Gaokao Mathematics cheat sheet

For sets A and B, A⊆B means every element of A is also in B; ∅⊆A for every set A, and A∩B⊆A⊆A∪B.

Cardinality: |A∪B|=|A|+|B|−|A∩B|; for three sets, |A∪B∪C|=|A|+|B|+|C|−|A∩B|−|A∩C|−|B∩C|+|A∩B∩C|.

De Morgan laws: (A∪B)^c=A^c∩B^c and (A∩B)^c=A^c∪B^c, where ^c denotes complement in the universal set.

An implication P⇒Q is false only when P is true and Q is false; it is true in all other truth cases.

The contrapositive ¬Q⇒¬P is equivalent to P⇒Q; the converse Q⇒P and inverse ¬P⇒¬Q are not equivalent to P⇒Q.

In P⇒Q, P is sufficient for Q and Q is necessary for P; P⇔Q means P is both necessary and sufficient for Q.

Negations: ¬(P∧Q)=¬P∨¬Q, ¬(P∨Q)=¬P∧¬Q, and ¬(P⇒Q)=P∧¬Q.

Quantifier negation: ¬(∀x P(x))≡∃x ¬P(x) and ¬(∃x P(x))≡∀x ¬P(x).

A set with n distinct elements has 2^n subsets, including ∅ and the whole set, and 2^n−1 proper subsets.

Domain restrictions: denominators must be nonzero; even-root radicands must be ≥0; log arguments must be >0 with base >0 and ≠1; tan x requires cos x≠0.

Derivative rules: (x^n)′=nx^{n−1}, (e^x)′=e^x, (ln x)′=1/x, (sin x)′=cos x, (cos x)′=−sin x.

Product, quotient, and chain rules: (uv)′=u′v+uv′, (u/v)′=(u′v−uv′)/v², and [f(g(x))]′=f′(g(x))g′(x).

If f′>0 on an interval, f is strictly increasing there; if f′<0, f is strictly decreasing; sign changes of f′ identify local extrema.

A local extremum can occur only at a critical point (f′=0 or f′ undefined) or at an endpoint of the considered domain.

Concavity: f″>0 gives concave up, f″<0 gives concave down; an inflection point requires a change of concavity.

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

Mean value theorem: if f is continuous on [a,b] and differentiable on (a,b), then some c in (a,b) satisfies f′(c)=[f(b)−f(a)]/(b−a).

On a closed interval, the maximum and minimum of a continuous function occur at endpoints or critical points; compare those values.

Pythagorean identity: sin²x+cos²x=1; tan x=sin x/cos x and 1+tan²x=sec²x where defined.

Addition formulas: sin(a±b)=sin a cos b±cos a sin b; cos(a±b)=cos a cos b∓sin a sin b.

Double-angle formulas: sin 2a=2sin a cos a; cos 2a=cos²a−sin²a=2cos²a−1=1−2sin²a.

Auxiliary angle: a sin x+b cos x=√(a²+b²) sin(x+φ), with cos φ=a/√(a²+b²) and sin φ=b/√(a²+b²).

Sine law: a/sin A=b/sin B=c/sin C=2R, where R is the circumradius of triangle ABC.

Cosine law: a²=b²+c²−2bc cos A; equivalently cos A=(b²+c²−a²)/(2bc).

Triangle area: K=(1/2)ab sin C=abc/(4R)=rs, where r is the inradius and s is the semiperimeter.

Angle sum A+B+C=π; the larger side lies opposite the larger angle, and each angle lies between 0 and π.

For SSA data, the sine law can give zero, one, or two triangles; check whether the candidate angles sum to less than π.

Arithmetic sequence: a_n=a₁+(n−1)d; S_n=n(a₁+a_n)/2=n[2a₁+(n−1)d]/2.

Geometric sequence: a_n=a₁q^{n−1}; for q≠1, S_n=a₁(1−q^n)/(1−q); for q=1, S_n=na₁.

Infinite geometric series converges exactly when |q|<1, and then its sum is a₁/(1−q).

