NECO SSCE Mathematics cheat sheet

Base conversion: expand a numeral d₃d₂d₁d₀ in base b as d₃b³ + d₂b² + d₁b + d₀ to obtain its base-10 value; to reverse, divide repeatedly by b and read the remainders upward.

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 for a ≠ 0; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a.

Laws of logarithms: log(MN) = log M + log N; log(M/N) = log M − log N; log Mⁿ = n log M; log_a a = 1; log_a 1 = 0; log_a b = log_c b ÷ log_c a.

Standard form: N = A × 10ⁿ with 1 ≤ A < 10; multiplying numbers adds the indices, dividing subtracts them.

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

Sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); n(A ∪ B ∪ C) = Σn(A) − Σn(pairwise intersections) + n(A ∩ B ∩ C); a set with n elements has 2ⁿ subsets.

Percentage and finance: profit% = (SP − CP)/CP × 100; loss% uses the same denominator CP; simple interest I = PRT/100 and amount A = P(1 + RT/100).

For any two positive integers, HCF × LCM = product of the two numbers; the number of factors of p^a q^b is (a + 1)(b + 1) for distinct primes p and q.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac > 0 gives two distinct real roots, = 0 gives equal roots, < 0 gives no real roots.

Roots: for ax² + bx + c = 0, sum of roots = −b/a and product of roots = c/a; the equation can be rebuilt as x² − (sum)x + (product) = 0.

Remainder theorem: dividing f(x) by (x − a) leaves remainder f(a); if f(a) = 0 then (x − a) is a factor (factor theorem).

AP: nth term = a + (n − 1)d; sum of n terms = n/2[2a + (n − 1)d] = n/2(a + l), where l is the last term.

GP: nth term = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1; S∞ = a/(1 − r) only when |r| < 1.

Variation: direct y = kx; inverse y = k/x; joint y = kxz; partial y = kx + c/x - substitute the given values to find k before answering.

Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign; solution sets are shown on a number line with open circles for < and > and closed circles for ≤ and ≥.

Simultaneous linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂: x = (c₁b₂ − c₂b₁)/(a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁)/(a₁b₂ − a₂b₁) when a₁b₂ − a₂b₁ ≠ 0.

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

Parallel lines cut by a transversal: alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°.

Polygons: interior angle sum = (n − 2)180°; exterior angle sum = 360°; for a regular polygon each exterior angle = 360°/n.

Circle theorems: angle at the centre = 2 × angle at the circumference; angle in a semicircle = 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; a tangent is perpendicular to the radius at the point of contact.

Areas: triangle = ½ab sin C; parallelogram = bh; trapezium = ½(a + b)h; circle = πr²; sector = (θ/360)πr²; segment = sector − triangle.

Volumes: cuboid lwh; cylinder πr²h; cone ⅓πr²h; sphere (4/3)πr³; pyramid ⅓ × base area × height; prism = base area × length.

Surface areas: cylinder 2πr(r + h); cone πr(l + r) with l = √(r² + h²); sphere 4πr²; cube 6l².

Similar figures: if corresponding lengths are in ratio k, areas are in ratio k² and volumes in ratio k³; equal angles are preserved.

Ungrouped data: mean x̄ = Σx/n; the median is the middle value of the ordered data (mean of the two middle values when n is even); the mode is the most frequent value.

Grouped data: mean = Σfx/Σf with x the class midpoint; the modal class has the highest frequency; the median class contains the (N/2)th value.

Dispersion: range = highest − lowest; interquartile range = Q₃ − Q₁; semi-interquartile range = ½(Q₃ − Q₁); variance = Σ(x − x̄)²/n; standard deviation = √variance.

Ogive: plot cumulative frequency against upper class boundaries; read the median at N/2, lower quartile at N/4 and upper quartile at 3N/4.

Probability: P(E) = favourable outcomes ÷ total equally likely outcomes; 0 ≤ P(E) ≤ 1; P(E′) = 1 − P(E); P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Mutually exclusive events cannot occur together, so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B); independent events satisfy P(A ∩ B) = P(A) × P(B).

Conditional probability: P(A | B) = P(A ∩ B)/P(B); for independent events P(A | B) = P(A).

Charts: pie chart angle for a category = (frequency ÷ total) × 360°; in a histogram, frequency = frequency density × class width.

Right triangles: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent (SOHCAHTOA); Pythagoras gives a² + b² = c².

Special angles: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.

Sine rule: a/sin A = b/sin B = c/sin C; use it with two angles and any side, or two sides and a non-included angle (the ambiguous case may give two triangles).

Cosine rule: a² = b² + c² − 2bc cos A; rearranged, cos A = (b² + c² − a²)/(2bc).

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

Identities: sin²θ + cos²θ = 1; tan θ = sin θ/cos θ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Bearings are measured clockwise from North and written with three digits, e.g. 072°; back bearing = bearing ± 180°.

Radians: π radians = 180°; arc length s = rθ and sector area = ½r²θ when θ is in radians; angular speed ω = θ/t.