HSC Mathematics Extension 2
HSC Mathematics Extension 2
Topic 1: Graphs
1.1 Basic Curves, linear, quadratic, cubic, quartic, rectangular hyperbola, circle, exponential, logarithmic, trigonometric, inverse trigonometric, roots
1.2 Drawing graphs by addition and subtraction of ordinates y = f (x) ± g (x)
1.3 Drawing graphs by reflecting functions in coordinate axes –f (x), |f (x)|, f (–x)
1.4 Sketching functions by multiplication of ordinates y = f(x) · g(x)
1.5 Sketching functions by division of ordinates y = f(x)/g(x)
1.6 Drawing graphs of the form exponencial [f (x)]n
1.7 Drawing graphs of the form roots vf (x)
1.8 General approach to curve sketching, use implicit differentiation, use the most appropriate method to graph
1.9 Using graphs: solve an inequality, find the number of solutions of an equation
Topic 2: Complex Numbers
2.1 Arithmetic of complex numbers and solving quadratic equations, complex number z = x + iy. Re(z), Im(z), add, subtract and multiply, divide, conjugate, solve quadratic equations
2.2 Geometric representation of a complex number as a point: Argand diagram, modulus (| z |) and argument (arg z),
2.3 Geometrical representations of a complex number as a vector on an Argand diagram,
2.4 Powers and roots of complex numbers: prove (cos q + i sin q)n = cos nq + i sin nq, sketch the nth roots
2.5 Curves and Regions: | z – z1 | = | z – z2 |, | z – z1 | = R Circles,
Topic 3: Conics
3.1 The Ellipse: x2/a2+y2/b2=1,auxiliary circle, the foci and equations of the directrices, the tangent and the normal, a chord, prove that the sum of the focal lengths is constant
3.2 The Hyperbola: x2/a2-y2/b2=1, length of major and minor axes and semi-major and semi-minor axes, focus-directrix, foci and directrices, the tangent and the normal, achord, the difference of the focal lengths, reflection property
3.3 The Rectangular Hyperbola: xy=1/2a2, vertices, foci, directrices and asymptotes, tangent, normal, intersection
3.4 General descriptive properties of conics: various conic sections (circle, ellipse, parabola, hyperbola
and pairs of intersecting lines), eccentricity e
Topic 4: Integration
4.1 Integration: using algebraic, trigonometric substitutions, integration by parts, recurrence relations, the square in a quadratic denominator
Topic 5: Volumes
5.1 Volumes: dividing a solid into a number of slices or shells, volume of a solid of revolution,
Topic 6: Mechanics
6.1 Mathematical Representation of a motion described in physical terms: motion of a projectile, displacement, velocity and acceleration, simple harmonic motion, Newton’s laws,
6.1.2 Physical explanations of mathematical descriptions of motion: function of second of derivative, resultant motion
6.2 Resisted motion
6.2.1 Resisted Motion along a horizontal line: under a resistance proportional to a power of the speed, velocity as a function of time, velocity as a function of displacement, displacement as a function of time
6.2.2 Motion of a particle moving upwards in a resisting medium and under the influence of gravity: resistance R proportional to the first or second power of its speed, velocity as a function of displacement (or vice versa)
6.2.3 Motion of a particle falling downwards in a resisting medium and under the influence of gravity: solve problems by using the expressions derived for acceleration, velocity and displacement.
6.3 Circular Motion
6.3.1 Motion of a particle around a circle: angular velocity w, instantaneous velocity, prove that the tangential and normal components of the force
6.3.2 Motion of a particle moving with uniform angular velocity around a circle: the formulae for a particle moving around a circle with uniform angular velocity
6.3.3 The bob of Conical Pendulum: tan q = v2/ag and h=g/w2
6.3.4 Motion around a banked circular track: calculate the optimum speed
Topic 7: Polynomials
7.1 Integer roots of polynomials with integer coefficients: prove that, if a polynomial has integer coefficients and if a is an integer root, then a is a divisor of the constant term
7.2 Multiple Roots: prove that if P(x) = (x – a)rS(x), where r > 1 and S(a) 0, then P'(x) has a root a of multiplicity (r – 1)
7.3 Fundamental Theorem of Algebra: deduce that a polynomial of degree n > 0, with real or complex coefficients, has exactly n complex roots, allowing for multiplicities.
7.4 Factoring Polynomials: recognise that a complex polynomial of degree n can be written as a product of n complex linear factors, factor a real polynomial into a product of real linear and real quadratic factors
7.5 Roots and Coefficients of a Polynomial Equation: write down the relationships between the roots and coefficients of polynomial equations of degrees 2, 3 and 4. multiple, reciprocals,
7.6 Partial Fractions: write f (x) = A(x)/B(x) , where deg A(x)>= deg B(x), in the form f(x) = Q(x) + R(x)/B(x), where deg R(x) < deg B(x)
