UEE Mathematics cheat sheet

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a); the discriminant Δ = b² − 4ac gives two real roots if Δ > 0, one double root if Δ = 0, and no real roots if Δ < 0.

Factor and remainder theorems: if P(x) is divided by (x − a), the remainder is P(a); (x − a) is a factor exactly when P(a) = 0. For division by (ax − b), the remainder is P(b/a).

Logarithm laws: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n·logₐx, logₐx = ln x / ln a, and a^(logₐx) = x for x > 0, a > 0, a ≠ 1.

Arithmetic progression: nth term aₙ = a + (n − 1)d; sum Sₙ = (n/2)(2a + (n − 1)d) = (n/2)(first + last). Geometric progression: aₙ = ar^(n−1); Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; infinite sum a/(1 − r) only when |r| < 1.

Binomial theorem: (a + b)ⁿ = Σ C(n,k)·a^(n−k)·b^k for k = 0 to n, where C(n,k) = n!/(k!(n−k)!); the general term is C(n,k)·a^(n−k)·b^k, and C(n,k) = C(n, n−k).

Laws of indices: aᵐ·aⁿ = a^(m+n), aᵐ/aⁿ = a^(m−n), (aᵐ)ⁿ = a^(mn), a^(−n) = 1/aⁿ, a^(1/n) = ⁿ√a, and a⁰ = 1 for a ≠ 0.

Solving simultaneous linear equations: elimination scales one equation so a coefficient matches, then subtracts; substitution isolates one variable. A unique solution exists when the coefficient determinant ad − bc ≠ 0 for ax + by = e, cx + dy = f.

Absolute value and inequalities: |x| < k (k > 0) means −k < x < k, while |x| > k means x < −k or x > k. Multiplying or dividing an inequality by a negative number reverses its direction.

Domain restrictions to check: denominators cannot be zero; even roots require a non-negative radicand; logarithms require a positive argument. State the domain before simplifying.

Transformations of y = f(x): y = f(x − h) + k shifts right by h and up by k; y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis; y = af(x) stretches vertically by factor |a|.

Composite and inverse functions: (f∘g)(x) = f(g(x)) with domain restricted to x in dom g where g(x) ∈ dom f. The inverse f⁻¹ exists only if f is one-to-one, and its graph is the reflection of y = f(x) in the line y = x.

Parabola y = ax² + bx + c: vertex at x = −b/(2a), axis of symmetry x = −b/(2a); opens upward if a > 0, downward if a < 0. Completed form y = a(x − h)² + k gives vertex (h, k).

Exponential and logarithmic graphs: y = aˣ (a > 1) increases with horizontal asymptote y = 0 and passes through (0, 1); y = logₐx is its reflection in y = x, has vertical asymptote x = 0, and passes through (1, 0).

Rational function asymptotes: for N(x)/D(x) in lowest terms, vertical asymptotes occur where D(x) = 0; if deg N < deg D the horizontal asymptote is y = 0, if degrees are equal it is y = (leading coefficient of N)/(leading coefficient of D), and if deg N = deg D + 1 there is an oblique asymptote.

Even and odd functions: f is even if f(−x) = f(x) for all x in its domain (graph symmetric about the y-axis); f is odd if f(−x) = −f(x) (graph symmetric about the origin).

Graph sketching checklist: intercepts, domain and range, asymptotes, turning points, and intervals of increase or decrease determined by the sign of the derivative.

Circle facts: circumference C = 2πr, area A = πr²; arc length s = rθ and sector area = ½r²θ with θ in radians. A tangent is perpendicular to the radius at the point of contact.

Triangle mensuration: area = ½ × base × height = ½ab·sin C; Heron's formula A = √(s(s−a)(s−b)(s−c)) with s = (a + b + c)/2; the angle sum is 180° and the exterior angle equals the sum of the two opposite interior angles.

Pythagoras and similar figures: in a right triangle a² + b² = c². If two figures are similar with linear scale factor k, areas scale by k² and volumes by k³.

Volume and surface area: cylinder V = πr²h; cone V = ⅓πr²h; sphere V = (4/3)πr³ and surface area 4πr²; prism or cylinder volume = cross-sectional area × length.

Coordinate geometry: distance between points = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2); gradient m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal gradients, perpendicular lines have gradients whose product is −1.