For an arithmetic sequence, a_n is linear in n and S_n is quadratic in n with zero constant term.

For a geometric sequence with nonzero terms, a_n²=a_{n−1}a_{n+1}; for an arithmetic sequence, 2a_n=a_{n−1}+a_{n+1}.

Partial sums satisfy a₁=S₁ and a_n=S_n−S_{n−1} for n≥2; this relation converts between sums and terms.

Arithmetic mean of a and b is (a+b)/2; geometric mean of positive a and b is √(ab); the arithmetic mean is at least the geometric mean.

Telescoping sums cancel consecutive terms; regrouping is valid for finite sums, and for infinite sums only under convergence conditions.

If a>b, then a+c>b+c; if c>0 then ac>bc; if c<0 then ac<bc, so multiplying by a negative reverses the inequality.

For ax²+bx+c with a≠0, the sign is controlled by the leading coefficient and the real roots; the discriminant Δ=b²−4ac determines the number of real roots.

For positive a and b, (a+b)/2≥√(ab), with equality iff a=b; for positive a_i, (Σa_i)/n≥(Πa_i)^{1/n}.

Cauchy-Schwarz: (Σa_i b_i)²≤(Σa_i²)(Σb_i²), with equality when the vectors (a_i) and (b_i) are proportional.

For a>0, |x|<a iff −a<x<a; |x|>a iff x<−a or x>a; |x|=a iff x=±a.

Triangle inequality: |a+b|≤|a|+|b|, and equality holds when a and b have the same sign or one is zero.

A rational inequality is solved by bringing all terms to one side and using a sign chart; denominator zeros are excluded.

Exponential a^x is increasing for a>1 and decreasing for 0<a<1; log_a x has the same monotonicity on its domain x>0.

For x>0, ln x≤x−1 with equality at x=1; for all real x, e^x≥x+1 with equality at x=0.

Vector coordinates: if a=(x₁,y₁) and b=(x₂,y₂), then a±b=(x₁±x₂,y₁±y₂) and λa=(λx₁,λy₁).

Dot product: a·b=|a||b|cos θ=x₁x₂+y₁y₂, where θ is the angle between the vectors.

Perpendicularity: a⊥b iff a·b=0; in coordinates x₁x₂+y₁y₂=0.

Parallelism in the plane: a∥b iff x₁y₂−x₂y₁=0, provided both vectors are nonzero.

Angle between nonzero vectors: cos θ=(a·b)/(|a||b|); the scalar projection of a on b is (a·b)/|b|.

Complex numbers: i²=−1; for z=a+bi, conjugate z̄=a−bi and modulus |z|=√(a²+b²).

Complex multiplication: (a+bi)(c+di)=(ac−bd)+(ad+bc)i; division multiplies numerator and denominator by the conjugate of the denominator.

In the complex plane, |z₁−z₂| is the distance between z₁ and z₂; |z| is the distance from z to the origin.

For z=a+bi, z z̄=a²+b²; if z≠0, 1/z=z̄/(z z̄).

A line is perpendicular to a plane if its direction vector is parallel to the plane's normal vector; it is parallel to the plane if its direction vector is perpendicular to the normal.

Angle between two lines uses cos θ=|u·v|/(|u||v|) for direction vectors u and v.

Line-plane angle θ satisfies sin θ=|u·n|/(|u||n|), where u is a line direction vector and n is a plane normal vector.

Dihedral angle between two planes is found from their normal vectors: cos θ=|n₁·n₂|/(|n₁||n₂|) for the acute angle.

Distance from point P(x₀,y₀,z₀) to plane Ax+By+Cz+D=0 is |Ax₀+By₀+Cz₀+D|/√(A²+B²+C²).

Volume formulas: prism/cylinder V=Bh; pyramid/cone V=(1/3)Bh; sphere V=(4/3)πr³.

Surface areas: sphere 4πr²; cylinder 2πr²+2πrh; cone πr²+πrl, where l is the slant height.

