HSC Maths Advanced

Trigonometry

100

cot²x + 1 = _____

cosec²x

200

What is the domain and range of the tan function?

(100 points for correct domain, 100 points for correct range)

Range: (-∞, ∞)

Domain: (-∞, ∞), except for odd multiples of 90°

300

Define 'angle of elevation'

The angle of elevation is the angle measured upwards from the horizontal

400

x/sin(53) = 3.5sin(75)

Solve for x

x = 2.89 units

500

y = ksin(a(x + b)) + c

Identify each of the transformations

(100 points for each correct transformation)

(100 points for identifying the equation to find period)

k: amplitude

a: period

b: phase shift (left or right)

c: vertical shift (up or down)

Period = 2π/a or 2π/n

Probability and statistics

100

Define 'sample space'

The set of all possible outcomes for an experiment

200

P(A | B) 

What does this notation mean?

Conditional probability: probability of Event A occurring given Event B has occurred

300

Define 'continuous random variable' and give two examples

(100 points for correct definition, 100 points for each correct example)

Continuous random variable: measured along a continuous scale

Examples: diameter, weight, height, time, age, temperature

400

True or False:

Var(X) = E(x²) - μ²

Variance is a measure of central tendency

σ = Var(X)²

The sample mean symbol is x̄

(100 points for each correct answer)

1. True: equation is correct

2. False: variance is a measure of spread

3. False: σ² = Var(X), variance is the square of standard deviation

4. True: x̄ is sample mean, μ is population mean

500

List five ways of displaying data

(100 points for each correct answer)

Sector graph (pie chart), two-way tables, bar graph, column graph, pareto chart

Calculus

100

True or false: acceleration is the double derivative of displacement

True

200

Give two formulas for determining gradient 

(100 points for each formula)

m = rise/run

m = (y₂ - y₁)/(x₂ - x₁)

300

Differentiate y = x²e^x, factorising your answer

dy/dx = xe^x(x + 2)

400

What does the 'n' in the trapezoidal rule represent?

The number of trapeziums (or subintervals)

500

Integrate x⁴ - 4x³ + 9x² - 6x + 5

Integral: x⁵/5 - x⁴ + 3x³ - 3x² + 5x +c

Functions

100

True or false: a piecewise function is a function made up of three or more functions defined on different domains

False.

A piecewise function is a function made up of TWO or more functions defined on different domains

200

How do you find the y-intercept and the x-intercept?

(100 points for each correct answer)

y-intercept: let x = 0

x-intercept: let y = 0

300

Give the domain and range for f(x) = x²

(100 points for correct domain values, 100 points for correct range values, 100 points for correct brackets)

Domain: (-∞, ∞)

Range: [0, ∞)

400

Describe the symmetry and test for odd and even functions

(100 points for each correct symmetry and test)

Even function: symmetry across the y-axis. Even if f(x) = f(-x)

Odd function: point symmetry around the origin. Odd if f(-x) = -f(x)

500

Solve 2(x - 1)³ -5 = 0 for x

x = ³√ (5/2) + 1

x = 2.357 . . .

Financial

100

What is the formula for compound interest?

A = P(1 + r)ⁿ​

FV = PV(1 + r)ⁿ

​​​

200

What does the 'P' in the formula A = P(1 + r)ⁿ stand for?

Principal

300

Describe what an annuity and give one example of one

An annuity is a form of investment that involves regular contributions of money. Interest compounds at the end of each period before the next contribution of money is made.

Example: superannuation

400

Describe a what a reducing-balance loan is and give three examples.

A loan, the value of which decreases over time as it is paid off

Examples: car (or any other) loan, mortgage, credit card

500

Find the amount that will be in the bank after one year if $10000 is invested at 4.5% p.a. with interest paid montly.

FV = $10459.20