EJU Mathematics Course 2 cheat sheet

For real numbers a, b: |a+b| ≤ |a| + |b|, with equality iff ab ≥ 0; also |ab| = |a||b| and |a/b| = |a|/|b|.

Quadratic formula: roots of ax² + bx + c = 0 (a ≠ 0) are x = (−b ± √(b² − 4ac)) / (2a); the discriminant D = b² − 4ac decides the number of real roots.

Rationalizing: 1/(√a + √b) = (√a − √b)/(a − b); for cube roots use a³ − b³ = (a − b)(a² + ab + b²).

Double radical: √(a + 2√b) = √m + √n when m + n = a and mn = b (m, n ≥ 0).

Factoring toolkit: a² − b² = (a − b)(a + b), a³ ± b³ = (a ± b)(a² ∓ ab + b²), and grouping for four-term polynomials.

Rational expressions: simplify only after factoring numerator and denominator; domain excludes zeros of the original denominator.

AM–GM inequality: for a, b > 0, a + b ≥ 2√(ab), equality iff a = b; useful for minimum-value problems without calculus.

Absolute-value equations: |X| = k (k > 0) gives X = ±k; |X| = |Y| gives X = ±Y; |X| < k means −k < X < k.

Vertex form y = a(x − p)² + q has vertex (p, q); complete the square via p = −b/(2a), q = c − b²/(4a) for y = ax² + bx + c.

If a > 0 the parabola opens upward and the minimum is q at x = p; if a < 0 it opens downward and q is the maximum.

Discriminant D = b² − 4ac: D > 0 gives two distinct real roots, D = 0 a repeated root, D < 0 no real roots (and the sign of a is then constant).

Sign chart rule: with roots α < β and a > 0, ax² + bx + c > 0 on x < α or x > β, and < 0 on α < x < β; reverse for a < 0.

Axis of symmetry x = −b/(2a); f(x₁) = f(x₂) implies x₁ + x₂ = −b/a.

Restricted extrema: compare the vertex value with endpoint values when the domain is a closed interval and the vertex lies inside it.

Vieta's formulas: for roots α, β of ax² + bx + c = 0, α + β = −b/a and αβ = c/a.

Intersections of y = ax² + bx + c with a line give a quadratic; the number of intersection points equals the number of real roots of the difference.

Inclusion–exclusion: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets subtract pairwise intersections and add back the triple intersection.

De Morgan: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ.

Contrapositive A → B is equivalent to ¬B → ¬A; the converse B → A is not equivalent.

Necessary vs sufficient: 'A is sufficient for B' means A → B; 'A is necessary for B' means B → A.

Permutations P(n, r) = n!/(n − r)!; combinations C(n, r) = n!/(r!(n − r)!) with C(n, r) = C(n, n − r).

Circular arrangements of n distinct objects: (n − 1)!; arrangements with identical items: n!/(p!q!…).

Binomial theorem: (a + b)ⁿ = Σ C(n, k) aⁿ⁻ᵏ bᵏ for k = 0…n; the general term is C(n, k) aⁿ⁻ᵏ bᵏ.

C(n, r) = C(n − 1, r − 1) + C(n − 1, r); total subsets of an n-element set is 2ⁿ.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); mutually exclusive events have P(A ∩ B) = 0.

Independent events satisfy P(A ∩ B) = P(A)P(B); for dependent events use P(A ∩ B) = P(A)P(B | A).

Conditional probability P(B | A) = P(A ∩ B)/P(A), defined when P(A) > 0.

Complement rule P(Ā) = 1 − P(A) is the standard route for 'at least one' problems.

Expected value E(X) = Σ xᵢ pᵢ; for a constant shift E(X + c) = E(X) + c and E(aX) = aE(X).

Repeated independent trials with success probability p: P(exactly k successes in n trials) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ.

Bayes-type reasoning: P(A | B) = P(B | A)P(A) / P(B), with P(B) split over a partition of the sample space.

Geometric probability: when outcomes are points in a region, probability equals the ratio of lengths, areas, or volumes.

Division algorithm: for integers a, b with b ≠ 0 there are unique q, r with a = bq + r and 0 ≤ r < |b|.

Euclidean algorithm: gcd(a, b) = gcd(b, r) where r is the remainder of a divided by b; iterate until remainder 0.

gcd(a, b) · lcm(a, b) = |ab|; use prime factorizations with minimum exponents for gcd and maximum exponents for lcm.

Linear Diophantine equation ax + by = c has integer solutions iff gcd(a, b) divides c; solutions differ by (b/d, −a/d) with d = gcd(a, b).

Congruence rules: a ≡ b and c ≡ d (mod m) imply a + c ≡ b + d and ac ≡ bd (mod m); division requires coprime modulus.

Base-b representation: digits 0 to b − 1; converting from base b evaluates the polynomial of digits in powers of b.

Divisibility tests: by 3 or 9 use the digit sum; by 4 use the last two digits; by 8 use the last three digits; by 11 alternate the digit sum.

Number of divisors: if N = p₁^e₁ p₂^e₂ … then the count is (e₁ + 1)(e₂ + 1)…; the sum uses (p^(e+1) − 1)/(p − 1) per factor.

Inscribed angle theorem: an inscribed angle equals half the central angle on the same arc; angles subtending the same arc are equal.

Thales: an angle inscribed in a semicircle is 90°; conversely, a right angle subtends a diameter.

Cyclic quadrilateral: opposite angles sum to 180°; an exterior angle equals the interior opposite angle.

Power of a point P: PA · PB = PC · PD for two secants through P; tangent case PT² = PA · PB.

Two chords intersecting inside a circle: PA · PB = PC · PD where P is the intersection.

