mathematics · • apply straight forward application of ... find the x and y intercepts of graphs...
TRANSCRIPT
2017 SACAI WINTER SCHOOL
MATHEMATICS
NOTES
Use the following list to tick what skills and knowledge you need to be competent in for the upcoming exam:
If you are working only to pass Mathematics only focus on Level 1 and 2.
If you are working to get more than 50% in the exam you will have to be competent with Level 1 and 2 FIRST,
then only focus on level 3.
Do not fool yourself: if you are not competent in level 1 and 2, working on level 3 is futile!
Paper 1
Algebra ±17% Skill Know Not Sure No clue
Level 1: I know how to Identify Number sets
Round real numbers to an appropriate degree of accuracy.
Multiply binomials
Factorise common factor and simple difference of two squares
factorisation.
Simple application of Exponential laws (including negative exponents)
solve one or two step linear equations.
Solving quadratic equations with simple factorization.
Simple applications of rational exponents.
Solving quadratic equations using the quadratic formula
Calculating the discriminant
Identify the number of solutions for any equation
Converting an expression from exponential form to logarithmic form and vice versa.
Level 2: I know how to • Simplify expressions with surds within a few steps.
• Rationalise 1 term denominators
• Complete the square
• apply straight forward application of the discriminant (nature of roots)
• Solve simultaneous equation with two unknowns.
• Solve quadratic inequalities.
• Rationalize denominators
• Solve equations that simplifies to quadratic equations
• Solve quadratic simultaneous equations
• Solve inequalities that simplify to quadratic inequalities
• Solve exponential equation that only requires same bases on either side
• Solve an unknown in the exponent of a standard equation that requires the use of logarithms.
Level 3: I know how to • Simplify algebraic fractions with exponential expressions that requires
factorization
• Unusual application of surds
• Solve equation with surds
• Rationalize two term denominators
• Use completing the square to produce a desired format (as in the case of the parabola vertex form)
• apply the nature of roots in unusual formats
• solve equations with fractions that simplify to quadratic equations.
• solve unusual quadratic equations.
• solving unusual simultaneous equations (for example exponential simultaneous equations)
• solve simultaneous equations graphically
• solve inequalities with an unknown in the denominator
• simplify algebraic fractions with exponential expressions that requires factorization
• simplify expressions with logarithms that require factorization.
• Equations that simplify to a format that requires logarithms to solve.
Functions and Graphs ±23% Skill Know Not Sure No clue
Level 1: I know how to Plot Points
Filling in table for functions
write down the asymptotes of given functions
write down the domain or range of given functions
find the x and y intercepts of graphs
find the equation of a function with 1 unknown parameter.
define a function
Read information off of a graph
Find the inverse function of functions of the form
o 𝑓(𝑥) = 𝑏𝑥
Find the equation of a line
Level 2: I know how to Draw graphs based on the effect of any two parameters at a time
o 𝑦 = 𝑎(𝑥 − 𝑝)2 + 𝑞
o 𝑦 =𝑎
𝑥−𝑝+ 𝑞
o 𝑦 = 𝑐𝑏𝑥−𝑝 + 𝑞 where 𝑏 > 0; 𝑏 ≠ 1; 𝑐 ≠ 0
o 𝑦 = 𝑎 sin(𝑘𝑥 − 𝑝) + 𝑞
o 𝑦 = 𝑎 cos(𝑘𝑥 − 𝑝) + 𝑞
o 𝑦 = log𝑎(𝑥)
Find the 𝑥 and 𝑦 intercepts of graphs
Find the equation of a function with 1 unknown parameter.
Find the Turning point of a parabola
Find the intersection of the asymptotes for a hyperbola
Find the equation of a parabola given
o the Turning Point and another point,
o the roots and another point
Find the average gradient between two points on a curve
Solve for 𝑓(𝑥) = 𝑔(𝑥) for simple standard functions
Show that a function is one-to-one
Find the inverse of basic functions
o 𝑦 = 𝑎𝑥 + 𝑞
o 𝑦 = 𝑎𝑥2
Sketch the functions and their inverses on the same set of axis.
Level 3: I know how to Sketch multiple functions on one set of axis given at least three non-
standard parameters
Find the equation of a parabola in a form that is different from the form that should be used with the given information.
