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Mathematics Advanced HSC practice questions

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Question 1Functions2 marks
Find the domain of the function $f(x) = \sqrt{x-4} + \frac{1}{x-7}$.
What a full answer covers
Students need to consider two restrictions: the argument of the square root must be non-negative, and the denominator of the fraction cannot be zero. They will then combine these conditions to define the domain.
Question 2Financial mathematics4 marks
An investment of $15,000 earns interest at a rate of 6% per annum, compounded quarterly. How many full years will it take for the investment to grow to at least $25,000?
What a full answer covers
Students need to use the compound interest formula, adjusting the interest rate and number of periods for quarterly compounding. They will then solve for the total number of compounding periods and convert this to the number of full years, rounding up to ensure the target amount is met.
Question 3Calculus6 marks
Consider the function $f(x) = x^3 - 6x^2 + 9x - 2$. (a) Find the coordinates of the stationary points of $f(x)$. (3 marks) (b) Determine the nature of each stationary point using the second derivative test. (3 marks)
What a full answer covers
For part (a), students will differentiate the function, set the derivative to zero to find the $x$-coordinates, and then substitute these back into the original function to find the $y$-coordinates. For part (b), they will find the second derivative and use it to classify each stationary point as a local maximum or minimum.
Question 4Statistical analysis5 marks
The weights of newborn babies in a particular hospital are normally distributed with a mean of 3.2 kg and a standard deviation of 0.4 kg. (a) What percentage of newborn babies weigh between 2.8 kg and 3.6 kg? (2 marks) (b) Find the weight, to one decimal place, below which 15% of newborn babies fall. (3 marks)
What a full answer covers
For part (a), students should recognise that the given range corresponds to one standard deviation from the mean, allowing them to use the empirical rule for normal distributions. For part (b), they will need to find the Z-score corresponding to the 15th percentile using a Z-table or calculator, and then use the Z-score formula to calculate the actual weight.
Question 5Trigonometric functions5 marks
Prove the identity $\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x} = 2\csc x$.
What a full answer covers
Students should start with the left-hand side of the identity, find a common denominator, and combine the fractions. They will then expand the numerator, apply the Pythagorean identity to simplify, and finally express the result in terms of $\csc x$.
Question 6Calculus6 marks
The velocity of a particle moving in a straight line is given by $v(t) = 3t^2 - 12t + 9$ m/s, where $t$ is the time in seconds. (a) Find the acceleration of the particle at $t = 2$ seconds. (2 marks) (b) Find the total distance travelled by the particle in the first 3 seconds. (4 marks)
What a full answer covers
For part (a), students will differentiate the velocity function to find the acceleration function and then substitute $t=2$. For part (b), they must first find when the velocity is zero to identify any changes in direction. Then, they will integrate the velocity function over appropriate intervals, taking the absolute value of each displacement to find the total distance travelled.
Question 7Functions3 marks
A function is defined by $f(x) = ax^2 + bx - 3$. Given that $f(1) = 2$ and $f(-2) = 11$, find the values of $a$ and $b$.
What a full answer covers
Students should set up two simultaneous linear equations using the given function values. They will then solve these equations to find the values for $a$ and $b$.
Question 8Trigonometric functions3 marks
Solve $2 \cos(x) - 1 = 0$ for $0 \le x \le 2\pi$.
What a full answer covers
Students need to first isolate $\cos(x)$. Then, they should find the principal angle and use the unit circle or graph of cosine to identify all other solutions within the specified domain.
Question 9Financial mathematics7 marks
Liam takes out a loan of $400,000 to purchase a house. The loan has an interest rate of 4.8% per annum, compounded monthly. He makes monthly repayments of $M$. (a) Show that the amount owing after $n$ months, $A_n$, can be expressed as $A_n = 400000(1.004)^n - M\frac{(1.004)^n - 1}{0.004}$. (3 marks) (b) If Liam plans to repay the loan over 30 years, calculate his monthly repayment $M$. (2 marks) (c) After 10 years, Liam decides to increase his monthly repayments to $2500. How much sooner will he repay the loan? (2 marks)
What a full answer covers
For part (a), students should derive the loan repayment formula by considering the compound interest on the initial amount and the geometric series of the repayments. Part (b) requires setting the final amount owing to zero and solving for the monthly repayment $M$. Part (c) involves first calculating the outstanding balance after 10 years, then using the loan formula again with the new repayment amount to find the remaining time and comparing it to the original schedule.
Question 10Statistical analysis3 marks
A set of data has a mean of 50 and a standard deviation of 8. If each data value is increased by 5 and then multiplied by 2, what are the new mean and standard deviation?
What a full answer covers
Students should apply the properties of mean and standard deviation under linear transformations. Adding a constant shifts the mean but not the standard deviation, while multiplying by a constant scales both the mean and the standard deviation.

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Mathematics Advanced HSC practice questions

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