HSC Mathematics Standard 1 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Gradient formula

The gradient of a line through two points is the difference in y-values divided by the difference in x-values: m = (y₂ − y₁)/(x₂ − x₁).

Gradient–intercept form

A linear equation written as y = mx + b, where m is the gradient of the line and b is the y-intercept, the y-value when x equals zero.

Perpendicular line gradients

Two lines are perpendicular when the product of their gradients equals −1. For example, a line of gradient 3 is perpendicular to one of gradient −1/3.

Simultaneous equations by substitution

Rearrange one equation to express a variable in terms of the other, substitute this into the second equation, solve for one variable, then back-substitute to find the other.

Quadratic formula

For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac))/(2a). The discriminant b² − 4ac shows how many real solutions exist.

Direct variation

A relationship of the form y = kx, where k is a constant found from a known pair of values. As x increases, y increases in the same proportion.

Inverse variation

A relationship of the form y = k/x, where k is constant. As x increases, y decreases, and the product xy always equals the same value k.

Solving linear inequalities

Solve like an equation, but reverse the inequality sign whenever both sides are multiplied or divided by a negative number.

Circle circumference and area

Circumference C = 2πr or πd, where d is the diameter. Area A = πr². Both use the radius r, half the diameter.

Triangle area formula

Area of a triangle equals half the base multiplied by the perpendicular height: A = ½ × b × h. The height must meet the base at right angles.

Trapezium area formula

Area of a trapezium is A = ½(a + b)h, where a and b are the lengths of the parallel sides and h is the perpendicular distance between them.

Prism volume

Volume of any prism equals the cross-sectional area multiplied by its length. A cylinder is a prism with circular cross-section, so V = πr²h.

Pythagoras' theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b².

Area unit conversions

Length units convert by factors of 100, but area units convert by the square: 1 m² = 10 000 cm² and 1 hectare = 10 000 m².

Volume and capacity conversions

1 cm³ holds 1 mL, 1000 cm³ equals 1 litre, and 1 m³ equals 1000 litres. Multiply or divide by these factors to convert between units.

Similar figure scaling

When a figure is enlarged by scale factor k, matching lengths multiply by k, areas multiply by k², and volumes multiply by k³.

Speed, distance, time

Speed equals distance divided by time. Rearranged, distance = speed × time and time = distance ÷ speed. To convert km/h to m/s, divide by 3.6.

Simple interest formula

Simple interest is I = PRN, where P is the principal, R the annual interest rate written as a decimal, and N the time in years.

Compound interest formula

Compound interest uses A = P(1 + r)ⁿ, where A is the final amount, P the principal, r the rate per period as a decimal, and n the number of periods.

Declining-balance depreciation

The value of an asset after n periods is P(1 − r)ⁿ, where r is the depreciation rate per period as a decimal and P the original value.

Straight-line depreciation

An asset loses the same dollar amount each year. Its value after n years is P − nD, where D is the annual depreciation amount.

Time-and-a-half overtime

Overtime paid at time-and-a-half equals 1.5 times the normal hourly rate. Double time equals 2 times the normal hourly rate.

Converting salary to weekly pay

Divide an annual salary by 52 for weekly pay, by 26 for fortnightly pay, or by 12 for monthly pay. These are the standard pay-period divisors.

Removing GST from a price

A GST-inclusive price includes 10% tax, so it equals 1.1 times the pre-tax price. Divide the inclusive price by 1.1 to find the original amount.

Reducing-balance loan interest

Interest for each period is calculated on the outstanding balance. As repayments reduce the balance, the interest charged in later periods becomes smaller.

Mean, median and mode

The mean is the total divided by the count, the median is the middle of ordered data, and the mode is the most frequent value.

Range and interquartile range

Range equals highest minus lowest value. Interquartile range equals Q₃ minus Q₁ and measures the spread of the middle half of the data.

Standard deviation meaning

Standard deviation measures how far values typically sit from the mean. A larger standard deviation means the data are more spread out.

Outlier test using IQR

A value is an outlier if it lies below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR, where IQR is the interquartile range.

z-score formula

A z-score equals (x − mean) ÷ standard deviation. It states how many standard deviations a value lies above or below the mean.

Normal distribution percentages

In a normal distribution roughly 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

Relative frequency

Relative frequency equals the frequency of an outcome divided by the total number of trials. It estimates probability and all categories should sum to 1.