Question
The number of cells, C, in a culture is given by the equation \(C = p \times 2^{0.5t} + q\), where t is the time in hours measured from 12:00 on Monday and p and q are constants.
The number of cells in the culture at 12:00 on Monday is 47.
The number of cells in the culture at 16:00 on Monday is 53.
a.Use the above information to write down two equations in p and q ;[2]
b.Use the above information to calculate the value of p and of q ;[2]
c.Use the above information to find the number of cells in the culture at 22:00 on Monday.[2]
▶️Answer/Explanation
Markscheme
p + q = 47 (A1)
4p + q = 53 (A1) (C2)[2 marks]
Reasonable attempt to solve their equations (M1)
p = 2, q = 45 (A1) (C2)
Note: Accept only the answers p = 2, q = 45.[2 marks]
C = 2 × 20.5(10) + 45 (M1)
C = 109 (A1)(ft) (C2)
Note: Award (M1) for substitution of 10 into the formula with their values of p and q.[2 marks]
Question
A store sells bread and milk. On Tuesday, 8 loaves of bread and 5 litres of milk were sold for $21.40. On Thursday, 6 loaves of bread and 9 litres of milk were sold for $23.40.
If \(b =\) the price of a loaf of bread and \(m =\) the price of one litre of milk, Tuesday’s sales can be written as \(8b + 5m = 21.40\).
a.Using simplest terms, write an equation in b and m for Thursday’s sales.[2]
b.Find b and m.[2]
Draw a sketch, in the space provided, to show how the prices can be found graphically.
[2]
▶️Answer/Explanation
Markscheme
Thursday’s sales, \(6b + 9m = 23.40\) (A1)
\(2b + 3m = 7.80\) (A1) (C2)[2 marks]
\(m = 1.40\) (accept 1.4) (A1)(ft)
\(b = 1.80\) (accept 1.8) (A1)(ft)
Award (A1)(d) for a reasonable attempt to solve by hand and answer incorrect. (C2)[2 marks]
(A1)(A1)(ft)
(A1) each for two reasonable straight lines. The intersection point must be approximately correct to earn both marks, otherwise penalise at least one line.
Note: The follow through mark is for candidate’s line from (a). (C2)[2 marks]
Question
\(10 000\) people attended a sports match. Let \(x\) be the number of adults attending the sports match and \(y\) be the number of children attending the sports match.
a.Write down an equation in \(x\) and \(y\) .[1]
b.The cost of an adult ticket was \(12\) Australian dollars (AUD). The cost of a child ticket was \(5\) Australian dollars (AUD).
Find the total cost for a family of 2 adults and 3 children.[2]
c.The total cost of tickets sold for the sports match was \(108800{\text{ AUD}}\).
Write down a second equation in \(x\) and \(y\) .[1]
d.Write down the value of \(x\) and the value of \(y\) .[2]
▶️Answer/Explanation
Markscheme
\(x + y = 10000\) (A1) (C1)[1 mark]
\(2 \times 12 + 3 \times 5\) (M1)
\(39{\text{ }}(39.0{\text{, }}39.00)\) (AUD) (A1) (C2)[2 marks]
\(12x + 5y = 108800\) (A1) (C1)[1 mark]
\(x = 8400\), \(y = 1600\) (A1)(ft)(A1)(ft) (C2)
Notes: Follow through from their equations. If \(x\) and \(y\) are both incorrect then award (M1) for attempting to solve simultaneous equations.[2 marks]