Finding vertical distances between points on two curves
Finding the 𝑥 values or intervals for the following algebraic relationship
o 𝑓(𝑥) = 𝑔(𝑥)
o 𝑓(𝑥) − 𝑔(𝑥) = 𝑐
o 𝑓(𝑥) × 𝑔(𝑥) > 0
o 𝑓(𝑥) × 𝑔(𝑥) < 0
Solve inequalities based on the graphs
Determine characteristics of a function produced from a transformation
o 𝑎𝑓(𝑥 − 𝑝) + 𝑞
o Reflections
Finding the equations of lines of symmetry given cryptic information
Evaluating and commenting on function properties and their graphs
Unusual inverse relations or functions
Calculus ±23% Skill Know Not Sure No clue
Level 1: I know how to Find the limit with straight forward substitution
Find the average gradient between two points on a function
Define the derivative of a function using limits
Find the value of a function at a certain point for a rate of change problem
Level 2: I know how to Find the limit of an expression that requires simplification first
Find the derivative of a function from first principles
o 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐
o 𝑦 = 𝑎𝑥3
o 𝑦 =𝑎
𝑥
o 𝑦 = 𝑐
Find the average gradient between two points on a function
Find the instantaneous gradient at a point using first principles
Find the derivative of polynomial functions (may require simplification)
Find the instantaneous gradient at a given point
Determine the equation of the tangent line to a simple polynomial function
Find the x-intercepts of polynomial functions
Find the turning points of a polynomial function
Draw a polynomial with calculated or given information
write certain aspect of the scenario in terms of the independent variable. (Calculus optimization application)
Find the rate of change of a function at a certain point. (Calculus rate of change application)
Level 3: I know how to
Find the derivative of expression with radicals or fractional exponents
Unusual expressions to be derived
Finding the intervals on which functions have positive or negative gradients
Find the equation of a tangent to a function that requires simplification first or only cryptic information is given.
Finding the roots of a polynomial function
Solve intervals problems related to polynomials and derivatives
For which values of 𝑘 will 𝑓(𝑥) = 𝑘 have o one root o two roots o three roots
show the expression that is to be optimized. (expression given). (Calculus optimization application)
comment on the rate of change at a certain point (Calculus rate of change application)
Number Patterns ±17% Skill Know Not Sure No clue
Level 1: I know how to
Recognise or show that a sequence or series is
o Arithmetic sequence/series
o Geometric Sequence/series
o Quadratic Sequence
how to find the next (few) term(s) in arithmetic, geometric and quadratic sequences and series.
how to find the 𝑛𝑡ℎ term of a Arithmetic and geometric sequences and series.
how to find a term number in the arithmetic sequence or series
all the formulas for sequences and series
how to evaluating whether or not a standard series will converge or diverge.
Level 2: I know how to Find the 𝑛𝑡ℎ term of a Quadratic number pattern given the first four
terms.
Find a term's number in a quadratic number pattern if the general term is already known or calculated.
Find the sum of the first n terms in an arithmetic or geometric series given routine information
Find the sum to infinite of the converging geometric series given routine information.
Find a single unknown value in a known sequence or series.
Find the general term or a specific term of a known type of sequence, given information about other terms.
Finding the number of terms in a sequence or series given the last term
Evaluating a sum notation
Finding an interval for an unknown in the constant ratio for a converging geometric series where the constant ratio is straight forward.
Level 3: I know how to find a term's number in a linear number pattern, given the first few terms.
find an arithmetic sequence or series with unknowns making up the pattern.
use inequalities to find a solution
find a term's number in a quadratic number pattern.
Find quadratic sequences with unknowns making up the pattern.
use inequalities in quadratic sequences to find a solution.
find the general term, specific term or sum of terms of a known type of series given cryptic information about the terms or the sums of terms.
find a single unknown value in an unknown type of sequence or series.
find specific information about a sequence given 𝑆𝑛
find the number of terms in a series given the sum of the series.
write a series in sum notation give the minimal information (first few terms and final term).
find an unknown in the terms of a quadratic sequence.
find an interval for an unknown in the constant ratio for a converging geometric series where the constant ratio is not straight forward or the compound inequality is slightly more complex.
apply this knowledge in unusual number patterns
Financial Mathematics ±10% Skill Know Not Sure No clue
Level 1: I know all the formulas
how to solve the future, present value in standard growth and decay problems.
