2020 VCE Maths Methods Mini Test 10

Question 1 [2020 Exam 2 Section B Q5]

Let \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = x^3 - x\).
Let \(g_a: \mathbb{R} \to \mathbb{R}\) be the function representing the tangent to the graph of \(f\) at \(x=a\), where \(a \in \mathbb{R}\).
Let \((b, 0)\) be the \(x\)-intercept of the graph of \(g_a\).

a. Show that \(b = \frac{2a^3}{3a^2-1}\). 3 marks

b. State the values of \(a\) for which \(b\) does not exist. 1 mark

c. State the nature of the graph of \(g_a\) when \(b\) does not exist. 1 mark

d.

i. State all values of \(a\) for which \(b = 1.1\). Give your answers correct to four decimal places. 1 mark

ii. The graph of \(f\) has an \(x\)-intercept at \((1, 0)\).
State the values of \(a\) for which \(1 \le b < 1.1\). Give your answers correct to three decimal places. 1 mark

The coordinate \((b, 0)\) is the horizontal axis intercept of \(g_a\).
Let \(g_b\) be the function representing the tangent to the graph of \(f\) at \(x=b\), as shown in the graph below.

e. Find the values of \(a\) for which the graphs of \(g_a\) and \(g_b\), where \(b\) exists, are parallel and where \(b \ne a\). 3 marks

Let \(p: \mathbb{R} \to \mathbb{R}\), \(p(x) = x^3 + wx\), where \(w \in \mathbb{R}\).

f. Show that \(p(-x) = -p(x)\) for all \(w \in \mathbb{R}\). 1 mark

A property of the graphs of \(p\) is that two distinct parallel tangents will always occur at \((t, p(t))\) and \((-t, p(-t))\) for all \(t \ne 0\).

g. Find all values of \(w\) such that a tangent to the graph of \(p\) at \((t, p(t))\), for some \(t > 0\), will have an \(x\)-intercept at \((-t, 0)\). 1 mark

h. Let \(T: \mathbb{R}^2 \to \mathbb{R}^2\), \(T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} m & 0 \\ 0 & n \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} h \\ k \end{pmatrix}\), where \(m, n \in \mathbb{R}\setminus\{0\}\) and \(h, k \in \mathbb{R}\).
State any restrictions on the values of \(m, n, h\) and \(k\), given that the image of \(p\) under the transformation \(T\) always has the property that parallel tangents occur at \(x=-t\) and \(x=t\) for all \(t \ne 0\). 1 mark