VCE Mathematical Methods/Exam One Practice One

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Reading Time: 15 minutes
Writing Time: 60 minutes

  • Students are permitted to use: pencils, pens, highlighters, erasers, sharpeners, rulers, protractors, set-squares, aids for curve sketching
  • Students are NOT permitted to use: blank sheets of paper, white-out, any type of technology
  • Any diagrams used are NOT drawn to scale unless otherwise indicated
  • Students must answer all the questions in the space provided
  • In questions where more than one mark is available, appropriate working MUST be shown
  • When instructed to use calculus, an appropriate derivative or anti-derivative MUST be shown


Question 1[edit]

(a) Given  e^{3x+1} - 1 = 0 \,, solve for x.

(b) If  f(x) = e^{3x+1} - 1 \, and  g(x) = e^x \, state the transformations required to change g into f

[1 + 2 = 3 marks]

Question 2[edit]

Let  P(x) = x^4 + 2x^3 - 9x^2  - 2x + 8 \, and  Q(x) = x - 1 \,.

(a) Evaluate  \frac{P(x)}{Q(x)}

(b) Hence factorise P(x) given that  P(2) = 0 .

(c) Hence sketch the graph of P

[2+2+2 = 6 marks]

Question 3[edit]

Let  f: [0, \pi) \to \mathbb{R}, f(x) = -cos(x) - x . Use calculus to find the co-ordinates of the stationary point.

[3 marks]

Question 4[edit]

A garden path can be modelled with the equation  y = sin(2x) + 1 \, where  x \in [0, 2\pi] .

(a)Sketch the garden path over the domain specified.

(b) If the x-axis represents a fence, use calculus to determine the area between the path and the fence.

[2+2 = 4 marks]

Question 5[edit]

State the equations of the tangent and the normal of the function  h: ( - \infty , -2) \cup (-2, \infty) \to \mathbb{R} , h(x) = log_e (x + 2) + 3 \, when  x = 1 \,

[2 marks]

Question 6[edit]

Shirley either eats lamingtons or a muesli bar for morning tea. If Shirley eats lamingtons one day, then the probability she will eat lamingtons the next day is 0.5. If Shirley eats a muesli bar one day, the probability that she will eat a muesli bar the next day is 0.3. If Shirley eats a muesli bar on Tuesday, what is the probability she will eat lamingtons on Thursday?

[2 marks]

Question 7[edit]