How to Solve 𝑖 and 𝑛 in the simple growth and decay problems.
How to calculate the effective interest rate
Level 2: I know how to Solve 𝑖 in the compound growth and decay problems
Calculate the future value, present value or recurring payment for regular annuities.
Solve 𝑛 in the compound growth and decay problems.
Level 3: I know how to find the unknowns of scenarios with multiple changes during the
investment period.
find the outstanding balance on a loan
changes during the investment period
derive new formulas given different information
use the basics in unusual scenarios
Probability ±10% Skill Know Not Sure No clue
Level 1: I know what is a frequency table
what is a contingency table
what is cardinality.
and understand what is probability
how to work out the probability in one step given the number of successes and total number of possibilities
what is means for events to be mutually exclusive
identify the mutually exclusive events from a description of the events
what it means for events to be complimentary
how to complete a frequency table
how to complete a contingency table
how to complete a Venn diagram
how to calculate probabilities of complimentary events in one step
how to find the relative frequency from a frequency table
how to calculate probabilities from a completed frequency tables
how to calculate probabilities from completed contingency tables
how to calculate probabilities from a completed Venn diagram
that for mutually exclusive events 𝑃(𝐴 and 𝐵) = 0
that 𝑃(𝐴 or 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 or 𝐵)
what is means for events to be independent
how to find the number of independent outcomes in a simple scenario
Level 2: I know how to draw a Venn diagram given all the information
use a Venn diagram to identify
o mutually exclusive events
o complimentary events
how to identify independent events with 𝑃(𝐴) × 𝑃(𝐵) = 𝑃(𝐴 and 𝐵) from
o given information about 𝑃(𝐴), 𝑃(𝐵) and 𝑃(𝐴 or 𝐵)
o a contingency table
use a Venn diagram to find the cardinality or probability
o A and B
o A or B
draw a tree diagram
find the number of dependent outcomes in simple scenarios
calculate the probability of an event occurring that requires simple application of counting principles
find the number of simple arrangements (of letters for examples)
find the probability of a certain simple arrangement occurring.
Level 3: I know how to I know how to find the probability of an event occurring from an
uncompleted
o frequency table
o contingency table
o Venn Diagram
o tree diagram
With limited information complete
o contingency table
o Venn diagram
calculate the number of dependent events (Especially word problems)
apply level 1 and level 2 knowledge and skills in real life scenarios
Paper 2 150 marks
Statistics ±13% Skill Know Not Sure No clue
Level 1: I know all the formulas
how to complete frequency Tables
how to complete cumulative frequency tables
how to calculate the mean from ungrouped data
how to calculate the median from ungrouped data
how to calculate the mode from ungrouped data
how to identify the modal interval for grouped date
how write down the five number summary
the difference between skewed to the left and skewed to the right and symmetric
how to calculate the range
how to calculate the variance using a calculator
how to calculate the standard deviation using a calculator
how to calculate the least square regression line using a calculator
how to calculate the correlation coefficient using a calculator
identify outliers in a scatter plot.
the difference between interpolation and extrapolation
Level 2: I know how to estimate the mean from ungrouped data
calculate the median from ungrouped data
identify the median interval for grouped data
calculate the quartiles
calculate the percentiles
calculate the interquartile range
calculate the semi-interquartile range
draw a box and whisker diagram
draw a histogram
draw a frequency polygon
draw an ogive (cumulative frequency curves)
identify outliers with a calculation
identify outliers using a box-and-whisker diagram
use an ogive to estimate quartiles
draw a scatter-plot
draw a least square regression line on a scatter plot
use the least square regression line to make an interpolated prediction
use the least square regression line to make an extrapolated prediction
Level 3: I know how to determine whether or not data is symmetric or skewed data using
measures of central tendency
determine whether or not data is symmetric or skewed using a box and whisker diagram
determine symmetric and skewed data using a histogram
determine symmetric and skewed data using an ogive
Find the percentage of observations that will lie within a given number of standard deviations from the mean
use statistical summaries and graphs to analyse and make meaningful comments on the context associated with given data, including discussions on skewness
use statistical summaries, scatterplots, regression and correlation to analyse and make meaningful comments on the context associated with given bivariate data, including interpolation, extrapolation and discussions on skewness.