Topic 8: Harder 3 Unit
8.1 Geometry of the Circle: solve more difficult problems in geometry.
8.2 Induction: carry out proofs by mathematical induction in which S(1), S(2)…S(k) are assumed to be true in order to prove S(k + 1) is true
8.3 Inequalities: prove simple inequalities by use of the definition of a > b for real a and b
Sample of Past Paper:
Topic 1: Graphs
1.1 Basic Curves, linear, quadratic, cubic, quartic, rectangular hyperbola, circle, exponential, logarithmic, trigonometric, inverse trigonometric, roots
1.2 Drawing graphs by addition and subtraction of ordinates y = f (x) ± g (x)
1.3 Drawing graphs by reflecting functions in coordinate axes –f (x), |f (x)|, f (–x)
1.4 Sketching functions by multiplication of ordinates y = f(x) · g(x)
1.5 Sketching functions by division of ordinates y = f(x)/g(x)
1.6 Drawing graphs of the form exponencial [f (x)]n
1.7 Drawing graphs of the form roots vf (x)
1.8 General approach to curve sketching, use implicit differentiation, use the most appropriate method to graph
1.9 Using graphs: solve an inequality, find the number of solutions of an equation
Topic 2: Complex Numbers
2.1 Arithmetic of complex numbers and solving quadratic equations, complex number z = x + iy. Re(z), Im(z), add, subtract and multiply, divide, conjugate, solve quadratic equations
2.2 Geometric representation of a complex number as a point: Argand diagram, modulus (| z |) and argument (arg z),
2.3 Geometrical representations of a complex number as a vector on an Argand diagram,
2.4 Powers and roots of complex numbers: prove (cos q + i sin q)n = cos nq + i sin nq, sketch the nth roots
2.5 Curves and Regions: | z – z1 | = | z – z2 |, | z – z1 | = R Circles,
Topic 3: Conics
3.1 The Ellipse: x2/a2+y2/b2=1,auxiliary circle, the foci and equations of the directrices, the tangent and the normal, a chord, prove that the sum of the focal lengths is constant
3.2 The Hyperbola: x2/a2-y2/b2=1, length of major and minor axes and semi-major and semi-minor axes, focus-directrix, foci and directrices, the tangent and the normal, achord, the difference of the focal lengths, reflection property
3.3 The Rectangular Hyperbola: xy=1/2a2, vertices, foci, directrices and asymptotes, tangent, normal, intersection
3.4 General descriptive properties of conics: various conic sections (circle, ellipse, parabola, hyperbola
and pairs of intersecting lines), eccentricity e
Topic 4: Integration
4.1 Integration: using algebraic, trigonometric substitutions, integration by parts, recurrence relations, the square in a quadratic denominator
Topic 5: Volumes
5.1 Volumes: dividing a solid into a number of slices or shells, volume of a solid of revolution,
Topic 6: Mechanics
6.1 Mathematical Representation of a motion described in physical terms: motion of a projectile, displacement, velocity and acceleration, simple harmonic motion, Newton’s laws,
6.1.2 Physical explanations of mathematical descriptions of motion: function of second of derivative, resultant motion
6.2 Resisted motion
6.2.1 Resisted Motion along a horizontal line: under a resistance proportional to a power of the speed, velocity as a function of time, velocity as a function of displacement, displacement as a function of time
6.2.2 Motion of a particle moving upwards in a resisting medium and under the influence of gravity: resistance R proportional to the first or second power of its speed, velocity as a function of displacement (or vice versa)
6.2.3 Motion of a particle falling downwards in a resisting medium and under the influence of gravity: solve problems by using the expressions derived for acceleration, velocity and displacement.
6.3 Circular Motion
6.3.1 Motion of a particle around a circle: angular velocity w, instantaneous velocity, prove that the tangential and normal components of the force
6.3.2 Motion of a particle moving with uniform angular velocity around a circle: the formulae for a particle moving around a circle with uniform angular velocity
6.3.3 The bob of Conical Pendulum: tan q = v2/ag and h=g/w2
6.3.4 Motion around a banked circular track: calculate the optimum speed
Topic 7: Polynomials
7.1 Integer roots of polynomials with integer coefficients: prove that, if a polynomial has integer coefficients and if a is an integer root, then a is a divisor of the constant term
7.2 Multiple Roots: prove that if P(x) = (x – a)rS(x), where r > 1 and S(a) 0, then P'(x) has a root a of multiplicity (r – 1)
7.3 Fundamental Theorem of Algebra: deduce that a polynomial of degree n > 0, with real or complex coefficients, has exactly n complex roots, allowing for multiplicities.
7.4 Factoring Polynomials: recognise that a complex polynomial of degree n can be written as a product of n complex linear factors, factor a real polynomial into a product of real linear and real quadratic factors
7.5 Roots and Coefficients of a Polynomial Equation: write down the relationships between the roots and coefficients of polynomial equations of degrees 2, 3 and 4. multiple, reciprocals,
7.6 Partial Fractions: write f (x) = A(x)/B(x) , where deg A(x)>= deg B(x), in the form f(x) = Q(x) + R(x)/B(x), where deg R(x) < deg B(x)
Topic 8: Harder 3 Unit
8.1 Geometry of the Circle: solve more difficult problems in geometry.
8.2 Induction: carry out proofs by mathematical induction in which S(1), S(2)…S(k) are assumed to be true in order to prove S(k + 1) is true
8.3 Inequalities: prove simple inequalities by use of the definition of a > b for real a and b
Sample of Past Paper: