Mie Entrance Mathematics flashcards

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

Distributive law

For all numbers a, b, c: a(b+c)=ab+ac. It also expands products of sums such as (x+2)(x−3)=x²−x−6 and is the basis for factoring.

Difference of squares

a²−b²=(a+b)(a−b). It factors expressions like 9x²−16 into (3x+4)(3x−4) and simplifies products of conjugate binomials.

Negative exponent rule

For a≠0 and positive integer n, a⁻ⁿ=1/aⁿ and 1/a⁻ⁿ=aⁿ. Negative exponents never make a nonzero value negative; they indicate a reciprocal.

Rationalising a denominator

Rewrite a fraction so no radical remains in the denominator. Multiply numerator and denominator by the conjugate, e.g. 1/(√3−1)=(√3+1)/2.

Quadratic discriminant

For ax²+bx+c=0 with a≠0, D=b²−4ac. If D>0 there are two distinct real roots, D=0 one repeated root, D<0 no real roots.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Adding or subtracting any number leaves the sign unchanged.

Vieta's formulas

For ax²+bx+c=0 with roots α and β: α+β=−b/a and αβ=c/a. They give root sums and products without solving the equation.

Extraneous solution

A value produced by squaring or clearing denominators that does not satisfy the original equation. Always substitute candidates back to check validity.

Slope of a line

For points (x₁,y₁) and (x₂,y₂) with x₁≠x₂, slope a=(y₂−y₁)/(x₂−x₁). It measures vertical change per unit horizontal change.

Perpendicular slope condition

Two non-vertical lines are perpendicular exactly when their slopes multiply to −1, so a₂=−1/a₁. A horizontal line is perpendicular to a vertical line.

Inverse proportion

y=a/x with constant a≠0, equivalently xy=a. The graph is a hyperbola in two quadrants, and y decreases as x increases.

Vertex form of a quadratic

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

Graph shift rules

Replacing x by x−h shifts the graph right by h; adding k outside shifts it up by k. Negating f(x) reflects in the x-axis.

Alternate interior angles

When two parallel lines are cut by a transversal, alternate interior angles are equal. The converse proves the lines are parallel.

Triangle exterior angle

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles, because it is supplementary to the third interior angle.

Similarity criteria for triangles

Two triangles are similar if two pairs of angles are equal (AA), or if all three pairs of corresponding sides are proportional (SSS), or two sides and the included angle (SAS).

Angle in a semicircle

An angle subtended by a diameter at any point on the circle is a right angle. This follows from the angle-at-centre theorem with a 180° central angle.

Tangent-radius theorem

A tangent to a circle is perpendicular to the radius drawn to the point of contact. Two tangents from one external point are equal in length.

Cyclic quadrilateral angles

Opposite angles of a quadrilateral inscribed in a circle sum to 180°. The converse shows four points lie on one circle.

Sector area formula

For a central angle θ in degrees, sector area =(θ/360)·πr² and arc length =(θ/360)·2πr. These are fractions of the whole circle.

Cone volume and lateral area

Cone volume V=⅓πr²h using perpendicular height h; lateral surface area =πrl using slant height l, where l²=r²+h².

Sphere formulas

A sphere of radius r has volume V=(4/3)πr³ and surface area S=4πr². Volume scales with the cube of the radius.

Similar solid scaling

If two similar solids have length ratio k, their surface areas have ratio k² and their volumes have ratio k³. Apply to models and scale drawings.

Mean, median, mode

The mean is the sum divided by the count; the median is the middle of ordered data; the mode is the most frequent value. Each describes a different aspect.

Interquartile range

IQR=Q3−Q1, the spread of the middle half of ordered data. It is resistant to extreme values, unlike the range.

Complement rule

P(not A)=1−P(A) for any event A. It is useful when the complement is easier to count than the event itself.

Independent events product rule

Events A and B are independent if P(A and B)=P(A)·P(B). For dependent events, multiply P(A) by P(B given A).

Proof by contradiction

Assume the statement is false, then derive a logical impossibility. Concluding the assumption fails proves the original statement must be true.

Counterexample

A single case where a general statement fails. One counterexample disproves a universal claim, while examples alone cannot prove one.

Converse statement

The converse of 'if P then Q' is 'if Q then P'. A true statement does not guarantee a true converse; each must be checked separately.

Necessary and sufficient

If P implies Q, then P is sufficient for Q and Q is necessary for P. 'P if and only if Q' means both implications hold.

Distance and midpoint formulas

For two points, distance=√((x₂−x₁)²+(y₂−y₁)²) and midpoint=((x₁+x₂)/2,(y₁+y₂)/2). Both follow from right-triangle and averaging ideas.

Combined work rate

If one worker finishes a job in a hours and another in b hours, together they complete 1/a+1/b of the job per hour, so the total time is ab/(a+b).