HSC Maths Ext 2 cheat sheet

Direct proof: assume the hypothesis, apply definitions and known theorems in logical steps, and reach the conclusion; every step must be justified by a definition, an axiom or a previously proved result.

Proof by contradiction: assume the negation of the statement, derive a contradiction with a known truth, and conclude the original statement; valid for statements of the form 'there is no x such that P(x)'.

Proof by contrapositive: proving 'not Q implies not P' establishes 'P implies Q'; the two statements are logically equivalent, so this is not a contradiction argument.

Counterexample: a single case where the hypothesis holds but the conclusion fails disproves a universal statement; existence statements are proved by producing one example.

Equivalence proofs require two separate implications: prove P implies Q and Q implies P independently, and label each direction clearly.

Negation rules: the negation of 'for all x, P(x)' is 'there exists x such that not P(x)'; the negation of 'P and Q' is 'not P or not Q' by De Morgan's laws.

Irrationality template: assume sqrt(n) = a/b in lowest terms, square to get n b^2 = a^2, deduce a prime divisor of n divides both a and b, contradicting coprimality.

Parity and divisibility notation: a | b means b = ak for some integer k; use a = 2k or a = 2k+1 for even/odd cases, and remember a^2 has the same parity as a.

A proof must not assume the conclusion; avoid circular reasoning and check that each variable introduced is quantified (for all, there exists) exactly once.

Dot product: a·b = |a||b|cos(theta) = a1b1 + a2b2 + a3b3; perpendicular vectors satisfy a·b = 0, and the projection of a onto b is (a·b/|b|^2) b.

Cross product: a x b is perpendicular to both a and b, with |a x b| = |a||b|sin(theta) equal to the area of the parallelogram spanned by a and b.

Determinant formula: a x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1); the scalar triple product a·(b x c) gives the signed volume of the parallelepiped.

Line in vector form: r = a + t d, where a is a point on the line and d is a direction vector; two lines are skew if they are not parallel and never intersect.

Plane equation: r·n = d with normal n, or ax + by + cz = d; the distance from point P to a plane is |(P - A)·n|/|n| for any point A on the plane.

Angle between vectors: cos(theta) = (a·b)/(|a||b|); the angle between a line and a plane is the complement of the angle between the line's direction and the plane's normal.

Vector equation of a sphere: |r - c| = R, or (x - h)^2 + (y - k)^2 + (z - l)^2 = R^2; the midpoint of AB is (a + b)/2 and divides segments in given ratios via section formula.

Scalar projections and components: the component of a in the direction of the unit vector u is a·u, and a vector is a unit vector exactly when its magnitude equals 1.

Cartesian and polar form: z = x + iy = r(cos(theta) + i sin(theta)) with r = |z| = sqrt(x^2 + y^2) and arg z = atan2(y, x), taken in (-pi, pi].

Euler's formula: e^(i theta) = cos(theta) + i sin(theta), so z = r e^(i theta); multiplication multiplies moduli and adds arguments, division divides moduli and subtracts arguments.

De Moivre's theorem: (cos(theta) + i sin(theta))^n = cos(n theta) + i sin(n theta) for integer n; use it to expand cos(n theta) and sin(n theta) in powers of cos and sin.

Conjugate properties: z + conj(z) = 2 Re(z), z conj(z) = |z|^2, conj(z w) = conj(z) conj(w); a polynomial with real coefficients has non-real roots occurring in conjugate pairs.

Roots of unity: the n-th roots of unity are e^(2 pi i k/n) for k = 0, 1, ..., n-1; they sum to zero and the product of all n roots equals (-1)^(n+1) times the constant term ratio.

Triangle inequality: |z + w| <= |z| + |w|, with equality when the arguments of z and w are equal; also |z - w| is the distance between points z and w in the Argand plane.

Loci: |z - a| = |z - b| is the perpendicular bisector of the segment joining a and b; |z - a| = r is a circle; arg((z - a)/(z - b)) = alpha is an arc of a circle.

Quadratic formula over C: z = (-b +/- sqrt(b^2 - 4ac))/(2a) with complex square roots; factorise cubics using the conjugate-root theorem and polynomial division.

Inverse trigonometric derivatives: d/dx arcsin x = 1/sqrt(1 - x^2), d/dx arccos x = -1/sqrt(1 - x^2), d/dx arctan x = 1/(1 + x^2), valid on the interior of each domain.

Implicit and parametric differentiation: dy/dx = (dy/dt)/(dx/dt) for parametric curves, and the second derivative is d/dt(dy/dx) divided by dx/dt.

Integration by parts: integral of u dv = uv - integral of v du; choose u by the LIATE order (logarithmic, inverse trig, algebraic, trig, exponential) to simplify the remaining integral.

Trigonometric substitutions: use x = a sin(theta) for sqrt(a^2 - x^2), x = a tan(theta) for a^2 + x^2, and x = a sec(theta) for sqrt(x^2 - a^2), then convert back using a right triangle.

Partial fractions: decompose proper rational functions into linear and irreducible quadratic factors before integrating; repeated factors need terms for every power up to the multiplicity.

Volumes of revolution: V = pi integral of y^2 dx about the x-axis, and V = 2 pi integral of x y dx (shells) about the y-axis; arc length is integral of sqrt(1 + (dy/dx)^2) dx.