Analytic Geometry ±27% Skill Know Not Sure No clue
Level 1: I know all the formulas
o distance formula
o gradient formula
o midpoint formula
o incline angle formula
how to calculate the distance between two coordinates
how to calculate the gradient between two coordinates
how calculate the Midpoint between two coordinates
how to identify lines that are parallel based on their gradients
how to identify lines that are perpendicular based on their gradient
how to find the equation of a circle if we are given the center and the radius
how to find the coordinates of the centre and the radius of a circle given the equation in standard format
Level 2: I know how to Find the equation of a line between two points
Find the equation of a line though one point and parallel to a given line
Find the equation of a line through one point and perpendicular to a given line
Find the inclination angle
o of a line
o of the line between two points
Find the equation of a circle given the centre and a coordinate on the circumference
Find the coordinates of the center and radius from the equation of a circle that is NOT in standard format.
Find the intersections of a line with a circle
Determine the equation of a tangent line to a circle (centre given) at a given point
Level 3: I know how to Use analytical geometry calculations to show that a set of coordinates
fits the minimum conditions for a certain polygon.
Find the angle between two lines that intersect.
Find the equation of a tangent to a circle not in standard format.
determine whether or not a point lies inside, outside of on a circle
determine whether or not two circles intersect in none, one or two points
Trigonometry ±27% Skill Know Not Sure No clue
Level 1: I know the definitions of
o sin 𝜃
o cos 𝜃
o tan 𝜃
Finding trigonometric ratios (including reciprocals) where information is directly available
Find the side length of a right angle triangle given
o an angle other than the right angle and a side
o two sides
the trigonometric ratios for
o 0°
o 30°
o 45°
o 60°
o 90°
that
o sin2 𝜃 + cos2 𝜃 = 1
o tan 𝜃 =sin 𝜃
cos 𝜃
how to apply the above identities
how to apply simple single step application of
o reduction formulae
o co-ratios
how to solve one step trigonometric equations for angles between 0° and 90°
how to find the angles of a right angle triangle given any two sides
what is the amplitude of a trigonometric function
what is the period of a function
how to find the range of a trigonometric function given the
o function expression
o graph
how to find the period of a trigonometric function
o function expression
o graph
how to draw each each of the following functions
o 𝑓(𝑥) = sin (𝑥)
o 𝑓(𝑥) = cos(𝑥)
o 𝑓(𝑥) = tan (𝑥)
know how to find the 𝑥 intercepts for the above functions given the
o function expression
o graph
Level 2: I know how to find simple trigonometric expressions given
o a different trigonometric ratio
o a point on the Cartesian plane
o an simple trigonometric equation
prove the following identities
o sin2 𝜃 + cos2 𝜃 = 1
o tan 𝜃 =sin 𝜃
cos 𝜃
o cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽
o cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽
o sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽
o sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽
o cos(2𝛼) = cos2 𝛼 − sin2 𝛼
o sin(2𝛼) = 2sin 𝛼 cos 𝛽
Simplifying standard trigonometric expressions using fundamental Identities, co-ratios, reduction formulas and/or double and compound identities.
Proving standard trigonometric identities using fundamental Identities, co-ratios, reduction formulas and/or compound and double angle identities.
Find the values for which an identity holds/does not hold.
Finding the General solution for standard trigonometric equations.
Finding the specific solution for standard trigonometric equations on a certain interval.
to prove the
o sine rule
o cosine rule
o area rule
apply the sine, cosine and area rules to find any angle or side given
o an angle and two sides
o two angles and a side
o three sides
how to draw each of the following functions (given any two of 𝑎, 𝑝, 𝑏 or 𝑐)
o 𝑓(𝑥) = 𝑎 sin(𝑝𝑥 − 𝑏) + 𝑐
o 𝑓(𝑥) = 𝑎 cos(𝑝𝑥 − 𝑏) + 𝑐
o 𝑓(𝑥) = 𝑎 tan(𝑝𝑥 − 𝑏) + 𝑐
know how to find the 𝑥 intercepts for the above functions given the
o function expression
o graph
know how to find the function expression for the above functions from a graph or given information
Level 3: I know how to simplify trigonometric expressions or prove trigonometric identities that
involves factorization
identify and apply inverse applications of the double and compound identities
simplify trigonometric equations of the form
o cos? ? ? = sin? ?
