HSC Maths Ext 2 flashcards

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

Proof by contradiction

Assume the negation of the statement you want to prove, then derive a logical impossibility with a known truth. Concluding the original statement follows is valid because the negation cannot hold.

Contrapositive equivalence

The statement 'if P then Q' is logically equivalent to 'if not Q then not P'. Proving the contrapositive therefore proves the original implication, and this differs from assuming the negation.

Disproof by counterexample

A universal statement claims something for all cases. One valid case where the hypothesis holds but the conclusion fails is enough to disprove it; no general argument is needed.

Irrationality of square root two

Assume sqrt(2) equals a fraction a/b in lowest terms. Squaring gives 2b^2 = a^2, so a is even; writing a = 2c forces b even too, contradicting lowest terms.

Dot product formula

For vectors a and b, the dot product equals |a||b|cos(theta) and also the sum of componentwise products. It is zero exactly when the vectors are perpendicular, since cos(90 degrees) is zero.

Cross product magnitude

The magnitude of a cross b equals |a||b|sin(theta), the area of the parallelogram spanned by a and b. The cross product vector is perpendicular to both a and b by the right-hand rule.

Vector equation of a line

A line is written r = a + t d, where a is the position vector of one point on the line, d is a direction vector, and t is a real parameter taking every value.

Distance from point to plane

For a plane with unit normal n through point A, the perpendicular distance from point P is the absolute value of (P - A) dot n. Without a unit normal, divide by the magnitude of n.

Scalar triple product

The quantity a dot (b cross c) equals the signed volume of the parallelepiped with edges a, b and c. It is zero exactly when the three vectors are coplanar.

Euler form of complex numbers

Every complex number can be written r e^(i theta), where r is its modulus and theta its argument. Products multiply moduli and add arguments; quotients divide moduli and subtract arguments.

De Moivre's theorem

For any integer n, (cos theta + i sin theta)^n equals cos(n theta) + i sin(n theta). It converts powers of complex numbers into trigonometric multiples of the argument and vice versa.

Roots of unity

The n solutions of z^n = 1 are e^(2 pi i k/n) for k from zero to n minus one. They lie equally spaced on the unit circle and their sum is zero.

Conjugate root theorem

If a polynomial has real coefficients and z is a root, then the conjugate of z is also a root. Non-real roots therefore occur in conjugate pairs, allowing real factorisation into quadratics.

Perpendicular bisector locus

The set of points z satisfying |z - a| = |z - b| is the perpendicular bisector of the segment joining the points a and b in the complex plane, because both distances are equal.

Derivative of arctangent

The derivative of arctan x is 1/(1 + x^2) for all real x. This standard result underlies the integral of 1/(x^2 + a^2), which equals (1/a) arctan(x/a).

Integration by parts

The rule integral of u dv equals uv minus the integral of v du. Choosing u as the logarithmic or inverse trigonometric factor first usually simplifies the remaining integral.

Trigonometric substitution

For radicals, substitute x = a sin theta for a^2 - x^2, x = a tan theta for a^2 + x^2, and x = a sec theta for x^2 - a^2. Convert back using a right triangle.

Parametric second derivative

For parametric curves, dy/dx equals (dy/dt) divided by (dx/dt). The second derivative is the t-derivative of dy/dx, then divided again by dx/dt; it is not dy'/dx'.

Volume of revolution

Rotating y = f(x) about the x-axis between a and b gives volume pi times the integral of y squared dx. The shell method uses 2 pi times the integral of x y dx.

Simple harmonic motion

Motion satisfying acceleration equals minus omega squared times displacement. The solution is x = A cos(omega t + phi), with period 2 pi/omega and maximum speed A omega at the centre.

Variable acceleration identity

Acceleration equals the time derivative of velocity, and also velocity times the displacement derivative of velocity. The second form is used when acceleration is given as a function of position.

Resisted motion

When resistance is proportional to speed, the equation m dv/dt = -kv gives exponential decay of speed. For resistance proportional to speed squared, integrate 1/v^2 instead.

Projectile range

For projection speed v at angle theta over level ground with no air resistance, the horizontal range is v squared times sin(2 theta) divided by g, maximised at 45 degrees.

Geometric series sum

A geometric series with first term a and ratio r has partial sum a(r^n - 1)/(r - 1) for r not equal to 1. It converges to a/(1 - r) exactly when |r| < 1.

Sum of squares formula

The sum of k squared from k = 1 to n equals n(n + 1)(2n + 1)/6. Together with the sum of k and sum of k cubed, it evaluates polynomial series term by term.

Telescoping series

Writing each term as a difference f(k) minus f(k + 1) makes intermediate terms cancel in the partial sum. The sum then reduces to the first term minus the limiting last term.

Ratio test

For a series with positive terms, compute the limit of the ratio of consecutive terms. A limit below 1 gives convergence, above 1 gives divergence, and equal to 1 is inconclusive.

General term of binomial expansion

In the expansion of (a + b)^n, the term containing b^k is C(n, k) a^(n-k) b^k. Setting exponents equal to a target value solves for k and hence the coefficient.

Pascal's identity

The binomial coefficient C(n, k) equals C(n-1, k-1) + C(n-1, k). This identity generates Pascal's triangle, where each interior entry is the sum of the two above it.

Sum of binomial coefficients

Setting both terms to 1 in the binomial theorem gives the sum of C(n, k) from k = 0 to n equal to 2^n. Setting one term to -1 gives an alternating sum of zero.

Greatest binomial coefficient

Consecutive coefficients have ratio (n - k)/(k + 1). The coefficients increase while this ratio exceeds 1 and decrease afterwards, locating the largest term in the expansion.

Principle of mathematical induction

If a statement holds for a base integer and holding for n = k always implies holding for n = k + 1, then it holds for every integer at or beyond the base.

Inductive step for sums

Assume the sum formula holds for k terms. Then add the next term to both sides and simplify algebraically until the expression matches the formula with k replaced by k + 1.