12 Methods Practice Topic Test 3 Ver. 2 PDF

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VCE Mathematical Methods Units 3 & 4 Practice Topic Test 3 Ver. 2 20 Name: ___________________ Date: _________________ Instructions Answer all questions in the space provided. In all questions where a numerical answer is required, an exact value must be given, unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 (9 marks) Solve the following equations for 𝑥. a. 81−2𝑥 = 24+𝑥.

2 marks

b. 32𝑥 − 8 × 3𝑥 = 9

2 marks

c. log 𝑒 (𝑥) + log 𝑒 (3𝑥 + 2) = 2 log 𝑒 (𝑥 + 1)

3 marks

1

d. log 5 (𝑥3 ) + 2 log 5 (𝑥) = 15

2 marks

Question 2 (3 marks)

1 𝑥 −2 0 𝑥 The transformation 𝑇 ∶ ℝ2 → ℝ2 is defined by ([𝑦 ]) = [ 2 ] [𝑦 ] + [ ]. 1 0 −3 𝑥−4

Under this transformation the image of the curve of the curve 𝑦 = 𝑒 2 − 1 has equation

𝑦 = 𝑎 + 𝑏𝑒 𝑥 . Find the values of 𝑎 and 𝑏.

2

Question 3 (8 marks)

Let 𝑔 ∶ (1, ∞) → ℝ, 𝑔(𝑥) = 2 log 𝑒 (𝑥 − 1).

The graph of 𝑔 is shown below.

a. Find the coordinates of the 𝑥-axis intercept.

1 mark

b. Find the rule of 𝑔−1 , the inverse of 𝑔.

2 marks

c. Sketch the graph of 𝑦 = 𝑔−1 (𝑥) on the axes above. Label the equation of the asymptote and the coordinate of the 𝑦-axis intercept.

3 marks

d. State the domain and range of 𝑔−1 .

2 marks

END OF QUESTION AND ANSWER BOOK

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