Gaokao Mathematics common mistakes
Gaokao Mathematics · Understand the traps and how to avoid them.
When asked for the parameter range that makes A⊆B, substitutes one convenient element of A into B's inequality and declares the range, without checking endpoints or the case A=∅.
Set inclusion is often learned through Venn diagrams, but exam problems encode it as an inequality system; one successful sample feels like proof, and the empty set is easy to overlook.
What to do instead
Rewrite A⊆B as 'for all x, x∈A ⇒ x∈B', solve the resulting inequality system including endpoint equality, then test A=∅ separately by checking whether the parameter range makes the set empty.
Finds f'(x)>0 and states the monotonic interval directly, without first writing the domain of f; for example, for f(x)=ln x−ax, discusses all real x.
Derivative calculations are algebraic and the domain check is treated as a formality, especially when the formula simplifies.
What to do instead
Before differentiating, record the domain (denominators nonzero, arguments of logarithms positive, square roots nonnegative); after solving f'(x)>0, intersect the solution set with the domain and list intervals with open or closed endpoints justified.
In a triangle with known a, b, A, applies the sine rule to get sin B, then takes only the acute value B0, even when a<b permits B=π−B0.
The inverse sine function returns a principal value in [−π/2, π/2], and the candidate forgets that sin B=sin(π−B) gives a second triangle candidate.
What to do instead
After sin B=k, write both B0 and π−B0; keep a candidate only if A+B<π and side ordering is consistent (a≥b forces B≤A, so only one; a<b may allow two). State the accepted triangles explicitly.
Given S_n, writes a_n=S_n−S_{n−1}=2n for all n≥1 without computing a_1=S_1 separately; when S_n has a constant term, the first term is wrong.
The subtraction identity is valid only for n≥2; when the resulting formula happens to work for n=1, the habit is reinforced, and constant terms are easy to miss.
What to do instead
Always compute a_1=S_1 first, then derive a_n=S_n−S_{n−1} for n≥2, and compare the n=1 value with the formula; if different, present a_n as a piecewise definition.
Solving (x−1)/(x+2)<1 by multiplying both sides by x+2 without splitting the sign, yielding a solution set that wrongly includes x<−2.
Clearing denominators is a habit from equations; in inequalities the direction reverses when the multiplier is negative, and the sign of a variable multiplier is unknown.
What to do instead
Move all terms to one side, combine into a single fraction, and use a sign chart of numerator and denominator; never multiply by a variable expression unless its sign is known, and exclude zeros of the denominator.
For complex numbers, replaces |z|^2 with z^2, or solves |z|=2 by z=±2 only; for vectors, cancels a·b=a·c to b=c.
The notation |z| resembles absolute value of a real number, and the vector dot product is treated like ordinary multiplication even though it is not cancellative.
What to do instead
Use |z|^2=z\bar z and write z=x+yi; convert modulus equations to x^2+y^2. For vectors, expand a·(b−c)=0 and conclude only that a is perpendicular to b−c, not b=c.
To prove l∥α, shows l is parallel to a line m inside α and stops, omitting the condition l⊄α; or computes a dihedral angle from normals and reports the supplement.
The line-plane parallel theorem has three conditions, but the visual picture suggests 'parallel lines mean parallel plane'; normal vectors give the angle between normals, which may be the supplement of the dihedral angle.
What to do instead
Write all conditions: m⊂α, l∥m, l⊄α. For angles, identify whether the required angle is acute or obtuse from the geometry, or use cos θ=|n1·n2|/(|n1||n2|) for line-plane and check the sign for dihedral.
For an ellipse uses c^2=a^2+b^2, and for a hyperbola uses c^2=a^2−b^2, swapping the two relations; or sets a line as y=kx+m and divides by k, losing the vertical line.
The two conics are studied together and the formulas look similar; the standard line form y=kx+m implicitly assumes a finite slope.
What to do instead
Remember: ellipse b^2=a^2−c^2 (so a^2=b^2+c^2), hyperbola b^2=c^2−a^2 (so c^2=a^2+b^2). For lines, discuss x=my+t separately or check the vertical line x=x0 before dividing by k.
For two events, writes P(A∪B)=P(A)+P(B) without subtracting P(A∩B), or concludes A and B are independent because they cannot occur together.
Everyday language treats 'no overlap' as 'unrelated', and the addition rule is remembered without its disjointness condition.
What to do instead
Always start from P(A∪B)=P(A)+P(B)−P(A∩B). Distinguish: mutually exclusive means P(A∩B)=0; independent means P(A∩B)=P(A)P(B). Test with a concrete two-way table or numeric example before concluding.
In (a+b)^n, calls C_n^k a^{n−k}b^k the k-th term; asked for the coefficient of b^3, uses k=2 and reports C_n^2 instead of C_n^3.
The formula is memorized with k starting at 0, but term number is counted from 1, so an off-by-one shift occurs under time pressure.
What to do instead
Write a small table: k=0 gives term 1, k=1 gives term 2, and so on; the term containing b^r has k=r and is term r+1. For coefficients, set the exponent equal to r and solve for k before evaluating C_n^k.