Maths Extension 2 flashcards

103 HSC Maths Extension 2 cards across 5 modules (11 definitions, 40 key facts, 13 fill-the-gap, 39 formulas). Each session gives you 20 cards, due cards first. Swipe right for easy, left for hard, down for again. Keyboard: Space to flip, 1 to 4 to rate.

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Definition
Complex number arithmetic and the Argand plane in HSC Mathematics Extension 2: Cartesian and polar form, modulus, argument, conjugate and geometric interpretation
Define Conjugation
$z \mapsto \bar z$ reflects the point in the real axis (it negates the imaginary part, and so negates the argument).
Fill the gap
Trigonometric integrals in HSC Mathematics Extension 2: powers of sine and cosine (odd and even strategies), powers of tangent and secant, the integral of sec x and sec cubed x, and products of sines and cosines via product-to-sum identities
When the integrand is a product of trigonometric functions of ___ multiples of $x$ , such as $\sin 3x\cos 2x$ or $\cos 5x\cos 3x$ , there is no useful Pythagorean factor to peel.
When the integrand is a product of trigonometric functions of different multiples of $x$ , such as $\sin 3x\cos 2x$ or $\cos 5x\cos 3x$ , there is no useful Pythagorean factor to peel.
Definition
Forces and Newton's laws in HSC Mathematics Extension 2: resolving weight, tension, normal and resistive forces, the resultant of concurrent forces, equilibrium, building the equation of motion from a free-body diagram, and the conical pendulum
Define First law (inertia)
A body stays at rest, or moves in a straight line at constant velocity, unless a resultant force acts on it. So constant velocity, including being at rest, means zero resultant force. This is exactly the condition for equilibrium.
Definition
Further proof by induction in HSC Mathematics Extension 2: series, divisibility, inequalities, strong induction with multiple base cases, and where ordinary induction fails
Define Statements proved by going down, not up
Occasionally it is cleaner to assume the result fails and take a smallest counterexample, then construct a smaller one, a contradiction. This "least counterexample" argument is logically equivalent to strong induction and is sometimes the neatest route.
Definition
Vector geometric proofs in HSC Mathematics Extension 2: proving collinearity, perpendicularity, midpoint and parallelogram properties, and properties of 3D figures using position vectors and the scalar product
Define Equal and parallel
Equal and parallel ( $\overrightarrow{XY} = \overrightarrow{UV}$ ): same direction and length, the test for a parallelogram.
Fill the gap
Curves and regions in the complex plane in HSC Mathematics Extension 2: loci from modulus and argument conditions, including circles, perpendicular bisectors and rays
Watch the sign: the centre is the value of $z_0$ that makes the bracket zero, which is the ___ sign to what appears after the minus.
Watch the sign: the centre is the value of $z_0$ that makes the bracket zero, which is the opposite sign to what appears after the minus.
Key fact
Integration by parts in HSC Mathematics Extension 2: the formula, choosing u and dv, repeated application, and the boomerang (recovery) technique
Tell me about Choosing the parts: LIATE
Whatever appears earlier in this list is usually the better choice for $u$ , and whatever is later becomes $\dfrac{dv}{dx}$ . The list is not a law of nature, it is a heuristic ranked by how the function behaves under differentiation.
Definition
Resisted motion in HSC Mathematics Extension 2: rectilinear motion under gravity with resistance proportional to velocity or velocity squared, and terminal velocity
Define Terminal velocity
Terminal velocity is whatever speed makes the acceleration zero: set $\dot{v} = 0$ and solve, giving $v_T = \dfrac{mg}{k}$ for linear resistance and $v_T = \sqrt{\dfrac{mg}{k}}$ for the square law. It is approached but never reached.
Definition
The nature of proof in HSC Mathematics Extension 2: direct proof, proof by contradiction, contrapositive and disproof by counterexample
Define Contrapositive
$\lnot Q \implies \lnot P$ . The contrapositive is logically equivalent to $P \implies Q$ : the two are true in exactly the same situations, so proving one proves the other.
Fill the gap
Three-dimensional vectors, lines and spheres in HSC Mathematics Extension 2: component form, magnitude, the scalar product, vector equations of lines and the equation of a sphere
The point $\mathbf{a}$ may be ___ point on the line and $\mathbf{d}$ any non-zero multiple of the direction, so a line has infinitely many vector equations.
The point $\mathbf{a}$ may be any point on the line and $\mathbf{d}$ any non-zero multiple of the direction, so a line has infinitely many vector equations.