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Skills available for Singapore higher 2 maths curriculum
IXL's higher 2 skills will be aligned to the Singapore Mathematics Syllabus soon! Until then, you can view a complete list of higher 2 objectives below.
Objectives are in black and IXL maths skills are in dark green. Hold your mouse over the name of a skill to view a sample question. Click on the name of a skill to practise that skill.
H2 Pure Mathematics
H2.1 Functions and graphs
H2.1.1 Functions, inverse functions and composite functions
Include:
H2.1.1.a concepts of function, domain and range
H2.1.1.b use of notations such as f(x) = x² + 5, f: x → x² + 5, f^{-1} (x) fg(x) and f²(x)
H2.1.1.c finding inverse functions and composite functions
H2.1.1.d conditions for the existence of inverse functions and composite functions
H2.1.1.e domain restriction to obtain an inverse function
H2.1.1.f relationship between a function and its inverse as reflection in the line y = x
Exclude:
H2.1.1.a the use of the relation (fg)^{-1} = g^{-1} f^{-1}
H2.1.2 Graphing techniques
Include:
H2.1.2.a use of a graphic calculator to graph a given function
H2.1.2.b relating the following equations with their graphs
H2.1.2.c x²/a² ± y²/b² = 1, y= (ax + b)/(cx +d), y = (ax²+bx+c)/(dx+e)
H2.1.2.d characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes
H2.1.2.e determining the equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y
H2.1.2.f effect of transformations on the graph of y = f(x) as represented by y = a f(x), y = f(x) + a, y = f(x + a) and y = f(ax), and combinations of these transformations
H2.1.2.g relating the graphs of y = |f(x)|, y = f(|x|), y = 1/f(x) and y² = f(x) to the graph of y = f(x)
H2.1.2.h simple parametric equations and their graphs
H2.1.3 Equations and inequalities
Include:
H2.1.3.a solving inequalities of the form f(x)/g(x)> 0 where f(x) and g(x) are quadratic expressions that are either factorisable or always positive
H2.1.3.b solving inequalities by graphical methods
H2.1.3.c formulating an equation or a system of linear equations from a problem situation
H2.1.3.d finding the numerical solution of equations (including system of linear equations) using a graphic calculator
H2.2 Sequences and series
H2.2.1 Summation of series
Include:
H2.2.1.a concepts of sequence and series
H2.2.1.b relationship between un (the nth term) and Sn (the sum to n terms)
H2.2.1.c sequence given by a formula for the nth term
H2.2.1.d sequence generated by a simple recurrence relation of the form x(n+1) = f(xn)
H2.2.1.e use of ∑ notation
H2.2.1.f summation of series by the method of differences
H2.2.1.g convergence of a series and the sum to infinity
H2.2.1.h binomial expansion of (1+ x)^{n} for any rational n
H2.2.1.i condition for convergence of a binomial series
H2.2.1.j proof by the method of mathematical induction
H2.2.2 Arithmetic and geometric series
Include:
H2.2.2.a formula for the n th term and the sum of a finite arithmetic series
H2.2.2.b formula for the n th term and the sum of a finite geometric series
H2.2.2.c condition for convergence of an infinite geometric series
H2.2.2.d formula for the sum to infinity of a convergent geometric series
H2.2.2.e solving practical problems involving arithmetic and geometric series
H2.3 Vectors
H2.3.1 Vectors in two and three dimensions
Include:
H2.3.1.a addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations
H2.3.1.b use of notations such as (x y), (x y z), xi + yj, xi + yj + zk, →AB, a
H2.3.1.c position vectors and displacement vectors
H2.3.1.d magnitude of a vector
H2.3.1.e unit vectors
H2.3.1.f distance between two points
H2.3.1.g angle between a vector and the x-, y- or z-axis
H2.3.1.h use of the ratio theorem in geometrical applications
H2.3.2 The scalar and vector products of vectors
Include:
H2.3.2.a concepts of scalar product and vector product of vectors
H2.3.2.b calculation of the magnitude of a vector and the angle between two directions
H2.3.2.c calculation of the area of triangle or parallelogram
H2.3.2.d geometrical meanings of |a.b| and |a×b|, where b is a unit vector
Exclude:
H2.3.2.a triple products a.b × c and a × b × c
