NSSCO Mathematics (6131) cheat sheet

Place value: each position is ten times the value of the position to its right; in 4 072 the digit 7 has value 70.

Standard form: a × 10^n where 1 ≤ a < 10 and n is an integer; 0.000 42 = 4.2 × 10^-4.

Rounding: to round to n decimal places look at the (n+1)th digit; if it is 5 or more, increase the nth digit by 1.

Percentage change = (change ÷ original value) × 100%; a 20% increase then a 20% decrease leaves 0.96 of the original, a 4% net loss.

Simple interest I = PRT/100; compound amount A = P(1 + r/100)^n where r is the rate per period and n the number of periods.

Ratio: dividing an amount in the ratio a : b uses parts a + b; a share equals (its part ÷ total parts) × amount.

Rules 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.

Surds: √(ab) = √a × √b and √(a/b) = √a ÷ √b; rationalise 1/√a by multiplying numerator and denominator by √a.

Limits of accuracy: for a value given to the nearest unit d, the maximum error is d/2, so 8 kg to the nearest kg lies between 7.5 and 8.5 kg.

Expansion: a(b + c) = ab + ac and (a + b)(c + d) = ac + ad + bc + bd; check by collecting like terms.

Difference of two squares: a² − b² = (a − b)(a + b); perfect square: (a ± b)² = a² ± 2ab + b².

Factorising quadratics x² + bx + c: find two numbers with product c and sum b, then write (x + p)(x + q).

Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a for ax² + bx + c = 0; the discriminant b² − 4ac gives two, one or no real roots as it is positive, zero or negative.

Solving linear equations: apply the same operation to both sides, clear fractions by multiplying by the LCM of the denominators.

Simultaneous equations: eliminate one variable by adding or subtracting the equations after matching its coefficients.

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

Laws of logarithms: log(xy) = log x + log y; log(x/y) = log x − log y; log x^n = n log x.

Changing the subject: perform inverse operations in reverse order; to remove a square root, square both sides.

Angle sums: angles on a straight line total 180°, angles at a point total 360°, and angles in a triangle total 180°.

Parallel lines: corresponding and alternate angles are equal; co-interior (allied) angles sum to 180°.

Polygon interior angle sum = (n − 2) × 180°; each exterior angle of a regular n-gon is 360°/n.

Circle: circumference = 2πr, area = πr²; arc length = (θ/360) × 2πr and sector area = (θ/360) × πr² for central angle θ.

Pythagoras: in a right-angled triangle a² + b² = c², where c is the hypotenuse; the converse tests for a right angle.

Volume: cuboid lbh, prism area of cross-section × length, cylinder πr²h, cone ⅓πr²h, sphere 4/3 πr³.

Surface area: cylinder 2πr² + 2πrh; sphere 4πr²; cone πr² + πrl with slant height l.

Similar figures: corresponding lengths are in ratio k, areas in ratio k², volumes in ratio k³.

Congruence tests for triangles: SSS, SAS, ASA (with AAS) and RHS.

Right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH-CAH-TOA).

Exact values: sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 45° = cos 45° = 1/√2, sin 60° = cos 30° = √3/2.

Sine rule: a/sin A = b/sin B = c/sin C; use it when a side is paired with its opposite angle.

Cosine rule: a² = b² + c² − 2bc cos A; use it with two sides and the included angle, or with all three sides.

Area of any triangle = ½ab sin C, using two sides and the angle between them.

Angles of elevation and depression are measured from the horizontal; draw a right-angled triangle and label the known side and angle.

In the unit circle, sin and cos repeat every 360° and tan every 180°; sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ.

Bearing: measured clockwise from north, written as three digits, e.g. 045°; the back bearing differs by 180°.

Mean = sum of values ÷ number of values; for grouped data use class midpoints and mean = Σfx ÷ Σf.

Median is the middle value of ordered data (average of the two middle values if n is even); mode is the most frequent value.

Range = largest − smallest; interquartile range = upper quartile − lower quartile and ignores extreme values.

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

Probability: P(event) = favourable outcomes ÷ total equally likely outcomes; 0 ≤ P ≤ 1 and P(not A) = 1 − P(A).

Combined events: P(A and B) = P(A) × P(B) for independent events; P(A or B) = P(A) + P(B) − P(A and B).

Mutually exclusive events cannot occur together, so P(A or B) = P(A) + P(B).

Cumulative frequency curves: plot cumulative frequency against upper class boundary; the median is read at half the total frequency.

Histogram with unequal class widths: frequency density = frequency ÷ class width, and area of a bar represents frequency.

Function notation: f(x) means the output for input x; f(3) is found by substituting 3 for every x.

Inverse function: y = f(x) rearranged to give x in terms of y, then x and y swapped; the graph of f⁻¹ is the reflection of f in y = x.

Composite function fg(x) means apply g first, then f; in general fg(x) ≠ gf(x).

Straight line y = mx + c: m is the gradient, c the y-intercept; parallel lines have equal gradients.

Gradient of a line through (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁); perpendicular gradients multiply to −1.

Quadratic graph y = ax² + bx + c is a parabola; it opens upwards if a > 0, and its line of symmetry is x = −b/2a.

Graph transformations: y = f(x) + k shifts up k; y = f(x + k) shifts left k; y = kf(x) stretches vertically by factor k.

Reciprocal graph y = k/x is a hyperbola with the axes as asymptotes; exponential y = a^x passes through (0, 1).

Roots of f(x) = 0 are the x-intercepts of y = f(x); intersections of two graphs solve the equations simultaneously.

Distance between two points = √[(x₂ − x₁)² + (y₂ − y₁)²], from Pythagoras on the coordinate differences.

Midpoint of a segment = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Equation of a line: y − y₁ = m(x − x₁) for gradient m through (x₁, y₁); general form ax + by + c = 0.

Vector notation: a = (x, y) column vector; magnitude |a| = √(x² + y²) and direction tan θ = y/x from the horizontal.

Vector addition is tip-to-tail; subtraction a − b = a + (−b); scalar multiplication ka changes length by factor |k|.

Position vectors: AB = OB − OA, and the midpoint of AB has position vector ½(OA + OB).

Parallel vectors are scalar multiples of each other; a point dividing AB in ratio m : n has position vector (n·OA + m·OB)/(m + n).

Collinear points A, B, C satisfy AB = k·BC for some scalar k.