Straight line forms: y = mx + c (gradient m, intercept c); y − y₁ = m(x − x₁); ax + by + c = 0 has gradient −a/b. Distance from point (x₀, y₀) to ax + by + c = 0 is |ax₀ + by₀ + c| / √(a² + b²).

Polygon angles: interior angle sum of an n-sided polygon = (n − 2) × 180°; exterior angle sum = 360°; a regular polygon has each exterior angle equal to 360°/n.

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

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent = sin θ/cos θ. Exact values include sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, cos 60° = ½, tan 45° = 1.

Radians and degrees: π radians = 180°, so degrees × π/180 = radians. Arc length s = rθ and sector area A = ½r²θ require θ in radians.

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

Compound and double angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1.

Sine and cosine rules: a/sin A = b/sin B = c/sin C = 2R; a² = b² + c² − 2bc·cos A. Use the sine rule for a matching side–angle pair, the cosine rule when three sides or two sides and the included angle are known.

General solutions: sin θ = sin α gives θ = α + 360°n or θ = 180° − α + 360°n; cos θ = cos α gives θ = ±α + 360°n; tan θ = tan α gives θ = α + 180°n, for integer n.

Graphs of sine and cosine have period 360° (2π) and amplitude 1; y = a sin(bx + c) + d has amplitude |a|, period 360°/|b|, phase shift −c/b, and vertical shift d. The tangent graph has period 180° and vertical asymptotes at 90° + 180°n.

Sign by quadrant (ASTC): all ratios positive in the first quadrant; only sine positive in the second; only tangent positive in the third; only cosine positive in the fourth. The reference angle is measured from the horizontal axis.

Probability axioms: 0 ≤ P(A) ≤ 1; P(A′) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); A and B are independent when P(A ∩ B) = P(A)P(B), and mutually exclusive when P(A ∩ B) = 0.

Conditional probability: P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. Multiplication rule P(A ∩ B) = P(A|B)P(B); Bayes' theorem reverses the conditioning using the law of total probability.

Counting: permutations nPr = n!/(n − r)! (order matters); combinations nCr = n!/(r!(n − r)!) (order does not matter); with repetition allowed, arrangements of n objects taken r at a time number n^r.

Binomial distribution: P(X = r) = C(n,r)·p^r·(1 − p)^(n−r) for n independent trials with constant success probability p; mean = np and variance = np(1 − p).

Measures of centre and spread: mean = Σx/n (grouped data uses class midpoints); median is the middle value of ordered data; mode is the most frequent value; range = max − min; variance = Σ(x − mean)²/n and standard deviation is its square root.

Normal distribution: approximately 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Standardise with z = (x − μ)/σ before using tables.

Linear regression and correlation: the least-squares line is y = a + bx with b = Σ((x − x̄)(y − ȳ))/Σ(x − x̄)²; the correlation coefficient r lies between −1 and 1, and r² is the proportion of variance explained.

Expected value and data types: E(X) = Σx·P(x) for a discrete variable; qualitative data are categorical and quantitative data are discrete or continuous, which determines whether a bar chart or histogram is appropriate.

Limit definition and continuity: f is continuous at x = a when lim(x→a) f(x) = f(a), requiring the left and right limits to agree. For rational functions, cancel common factors before substituting; lim(x→0) (sin x)/x = 1.

Differentiation rules: d/dx(xⁿ) = nx^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v²; chain rule dy/dx = dy/du · du/dx.

Standard derivatives: d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec²x; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x for x > 0; d/dx(aˣ) = aˣ ln a.

Tangents and normals: the gradient at x = a is f′(a); the tangent is y − f(a) = f′(a)(x − a); the normal gradient is −1/f′(a) where f′(a) ≠ 0.

Stationary points: solve f′(x) = 0, then classify with the second derivative - f″(x) > 0 gives a local minimum, f″(x) < 0 gives a local maximum, and f″(x) = 0 is inconclusive (use a sign change test).

Integration: ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C; ∫cos x dx = sin x + C; ∫sin x dx = −cos x + C.

Definite integrals and area: ∫ from a to b of f(x) dx = F(b) − F(a) by the fundamental theorem. Area between curves is ∫|f(x) − g(x)| dx over the interval, with intersection points as limits.

Kinematics and rates: if displacement is s(t), velocity v = ds/dt and acceleration a = dv/dt = d²s/dt². Related rates use the chain rule; optimisation sets the derivative to zero and checks endpoints for a global extremum.