Similar triangles: corresponding sides are proportional and areas are in the ratio of the squares of corresponding lengths.

Angle bisector theorem: the bisector of angle A in triangle ABC meets BC at D with BD/DC = AB/AC.

Area formulas: S = ½ab sin C for a triangle; S = ½r(a + b + c) with inradius r; Brahmagupta for a cyclic quadrilateral.

Reciprocal and quotient identities: tan θ = sin θ / cos θ, and sin²θ + cos²θ = 1 gives 1 + tan²θ = 1/cos²θ.

Addition formulas: sin(α ± β) = sin α cos β ± cos α sin β; cos(α ± β) = cos α cos β ∓ sin α sin β.

Double angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.

Symmetric reduction: sin(θ + 90°) = cos θ, cos(θ + 90°) = −sin θ, tan(θ + 180°) = tan θ; sin(−θ) = −sin θ, cos(−θ) = cos θ.

Sinusoidal form a sin θ + b cos θ = √(a² + b²) sin(θ + φ) with cos φ = a/√(a² + b²), sin φ = b/√(a² + b²).

Trigonometric equation sin θ = k on 0 ≤ θ < 2π has solutions θ = α and θ = π − α for α = arcsin k (when |k| ≤ 1).

Law of sines a/sin A = b/sin B = c/sin C = 2R; law of cosines a² = b² + c² − 2bc cos A.

Triangle area S = ½bc sin A; the ambiguous SSA case can give 0, 1, or 2 triangles.

Laws of exponents: aᵐaⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, and a^(m/n) = ⁿ√(aᵐ) for a > 0.

Definition: log_a b = c ⇔ aᶜ = b, requiring a > 0, a ≠ 1, b > 0.

Log laws: log_a(xy) = log_a x + log_a y, log_a(x/y) = log_a x − log_a y, log_a xⁿ = n log_a x.

Base change: log_a b = log_c b / log_c c, so log_a b · log_b a = 1; also a^(log_a b) = b.

Monotonicity: aˣ is increasing for a > 1 and decreasing for 0 < a < 1; log_a x has the same monotonicity, so inequalities preserve or reverse direction accordingly.

Domain check: equations with logarithms require every argument positive; discard extraneous roots.

Common logarithms use base 10 and natural logarithms base e; log 2 ≈ 0.3010 and log 3 ≈ 0.4771 support digit-count estimates.

Number of digits of a positive integer N is ⌊log₁₀ N⌋ + 1; the leading digits come from the fractional part of log₁₀ N.

Definition: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; differentiability implies continuity but not conversely.

Rules: (xⁿ)′ = nxⁿ⁻¹, product (fg)′ = f′g + fg′, quotient (f/g)′ = (f′g − fg′)/g², chain rule dy/dx = dy/du · du/dx.

Tangent line at x = a: y − f(a) = f′(a)(x − a); normal slope is −1/f′(a) when f′(a) ≠ 0.

Monotonicity: f′ > 0 on an interval means increasing there, f′ < 0 decreasing; f′ = 0 with sign change gives an extremum.

Second derivative test: f′(a) = 0 and f″(a) > 0 gives a local minimum; f″(a) < 0 gives a local maximum.

Indefinite integrals: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫1/x dx = log|x| + C.

Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a) where F′ = f; areas between curves use ∫(upper − lower) dx.

Even/odd symmetry: for odd f, ∫₋ₐᵃ f = 0; for even f, ∫₋ₐᵃ f = 2∫₀ᵃ f.

Arithmetic sequence: aₙ = a₁ + (n − 1)d; sum Sₙ = n(a₁ + aₙ)/2 = n(2a₁ + (n − 1)d)/2.

Geometric sequence: aₙ = a₁rⁿ⁻¹; sum Sₙ = a₁(rⁿ − 1)/(r − 1) for r ≠ 1, and Sₙ = na₁ for r = 1.

Infinite geometric series converges to a₁/(1 − r) exactly when |r| < 1.

Two-term recurrence aₙ₊₁ = paₙ + q: set aₙ₊₁ − α = p(aₙ − α) with α = q/(1 − p) when p ≠ 1, giving a geometric sequence.

Characteristic equation for aₙ₊₂ = paₙ₊₁ + qaₙ: solve t² = pt + q; distinct roots give aₙ = Aαⁿ + Bβⁿ, a double root gives (A + Bn)αⁿ.

Summation formulas: Σk = n(n + 1)/2, Σk² = n(n + 1)(2n + 1)/6, Σk³ = [n(n + 1)/2]².

Telescoping: if aₙ = bₙ − bₙ₊₁ then the partial sum collapses to b₁ − bₙ₊₁.

Induction: prove the base case n = 1, assume the statement for n = k, then derive it for n = k + 1.

Components: a = (a₁, a₂); magnitude |a| = √(a₁² + a₂²); a ∥ b iff a₁b₂ − a₂b₁ = 0 (nonzero vectors).

Inner product a · b = a₁b₁ + a₂b₂ = |a||b| cos θ; perpendicularity is a · b = 0.

Angle formula cos θ = (a · b)/(|a||b|); projection of b onto a has length (a · b)/|a|.

Position vectors: the point dividing AB internally in ratio m : n has position vector (n a + m b)/(m + n).

Line through point A with direction d: p = a + t d; in the plane a normal n gives n · (p − a) = 0.

Distance from point P to line through A with direction d is |AP × d|/|d|, computed in the plane as |AP₁d₂ − AP₂d₁|/|d|.

Triangle area via vectors: S = ½|a₁b₂ − a₂b₁| for two sides a, b from one vertex.

Space vectors: a · b = a₁b₁ + a₂b₂ + a₃b₃; a plane has equation n · (p − a) = 0 with normal n.