Methods2025 Exam 1 · Q9Tech-free7 marks

Intersections of a rational function and a quadratic with parameter w

Consider the functions

$f: \R \setminus \{1\} \to \R,\ f(x) = \frac{w^2}{(x-1)^2}$

If $w = -3$ , find the four solutions to $f(x) = g(x)$ .

3 marks

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b.i.

Consider the case where $w > 0$ .

Find, in terms of $w$ , the coordinates of the minimum point of the graph of $y = (x-1)(x-w)$ .

2 marks

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b.ii.

Hence, or otherwise, find the positive values of $w$ for which $f(x) = g(x)$ has exactly three solutions.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.