HSC 2023 Maths Advanced

How well did you do in the HSC Maths Advanced exam? In this blog, let’s break down the solution and strategies from this year’s exam. Whether you’re aiming for a perfect score or striving to overcome specific challenges, this detailed review will provide you with the guidance and knowledge you need. This can also help you sharpen your Maths skills in preparation for next year.

Section I – Multiple Choice

Before we begin, you should already know:

1. This question requires you to find Pearson’s correlation coefficient from a scatter plot.

D – 0.8. No need to do calculations – almost a straight line (so strong correlation) and a positive gradient

2. This question requires you to find the probability of getting a score of 7 or more in a game involving throwing a die and spinning a spinner.

D – ⁵⁄₁₂ There are a total 10 favourable outcomes [scores of 7,8,9,10] and total possible outcomes are 24 therefore ¹⁰ / ₂₄ = ⁵ / ₁₂

3. This question requires you to find the domain of the reciprocal square-root function

A – x < 1. Denominator has be greater than 0.

4. This question requires you to find the correct information for a polynomial function, including the polynomial function’s equation and the value of its coefficients.

B – Equation?: y = – (x – b) (x – c)² , b <0, c>0
The function is not passing through the origin – therefore can’t be c or d
There is a double root therefore c>0

5. This question requires you to evaluate a definite integral.

A – -1

6. This question requires you to choose the correct sketch of a function given the value of its first and second derivatives
C
Positive gradient at x=2 and x=-2, eliminate B&D
0 gradient at x=0, eliminate A

7. This question requires you to evaluate the value of the first derivative at a given value for  when given other information.
A – -8

8. This question requires you to find the solution to a logarithmic equation given that two values are positive constants.

9. This question requires you to choose the even function from the options given.
D – f(x)f(-x)
When substituting -x, only D gives the original equation

10. This question requires you to find the length of ST, which is the interval formed by

intersecting with each other.
C – aL

Section II – Short Answers

Q11. This question requires you to find the 15th term of an arithmetic sequence where the first three terms are 3, 7 and 11.

a = 3
d = 4 [(7 – 3 = 4) and (11 – 7 = 4}]
T = a + (n – 1)d
T₁₅ = 3 + (15 – 1)(4)
T₁₅ = 3 +56 = 59

Q12. The table shows the probability distribution of a discrete random variable.

x	0	1	2	3	4
P (X = x)	0	0.3	0.5	0.1	0.1

Part a asks you to find the expected value E(X) = 2
Show that the expected value E(X). (1 mark)
E(X) = 0 + 0.3 + 1 + 0.3 + 0.4 = 2

Part b asks you to calculate the standard deviation to one decimal place.
Calculate the standard deviation, correct to one decimal place. (2 marks)

Q13. This question asks you to find an expression for the exponential function P(t) given that

Q14. This question asks you to find the equation of the tangent to y = (2x + 1)³ at (0,1)

y = 6x + 1 (equation of tangent)

Q15: This question provides you with the future value table. Micky wants to save $450,000 over the next 10 years.
Part a) asks you to find how much Mickey should contribute to his savings each year to reach his goal. You are given that the interest rate is 6% p.a. You also need to round to the nearest dollar.

A. A = p(rate)

450000 = p(13.181)
P= ⁴⁵⁰⁰⁰⁰ / ₁₃.₁₈₁ = $34140

Part b) provides that Micky contributes $8 535 every 3 months for 10 years to an annuity paying 6% per year, compounding quarterly. You are asked to find how much Micky will have at the end of 10 years.

A = $463177.38

Q16: You are provided with compound shape APQBCD, and are asked to find the shape’s perimeter correct to one decimal place.

Perimeter = AD + DC + CB + BQ + QP + PA
PA + BQ = 2 * 2.1 * sin (55)
PA + BQ =3.4404…m

QP = 110/360 *2PI*2.1
Substituting these values gets us,

Perimeter = 3.6 + 8.0 + 3.6 + 3.44 + 11-/360 * 2 * PI * 2.1
Perimeter = 28.7 m to 1d.p

Q17: This question asks you to evaluate

Q18: You are provided with a series of data for the average daily temperature across a series of days  and the total daily gas usage across those days (x) and the total daily gas usage acrpss those days (y).

