In this post, our Senior Maths Coordinator Oak Ukrit shares 7 HSC Maths Advanced questions that are likely to appear in your upcoming HSC Maths Trial Exam.

Have you switched on your brain? Alright, let’s do this!

## Question 1 – Laws of Logarithms

Given that

$$x^{a}=y^{b}=(xy)^{c}$$

Prove that

$$ab=c(a+b)$$

## Question 2 – Trigonometric Functions

Evaluate

$$\int_{0}^{\pi/2 } sin^{2} \ x \ dx + \int_{0}^{\pi/2 } cos^{2} \ x \ dx$$

$$\frac{\pi }{2}$$

## Question 3 – Application of Arithmetic/ Geometric Progression

Sam borrows $P to fund his new car. The term of the loan is 10 years with an interest rate of 6% p.a., reducible monthly. He repays the loan in equal monthly instalments of$750.

a.   Show that at the end of n months, the amount owing is given by

= P(1.005)– 150000(1.005)n  + 150000

b. If at the end of 10 years, the loan has been repaid in full, find the amount Sam originally borrowed, to the nearest dollar.

\$67, 555

## Question 4 – Differentiation

Show that $$\frac{d}{dx}ln(f(x)) = \frac{\mathrm{\textit{f} \ ^ \prime }(x) }{\mathrm{\textit{f}} \ (x)}$$. Hence, find an expression for $$\textit{f}(x) \$$ if $$\ \frac{\mathrm{\textit{f} \ ^ \prime }(x) }{\mathrm{\textit{f}} \ (x)} = \ – \frac{x}{2a}\\$$

and $$\ f(0)=\sqrt{\frac{\pi }{a}}$$

$$f(x) = \sqrt{\frac{\pi }{a}} \ e ^-{\frac {x^{2}}{4a}}$$

## Question 5 – Volumes by Integration

The region bounded by the curves y = ln(x), the co-ordinate axes, and the line y = ln(3) is rotated about the y-axis. Find the volume of the solid formed by the rotation in exact form.

## Question 6 – Probability

A game is played where an n-sided die is rolled repeatedly. (An -sided die has the numbers 1, 2, 3, …, n on the different faces). The game ends when the number facing up after rolling the die is n.

a. Using a tree diagram or otherwise, show that the probability that the game finishes 3 rolls or less is: $$\frac{1}{n} + (\frac {n-1}{n}) \frac{1}{n} + (\frac {n-1}{n})^{2} \frac{1}{n}$$

b. Hence, show that the probability that the game finishes in n rolls or less is given by $$1-(1- \frac{1}{n})^{n}$$

c. What is the probability that the game takes longer than n rolls to finish?

$$\left (1 -\frac{1}{n}\right)^{^{n}}$$

## Question 7 – Fundamentals of Calculus

Differentiate $$f(x) = \sqrt{x^3}$$ from first principles

$$f^ \prime (x) =\frac{3}{2} \sqrt{x}$$

How did you go? Did you get them all right? They were some pretty tough questions.