Methods Mega Quiz

Tech Free §

Calculus §

Question 1 (2016 MAV Trial Exam 1, Q1b) §

a) Factorise $x^3-3x^2+3x-1$.

b) Hence, anti-differentiate $\frac{1}{(1-x)\left(x^3-3x^2+3x-1\right)}$.

Question 2 (2016 MAV Trial Exam 1, Q5) §

Let $f(x)=x{e}^{2x}$.

a) Find $f^{\prime }(x)$.

b) Hence, find the average value of $f$ over the interval $[0,\frac{1}{2}]$.

Functions & Graphs §

Question 1 (2016 MAV Trial Exam 1, Q3) §

Consider the function $g:[0,2]\to R,g(x)=2x^5-10x^4+20x^3-20x^2+10x+2$.

a) Find $A,B$ and $C$ given $g(x)=A(x+B{)}^5+C$, where $A,B$ and $C$ are real constants.

b) Sketch the graph of $g$ on the set of axes below. Label the endpoints and any stationary points with their coordinates.

Algebra §

Question 1 (2016 MAV Trial Exam 1, Q4) §

a) Show that $x={\log }_e(3)$ is a solution of the equation $\frac{9}{e^{2x}}-\frac{6}{e^x}+2e^x=5$.

b) Hence, or otherwise, solve the equation $\frac{9}{e^{2x}}-\frac{6}{e^x}+2e^x=5$ for $x$, given that there are only two real solutions.

Question 2 (2016 MAV Trial Exam 1, Q6) §

Find the values of $A$ and $B$, where $A$ and $B$ are real constants, if the graph of $y=A{\log }_e(x-B)$ passes through the points $(2,10)$ and $(8,20)$.

Tech Active §

Probability §

Question 1 §

The time (in minutes) it takes Jerry to cycle to school is given by the random variable $T$ with the pdf:

$$f(t)=\{\begin{array}{ll}\frac{1}{24}(t-6)& 6\le t\le 10\\ \frac{1}{48}(K-t)& 10\le t\le K\end{array}$$

a) Find $K$. Hence state the minimum and maximum times it could take Jerry to cycle to school.

b) Sketch the pdf on the set of axes below.

c) Find the median time it takes Jerry to cycle to school.

d) Find the expected time it takes Jerry to cycle to school.

e) Find, correct to three decimal places, the probability that Jerry takes longer than 10 minutes at least twice in a 5-day school week.