FY13CE Pure Mathematics flashcards

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

Contrapositive equivalence

The statement P implies Q is logically equivalent to not-Q implies not-P. Proving the contrapositive is a valid proof of the original implication and is often easier when the conclusion is negative.

Disproof by counterexample

A single case in which the hypothesis holds but the conclusion fails disproves a universal statement. One counterexample suffices for disproof but can never establish the truth of a general claim.

Quantifier negation rules

The negation of 'for all x, P(x)' is 'there exists x such that not P(x)'. The negation of 'there exists x with P(x)' is 'for all x, not P(x)'. Apply the negation to the predicate as well.

Discriminant of a quadratic

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

Factor and remainder theorems

For a polynomial f, division by (x − a) leaves remainder f(a), and (x − a) is a factor exactly when f(a) = 0. For (ax − b) use f(b/a).

Modulus inequality interval

For b > 0, the inequality |x − a| < b is equivalent to a − b < x < a + b. The reversed inequality |x − a| > b gives x < a − b or x > a + b.

Sum of a geometric series

For first term a and common ratio r not equal to 1, the sum of n terms is a(1 − rⁿ)/(1 − r). The infinite sum a/(1 − r) exists only when |r| < 1.

Telescoping by differences

If the general term equals f(r) − f(r+1), summing from r = 1 to n cancels all intermediate terms, leaving f(1) − f(n+1). This method of differences evaluates many finite series.

Binomial general term

In the expansion of (a + b)ⁿ, the term containing b^k is C(n,k) a^(n−k) b^k, where C(n,k) = n!/(k!(n−k)!). This locates a specific coefficient without full expansion.

Circle in expanded form

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

Point-to-line distance

The perpendicular distance from the point (x₁, y₁) to the line ax + by + c = 0 is |ax₁ + by₁ + c| divided by the square root of a² + b².

Perpendicular gradient condition

Two straight lines are perpendicular exactly when the product of their gradients equals −1, provided neither is vertical. A horizontal line is perpendicular to a vertical line.

Arc length and sector area

With angle θ measured in radians, an arc of a circle of radius r has length rθ, and the corresponding sector has area ½r²θ. Degrees must be converted first.

Compound angle formulas

sin(A ± B) equals sinA cosB ± cosA sinB, while cos(A ± B) equals cosA cosB ∓ sinA sinB. Note the sign reversal in the cosine expansion.

Harmonic form R sin(theta + alpha)

For a sinθ + b cosθ, write R sin(θ + α) with R equal to the square root of a² + b² and tanα equal to b/a. The maximum value of the expression is R.

Change of base formula

The logarithm of b to base a equals log b divided by log a, using any common base. A consequence is that log to base a of b times log to base b of a equals 1.

Exponential growth half-life

For y = Ae^(kt) with k negative, the half-life is ln2 divided by the modulus of k. For positive k the corresponding doubling time is ln2 divided by k.

Linearising a power law

If y = axⁿ then taking logarithms gives log y = log a + n log x. Plotting log y against log x yields a straight line of gradient n and intercept log a.

Quotient rule

For a quotient u/v, the derivative is (u′v − uv′)/v². The numerator order matters: the derivative of the top multiplies the bottom, minus the top times the derivative of the bottom.

Second derivative test

At a stationary point where the first derivative vanishes, a positive second derivative indicates a local minimum and a negative second derivative indicates a local maximum. Zero is inconclusive.

Implicit differentiation

Differentiate both sides with respect to x, treating y as a function of x. Every derivative of a y-term is multiplied by dy/dx through the chain rule, then solve for dy/dx.

Integration by parts

The rule is the integral of u dv equals uv minus the integral of v du. Choose u as the factor that simplifies when differentiated, such as a power of x or a logarithm.

Volume of revolution

Rotating y = f(x) about the x-axis between x = a and x = b gives volume π times the integral of y² dx. About the y-axis the corresponding integral uses x² dy.

Trapezium rule formula

The integral is approximated by (h/2) times the sum of the first and last ordinates plus twice all interior ordinates, where h is the strip width. Accuracy improves as strips increase.

Scalar product and angle

The scalar product of a and b equals a₁b₁ + a₂b₂ + a₃b₃ and also the product of the moduli times the cosine of the angle between them. Perpendicular vectors give zero.

Vector equation of a line

A line has equation r = a + λd, where a is the position vector of a point on the line, d is a direction vector and λ is a scalar parameter taking all real values.

Skew lines test

Two lines in three dimensions are skew when their direction vectors are not parallel and the equations a + λd = b + μe have no solution for λ and μ. They never meet.

De Moivre's theorem

For a complex number in modulus-argument form r(cosθ + i sinθ), raising to the power n gives rⁿ(cos nθ + i sin nθ). Multiplying arguments and powers of the modulus follow directly.

Roots of a complex number

The nth roots of re^(iθ) have modulus r^(1/n) and arguments (θ + 2kπ)/n for k from 0 to n − 1. They form a regular polygon centred at the origin.

Complex locus of a circle

The locus of points z satisfying |z − a| = r is a circle with centre a and radius r. The condition |z − a| = |z − b| gives the perpendicular bisector of the segment joining a and b.

Determinant and inverse

A 2×2 matrix has an inverse exactly when its determinant ad − bc is non-zero. The inverse is the reciprocal of the determinant times the matrix with a and d swapped and b and c negated.

Newton-Raphson iteration

The iteration x_{n+1} = xₙ − f(xₙ)/f′(xₙ) converges quickly to a root when the starting value is close. It fails where the derivative is zero or nearly zero.