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NSW HSC Mathematics Extension 2

HSC · NESAMathematics Extension 2359 cards

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Flashcards for the NSW HSC Mathematics Extension 2 11–12 Syllabus (2024, first examined 2027), covering the Year 12 content it adds beyond Mathematics Advanced and Extension 1: proof, vectors, complex numbers, further integration and mechanics, split into topics in the syllabus's order. Each formula card says whether it is on the HSC reference sheet.

Adding it gives you your own copy, with every subdeck below. Each card then comes back just before you would forget it, and every one you get right or wrong counts towards your mastery of its topic. You can delete or suspend the parts you are not studying once it is yours.

What is inside (27 subdecks)

  • 1. Proof76 cards
    • 1.1 The language and notation of proof28 cards
    • 1.2 Illustrations of proof10 cards
    • 1.3 Proof of inequalities22 cards
    • 1.4 Further proof by mathematical induction16 cards
  • 2. Vectors49 cards
    • 2.1 Vector equations of lines and curves28 cards
    • 2.2 Vectors and geometry21 cards
  • 3. Complex numbers96 cards
    • 3.1 Arithmetic of complex numbers26 cards
    • 3.2 Geometric representation of complex numbers20 cards
    • 3.3 Solving equations with complex numbers11 cards
    • 3.4 Powers and roots of complex numbers25 cards
    • 3.5 Describing lines, curves and regions14 cards
  • 4. Calculus: further integration51 cards
    • 4.1 Trigonometric products as sums10 cards
    • 4.2 The t-formulas8 cards
    • 4.3 Integration by substitution8 cards
    • 4.4 Partial fractions and other rational integrals13 cards
    • 4.5 Integration by parts and reduction formulae12 cards
  • 5. Mechanics87 cards
    • 5.1 Forces and further motion in a straight line18 cards
    • 5.2 Simple harmonic motion21 cards
    • 5.3 Modelling motion without resistance13 cards
    • 5.4 Rectilinear resisted motion13 cards
    • 5.5 Vertical resisted motion12 cards
    • 5.6 Projectiles and resisted motion10 cards

Some of its cards

  • Derive the identity for cos⁡Acos⁡B\cos A \cos B as a sum, starting from cos⁡(A−B)\cos(A-B) and cos⁡(A+B)\cos(A+B).
    cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A-B)=\cos A\cos B+\sin A\sin B and cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B)=\cos A\cos B-\sin A\sin B. Adding gives cos⁡(A−B)+cos⁡(A+B)=2cos⁡Acos⁡B\cos(A-B)+\cos(A+B)=2\cos A\cos B, so cos⁡Acos⁡B=12(cos⁡(A−B)+cos⁡(A+B))\cos A\cos B = \frac{1}{2}\left(\cos(A-B)+\cos(A+B)\right)
  • What is the tt-formula for sin⁡A\sin A in terms of t=tan⁡A2t = \tan\dfrac{A}{2}?
    sin⁡A=2t1+t2\sin A = \dfrac{2t}{1+t^2}
  • What is a direction vector of a straight line?
    Any non-zero vector parallel to the line
  • What shape is the trajectory of a projectile under gravity alone, with no resistance?
    A parabola, symmetric about its highest point
  • How is the inequality a>ba > b defined for real numbers aa and bb?
    a>ba > b if and only if a−b>0a - b > 0
  • What is the structure of a proof by contradiction?
    Assume the statement is false, then show this assumption leads to something impossible
  • What is the general method of integration by substitution?
    Replace part of the integrand with a new variable uu, rewrite dxdx in terms of dudu, then integrate with respect to uu before converting back
  • What does the equation Re⁡(z)=c\operatorname{Re}(z) = c describe on the complex plane, for a real constant cc?
    A vertical line through (c,0)(c,0)
  • Besides sums and divisibility results, what other kinds of identities can mathematical induction prove?
    Trigonometric, logarithmic, exponential and polynomial identities, including ones involving the binomial theorem
  • Derive the formula for integration by parts, starting from the product rule ddx(uv)=udvdx+vdudx\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx}+v\dfrac{du}{dx}.
    Integrating both sides gives uv=∫udvdx dx+∫vdudx dxuv = \int u\dfrac{dv}{dx}\,dx + \int v\dfrac{du}{dx}\,dx, so ∫udvdx dx=uv−∫vdudx dx\int u\dfrac{dv}{dx}\,dx = uv - \int v\dfrac{du}{dx}\,dx

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