Source:** MOE 2019 JC H2 Maths Syllabus **

H2 Mathematics is designed to prepare students for a range of university courses, including mathematics, sciences, engineering and related courses, where a good foundation in mathematics is required. It develops mathematical thinking and reasoning skills that are essential for further learning of mathematics. Through applications of mathematics, students also develop an appreciation of mathematics and its connections to other disciplines and to the real world.

1.1 | Functions | Include: · concepts of function, domain and range · use of notations such as f(x) = x ^{2} + 5 , f : x a x ^{2} + 5 , f ^{–}^{1}(x) , fg(x) and f ^{2} (x)· finding inverse functions and composite functions · conditions for the existence of inverse functions and composite functions · domain restriction to obtain an inverse function · relationship between a function and its inverse Exclude the use of the relation (fg) ^{–}^{1} = g^{–}^{1}f ^{–}^{1} , and restriction of domain to obtain a composite function. | |||||||||||||||

1.2 | Graphs and transformations | Include: · use of a graphing calculator to graph a given function · important characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes. · determining the equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y · effect of transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a) and y = f(ax), and combinations of these transformations · relating the graphs of y = f ^{–}^{1}(x ), y = f (x ),y = f (x ), and y = ^{1 }/ f(x) to the graph of y = f(x)· simple parametric equations and their graphs | |||||||||||||||

1.3 | Equations and inequalities | Include: · formulating an equation, a system of linear equations, or inequalities from a problem situation · solving an equation exactly or approximately using a graphing calculator · solving a system of linear equations using a graphing calculator · solving inequalities of the form ^{ f}^{(}^{x}^{)} > 0 whereg(x ) f(x) and g(x) are linear expressions or quadratic expressions that are either factorisable or always positive · concept of |x|, and use of relations | x – a | < b Û a – b < x < a + b and | x – a | > b Û x < a – b or x > a + b, in the course of solving inequalities · solving inequalities by graphical methods |

2.1 | Sequences and series | Include: · concepts of sequence and series for finite and infinite cases · sequence as function y = f(n) where n is a positive integer · relationship between u _{n} (the nth term) and S_{n} (the sum to n terms)· sequence given by a formula for the nth term · use of Σ notation · sum and difference of two series · summation of series by the method of differences · convergence of a series and the sum to infinity · formula for the nth term and the sum of a finite arithmetic series · formula for the nth term and the sum of a finite geometric series · condition for convergence of an infinite geometric series · formula for the sum to infinity of a convergent geometric series |

3.1 | Basic properties of vectors in two- and three dimensions | Include: · addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations · position vectors, displacement vectors and direction vectors · magnitude of a vector · unit vectors · distance between two points · concept of direction cosines · collinearity · use of the ratio theorem in geometrical applications | |||||||||||||||

3.2 | Scalar and vector products in vectors | Include: · concepts of scalar product and vector product of vectors and their properties · calculation of the magnitude of a vector and the angle between two vectors · geometrical meanings of | a · n ^{ˆ }| and | a ´ n^{ˆ }|,where n ^{ˆ }is a unit vectorExclude triple products a · b ´ c and a ´ b ´ c . | |||||||||||||||

3.3 | Three-dimensional vector geometry | Include: · vector and cartesian equations of lines and planes · finding the foot of the perpendicular and distance from a point to a line or to a plane · finding the angle between two lines, between a line and a plane, or between two planes · relationships between (i) two lines (coplanar or skew) (ii) a line and a plane (iii) two planes Exclude: · finding the shortest distance between two skew lines · finding an equation for the common perpendicular to two skew lines |

4.1 | Complex numbers expressed in cartesian form | Include: · extension of the number system from real numbers to complex numbers · complex roots of quadratic equations · conjugate of a complex number · four operations of complex numbers · equality of complex numbers · conjugate roots of a polynomial equation with real coefficients | |||||||||||||||

4.2 | Complex numbers expressed in polar form | Include: · representation of complex numbers in the Argand diagram · complex numbers expressed in the form r(cos θ + i sin θ), or re ^{i}^{θ} where r > 0 and– π < θ ⩽ π · calculation of modulus (r) and argument (θ) of a complex number · multiplication and division of two complex numbers expressed in polar form |

5.1 | Differentiation | Include: · graphical interpretation of (i) f ′(x) > 0, f ′(x) = 0 and f ′(x) < 0 (ii) f ″(x) > 0 and f ″(x) < 0 · relating the graph of y = f ′(x) to the graph of y = f(x) · differentiation of simple functions defined implicitly or parametrically · determining the nature of the stationary points (local maximum and minimum points and points of inflexion) analytically, in simple cases, using the first derivative test or the second derivative test · locating maximum and minimum points using a graphing calculator · finding the approximate value of a derivative at a given point using a graphing calculator · finding equations of tangents and normals to curves, including cases where the curve is defined implicitly or parametrically · local maxima and minima problems · connected rates of change problems Exclude finding non-stationary points of inflexion and finding second derivatives of functions defined parametrically. | |||||||||||||||

