VCE Mathematical Methods/Exam One Practice One

Instructions

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

Questions

Question 1

(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

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

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

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

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

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]