Mie Entrance Mathematics cheat sheet

Sign rules for multiplication: (+)(+)=+, (−)(−)=+, (+)(−)=−; the same rule applies to division of nonzero numbers.

Distributive law: a(b+c)=ab+ac and (a+b)(a−b)=a²−b²; these two expansions cover most simplification tasks.

Perfect-square expansions: (a+b)²=a²+2ab+b² and (a−b)²=a²−2ab+b².

Laws of exponents (a>0): aᵐ·aⁿ=aᵐ⁺ⁿ, aᵐ÷aⁿ=aᵐ⁻ⁿ, (aᵐ)ⁿ=aᵐⁿ, (ab)ⁿ=aⁿbⁿ, a⁰=1, a⁻ⁿ=1/aⁿ.

Radicals: √(a²)=|a|, √(ab)=√a·√b for a,b≥0, and √a/√b=√(a/b) for a≥0, b>0; rationalize by multiplying by the conjugate.

Factoring order: common factor first, then difference of squares, then sum/product pattern x²+(p+q)x+pq=(x+p)(x+q).

Absolute value: |x|=x if x≥0 and |x|=−x if x<0; |x| equals the distance from x to 0 on the number line.

Comparing irrationals: square both positive quantities to compare them, e.g. 2√3 vs 3√2 becomes 12 vs 18.

Linear equation ax+b=0 with a≠0 has the single solution x=−b/a.

A quadratic ax²+bx+c=0 (a≠0) has discriminant D=b²−4ac: D>0 two distinct real roots, D=0 one repeated root, D<0 no real roots.

Quadratic formula: x=(−b±√(b²−4ac))/(2a); Vieta's relations give sum = −b/a and product = c/a.

Multiplying or dividing an inequality by a negative number reverses the inequality sign; adding or subtracting a number never does.

A compound inequality a<x<b is solved by applying the same operation to all three parts; the solution is an interval on the number line.

Systems of two linear equations: solve by substitution or elimination; parallel lines (same slope, different intercept) mean no solution, identical lines mean infinitely many.

Always check a candidate solution in the original equation, especially after squaring both sides (extraneous roots) or when a denominator could be zero.

An equation with a variable in a denominator is undefined where the denominator is 0; exclude those values from the solution set.

Linear function y=ax+b: a is the slope (rate of change) and b is the y-intercept; slope a=(y₂−y₁)/(x₂−x₁).

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1 (a₂=−1/a₁).

Direct proportion y=ax passes through the origin; inverse proportion y=a/x has the product xy constant and a hyperbola graph.

Quadratic y=a(x−p)²+q has vertex (p,q) and axis of symmetry x=p; a>0 opens upward, a<0 opens downward.

Vertex x-coordinate of y=ax²+bx+c is x=−b/(2a); the vertex is a minimum when a>0 and a maximum when a<0.

Graph transformations: y=f(x−h) shifts right by h, y=f(x)+k shifts up by k, y=−f(x) reflects in the x-axis, y=f(−x) reflects in the y-axis.

Domain is the set of valid inputs, range the set of resulting outputs; for y=√x, domain x≥0 and range y≥0.

Rate of change of a function on an interval equals (change in y)/(change in x); a steeper graph means a larger absolute slope.

Angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal.

With two parallel lines cut by a transversal: corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°.

Triangle angle sum is 180°; an exterior angle equals the sum of the two non-adjacent interior angles.

Congruence criteria: SSS, SAS, ASA (with AAS) and RHS for right triangles; AAA proves similarity, not congruence.

Similarity criteria: AA, SSS ratio, SAS ratio; corresponding sides are proportional and areas scale by the square of the ratio.

Pythagorean theorem: in a right triangle a²+b²=c² with c the hypotenuse; the converse identifies a right angle.

Circle: the angle at the centre is twice the angle at the circumference on the same arc; an angle in a semicircle is 90°.

Tangent is perpendicular to the radius at the point of contact; tangents from an external point are equal in length.

Inscribed angle theorem consequences: angles subtended by the same chord in the same segment are equal, and opposite angles of a cyclic quadrilateral sum to 180°.

Triangle area = ½·base·height; trapezium area = ½(a+b)h; parallelogram area = base·height.

Circle: circumference C=2πr, area A=πr²; arc length = (θ/360)·2πr and sector area = (θ/360)·πr² for central angle θ.

Prism or cylinder volume V=(base area)·h; cylinder lateral surface area = 2πrh, total = 2πrh+2πr².

Pyramid or cone volume V=⅓·(base area)·h; cone volume = ⅓πr²h and lateral area = πrl with slant height l.

Sphere: volume V=(4/3)πr³, surface area S=4πr².

Similar solids: lengths scale by k, surface areas by k², volumes by k³.

Unit conversion: 1 m²=10 000 cm², 1 m³=1 000 000 cm³, 1 L=1000 cm³; convert lengths before computing areas or volumes.

Euler's relation for a convex polyhedron: vertices − edges + faces = 2.

Mean = sum of values ÷ number of values; median is the middle value of ordered data; mode is the most frequent value.

For grouped data, the median lies in the class where cumulative frequency first reaches half the total.

Range = maximum − minimum; interquartile range = Q3 − Q1 and measures spread of the middle half.

Probability of an event P(A)= (number of favourable outcomes)/(total equally likely outcomes); 0≤P(A)≤1.

Complement rule: P(not A)=1−P(A); for mutually exclusive events P(A or B)=P(A)+P(B).

Independent events: P(A and B)=P(A)·P(B); for dependent events multiply by the conditional probability.

Expected frequency = probability × number of trials; relative frequency from data estimates the probability.

A histogram uses area, not height alone, to represent frequency when class widths differ.

A statement and its contrapositive are logically equivalent; the converse and inverse are not automatically true.

Proof by contradiction: assume the negation of the claim, derive a contradiction, conclude the claim must hold.

Counterexample disproves a universal statement; one example never proves a universal statement.

Proof by exhaustion checks all finitely many cases; proof by induction has a base case plus an inductive step.

In a geometric proof, each step must cite a definition, postulate, or previously proved theorem; mark given information on the diagram.

Necessary versus sufficient: 'P implies Q' makes P sufficient for Q and Q necessary for P.

Even and odd parity rules: even+even=even, odd+odd=even, odd·odd=odd; use these to prove divisibility claims.

To prove two triangles congruent or similar, state the criterion explicitly and list the three corresponding matched parts.

Translate word problems into an equation by naming the unknown, then check that the answer satisfies every stated condition.

Rate problems: distance = speed × time; for combined work, add rates (1/a+1/b per unit time).

Optimisation with quadratics: complete the square or use x=−b/(2a) to find the maximum area or minimum cost.

Coordinate geometry links algebra and geometry: distance =√((x₂−x₁)²+(y₂−y₁)²), midpoint = ((x₁+x₂)/2,(y₁+y₂)/2).

A line through two points: slope from the two points, then y−y₁=a(x−x₁); intersection of two graphs solves the simultaneous equations.

Similar-figure scaling converts a length ratio into area ratio k² and volume ratio k³, useful in map and model problems.

Probability combined with counting: list outcomes systematically or use the multiplication principle before computing ratios.

When a problem mixes units or scales, convert to a single unit at the start and state the unit in the final answer.