o that requires factorisation
o that first requires recognising the application of identities
apply trigonometry in other areas in the Mathematics syllabus.
apply the sine, cosine and area rules in problems that
o requires more than one step before applying the rule
o where mostly unknown variables are given rather than known value
o requires you to prove a certain expression
draw multiple trigonometric graphs on one set of axis
find the new equation for any trigonometric graph give a transformation
o vertical translations
o horizontal translations
o vertical enlargements
o horizontal enlargements
o vertical reflections
o horizontal reflections
Finding the points of intersections between two trigonometric graphs
Finding the 𝑥 values or intervals for the following algebraic relationship
o 𝑓(𝑥) = 𝑔(𝑥)
o 𝑓(𝑥) − 𝑔(𝑥) = 𝑐
o 𝑓(𝑥) × 𝑔(𝑥) > 0
o 𝑓(𝑥) × 𝑔(𝑥) < 0
Geometry ±33% Skill Know Not Sure No clue
Level 1: I know
all the definitions from Euclidean geometry
o circle
o chord
o radius
o diameter
o parallel lines
o vertically opposite angles
o co-interior angles
o alternate angles
o corresponding angles
o equilateral triangle
o isosceles triangle
o cyclic quadrilateral
o similar triangles
how to use fundamental Euclidean Geometry theorems
o angles on a straight line are supplementary
o vertically angles are equal
o co-interior angles are supplementary
o corresponding angles are equal
o alternative angles are equal
o interior angles of triangle supplementary
o base angles opposite equal sides are equal
o sides opposite equal angles are equal
the formulation of all the grade 11 and 12 theorems
o The line drawn from the centre of a circle perpendicular to a chord bisects the chord;
o The perpendicular bisector of a chord passes through the centre of the circle;
o The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
o Angles subtended by a chord of the circle, on the same side of the chord, are equal;
o Angles opposite equal chords are equal
o Angle in a semi-circle is a right angle
o The opposite angles of a cyclic quadrilateral are supplementary;
o Exterior angle of a cyclic quadrilateral is equal to the opposite interior angle
o Two tangents drawn to a circle from the same point outside the circle are equal in length;
o The radius is perpendicular to the tangent line
o The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
o that a line drawn parallel to one side of a triangle divides the other two sides proportionally (and the Midpoint Theorem as a special case of this theorem);
o that equiangular triangles are similar;
o that triangles with sides in proportion are similar; and
o A perpendicular drawn from a right angle on the hypotenuse divides the triangle into three similar triangles
How to apply the basic parallel theorems and triangle theorems
How to apply straight forward definition and theorems
How to fill in missing steps from proofs
Level 2: I know how to
apply results from prior grades to solve simple rider problems
prove two triangles are congruent if
o three corresponding sides are equal
o two corresponding angles and one corresponding side are equal
o two corresponding sides and the inclusive angle are equal
o they are right angle triangles with equal hypotenuses and another side equal
prove
o The line drawn from the centre of a circle perpendicular to a chord bisects the chord;
o The perpendicular bisector of a chord passes through the centre of the circle;
o The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre);
o Angles subtended by a chord of the circle, on the same side of the chord, are equal;
o Angle in a semi-circle is a right angle
o The opposite angles of a cyclic quadrilateral are supplementary;
o Exterior angle of a cyclic quadrilateral is equal to the opposite interior angle
o Two tangents drawn to a circle from the same point outside the circle are equal in length;
o The radius is perpendicular to the tangent line
o The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
o that a line drawn parallel to one side of a triangle divides the other two sides proportionally (and the Mid-point Theorem as a special case of this theorem) ;
o that equiangular triangles are similar;
o that triangles with sides in proportion are similar; and
o the Pythagorean Theorem by similar triangles.
Prove that two triangles are congruent
Prove that two triangles are similar
o by showing they are equiangular
o by showing that their sides are in proportion
Use the above theorems in to solve simple riders
Use the converse of the above theorems in straight forward application
find or prove the relationship between sides in similar triangles or triangles with parallel lines.
Level 3: I know how to
use a combination of theorems with at least one theorem from current circle theorems of similar triangle theorems
Solving riders where there is not an obvious path to the solution.
Level 2 procedures without a sketch given.