H2.3.3 Three-dimensional geometry
Include:
H2.3.3.a vector and cartesian equations of lines and planes
H2.3.3.b finding the distance from a point to a line or to a plane
H2.3.3.c finding the angle between two lines, between a line and a plane, or between two planes
H2.3.3.d relationships between
H2.3.3.d.1 two lines (coplanar or skew)
H2.3.3.d.2 a line and a plane
H2.3.3.d.3 two planes
H2.3.3.d.4 three planes
H2.3.3.e finding the intersections of lines and planes
Exclude:
H2.3.3.a finding the shortest distance between two skew lines
H2.3.3.b finding an equation for the common perpendicular to two skew lines
H2.4 Complex numbers
H2.4.1 Complex numbers expressed in cartesian form
Include:
H2.4.1.a extension of the number system from real numbers to complex numbers
H2.4.1.b complex roots of quadratic equations
H2.4.1.c four operations of complex numbers expressed in the form (x + iy)
H2.4.1.d equating real parts and imaginary parts
H2.4.1.e conjugate roots of a polynomial equation with real coefficients
H2.4.2 Complex numbers expressed in polar form
Include:
H2.4.2.a complex numbers expressed in the form r(cosθ + isinθ) or r e to the iθ power, where r >0 and –π < θ ≤ π
H2.4.2.b calculation of modulus (r) and argument (θ) of a complex number
H2.4.2.c multiplication and division of two complex numbers expressed in polar form
H2.4.2.d representation of complex numbers in the Argand diagram
H2.4.2.e geometrical effects of conjugating a complex number and of adding, subtracting, multiplying, dividing two complex numbers
H2.4.2.f loci such as |z – c| ≤ r, |z – a| = |z – b| and arg(z – a) = α
H2.4.2.g use of de Moivre's theorem to find the powers and nth roots of a complex number
Exclude:
H2.4.2.a loci such as |z – a| = k| z – b|, where k ≠ 1 and arg(z – a) – arg(z – b) = α
H2.4.2.b properties and geometrical representation of the nth roots of unity
H2.4.2.c use of de Moivre's theorem to derive trigonometric identities
H2.5 Calculus
H2.5.1 Differentiation
Include:
H2.5.1.a graphical interpretation of
H2.5.1.a.1 f '(x) > 0, f '(x) = 0 and f '(x) < 0
H2.5.1.a.2 f ''(x) > 0 and f ''(x) < 0
H2.5.1.b relating the graph of y = f '(x) to the graph of y = f(x)
H2.5.1.c differentiation of simple functions defined implicitly or parametrically
H2.5.1.d finding the numerical value of a derivative at a given point using a graphic calculator
H2.5.1.e finding equations of tangents and normals to curves
H2.5.1.f solving practical problems involving differentiation
Exclude:
H2.5.1.a finding non-stationary points of inflexion
H2.5.1.b problems involving small increments and approximation
H2.5.2 Maclaurin's series
Include:
H2.5.2.a derivation of the first few terms of the series expansion of (1+ x)^{n} , e to the x power, sin x, ln(1+ x), and other simple functions
H2.5.2.b finding the first few terms of the series expansions of sums and products of functions, e.g. excos2x, using standard series
H2.5.2.c summation of infinite series in terms of standard series
H2.5.2.d sin x ≈ x, cos x ≈1– 1/2 x², tan x ≈ x
H2.5.2.e concepts of 'convergence' and 'approximation'
Exclude:
H2.5.2.a derivation of the general term of the series
H2.5.3 Integration techniques
Include:
H2.5.3.a integration of f'(x)/f(x) sin² x, cos² x, tan² x, 1/(a² + x²), 1/(√(a² - x²)), 1/(a² - x²) and 1/(x² - a²)
H2.5.3.b integration by a given substitution
H2.5.3.c integration by parts
Exclude:
H2.5.3.a reduction formulae
H2.5.4 Definite integrals
Include:
H2.5.4.a concept of definite integral as a limit of sum
H2.5.4.b definite integral as the area under a curve
H2.5.4.c evaluation of definite integrals
H2.5.4.d finding the area of a region bounded by a curve and lines parallel to the coordinate axes, between a curve and a line, or between two curves
H2.5.4.e area below the x-axis
H2.5.4.f finding the area under a curve defined parametrically
H2.5.4.g finding the volume of revolution about the x- or y-axis
H2.5.4.h finding the numerical value of a definite integral using a graphic calculator
Exclude:
H2.5.4.a approximation of area under a curve using the trapezium rule
H2.5.5 Differential equations
Include:
