MSCE Mathematics cheat sheet

Prime factorisation: every integer >1 is a unique product of primes; use to find HCF (product of common prime factors) and LCM (product of highest powers of all primes).

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≠0); a^(-n) = 1/a^n; a^(1/n) = n√a; a^(m/n) = (n√a)^m.

Surds: √(ab) = √a × √b; √(a/b) = √a/√b; rationalise denominator: 1/√a = √a/a; 1/(a+√b) = (a-√b)/(a²-b).

Standard form: a × 10^n where 1 ≤ a < 10 and n is an integer; to multiply, multiply the a-parts and add the indices; to divide, divide the a-parts and subtract the indices.

Percentages: percentage change = (change ÷ original) × 100%; multiplier for increase p% is (1 + p/100); for decrease p% is (1 - p/100); compound interest: A = P(1 + r/100)^n.

Ratio: to share an amount in ratio a:b:c, total parts = a+b+c; each part = amount ÷ total parts; then multiply by each ratio number.

Rounding and estimation: round to given decimal places or significant figures; for estimation, round each number to 1 significant figure then compute.

Upper and lower bounds: for a rounded value x to the nearest unit, UB = x + 0.5, LB = x - 0.5; for division, maximum = UB ÷ LB, minimum = LB ÷ UB.

Expanding: (a+b)(c+d) = ac + ad + bc + bd; (a+b)² = a² + 2ab + b²; (a-b)² = a² - 2ab + b²; (a+b)(a-b) = a² - b².

Factorising quadratics x² + bx + c: find two numbers that multiply to c and add to b; for ax² + bx + c, split middle term and factor by grouping.

Quadratic formula: x = [-b ± √(b² - 4ac)] / (2a); discriminant D = b² - 4ac; D>0 two real roots, D=0 one repeated root, D<0 no real roots.

Solving simultaneous equations: elimination-multiply to make coefficients equal, then add/subtract; substitution-express one variable from one equation and substitute into the other.

Rearranging formulae: perform the same operation on both sides; to make a variable the subject when it appears twice, collect its terms on one side and factorise.

Inequalities: when multiplying or dividing by a negative number, reverse the inequality sign; solution sets can be shown on a number line.

Algebraic fractions: simplify by factorising numerator and denominator and cancelling common factors; add/subtract by finding a common denominator.

Sequences: nth term of arithmetic sequence a, a+d, a+2d,... is a + (n-1)d; sum of first n terms S_n = n/2[2a + (n-1)d]; for quadratic sequences, second differences are constant.

Angles on a straight line sum to 180°; angles at a point sum to 360°; vertically opposite angles are equal.

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

Triangle angle sum = 180°; exterior angle of a triangle equals the sum of the two opposite interior angles.

Polygons: sum of interior angles of an n-sided polygon = (n-2)×180°; each interior angle of a regular n-gon = (n-2)×180°/n; sum of exterior angles = 360°.

Congruence tests: SSS, SAS, ASA, RHS; similar triangles have equal corresponding angles and proportional corresponding sides.

Circle theorems: angle in a semicircle is 90°; angle at centre is twice angle at circumference; angles in the same segment are equal; opposite angles of cyclic quadrilateral sum to 180°; tangent is perpendicular to radius.

Pythagoras' theorem: in a right-angled triangle, a² + b² = c² where c is the hypotenuse; common triples: (3,4,5), (5,12,13), (8,15,17).

Area and circumference: circle area = πr², circumference = 2πr; arc length = (θ/360)×2πr; sector area = (θ/360)×πr².

Right-angled triangle: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent (SOHCAHTOA).

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

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

Cosine rule: a² = b² + c² - 2bc cos A; use when given two sides and included angle, or all three sides.

Area of any triangle = ½ab sin C, where C is the included angle between sides a and b.

Trigonometric identities: sin²θ + cos²θ = 1; tan θ = sin θ / cos θ.

Graphs: y = sin x and y = cos x have period 360° and range -1 to 1; y = tan x has period 180° and asymptotes at 90°, 270°, ...

Bearings: measured clockwise from north, written as three figures (e.g., 045°); back bearing = bearing ± 180°.

Rectangle: area = length × width; perimeter = 2(l + w); parallelogram: area = base × perpendicular height.

Triangle: area = ½ × base × perpendicular height; trapezium: area = ½(a + b)h where a and b are parallel sides.

Circle: area = πr²; circumference = 2πr; semicircle area = ½πr²; perimeter of semicircle = πr + 2r.

Cuboid: volume = l × w × h; surface area = 2(lw + lh + wh); prism: volume = area of cross-section × length.

Cylinder: volume = πr²h; curved surface area = 2πrh; total surface area = 2πr² + 2πrh.

Cone: volume = ⅓πr²h; curved surface area = πrl where l is slant height; l² = r² + h².

Sphere: volume = (4/3)πr³; surface area = 4πr².

Pyramid: volume = ⅓ × base area × perpendicular height.

Compound measures: density = mass/volume; speed = distance/time; pressure = force/area; use consistent units.

Mean = sum of values ÷ number of values; median = middle value when ordered; mode = most frequent value; range = highest - lowest.

For grouped data, estimate the mean using midpoints of class intervals: mean ≈ Σ(f × midpoint) / Σf.

Cumulative frequency: plot points at upper class boundaries; median is value at cumulative frequency n/2; interquartile range = upper quartile - lower quartile.

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

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

Tree diagrams: multiply along branches for successive events; add probabilities of different branches for mutually exclusive outcomes.

Scatter graphs: line of best fit shows correlation; positive, negative, or no correlation; correlation does not imply causation.

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

Straight line: y = mx + c, where m is gradient and c is y-intercept; gradient m = (y₂ - y₁)/(x₂ - x₁).

Parallel lines have equal gradients; perpendicular lines have gradients whose product is -1 (m₁ × m₂ = -1).

Quadratic graph y = ax² + bx + c is a parabola; axis of symmetry x = -b/(2a); turning point at x = -b/(2a).

Completing the square: y = a(x + p)² + q; vertex at (-p, q); used to find maximum or minimum values.

Transformations: y = f(x) + a translates graph by a units up; y = f(x + a) translates left by a units; y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a.

Reciprocal graph y = k/x has two branches with asymptotes x = 0 and y = 0; exponential graph y = a^x passes through (0,1) and has horizontal asymptote y = 0.

Inverse function: swap x and y then rearrange; graph of f⁻¹ is reflection of f in the line y = x.

Composite function: fg(x) means f(g(x)); apply g first, then f.