Space vector distance: |PQ|=√[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²]; midpoint coordinates are averages of corresponding coordinates.

If a plane has equation Ax+By+Cz+D=0, then (A,B,C) is a normal vector; a line with direction vector (m,n,p) has parametric equations x=x₀+mt, y=y₀+nt, z=z₀+pt.

Circle: center (h,k), radius r gives (x−h)²+(y−k)²=r²; general form x²+y²+Dx+Ey+F=0 has center (−D/2,−E/2) and radius √(D²+E²−4F)/2 when positive.

Ellipse x²/a²+y²/b²=1 with a>b>0 has c²=a²−b², foci (±c,0), eccentricity e=c/a<1.

Hyperbola x²/a²−y²/b²=1 has c²=a²+b², foci (±c,0), eccentricity e=c/a>1, and asymptotes y=±(b/a)x.

Parabola y²=2px has focus (p/2,0), directrix x=−p/2, and every point is equidistant from focus and directrix.

For a line intersecting a conic, substitute the line equation into the conic equation; the discriminant determines 0, 1, or 2 intersections.

Chord length from intersection parameters t₁,t₂ on a line is |t₁−t₂|√(1+k²) when the line has slope k.

Midpoint chord problems often use the point-difference method: subtract the two conic equations at the endpoints and use the midpoint coordinates.

Eccentricity: ellipse 0<e<1, parabola e=1, hyperbola e>1; e=c/a relates focal distance to semi-major axis.

Distance from point (x₀,y₀) to line Ax+By+C=0 is |Ax₀+By₀+C|/√(A²+B²).

Probability axioms: 0≤P(A)≤1, P(Ω)=1, and for mutually exclusive events P(A∪B)=P(A)+P(B).

Complement rule: P(A^c)=1−P(A); addition rule: P(A∪B)=P(A)+P(B)−P(A∩B).

Conditional probability: P(A|B)=P(A∩B)/P(B) for P(B)>0; multiplication rule P(A∩B)=P(A|B)P(B).

Events A and B are independent iff P(A∩B)=P(A)P(B); this differs from mutual exclusivity, which gives P(A∩B)=0.

Total probability: if B_i partition Ω, P(A)=ΣP(A|B_i)P(B_i); Bayes' theorem gives P(B_j|A)=P(A|B_j)P(B_j)/ΣP(A|B_i)P(B_i).

Binomial distribution: P(X=k)=C(n,k)p^k(1−p)^{n−k}, mean E(X)=np, variance D(X)=np(1−p).

For data, mean x̄=(Σx_i)/n and variance s²=[Σ(x_i−x̄)²]/n; linear transformation Y=aX+b gives E(Y)=aE(X)+b and D(Y)=a²D(X).

Normal distribution: standardization Z=(X−μ)/σ; approximately 68% of values lie within μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ.

Regression line ŷ=b̂x+â passes through (x̄,ȳ); correlation r measures the strength and direction of linear association, with |r|≤1.

Addition principle: if a task can be done in m ways or n mutually exclusive ways, the total is m+n.

Multiplication principle: if a task consists of successive independent stages with m and n choices, the total is mn.

Permutations: P(n,k)=n!/(n−k)! counts ordered arrangements of k objects from n distinct objects.

Combinations: C(n,k)=n!/[k!(n−k)!] counts unordered selections; C(n,k)=C(n,n−k).

Binomial theorem: (a+b)^n=Σ_{k=0}^n C(n,k)a^{n−k}b^k; the general term is C(n,k)a^{n−k}b^k.

The sum of all binomial coefficients is 2^n; the alternating sum Σ(−1)^kC(n,k) is 0 for n≥1.

For circular arrangements of n distinct objects, the number is (n−1)!; reflections are distinct unless the problem identifies them.

When identical objects are distributed into distinct groups, divide by factorials of identical group sizes if groups are unordered.

Inclusion-exclusion: |A∪B|=|A|+|B|−|A∩B|; for three sets add single sizes, subtract pairwise intersections, add the triple intersection.