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Mathematics Standard 2 HSC practice questions

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Question 1Algebra2 marks
Simplify the expression $\frac{x^{2} - 9}{x^{2} - 6x + 9}$.
What a full answer covers
Factor both numerator and denominator, cancel the common factor and write the reduced form.
Question 2Financial mathematics9 marks
A loan of $\$20,000$ is to be repaid over $5$ years with monthly payments at an annual nominal interest rate of $6\%$ compounded monthly. Calculate: a) the monthly payment amount, b) the total interest paid over the life of the loan.
What a full answer covers
Find the monthly rate, apply the amortisation formula to get the payment, then multiply by the number of payments and subtract the principal.
Question 3Financial mathematics3 marks
A sum of money is invested at $4\%$ per annum simple interest for $3$ years and then at $5\%$ per annum for a further $2$ years. If the total interest earned is $\$650$, find the principal amount.
What a full answer covers
Set up the interest equation for the two periods, combine terms, solve for the principal and round to the nearest dollar.
Question 4Networks3 marks
In a weighted graph the vertices are connected by the following edges: $AB=4$, $AC=2$, $BD=5$, $CD=1$, $CE=3$, $DE=2$. Determine the length of the shortest path from $A$ to $E$.
What a full answer covers
Consider each possible route, add the edge weights and pick the smallest total.
Question 5Algebra6 marks
Solve the system \[ \begin{cases} 3x + 2y = 7 \\ 5x - y = 4 \end{cases} \] Then verify the solution by substitution.
What a full answer covers
Eliminate one variable, solve for the other, back‑substitute to find the remaining variable, then check both original equations.
Question 6Measurement8 marks
Water is being pumped into a conical tank with radius $5\,\text{m}$ and height $12\,\text{m}$. The water depth $h$ is increasing at a rate of $0.3\,\text{m}\,\text{min}^{-1}$ when $h = 6\,\text{m}$. Find the rate at which the water volume is increasing at that instant.
What a full answer covers
Express volume in terms of depth, differentiate, substitute the given depth and rate, then compute the numeric value.
Question 7Algebra3 marks
Solve the quadratic equation $2x^2 - 5x - 3 = 0$ and state the solution(s) as exact values.
What a full answer covers
Use the quadratic formula, simplify the square root and give the two exact solutions.
Question 8Measurement4 marks
A solid consists of a cylinder of radius $3\,\text{cm}$ and height $8\,\text{cm}$ attached to a hemisphere of the same radius on one end. Find the total volume of the solid, giving your answer in cubic centimetres.
What a full answer covers
Calculate the cylinder volume, the hemisphere volume, add them and give the final numeric answer.
Question 9Statistical analysis2 marks
For the data set \{12, 15, 15, 18, 22, 22, 22, 27\}, find the mean, median and mode.
What a full answer covers
Compute the mean using the sum, then identify the median and the most frequent value.
Question 10Statistical analysis7 marks
The following data represent the ages (in years) of participants in a study: \{22, 25, 27, 30, 31, 33, 34, 36, 38, 45, 48, 52, 58, 60, 62\}. Construct a box plot and determine the interquartile range (IQR). Identify any outliers using the $1.5\times\text{IQR}$ rule.
What a full answer covers
Find the quartiles, compute the IQR, apply the outlier rule and state whether any values lie beyond the fences.

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Mathematics Standard 2 HSC practice questions

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