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

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]