JCE Mathematics flashcards

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

Standard form

A number written as a × 10^n where 1 ≤ a < 10 and n is an integer. Used to express very large or very small quantities compactly.

Rational number

Any number expressible as p/q where p and q are integers and q ≠ 0. Terminating and recurring decimals are rational; surds and π are not.

Upper and lower bounds

The range within which a rounded value truly lies. For a value rounded to unit u, the error is u/2, giving lower = value − u/2 and upper = value + u/2.

Compound interest formula

A = P(1 + r/100)^n gives the total after n periods at r percent per period, where P is the principal. A negative sign models depreciation or decay.

Rationalising a denominator

Removing a surd from a denominator by multiplying numerator and denominator by the surd or its conjugate, e.g. 1/(√a + √b) = (√a − √b)/(a − b).

Quadratic formula

x = [−b ± √(b² − 4ac)]/(2a) solves ax² + bx + c = 0 with a ≠ 0. The discriminant b² − 4ac determines the number of real roots.

Discriminant

The quantity b² − 4ac in a quadratic. Positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots.

Completing the square

Rewriting x² + bx + c as (x + b/2)² + c − b²/4. It reveals the vertex of the quadratic graph and can solve equations without factorising.

Geometric series sum

For a geometric series with first term a and ratio r ≠ 1, the sum of n terms is Sₙ = a(1 − rⁿ)/(1 − r).

Laws of logarithms

log(ab) = log a + log b, log(a/b) = log a − log b, and log(aⁿ) = n log a. These convert multiplication into addition and powers into coefficients.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign, because order on the number line flips.

Perpendicular gradient condition

Two straight lines are perpendicular exactly when the product of their gradients equals −1, so m₂ = −1/m₁. Parallel lines instead share the same gradient.

Inverse function

The function f⁻¹ that reverses f, found by swapping x and y and solving for y. Its graph is f reflected in the line y = x.

Composite function

The function fg(x) = f(g(x)), formed by applying g first and then f to the result. In general fg(x) does not equal gf(x).

Exponential graph properties

The graph of y = a^x always passes through (0, 1). It rises for a > 1, falls for 0 < a < 1, and approaches but never meets the x-axis.

Circle equation

(x − a)² + (y − b)² = r² describes a circle with centre (a, b) and radius r. Expanding it gives the general form x² + y² + 2fx + 2gy + c = 0.

Graph transformation rules

For y = f(x): f(x) + a shifts up a, f(x + a) shifts left a, af(x) stretches vertically by a, and f(ax) stretches horizontally by 1/a.

Sine rule

a/sin A = b/sin B = c/sin C in any triangle. It applies when a side and its opposite angle are known alongside one further side or angle.

Cosine rule

a² = b² + c² − 2bc cos A. Use it when two sides and the included angle are known, or when all three sides are known and an angle is required.

Triangle area with sine

Area = ½ab sin C, where a and b are two sides and C is the angle between them. It works for any triangle, not only right-angled ones.

Angle in a semicircle

Any angle subtended by a diameter at a point on the circumference is a right angle, 90°. This follows from the angle-at-centre theorem with a straight angle of 180°.

Sector arc length

Arc length = (θ/360) × 2πr for an angle θ in degrees. In radians the same result is written s = rθ.

Similar shape scale factors

If corresponding lengths scale by factor k, then areas scale by k² and volumes by k³. This links enlargement to surface area and capacity problems.

Interquartile range

IQR = upper quartile − lower quartile. It measures the spread of the middle half of the data and is less affected by extreme values than the range.

Estimated mean of grouped data

For grouped frequency tables, estimate the mean as Σ(fx)/Σf, where x is the midpoint of each class interval and f its frequency.

Mutually exclusive events

Events that cannot occur together, so P(A and B) = 0 and P(A or B) = P(A) + P(B). The probabilities of all outcomes sum to 1.

Independent events

Events where one outcome does not affect the other, so P(A and B) = P(A) × P(B). Tree diagram branches are multiplied along a path.

Conditional probability

P(A|B) = P(A ∩ B)/P(B), the probability of A given that B has occurred, defined when P(B) > 0. It restricts the sample space to event B.

Cumulative frequency curve

A graph of cumulative frequency against upper class boundary. Quartiles are read at ¼, ½ and ¾ of the total frequency, giving median and interquartile range.

Modelling cycle

Formulate a real problem mathematically, solve the model, interpret results in context, then validate and refine assumptions. Real-world modelling rarely succeeds in a single pass.

Rate formulas

Speed = distance/time, density = mass/volume, pressure = force/area. Each can be rearranged to find any of the three quantities when the other two are known.

Gradient and area in context

On a real-world graph, the gradient usually represents a rate of change and the area under the curve represents a total accumulated quantity over the interval.