MATSEC SEC Mathematics cheat sheet

Order of operations: brackets, indices, division and multiplication (left to right), addition and subtraction (left to right).

Standard form: a × 10^n where 1 ≤ a < 10; multiply by adding indices, divide by subtracting indices.

Laws of indices: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1; a^(−n) = 1/a^n.

Percentage change = (change ÷ original) × 100; a multiplier of 1.15 increases by 15%, 0.85 decreases by 15%.

Simple interest I = PRT/100; compound amount A = P(1 + r/100)^n.

Expand (a + b)^2 = a^2 + 2ab + b^2 and (a + b)(a − b) = a^2 − b^2; factorise by taking out the highest common factor first.

Quadratic formula: x = [−b ± √(b^2 − 4ac)] / 2a for ax^2 + bx + c = 0.

Direct proportion y = kx; inverse proportion y = k/x; find k from a given pair of values before solving.

Angle sum of a polygon with n sides = (n − 2) × 180°; exterior angle of a regular n-gon = 360°/n.

Circle: circumference = 2πr, area = πr^2; arc length = (θ/360) × 2πr, sector area = (θ/360) × πr^2.

Pythagoras' theorem a^2 + b^2 = c^2 applies only to right-angled triangles, with c the hypotenuse.

Volume: cuboid lwh, prism area of cross-section × length, cylinder πr^2h, cone (1/3)πr^2h, sphere (4/3)πr^3.

Similar shapes: corresponding lengths scale by k, areas by k^2, volumes by k^3.

Trigonometry in right triangles: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj; use sine rule a/sin A = b/sin B and cosine rule a^2 = b^2 + c^2 − 2bc cos A for non-right triangles.

Bearings are measured clockwise from north, written with three digits (e.g. 047°), and must lie between 000° and 360°.

Enlargement by scale factor k about a centre: multiply coordinates' distances from the centre by k; negative k rotates the image 180°.

Mean = sum of values ÷ number of values; for grouped data use midpoints of class intervals.

Median = middle value of ordered data; for n even, average the two middle values. Mode = most frequent value.

Range = largest − smallest; interquartile range IQR = upper quartile − lower quartile, containing the middle 50% of data.

For n ordered values, lower quartile is at position (n + 1)/4 and upper quartile at 3(n + 1)/4.

Probability scale runs from 0 to 1; P(not A) = 1 − P(A); P(A or B) = P(A) + P(B) for mutually exclusive events.

P(A and B) = P(A) × P(B) for independent events; for without-replacement draws, adjust the denominator after each pick.

Relative frequency = number of successful trials ÷ total trials; it estimates probability as trials increase.

On a cumulative frequency curve, the median is at 50% of the total frequency, quartiles at 25% and 75%.

Rounding: identify the required place value, then round up if the next digit is 5 or more.

Upper and lower bounds: for a value rounded to the nearest unit, add and subtract half a unit to find bounds.

Ratio a : b shares a quantity into a + b parts; one part = total ÷ (a + b).

Rearrange formulas by performing the same inverse operation on both sides, isolating the required variable.

Simultaneous equations: eliminate one variable by adding or subtracting scaled equations, then substitute back.

Straight line y = mx + c has gradient m = (y2 − y1)/(x2 − x1) and y-intercept c.

Sequences: nth term of an arithmetic sequence = a + (n − 1)d, where a is the first term and d the common difference.

Area of a triangle = (1/2)ab sin C; area of a trapezium = (1/2)(a + b)h.

Sum of angles in a triangle is 180°; angles on a straight line sum to 180°; angles at a point sum to 360°.

Vertically opposite angles are equal; alternate angles are equal and co-interior angles sum to 180° between parallel lines.

Density = mass ÷ volume; speed = distance ÷ time; pressure = force ÷ area - use consistent units.

Convert units: 1 km = 1000 m, 1 m = 100 cm, 1 cm^2 = 100 mm^2, 1 m^2 = 10 000 cm^2, 1 litre = 1000 cm^3.

Surface area of a cuboid = 2(lw + lh + wh); total surface area of a cylinder = 2πr^2 + 2πrh.

Congruent triangles can be proved by SSS, SAS, ASA or RHS.

Averages from a frequency table: mean = Σ(f × x) ÷ Σf; the median lies in the class where cumulative frequency reaches half the total.

Scatter graphs: positive correlation means both variables increase together; the line of best fit passes through the mean point.

Two-way tables: read row and column totals carefully, and check that all cells sum to the grand total.

Pie charts: angle for a category = (frequency ÷ total) × 360°.

Sample space diagrams list all equally likely outcomes; count favourable outcomes to find probability.

Tree diagrams: multiply along branches for combined events and add probabilities of mutually exclusive outcomes.

A histogram with unequal class widths uses frequency density = frequency ÷ class width on the vertical axis.

Reciprocal of a number a is 1/a; the reciprocal of a fraction is found by turning it upside down.

Surds: √(ab) = √a × √b and √(a/b) = √a ÷ √b; rationalise a denominator by multiplying by the conjugate.

Inequalities: multiplying or dividing by a negative number reverses the inequality sign.

Prime factorisation: divide repeatedly by the smallest prime factor until the quotient is 1.

HCF and LCM from prime factors: HCF uses the lowest powers of common primes, LCM uses the highest powers of all primes.

Distance between two points = √[(x2 − x1)^2 + (y2 − y1)^2]; midpoint = ((x1 + x2)/2, (y1 + y2)/2).

Function notation f(x): substitute the given value for x, and find the inverse by swapping x and y and solving.

Rotational symmetry order = number of positions in which a shape looks unchanged in a full turn.

Line symmetry: a shape has reflective symmetry if it can be folded onto itself along a line.

Transformations: translation moves without rotating or resizing; reflection flips across a mirror line; rotation turns about a centre through a given angle.

Volume of a frustum = volume of the whole cone or pyramid minus the volume of the removed similar top.

1 hectare = 10 000 m^2; 1 m^3 = 1000 litres.

Angles of elevation and depression are measured from the horizontal; alternate angles make the angle of elevation equal the angle of depression between two points.

Circle theorems: the angle in a semicircle is 90°, and the angle at the centre is twice the angle at the circumference subtended by the same arc.

Standard deviation measures spread around the mean; a larger value means data are more spread out.

Outliers are values far from the rest of the data; they can be identified using quartiles and the IQR.

Box-and-whisker plots show minimum, lower quartile, median, upper quartile and maximum.

Comparing distributions: comment on both a measure of average (mean or median) and a measure of spread (range or IQR).

Mutually exclusive events cannot happen at the same time; exhaustive events cover all possible outcomes with total probability 1.

Expected frequency = probability × number of trials.

Correlation does not imply causation; a scatter graph shows association only.