GCE Advanced Level Maths Curriculum For 11th Grade And 12th Grade / GCE A LevelOur Singapore Maths books for 11th Grade And 12th Grade / GCE A Level are written in English and based on GCE A Level Maths curriculum for 11th Grade And 12th Grade / GCE A Level, which covers the following topics. If your child uses our Singapore Maths books for 11th Grade And 12th Grade / GCE A Level, he will be able to: Pure mathematics 
Functions and graphs  understand the terms function, domain, range and oneone function
 find composite functions and inverses of functions, including conditions for their existence
 understand and use the relation (fg)^{1} = g^{1}f^{1 }where appropriate
 illustrate in graphical terms the relation between a oneone function and its inverse
 understand the relationship between a graph and an associated algebraic equation, and in particular show familiarity with the forms of the graphs of:
 y = kx^{n} where n is a positive or negative integer or a simple rational number
 ax + by = c
 (knowledge of geometrical properties of conics is not required)
 understand and use the relationships between the graphs of y = f(x), y = af(x), y = f(x) + a, y = f(x+a), y = f(ax), where a is a constant, and express the transformations involved in terms of translations, reflections and scalings
 relate the equation of a graph to its symmetries
 understand, and use in simple cases, the expression of the coordinates of a point on a curve in terms of a parameter

Partial fractions  recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than:
 (ax + b)(cx + d)(ex + f)
 (ax + b)(cx + d)^{2}
 (ax + b)(x^{2} + c^{2})
 including cases where the degree of the numerator exceeds that of the denominator

Inequalities; the modulus function  use properties of inequalities, and in particular understand that x > y and z > 0 imply that xz > yz while x > y and z < 0 imply xz < yz
 find the solution set of inequalities that are reducible to the form f(x) > 0, where f(x) can be factorised, and illustrate such solutions graphically
 understand the meaning of x and sketch the graph of functions of the form y = lax+ b
 use relations such as lx  al < b a  b < x < a + b and al = b a^{2} = b^{2} in the course of solving equations and inequalities

Logarithmic and exponential functions  recall and use the laws of logarithms (including change of base) and sketch graphs of simple logarithmic and exponential functions
 recall and use the definition a^{x} = e^{x ln a}
 use logarithms to solve equations reducible to the form a^{x} = b, and similar inequalities

Sequences and series  understand the idea of a sequence of terms, and use notations such as u_{n} to denote the nth term of a sequence
 recognise arithmetic and geometric progressions
 use formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions
 recall the condition for convergence of a geometric series, and use the formula for the sum to infinity of a convergent geometric series
 use notation
 use the binomial theorem to expand (a + b)^{n}, where n is a positive integer
 use the binomial theorem to expand (1 + x)^{n}, where n is rational, and recall the condition x< 1 for the validity of this expansion
 recognise and use the notations n! (with 0! = 1) and

Permutations and combinations  understand the terms 'permutation' and 'combination'
 solve problems involving arrangements (of objects in a line or in a circle), including those involving
 repetition (e.g. the number of ways of arranging the letters of the word NEEDLESS)
 restriction (e.g. the number of ways several people can stand in a line if 2 particular people must  or must not  stand next to each other)

Trigonometry  use the sine and cosine formulae
 calculate the angle between a line and a plane, the angle between two planes, and the angle between two skew lines in simple cases

Trigonometrical functions  understand the definition of the six trigonometrical functions for angles of any magnitude
 recall and use the exact values of trigonometrical functions of
 use the notations sin^{1}x, cos^{1}x and tan^{1}x to denote the principal values of the inverse trigonometrical relations
 relate the periodicity and symmetries of the sine, cosine and tangent functions to the form of their graphs, and use the concepts of periodicity and / or symmetry in relation to these functions and their inverses
 use trigonometrical identities for the simplification and exact evaluation of expressions, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of:
 and equivalent statements
 the expansion of sin(A ± B), cos(A ± B) and tan(A + B)
 the formulae for sin2A, cos2A and tan2A
 the formulae for sin A ± sin B and cos A ± cos B
 the expression of a in the forms
 find the general solution of simple trigonometrical equations, including graphical interpretation
 use the smallangle approximations

Differentiation  understand the idea of a limit and the derivative defined as a limit, including geometrical interpretation in terms of the gradient of a curve at a point as the limit of the gradient of a suitable sequence of chords
 use the standard notations for derived functions
 use the derivatives of x^{n} (for any rational n), sin x, cos x, tan x, e^{x}, a^{x}, ln x, sin^{1}x, cos^{1}x and tan^{1}x; together with constant multiples, sums, differences, products, quotients and composites
 find and use the first derivative of a function which is defined implicitly or parametrically
 locate stationary points, and distinguish between maxima, minima and stationary points of inflexion (knowledge of conditions for general points of inflexion is not required)
 find equations of tangents and normals to curves, and use information about gradients for sketching graphs
 solve problems involving maxima and minima, connected rates of change, small increments and approximations
 derive and use the first few terms of the Maclaurin series for a function