5.2 | Maclaurin series | Include: · standard series expansion of (1 + x) ^{n} for any rational n, e^{x}, sin x, cos x and In(1 + x)· derivation of the first few terms of the Maclaurin series by – repeated differentiation, e.g. sec x – repeated implicit differentiation, e.g. y ^{3} + y^{2} + y = x^{2} – 2x– using standard series, e.g. e ^{x} cos 2x,_{ }· range of values of x for which a standard series converges· concept of “approximation” · small angle approximations: sin x » x, cos x » 1- ^{1 }x ^{2}, tan x » x2 Exclude derivation of the general term of the series. | |||||||||||||||

5.3 | Integration techniques | Include: · integration of f ^{¢}(x )[f(x )]^{n }(including n = –1), f ′(x)e^{f(}^{x}^{) }sin^{2} x, cos^{2} x, tan^{2} x,sin mx cos nx, cos mx cos nx and sin mx sin nx ^{ }a^{2} + x ^{2} a2 – x 2 a^{2} – x ^{2} x ^{2} – a^{2 }· integration by a given substitution· integration by parts | |||||||||||||||

5.4 | Definite integrals | Include: · concept of definite integral as a limit of sum · definite integral as the area under a curve · evaluation of definite integrals · 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 · area below the x-axis · finding the area under a curve defined parametrically · finding the volume of revolution about the x- or y-axis · finding the approximate value of a definite integral using a graphing calculator Exclude finding the volume of revolution about the x-axis or y-axis where the curve is defined parametrically. | |||||||||||||||

5.5 | Differential equations | Include: · solving for the general solutions and particular solutions of differential equations of the forms including those that can be reduced to (i) and (ii) by means of a given substitution · formulating a differential equation from a problem situation · interpreting a differential equation and its solution in terms of a problem situation |

6.1 | Probability | Include: · addition and multiplication principles for counting · concepts of permutation ( ^{n}P_{r}) and combination (^{n}C_{r})· arrangements of objects in a line or in a circle, including cases involving repetition and restriction · addition and multiplication of probabilities · mutually exclusive events and independent events · use of tables of outcomes, Venn diagrams, tree diagrams, and permutations and combinations techniques to calculate probabilities · calculation of conditional probabilities in simple cases · use of: P (A’ ) = 1 – P ( A) P (A È B) = P ( A) + P (B) – P (A Ç B) P (A | B) = ^{P (}^{A }^{Ç} ^{B}^{) }P (B) | |||||||||||||||

6.2 | Discrete random variables | Include: · concept of discrete random variables, probability distributions, expectations and variances · concept of binomial distribution B(n, p) as an example of a discrete probability distribution and use of B(n, p) as a probability model, including conditions under which the binomial distribution is a suitable model · use of mean and variance of binomial distribution (without proof) Exclude finding cumulative distribution function of a discrete random variable. | |||||||||||||||

6.3 | Normal distribution | Include: · concept of a normal distribution as an example of a continuous probability model and its mean and variance; use of N(µ, σ ^{2}) as a probability model· standard normal distribution · finding the value of P(X < x _{1}) or a related probability, given the values of x_{1}, µ, σ· symmetry of the normal curve and its properties · finding a relationship between x _{1}, µ, σ given the value of P(X < x_{1}), or a related probability· solving problems involving the use of E(aX + b) and Var (aX + b) · solving problems involving the use of E(aX + bY) and Var (aX + bY), where X and Y are independent Exclude normal approximation to binomial distribution. | |||||||||||||||

6.4 | Sampling | Include: · concepts of population, random and non-random samples · concept of the sample mean X as a random _{σ }2variable with E(X ) = µ and Var (X ) = n · distribution of sample means from a normal population · use of the Central Limit Theorem to treat sample means as having normal distribution when the sample size is sufficiently large · calculation and use of unbiased estimates of the population mean and variance from a sample, including cases where the data are given in summarised form Σx and Σx ^{2}, or Σ(x – a) and Σ(x – a)^{2} | |||||||||||||||

6.5 | Hypothesis testing | Include: · concepts of null hypothesis (H _{0}) and alternative hypotheses (H_{1}), test statistic, critical region, critical value, level of significance and p-value· formulation of hypotheses and testing for a population mean based on: – a sample from a normal population of known variance – a large sample from any population · 1-tail and 2-tail tests · interpretation of the results of a hypothesis test in the context of the problem Exclude the use of the term ‘Type I’ error, concept of Type II error and testing the difference between two population means. | |||||||||||||||

6.6 | Correlation and Linear regression | Include: · use of scatter diagram to determine if there is a plausible linear relationship between the two variables · correlation coefficient as a measure of the fit of a linear model to the scatter diagram · finding and interpreting the product moment correlation coefficient (in particular, values close to -1, 0 and 1) · concepts of linear regression and method of least squares to find the equation of the regression line · concepts of interpolation and extrapolation · use of the appropriate regression line to make prediction or estimate a value in practical situations, including explaining how well the situation is modelled by the linear regression model · use of a square, reciprocal or logarithmic transformation to achieve linearity Exclude: · derivation of formulae · relationship r ^{2} = b_{1}b_{2}, where b_{1} and b_{2} are regression coefficients· hypothesis tests |