BCS Written Mathematics flashcards

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

HCF-LCM product rule

For two positive integers, HCF times LCM equals the product of the two numbers. The rule fails for three or more numbers, so use prime factorisation there.

Successive percentage change

A change of a percent followed by b percent gives a net change of a + b + ab/100 percent. A 20 percent rise then 20 percent fall leaves a net 4 percent decrease.

Compound interest formula

Amount A = P(1 + R/100)^T when interest compounds annually. For n conversions per year use A = P(1 + R/(100n))^(nT), then subtract P for interest earned.

Harmonic mean average speed

For equal distances travelled at speeds x and y, average speed is 2xy/(x + y). The arithmetic mean overstates speed because more time is spent at the slower speed.

Combined work rate

Rates add, times do not. If A finishes in a days and B in b days, working together they need ab/(a + b) days, which is less than either individual time.

Quadratic formula

The roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac))/(2a). The discriminant b² − 4ac determines whether roots are real, repeated, or complex.

Sum and product of roots

For ax² + bx + c = 0 with roots α and β, α + β = −b/a and αβ = c/a. The equation can be rebuilt as x² − (sum)x + (product) = 0.

Remainder theorem

Dividing polynomial p(x) by (x − a) leaves remainder p(a). If p(a) = 0, then (x − a) is a factor of p(x) and a is a root.

Arithmetic progression sums

An AP has nth term a + (n − 1)d and sum of n terms (n/2)[2a + (n − 1)d]. A geometric progression has nth term ar^(n−1).

Logarithm laws

Logarithms turn products into sums: log(xy) = log x + log y, log(x/y) = log x − log y, log x^n = n log x. Also log_b a = log_c a / log_c b.

Binomial general term

In (x + y)^n the term containing y^r is C(n,r) x^(n−r) y^r, where C(n,r) = n!/(r!(n−r)!). This locates any single coefficient without full expansion.

Pythagorean theorem

In a right triangle, the square on the hypotenuse equals the sum of squares on the other two sides: a² + b² = c². Common triples include 3-4-5 and 5-12-13.

Sector and arc formulas

For a circle of radius r and central angle θ degrees, arc length is (θ/360) × 2πr and sector area is (θ/360) × πr². Circumference is 2πr.

Heron's area formula

A triangle with sides a, b, c and semiperimeter s = (a + b + c)/2 has area √(s(s−a)(s−b)(s−c)). It needs only the three side lengths.

Similar figure scaling

When two figures are similar with length ratio k, areas scale by k² and volumes by k³. Doubling every length multiplies area by four and volume by eight.

Polygon angle sums

The interior angles of an n-sided polygon sum to (n − 2) × 180°. In a regular n-gon each exterior angle is 360°/n and each interior angle is 180° − 360°/n.

Cone and sphere volume

A cone of radius r and height h has volume (1/3)πr²h and slant height √(r² + h²). A sphere has volume (4/3)πr³ and surface area 4πr².

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C, each equal to twice the circumradius. Use it when a side and its opposite angle are known.

Cosine rule

In any triangle, a² = b² + c² − 2bc cos A. Rearranged, cos A = (b² + c² − a²)/(2bc), which finds an angle from three known sides.

Pythagorean trigonometric identity

For any angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Double angle formulas

sin 2A = 2 sin A cos A, while cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1. The three cosine forms suit different rearrangement needs.

Radian arc and sector

With the angle θ measured in radians, arc length is s = rθ and sector area is ½r²θ. One full turn equals 2π radians, or 180° equals π radians.

Triangle area with sine

The area of any triangle equals ½ab sin C, using two sides a and b and the angle C between them. This avoids needing a height.

Chain rule

For a composite function, dy/dx = dy/du × du/dx. Differentiate the outer function, keep the inner unchanged, then multiply by the inner derivative.

Quotient rule

For y = u/v, the derivative is (u′v − uv′)/v². The numerator subtracts in that order, so reversing it flips the sign of the result.

Second derivative test

At a stationary point where f′(x) = 0, f″(x) > 0 indicates a local minimum and f″(x) < 0 a local maximum. If f″(x) = 0, test sign change.

Definite integral evaluation

For an antiderivative F of f, the definite integral from a to b equals F(b) − F(a). The constant of integration cancels, so it is omitted.

Volume of revolution

Rotating the curve y = f(x) about the x-axis between x = a and x = b sweeps a volume V = π∫ y² dx over those limits.

Grouped data standard deviation

For grouped data with midpoints x and frequencies f, variance equals Σfx²/Σf − (mean)², where mean = Σfx/Σf. Standard deviation is its positive square root.

Conditional probability

P(A|B) = P(A ∩ B)/P(B), valid when P(B) > 0. It rescales the sample space to outcomes where B has occurred, so the denominator is P(B), not 1.

Binomial distribution

For n independent trials each with success probability p, P(X = r) = C(n,r) p^r (1 − p)^(n−r). Mean is np and variance is np(1 − p).

Matrix inverse and determinant

A 2 × 2 matrix [[a, b], [c, d]] has determinant ad − bc and inverse (1/det)[[d, −b], [−c, a]]. No inverse exists when the determinant is zero.