Improper integrals: replace the infinite limit or singularity by a parameter and take a limit; the integral converges only if that limit exists and is finite.

Standard results: integral of 1/(x^2 + a^2) dx = (1/a) arctan(x/a) + C, and integral of 1/sqrt(a^2 - x^2) dx = arcsin(x/a) + C for a > 0.

Newton's second law in vector form: F_net = m a = m dv/dt = m d^2x/dt^2; resolve forces along and perpendicular to the direction of motion before integrating.

Motion with variable acceleration: a = dv/dt = v dv/dx; the second form is used when acceleration is given as a function of displacement x rather than time t.

Simple harmonic motion: x = A cos(omega t + phi) with a = -omega^2 x; period T = 2 pi/omega, and maximum speed is A omega occurring at the equilibrium position.

Projectile motion with no air resistance: horizontal velocity is constant, vertical acceleration is -g; range R = v^2 sin(2 theta)/g on level ground.

Resisted motion: for resistance proportional to speed, m dv/dt = -k v gives exponential decay of speed; for resistance proportional to v^2, use dv/dt = -k v^2 and integrate 1/v^2.

Circular motion: centripetal acceleration is v^2/r = r omega^2 directed toward the centre; the net force toward the centre equals m v^2/r.

Momentum and impulse: impulse equals the change in momentum, integral of F dt = m v_final - m v_initial; total momentum is conserved in collisions with no external force.

Equilibrium of a particle: the vector sum of all forces is zero; resolve into perpendicular components and solve the resulting simultaneous equations.

Arithmetic sequence: t_n = a + (n - 1)d and S_n = (n/2)(2a + (n - 1)d); geometric sequence: t_n = a r^(n-1) and S_n = a(r^n - 1)/(r - 1) for r not equal to 1.

Sum of an infinite geometric series converges to a/(1 - r) exactly when |r| < 1; otherwise the series diverges.

Sigma notation: sum from k = 1 to n of k = n(n + 1)/2, sum of k^2 = n(n + 1)(2n + 1)/6, and sum of k^3 = [n(n + 1)/2]^2.

Telescoping series: write each term as a difference f(k) - f(k + 1) so intermediate terms cancel, leaving only the first and last terms.

Method of differences and partial fractions: decompose the general term into partial fractions before summing, then evaluate the limit of the partial sums to test convergence.

Ratio test for series: if the limit of |t_(n+1)/t_n| is less than 1 the series converges absolutely, greater than 1 it diverges, and equal to 1 the test is inconclusive.

Recurrence relations: a linear recurrence t_n = a t_(n-1) + b has a closed form found by solving the characteristic equation or by finding a fixed point and substituting u_n = t_n - L.

Binomial series for non-integer index: (1 + x)^n = 1 + nx + n(n-1)x^2/2! + ..., convergent for |x| < 1 when n is not a positive integer.

Binomial theorem: (a + b)^n = sum from k = 0 to n of C(n, k) a^(n-k) b^k, where C(n, k) = n!/(k!(n-k)!); the expansion has n + 1 terms.

General term: T_(k+1) = C(n, k) a^(n-k) b^k; to find the coefficient of a specific power, set the exponent of the variable equal to that power and solve for k.

Pascal's identity: C(n, k) = C(n-1, k-1) + C(n-1, k); each interior entry of Pascal's triangle is the sum of the two entries above it.

Symmetry and boundary values: C(n, k) = C(n, n-k), C(n, 0) = C(n, n) = 1; setting a = b = 1 gives the sum of all binomial coefficients equal to 2^n.

Alternating sums: setting a = 1 and b = -1 gives the sum of C(n, k)(-1)^k equal to 0 for n >= 1, so even and odd coefficient sums are equal.

Greatest coefficient: the ratio of consecutive coefficients C(n, k+1)/C(n, k) = (n - k)/(k + 1) determines where the coefficients increase and decrease, locating the maximum term.

Identities by differentiation: differentiating (1 + x)^n and setting x = 1 gives the sum of k C(n, k) equal to n 2^(n-1); integrating gives sums involving 1/(k + 1).

Approximation: for small x, (1 + x)^n is approximately 1 + nx, with error bounded by the next term n(n-1)x^2/2 when the series is alternating or rapidly decreasing.

Structure: prove the base case (usually n = 1), assume the statement true for n = k, then use that assumption to prove it true for n = k + 1, and conclude by the principle of induction.

Summation induction: in the inductive step write S_(k+1) = S_k + t_(k+1), substitute the assumed formula for S_k, and simplify algebraically to the target form with k replaced by k + 1.

Divisibility induction: express f(k+1) - f(k) as a multiple of the divisor, or write f(k+1) = f(k) + (a multiple of the divisor), then conclude divisibility.

Strong induction: assume the statement holds for all values up to k and prove it for k + 1; required when the (k+1)-th case depends on several earlier cases.

Inequality induction: prove the base case, then multiply or add the inductive assumption by a positive quantity and show the result exceeds or falls below the required bound.

Induction with a general starting point: the base case may be n = m for any integer m, and the conclusion then holds for all integers n >= m.

Common error: the inductive step must use the assumption for n = k; proving the case n = k + 1 independently does not establish the statement for all n.

Recurrence induction: for sequences defined by a recurrence, verify the closed form for enough initial terms and then substitute both the recurrence and the assumed form in the inductive step.