Part a asks you to plot

and the y-intercept of the least-squares regression line on the grid shown.

y – intercept: y = 236 MW
Sample mean for x = (0 + 0 + 0 + 2 + 5 + 7 + 8 + 9 + 9 + 10)/10
= 5 degrees celsius Sample mean for y = 1840MW/10 = 184MW
Therefore, the two points that need to be plotted are (0, 236 MW) and (5,184)

Part b) asks you to find the equation of the least squares regression line
b) Start off with the general equation: y = mx + b, where b = 236 (y – intercept)
To find the gradient, we can use

Therefore, the question of the regression line is: y = -10.4x + 236

Part c) asks you to identify one problem with using the regression line to predict gas usage when the average outside temperature is 23°C

c) Data sets are not wide enough to cover an outside average temperature of 23 degrees. This is reflected in the graph wherein for temperatures more than 22.692 degrees celsius, the average gas usage is suggested to be less than zero.

Q19: Part a) asks you to sketch f(x) = x – 1 and g(x) = (1 – x)(3+x) showing the x-intercepts.

a.)

Part b) asks to to solve the inequality x – 1 < (1 – x)(3+x)

Solution includes all the x-values for which (y = x – 1) is less than the parabola. From the graph this can be deduced: x => (-4, 1)

Q21: This question asks you to find the possible value/s of the common ratio and corresponding first term/s for a geometric sequence with a fourth term of 48 and an eighth term of ³ / ₁₆.

Q23: This question provides you with a z-score table for a normal distribution. You are told that the weights of adult male koalas form a normal distribution with a mean

Q24: You are provided with a diagram of a garden with dimensions marked.

Part b) asks you to find the value of  such that the area of the concrete path is a minimum.

x	-10	-8	0	12	15
A’	11/36	0	-24	0	69/169

Therefore, x = 12 is the minimum as the gradient is negative before, but positive after it!

Therefore, this proves that when ‘x’ is equal to 12, we get the minimum area!

Q25: You are told that on the first day of November Jia deposits $10,000 into a new account which earns 0.4% interest per month, compounded monthly. You are then told that Jia intends to withdraw $M from the account at the end of each month after interest is added to the account.

Part a) asks you to show that the amount at the end of the second month (A₂) is 10 000(1.004)² – M(1,004) – M.

Part b) asks you to show that the amount at the end of n months is

Part c) asks you to find the largest value of M that will allow Jia to make at least 100 withdrawals.

Q26 provides a diagram to show how a camera is filming the motion of a swing in a park x(t)  is the horizontal distance (m) from the camera to the set of the swing at t seconds, and the seat is said to be released from rest at a horizontal distance of 11.2m from the camera.

Part a) This part asks you to find an expression for x(t) given that the rate of change

Part b) asks you to find how many times the swing reaches the closest point to the camera during the first 10 seconds.

Q27: This question provides you with a diagram of the graph f(x) = a |x-b|+c  which passes through the points (3,-5), (6,7) and (9,-5).

Part a) asks you to find the values of a, b and c.
b = 6 (shifted 6 units to the right)

Part b) provides that the line y = mx cuts the graph of y = f(x) in two distinct places. You are then asked to find all possible values of m.

b. The maximum possible gradient must be smaller than the line passing through (6,7)

Similarly, the line cannot be steeper than the right hand side of the graph i. e

Q28: You are provided with a diagram of y = f(x) and that the equation of the tangent to y = f(x) at T( – 1,6) is y = x + 7. You are told that another tangent parallel to the tangent at T passes through the curve at point R. You are also told that the gradient

Part b) This part asks you to find the cumulative distribution function for the probability density function provided.

Part c) asks you to show the mode is greater than the median without calculating the median.

Part b) asks you to sketch the graph of y = f(x) for the domain provided, showing stationary points and intercepts. You are told that you are not to use any further calculus f(0) = f(π) = f(2π) = 0

Q31: You are given that four Year 12 students want to organise a graduation party, and that all four students have the same probability, P(F), of being available next Friday. You are also told that all students have the same probability, P(S) of being available next Saturday.

Part a) asks you whether Kim’s availability next Friday (Kim is one of the students) is independent from his availability next Saturday? You are also asked to justify your answer.

Part c) asks you to find the probability that at least one of the four students is not available next Saturday.

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