EJU Mathematics Course 1 cheat sheet

Quadratic formula: roots of ax²+bx+c=0 are x = (−b ± √(b²−4ac))/(2a); discriminant D = b²−4ac gives two real roots if D>0, one if D=0, none (real) if D<0.

Vertex of y = ax²+bx+c is at x = −b/(2a), y = c − b²/(4a); axis of symmetry x = −b/(2a).

Sum and product of roots: α+β = −b/a, αβ = c/a; a quadratic with roots α, β is x² − (α+β)x + αβ = 0.

Factor theorem: (x−k) divides polynomial P(x) exactly when P(k)=0; remainder of P(x)÷(x−k) is P(k).

Absolute value: |x| < a ⇔ −a < x < a; |x| > a ⇔ x < −a or x > a (a>0).

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

Simultaneous linear equations: solve by elimination or substitution; number of solutions is one if determinant ad−bc ≠ 0, none or infinitely many if ad−bc = 0.

Irrational equations: after squaring both sides, always check candidate solutions in the original equation to discard extraneous roots.

Permutations: nPr = n!/(n−r)!; combinations: nCr = n!/(r!(n−r)!); nCr = nC(n−r).

Addition rule: P(A∪B) = P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0.

Conditional probability: P(B|A) = P(A∩B)/P(A) when P(A)>0; independent events satisfy P(A∩B)=P(A)P(B).

Complement: P(not A) = 1 − P(A); use it when 'at least one' events are easier counted by complement.

Expected value of a random variable X: E(X) = Σ x·P(X=x); variance V(X) = E(X²) − {E(X)}².

Mean of data with frequencies: x̄ = Σ(f·x)/Σf; weighted average uses the same formula with weights in place of frequencies.

Standard deviation: s = √( (1/n)Σ(xᵢ−x̄)² ); for grouped data use class midpoints in place of xᵢ.

Median and quartiles: order the data first; the median splits the set into two halves whose medians are Q1 and Q3.

Sine rule: a/sin A = b/sin B = c/sin C = 2R; cosine rule: a² = b² + c² − 2bc·cos A.

Triangle area: S = ½ab·sin C; also S = ½ × base × height, and S = √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2.

Trigonometric identities: sin²θ + cos²θ = 1; tan θ = sin θ/cos θ; sin(180°−θ)=sin θ, cos(180°−θ)=−cos θ.

Circle: circumference 2πr, area πr²; sector arc length rθ and area ½r²θ when θ is measured in radians.

Sphere: volume (4/3)πr³, surface area 4πr²; cone volume (1/3)πr²h; cylinder volume πr²h.

Exponential and logarithmic relations: if a^p = M (a>0, a≠1) then p = log_a M; log_a(MN) = log_a M + log_a N.

Function composition: (f∘g)(x) = f(g(x)); inverse function f⁻¹ satisfies f(f⁻¹(x)) = x and reflects the graph across y = x.

Coordinate geometry: distance √((x₂−x₁)²+(y₂−y₁)²); midpoint ((x₁+x₂)/2,(y₁+y₂)/2); line slope (y₂−y₁)/(x₂−x₁).