2016 VCE Maths Methods Mini Test 8

Question 1 [2016 Exam 2 Section B Q4]

a. Express \(\frac{2x+1}{x+2}\) in the form \(a + \frac{b}{x+2}\), where \(a\) and \(b\) are non-zero integers. 2 marks

b. Let \(f: R \setminus \{-2\} \to R, f(x) = \frac{2x+1}{x+2}\).

i. Find the rule and domain of \(f^{-1}\), the inverse function of \(f\). 2 marks

ii. Part of the graphs of \(f\) and \(y = x\) are shown in the diagram below.

Find the area of the shaded region. 1 mark

iii. Part of the graphs of \(f\) and \(f^{-1}\) are shown in the diagram below.

Find the area of the shaded region. 1 mark

c. Part of the graph of \(f\) is shown in the diagram below.

The point \(P(c, d)\) is on the graph of \(f\).
Find the exact values of \(c\) and \(d\) such that the distance of this point to the origin is a minimum, and find this minimum distance. 3 marks

d. Let \(g: (-k, \infty) \to R, g(x) = \frac{kx+1}{x+k}\), where \(k > 1\).
Show that \(x_1 < x_2\) implies that \(g(x_1) < g(x_2)\), where \(x_1 \in (-k, \infty)\) and \(x_2 \in (-k, \infty)\). 2 marks

e.

i. Let \(X\) be the point of intersection of the graphs of \(y = g(x)\) and \(y = -x\).
Find the coordinates of \(X\) in terms of \(k\). 2 marks

ii. Find the value of \(k\) for which the coordinates of \(X\) are \((-\frac{1}{2}, \frac{1}{2})\). 2 marks

iii. Let \(Z(-1, -1)\), \(Y(1, 1)\) and \(X\) be the vertices of the triangle \(XYZ\). Let \(s(k)\) be the square of the area of triangle \(XYZ\).

Find the values of \(k\) such that \(s(k) \ge 1\). 2 marks

f. The graph of \(g\) and the line \(y = x\) enclose a region of the plane. The region is shown shaded in the diagram below.

Let \(A(k)\) be the rule of the function \(A\) that gives the area of this enclosed region. The domain of \(A\) is \((1, \infty)\).

i. Give the rule for \(A(k)\). 2 marks

ii. Show that \(0 < A(k) < 2\) for all \(k > 1\). 2 marks