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Mathematics Extension 2

7 Must Answer Questions for your Maths Advanced HSC Exam

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.

Are you ready for your Maths Advanced HSC exam? I mean, are you sure you’re ready? Well, now’s your chance to see how ready you really are. Our head of Maths, Oak Ukrit has compiled 7 must answer questions for your Maths Advanced HSC Exam. There’s one for each fundamental skill. So, let’s see how ready you really are.

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

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

 

Answer:

\(\frac{\pi }{2}\)

 

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.

 

 

Answer:

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

 

 

 

Answer:

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

 

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.

 

Answer:

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?

Answer:

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

Question 7 – Fundamentals of Calculus

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

Answer:

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

 

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

 

 

 

Written by Oak Ukrit

Oak is the Senior Mathematics Teacher at Matrix Education and has been teaching for over 10 years. He has 1st class honours in Aeronautical Engineering from UNSW where he taught for over 4 years while he was undertaking a PhD. When not plane spotting he enjoys landscape photography.

 

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