GCE A/L Combined Mathematics flashcards

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

Quadratic discriminant

For ax²+bx+c=0 the discriminant is b²−4ac. Positive gives two distinct real roots, zero a repeated real root, negative a conjugate pair of non-real roots.

Sum and product of roots

For ax²+bx+c=0 with roots α and β, the sum α+β equals −b/a and the product αβ equals c/a, allowing equations to be built from given roots.

Remainder and Factor Theorems

Dividing polynomial p(x) by (x−a) leaves remainder p(a). The linear factor (x−a) divides p(x) exactly when p(a)=0, giving a root at x=a.

Modulus and argument

For z=a+bi, modulus |z|=√(a²+b²) is the distance from the origin and argument is the angle from the positive real axis, with tan(arg z)=b/a.

De Moivre's theorem

For integer n, (cosθ+i sinθ)ⁿ=cos nθ+i sin nθ. It converts powers and roots of complex numbers into simple multiplication or division of angles.

Geometric series sum

A geometric series with first term a and ratio r has sum a(1−rⁿ)/(1−r) for n terms, and infinite sum a/(1−r) only when |r|<1.

Pythagorean identity

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

Compound angle formulas

sin(A±B)=sinA cosB±cosA sinB and cos(A±B)=cosA cosB∓sinA sinB. These expand trigonometric functions of sums and differences into products of single angles.

Double angle formulas

sin2θ=2sinθcosθ and cos2θ=cos²θ−sin²θ, also written 2cos²θ−1 or 1−2sin²θ. They are used to simplify expressions and integrate products.

R-form transformation

a sinθ+b cosθ can be written R sin(θ+α) where R=√(a²+b²) and tanα=b/a. The expression then ranges between −R and R.

Cosine rule

In a triangle, a²=b²+c²−2bc cosA, where A is the angle opposite side a. Use it when two sides and the included angle or all three sides are known.

Triangle area formula

The area of a triangle equals ½ab sinC using two sides and the included angle. Heron's formula √(s(s−a)(s−b)(s−c)) uses all three sides.

Perpendicular distance to a line

The perpendicular distance from point (x₁,y₁) to line ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²), obtained by substituting the point into the normalised equation.

Circle in general form

The equation x²+y²+2gx+2fy+c=0 represents a circle with centre (−g,−f) and radius √(g²+f²−c), provided that quantity is positive.

Parabola y²=4ax

This parabola has vertex at the origin, focus (a,0) and directrix x=−a. Every point on it is equidistant from the focus and the directrix.

Ellipse eccentricity

For x²/a²+y²/b²=1 with a>b, foci lie at (±c,0) where c²=a²−b². Eccentricity e=c/a satisfies 0<e<1.

Hyperbola asymptotes

For x²/a²−y²/b²=1 the asymptotes are y=±(b/a)x. The foci are at (±c,0) with c²=a²+b² and eccentricity e=c/a>1.

Chain rule

If y depends on u and u depends on x, then dy/dx=(dy/du)(du/dx). This differentiates composite functions such as sin(3x²) or e^(kx).

Quotient rule

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

Second derivative test

At a stationary point where dy/dx=0, a positive second derivative means a local minimum, a negative one a local maximum, and zero requires further investigation.

Parametric gradient

For a curve given as x=f(t), y=g(t), the gradient is dy/dx=(dy/dt)/(dx/dt), valid where dx/dt is not zero.

Equation of a normal

The normal to y=f(x) at x=a passes through (a,f(a)) with gradient −1/f′(a), the negative reciprocal of the tangent gradient.

Integration by substitution

Replacing a variable by u=g(x) converts ∫f(g(x))g′(x)dx into ∫f(u)du. For definite integrals, also convert the limits to u-values.

Integration by parts

The rule ∫u dv=uv−∫v du reverses the product rule. Choose u to simplify on differentiation, commonly ln x, xⁿ or an inverse trigonometric function.

Area between curves

The area between y=f(x) and y=g(x) from a to b is ∫ₐᵇ(f−g)dx where f is the upper curve, or ∫|f−g|dx if they cross.

Volume of revolution

Rotating y=f(x) about the x-axis from a to b sweeps a volume V=π∫ₐᵇy²dx, because each slice is a disc of radius y.

Trapezium rule

With n equal strips of width h=(b−a)/n, the integral is approximated by (h/2)[f₀+2f₁+…+2fₙ₋₁+fₙ], using the end ordinates once and interior ordinates twice.

Dot product

a·b=|a||b|cosθ equals a₁b₁+a₂b₂+a₃b₃ in components. It is zero exactly when the vectors are perpendicular, and it gives the angle between them.

Cross product

a×b has magnitude |a||b|sinθ and direction perpendicular to both, given by the right-hand rule. Its magnitude equals the area of the parallelogram spanned.

Equation of a plane

A plane through point a with normal vector n satisfies (r−a)·n=0. The perpendicular distance from point p to the plane is |(p−a)·n|/|n|.

Limiting friction

When a body is on the point of slipping, friction equals μR, where μ is the coefficient of friction and R the normal reaction. Otherwise friction is at most μR.

Projectile range

For projection speed u at angle α on level ground, time of flight is 2u sinα/g and horizontal range is u²sin2α/g, greatest at α=45°.

Binomial distribution

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

Standardising a normal variable

If X is normal with mean μ and standard deviation σ, then z=(x−μ)/σ follows the standard normal distribution, so tables give P(X<x) as P(Z<z).