Calculators are allowed for all questions unless indicated by the question. There are 5 questions for this topic.

Question 1:

Calculate the equation of the line perpendicular to the line $2y +3x = 8$ and passes through the point $(6, 1)$ (3 marks)

Rearranging the equation of line in the form $y = mx + c$:

$y =-1.5x + 4$

The new gradient is $\frac{2}{3}$ (1 mark)

Using the given coordinates:

$1 = \frac{2}{3}(6) + c$

$1 = 4 + c$

$c = -3$ (1 mark)

$y = \frac{2}{3}x -3$ (1 mark)

Question 2: (Non-calculator)

Write the following expression as a single fraction:

$\frac{x+1}{3} + \frac{2x+2}{6}$ (3 marks)

Multiply the first term by 2 in the numerator and the denominator:

$\frac{x+1}{3} = \frac{2x+2}{6}$ (1 mark)

Multiply the second term by 3 in the numerator and the denominator:

$\frac{x-6}{2}= \frac{3x-18}{6}$ (1 mark)

Now they both have a common denominator we can add them together:

$\frac{2x+2}{6} + \frac{3x-18}{6} = \frac{5x-16}{6}$ (1 mark)

Question 3:

$f(x) = 3x +2, g(x) = 1-3x$

A. Calculate $g(3)$(1 mark)

B. Calculate $gf(3)$ (3 marks)

$g(3) = 1-3(3) = -8$ (1 mark)

$f(3) = 3(3) +2 = 11$ (1 mark)

$g(11) = 1-3(11) = -32$ (2 marks)

Question 4:

Josh has 500ml of paint to paint a model house. The real house is 40 times larger than the model.

Calculate the amount of paint needed for the real house in litres. (2 marks)

Paint needed is $500 \times 40^2 = 800000ml = 800$ litres (2 marks)

40 is squared because the relationship between the surface area of the real house and the model house is not linear, it is squared. If it was the volume, it would be cubed.

Question 5:

A colony of bacteria grows by 9% every hour. The mass at the start point is $4.5g$

A. Determine the formula for mass ($M$) in $g$, at time $t$ hours since the start point. (2 marks)

B. Estimate the mass of the bacteria after 4 hours and 30 minutes to 2dp. (2 marks)

C. Why can you not calculate the exact value for part B? (1 mark)

The mass increases by 9% every hour.

The original mass is $4.5g$, as the colony is growing exponentially. The rate of increase is $1.09^t$ (1 mark)

Therefore $M = 4.5 \times 1.09^t$ (1 mark)

$M = 4.5 \times 1.09^t , t = 4.5$ (1 mark)

Therefore $M = 4.5 \times 1.09^{4.5} = 6.63g$ (1 mark)

The colony grows 9% every hour. This doesn’t imply the colony grows 4.5% every half hour. If it did, then that colony would grow 4.5% for the second half hour, which results in 9.2% growth an hour, not 9%. We don’t know the rate of growth for half an hour, only the growth of an hour. (1 mark)

Accept any reasonable answer showing understanding of exponential growth.