Practice section 2 of 4

[Graphs and Applications of Differentiation] The curve $y=f(x)$, where $f$ is a cubic polynomial, has stationary points at $(-2,8)$ and $(\tfrac12,\tfrac{189}{8})$. Determine the equation of the curve.

A blood donation drive collected 520 litres from 1,200 donors, each with a unique type A, B or O. The volume collected per donor is:

By writing down a system of equations, find the range of possible numbers of type O donors.

Ace borrowed 10 books and 3 audio-visual items. Overdue fines per day are $x$ cents per book and $y$ cents per audio-visual item, each a multiple of 5 cents. Four days after the due date, he realises at school that none has been returned. He has $k$ books and no audio-visual items with him.

Option A: Return these $k$ books on his way home and the remaining books and audio-visual items the following day; total fine $14.55.

Option B: Return all items the following day; total fine $15.

Using this information, find $x$ and $y$.

A cinema sells tickets at three different prices, depending on the age of the customers. The age categories are under 16 years, between 16 and 65 years, and over 65 years.

Three groups of people A, B and C go to the cinema on the same day.

The numbers in each age category for each group, together with the total cost of the tickets for each group, are given in the following table.

Write down and solve equations to find the cost of a ticket for each of the age categories. [4]

Four housewives bought three different types of meat from the same meat seller. The mass of meat and the total each housewife paid are shown in the table below.

Find the value of $a$. [4]

A shop selling biscuits gave the following offers. A customer who bought a total of 5 or more packets of biscuits was given a discount, regardless of the types of biscuits bought. A 10% discount was given if 5 to 8 packets were bought and a 20% discount if more than 8 packets were bought.

The following table shows the number of packets bought and the total amounts that three customers paid.

Calculate the price per packet for each type of biscuit before the discount. [4]