Gaokao Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Union cardinality

For finite sets A and B, |A∪B| equals |A|+|B|−|A∩B|; this corrects double counting of elements lying in both sets.

Contrapositive equivalence

The statement P⇒Q is logically equivalent to ¬Q⇒¬P, so proving the contrapositive proves the original implication; the converse Q⇒P is not equivalent.

Negating quantifiers

The negation of “for all x, P(x)” is “there exists x such that not P(x)”; the negation of “there exists x, P(x)” is “for all x, not P(x)”.

Derivative monotonicity test

On an interval, f′>0 implies f is strictly increasing, f′<0 implies strictly decreasing; f′≥0 or ≤0 gives nondecreasing or nonincreasing with equality allowed at isolated points.

Chain rule

For a composite function f(g(x)), the derivative is f′(g(x))·g′(x); differentiate the outer function at the inner value, then multiply by the inner derivative.

Tangent line equation

At x=a, the tangent to y=f(x) is y−f(a)=f′(a)(x−a), provided f′(a) exists; this is the linear approximation near a.

Critical point

A point x=c in the domain where f′(c)=0 or f′(c) does not exist; local extrema can occur only at critical points or endpoints.

Sine law

In triangle ABC, a/sin A=b/sin B=c/sin C=2R, where R is the circumradius; it relates sides to opposite angles and the circumcircle.

Cosine law

In triangle ABC, a²=b²+c²−2bc cos A; equivalently cos A=(b²+c²−a²)/(2bc). It solves triangles from two sides and included angle or three sides.

Auxiliary angle identity

For constants a and b not both zero, a sin x+b cos x=R sin(x+φ), where R=√(a²+b²) and φ satisfies cos φ=a/R and sin φ=b/R.

Arithmetic sequence sum

For an arithmetic sequence with first term a₁ and common difference d, the sum of the first n terms is S_n=n(a₁+a_n)/2=n[2a₁+(n−1)d]/2.

Geometric sequence sum

For a geometric sequence with first term a₁ and ratio q≠1, S_n=a₁(1−q^n)/(1−q); if q=1 then S_n=n a₁.

Sequence partial sum relation

For a sequence with partial sums S_n, a₁=S₁ and a_n=S_n−S_{n−1} for n≥2; this converts sum information into term information.

AM-GM inequality

For positive real numbers a and b, (a+b)/2≥√(ab), with equality exactly when a=b; for n positive numbers, the arithmetic mean is at least the geometric mean.

Cauchy-Schwarz inequality

For real numbers a_i and b_i, (Σ a_i b_i)²≤(Σ a_i²)(Σ b_i²), with equality when the vectors (a_i) and (b_i) are proportional.

Absolute value inequality

For a>0, |x|<a is equivalent to −a<x<a, while |x|>a is equivalent to x<−a or x>a; these describe intervals centered at zero.

Dot product formula

For vectors a and b, a·b=|a||b|cos θ, where θ is the angle between them; in coordinates a·b=x₁x₂+y₁y₂.

Vector perpendicularity

Two nonzero vectors are perpendicular exactly when their dot product is zero; in coordinates, (x₁,y₁)·(x₂,y₂)=x₁x₂+y₁y₂=0.

Complex modulus

For z=a+bi, the modulus |z|=√(a²+b²) gives the distance from z to the origin in the complex plane; |z₁−z₂| is the distance between points.

Line-plane angle

If u is a direction vector of a line and n is a normal vector of a plane, the sine of the line-plane angle θ is |u·n|/(|u||n|).

Point-plane distance

In three-dimensional Cartesian coordinates, the distance from point (x₀,y₀,z₀) to the plane Ax+By+Cz+D=0 is |Ax₀+By₀+Cz₀+D| divided by √(A²+B²+C²).

Pyramid volume

The volume of a pyramid or cone is one third of the base area times the perpendicular height: V=(1/3)Bh; for a prism or cylinder it is V=Bh.

Ellipse eccentricity

For an ellipse x²/a²+y²/b²=1 with a>b>0, c²=a²−b² and eccentricity e=c/a lies between 0 and 1; larger e means a more elongated ellipse.

Hyperbola asymptotes

For x²/a²−y²/b²=1, the asymptotes are y=±(b/a)x; the hyperbola approaches these lines as |x| grows without bound.

Parabola focus-directrix

A parabola is the set of points equidistant from a fixed focus and a fixed directrix; for y²=2px, the focus is (p/2,0) and directrix is x=−p/2.

Circle equation

The circle centered at (h,k) with radius r has equation (x−h)²+(y−k)²=r²; expanding gives x²+y²+Dx+Ey+F=0 with center (−D/2,−E/2).

Conditional probability

For events A and B with P(B)>0, P(A|B)=P(A∩B)/P(B); it measures the probability of A after restricting the sample space to B.

Independent events

Events A and B are independent exactly when P(A∩B)=P(A)P(B); then P(A|B)=P(A) and P(B|A)=P(B), provided the conditional probabilities exist.

Binomial distribution

If X counts successes in n independent trials each with success probability p, then P(X=k)=C(n,k)p^k(1−p)^{n−k}, with mean np and variance np(1−p).

Combination formula

The number of unordered selections of k objects from n distinct objects is C(n,k)=n!/[k!(n−k)!]; order is not counted, and C(n,k)=C(n,n−k).

Binomial theorem

For nonnegative integer n, (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.

Permutation formula

The number of ordered arrangements of k objects chosen from n distinct objects is P(n,k)=n!/(n−k)!; when k=n it equals n!.