Practice section 3 of 4

Three brands of apple juice, A, B and C, each contain a certain percentage of pure juice and some artificial flavouring. The breakdown of the amount of juice and flavouring for each brand is summarised in the table below.

How many litres of each brand should be mixed to produce 300 litres of juice with 41.1% pure juice and 0.647% artificial flavouring? [3]

In basketball, points are scored by a successful free throw or a field goal. A successful free throw is worth 1 point while a field goal is worth either 2 or 3 points. An attempt can either be a free throw attempt or a field goal attempt.

In the recently concluded basketball season, Stephen had 1695 attempts in total. 60% of these attempts were successful and he scored a total of 1917 points. 40% of his successful free throws and 25% of his field goal points were scored during the second half of the season, where Stephen scored a total of 534 points.

Calculate the number of 2-point field goals during the season. [5]

The equation of a curve $C$ is given by $y=ax+b+c/x$, where $a,b,c$ are constants. The gradient of $C$ at the point $(1,1)$ is 3. It is also given that $C$ passes through the point $(2,2)$. Find the values of $a,b,c$. Hence state the equations of the asymptotes of $C$. [4]

The graph of $y=f(x)$, where $f(x)$ is a cubic polynomial, passes through the points $(1,6)$, $(-2,15)$ and has two stationary points at $x=1/3$ and $x=-2$.

Find the equation of the curve and hence find its $x$-intercept. [5]

A cubic polynomial $p(x)$ leaves a remainder of 52 when divided by $x-2$. Given that the curve with equation $y=p(x)$ passes through the point $(1.5,32.375)$ and has a stationary point at $(-1,-2)$, find $p(x)$. [4]

A brick manufacturer has a depot with a stock of 30 000 bricks. He receives orders from three builders P, Q and R for a total of 30 000 bricks and the total order from builders P and R is twice that of builder Q.

(a) If the profit earned from each brick is $3 and the manufacturer earns $9 600 more in profit from builder R compared to builder P, find the number of orders from each builder. [4]

(b) Instead of using the information in part (a), it is now given that the cost of transporting each brick to builders P, Q and R from the depot is $0.04, $0.02 and $0.03 respectively. If there is a minimum order of 1000 bricks from each builder, what are the orders that the manufacturer should accept so as to keep the cost of transportation a minimum? [3]