H2.5.5.a solving differential equations of the forms dy/dx = f(x), dy/dx = f(y), d²y/dx²=f (x)
H2.5.5.b formulating a differential equation from a problem situation
H2.5.5.c use of a family of solution curves to represent the general solution of a differential equation
H2.5.5.d use of an initial condition to find a particular solution
H2.5.5.e interpretation of a solution in terms of the problem situation
H2 Statistics
H2.6 Permutations, combinations and probability
H2.6.1 Permutations and combinations
Include:
H2.6.1.a addition and multiplication principles for counting
H2.6.1.b concepts of permutation (n! or ^{n} Pr) and combination (^{n} Cr)
H2.6.1.c arrangements of objects in a line or in a circle
H2.6.1.d cases involving repetition and restriction
H2.6.2 Probability
Include:
H2.6.2.a addition and multiplication of probabilities
H2.6.2.b mutually exclusive events and independent events
H2.6.2.c use of tables of outcomes, Venn diagrams, and tree diagrams to calculate probabilities
H2.6.2.d calculation of conditional probabilities in simple cases
H2.6.2.e use of: P(A') = 1–P(A), P(A∪B) = P(A) + P(B) –P(A∩B), P(A|B) = P(A∪B)/P(B)
H2.7 Binomial, Poisson and normal distributions
H2.7.1 Binomial and Poisson distributions
Include:
H2.7.1.a concepts of binomial distribution B(n,p) and Poisson distribution Po(μ) ; use of B(n,p) and Po(μ) as probability models
H2.7.1.b use of mean and variance of binomial and Poisson distributions (without proof)
H2.7.1.c solving problems involving binomial and Poisson variables
H2.7.1.d additive property of the Poisson distribution
H2.7.1.e Poisson approximation to binomial
Exclude:
H2.7.1.a calculation of mean and variance for other probability distributions
H2.7.2 Normal distribution
Include:
H2.7.2.a concept of a normal distribution and its mean and variance; use of N(μ,∑²) as a probability model
H2.7.2.b standard normal distribution
H2.7.2.c finding the value of P(X < x_{1} ) given the values of x_{1} , μ, ∑
H2.7.2.d use of the symmetry of the normal distribution
H2.7.2.e finding a relationship between x_{1} , μ, ∑ given the value of P(X < x_{1} )
H2.7.2.f solving problems involving normal variables
H2.7.2.g solving problems involving the use of E(aX + b) and Var(aX + b)
H2.7.2.h solving problems involving the use of E(aX + bY) and Var(aX + bY), where X and Y are independent
H2.7.2.i normal approximation to binomial
H2.7.2.j normal approximation to Poisson
Exclude:
H2.7.2.a finding probability density functions and distribution functions
H2.7.2.b calculation of E(X) and Var(X) from other probability density functions
H2.8 Sampling and hypothesis testing
H2.8.1 Sampling
Include:
H2.8.1.a concepts of population and sample
H2.8.1.b random, stratified, systematic and quota samples
H2.8.1.c advantages and disadvantages of the various sampling methods
H2.8.1.d distribution of sample means from a normal population
H2.8.1.e use of the Central Limit Theorem to treat sample means as having normal distribution when the sample size is sufficiently large
H2.8.1.f calculation of unbiased estimates of the population mean and variance from a sample
H2.8.1.g solving problems involving the sampling distribution
H2.8.2 Hypothesis testing
Include:
H2.8.2.a concepts of null and alternative hypotheses, test statistic, level of significance and p-value
H2.8.2.b tests for a population mean based on:
H2.8.2.b.1 a sample from a normal population of known variance
H2.8.2.b.2 a sample from a normal population of unknown variance
H2.8.2.b.3 a large sample from any population
H2.8.2.c 1-tail and 2-tail tests
H2.8.2.d use of t-test
Exclude:
H2.8.2.a testing the difference between two population means
H2.9 Correlation and Regression
H2.9.1 Correlation coefficient and linear regression
Include:
H2.9.1.a concepts of scatter diagram, correlation coefficient and linear regression
H2.9.1.b calculation and interpretation of the product moment correlation coefficient and of the equation of the least squares regression line
H2.9.1.c concepts of interpolation and extrapolation
H2.9.1.d use of a square, reciprocal or logarithmic transformation to achieve linearity
Exclude:
H2.9.1.a derivation of formulae
H2.9.1.b hypothesis tests