# VCE Mathematical Methods/Exam One Practice One

## Instructions

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 ${\displaystyle e^{3x+1}-1=0\,}$ , solve for x.

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

[1 + 2 = 3 marks]

### Question 2

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

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

(b) Hence factorise P(x) given that ${\displaystyle P(2)=0}$ .

(c) Hence sketch the graph of P

[2+2+2 = 6 marks]

### Question 3

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

[3 marks]

### Question 4

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

(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 ${\displaystyle h:(-\infty ,-2)\cup (-2,\infty )\to \mathbb {R} ,h(x)=log_{e}(x+2)+3\,}$  when ${\displaystyle x=1\,}$

[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]