NCEA Level 2 Mathematics and Statistics cheat sheet

Quadratic formula: if ax² + bx + c = 0 and a ≠ 0, then x = (−b ± √(b² − 4ac))/(2a).

Discriminant Δ = b² − 4ac: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots.

Index laws: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1 for a ≠ 0; a^(−n) = 1/a^n.

Log laws: log_b(xy) = log_b x + log_b y; log_b(x/y) = log_b x − log_b y; log_b(x^n) = n log_b x; log_b b = 1 and log_b 1 = 0.

Arithmetic sequence: nth term a_n = a + (n − 1)d; sum of first n terms S_n = n/2[2a + (n − 1)d].

Geometric sequence: nth term a_n = a r^(n−1); sum of first n terms S_n = a(1 − r^n)/(1 − r) for r ≠ 1.

Compound interest: A = P(1 + i)^n, where P is principal, i is interest rate per period as a decimal, and n is number of periods.

Completing the square: x² + bx + c = (x + b/2)² + c − b²/4; this gives the vertex (−b/2, c − b²/4) of the parabola.

Solving inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign; check the solution by testing a value.

Pythagoras: for a right triangle with hypotenuse c, c² = a² + b². Use it to find a side or test whether a triangle is right-angled.

Trig ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Sine rule: a/sin A = b/sin B = c/sin C. Use when given two angles and a side, or two sides and a non-included angle.

Cosine rule: c² = a² + b² − 2ab cos C. Use when given two sides and the included angle, or all three sides.

Triangle area: area = ½ ab sin C, where C is the included angle between sides a and b.

Circle formulas: circumference C = 2πr; area A = πr²; arc length = (θ/360) × 2πr; sector area = (θ/360) × πr².

Volume formulas: cylinder V = πr²h; cone V = ⅓πr²h; sphere V = 4/3 πr³; pyramid V = ⅓ × base area × height.

Similar shapes: if length scale factor is k, area scale factor is k² and volume scale factor is k³.

Coordinate geometry: distance d = √((x₂−x₁)² + (y₂−y₁)²); midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2); gradient m = (y₂−y₁)/(x₂−x₁). Parallel lines have equal gradients; perpendicular gradients multiply to −1.

Mean = sum of values ÷ number of values. Median = middle ordered value, or average of the two middle values if n is even. Mode = most frequent value.

Quartiles: Q1 is the median of the lower half and Q3 is the median of the upper half; IQR = Q3 − Q1; values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are often flagged as outliers.

Standard deviation measures spread around the mean; a larger standard deviation means data are more variable.

Probability range: 0 ≤ P(A) ≤ 1; P(not A) = 1 − P(A); P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A) × P(B). Conditional probability: P(A | B) = P(A and B)/P(B), provided P(B) > 0.

Expected value E(X) = Σ x·P(x); it is the long-run average outcome, not necessarily a possible single outcome.

Normal distribution: approximately 68% of values lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.

Z-score z = (x − μ)/σ measures how many standard deviations a value lies from the mean; use it to compare values from different distributions.

Correlation coefficient r ranges from −1 to 1; r near 0 means weak linear association, and correlation does not imply causation.