Integration  understand indefinite integration as the reverse process of differentiation
 integrate x^{n} (including the case where n = 1), e^{x}, sin x, cos x, sec^{2}x, together with
 sums, differences and constant multiples of these
 expressions involving a linear substitution (e.g. e^{2x1})
 applications involving the use of partial fractions
 applications involving the use of trigonometrical identities (e.g. )
 recognise an integrand of the form and integrate, e.g. or tan x
 integrate
 recognise when an integrand can usefully be regarded as a product, and use integration by parts to integrate, e.g., x sin 2x, x^{2}e^{x}, ln x
 use the in method of integration by substitution to simplify and evaluate either a definite or an indefinite integral (including simple cases in which the candidates have to select the substitution themselves, e.g. )
 evaluate definite integrals (including e.g.
 understand the idea of the area under a curve as the limit of a sum of the areas of rectangles and use simple applications of this idea
 use integration to find plane areas and volumes of revolution in simple cases
 use the trapezium rule to estimate the values of definite integrals, and identify the sign of the error in simple cases by graphical considerations

Vectors  use rectangular cartesian coordinates to locate points in three dimensions, and use standard notations for vectors, i.e. xi + yj + zk, a
 carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms
 use unit vectors, position vectors and displacement vectors
 recall the definition of and calculate the magnitude of a vector and the scalar product of two vectors
 use the scalar product to determine the angle between two directions and to solve problems concerning perpendicularity of vectors
 understand the significance of all the symbols used when the equation of a straight line is expressed in either of the forms r = a + tb and and convert equations of lines from vector to Cartesian form and vice versa
 solve simple problems involving finding and using either form of the equation of a line
 use equations of lines to solve problems concerning distances, angles and intersections, and in particular:
 determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists
 find the perpendicular distance from a point to a line
 find the angle between two lines
 use the ratio theorem in geometrical applications

Mathematical induction  understand the steps needed to carry out a proof by the method of induction
 use the method of mathematical induction to establish a given result e.g. the sum of a finite series, or the form of an nth derivative

Complex numbers  understand the idea of a complex number, recall the meaning of the terms 'real part', 'imaginary part', 'modulus', 'argument', 'conjugate', and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal
 carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in cartesian form (x + iy)
 recall and use the relation zz^{*} = z^{2}
 use the result that, for a polynomial equation with real coefficients, any nonreal roots occur in conjugate pairs
 represent complex numbers geometrically by means of an Argand diagram
 carry out operations of multiplication and division of two complex numbers expressed in polar form ()
 understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying, dividing two complex numbers
 illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram (e.g.z  a < k, z  a = z  b, arg(z  a) = , but excluding arg(za)arg(zb) = )

Curve sketching  understand the relationships between the graphs of y = f(x), y^{2} = f(x) and y = f(x)
 determine, in simple cases, the equations of asymptotes parallel to the axes
 use the equation of a curve, in simple cases, to make deductions concerning symmetry or concerning any restrictions on the possible values of x and / or y that there may be
 sketch curves of the form y = f(x), y^{2} = f(x) or y = f(x) (detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as turning points, asymptotes and intersections with the axes)

First order differential fquations  formulate a simple statement involving a rate of change as a differential equation, including the introduction if necessary of a constant of proportionality
 find by integration a general form of solution for a first order differential equation in which the variables are separable
 find the general solution of a first order linear differential equation by means of an integrating factor
 reduce a given first order differential equation to one in which the variables are separable or to one which is linear by means of a given simple substitution
 understand that the general solution of a differential equation is represented in graphical terms by a family of curves, and sketch typical members of a family in simple cases
 use an initial condition to find a particular solution to a differential equation, and interpret a solution in terms of a problem modelled by a differential equation

Numerical methods  locate approximately a root of an equation by means of graphical considerations and / or searching for a sign change
 use the method of linear interpolation to find an approximation to a root of an equation
 understand the idea of, and use the notation for, a sequence of approximations which converges to the root of an equation
 understand how a given simple iterative formula of the form relates to the equation being solved, and use a given iteration to determine a root to a prescribed degree of accuracy (conditions for convergence are not included)
 understand, in geometrical terms, the working of the NewtonRaphson method, and derive and use iterations based on this method
 appreciate that an iterative method may fail to converge to the required root
Particle mechanics 
Forces and equilibrium  identify the forces acting in a given situation
 understand the representation of forces by vectors, and find and use resultants and components
 solve problems concerning the equilibrium of a particle under the action of coplanar forces (using equations obtained by resolving the forces, or by using a triangle or polygon of forces)
 recall that the contact force between two surfaces can be represented by two components (the 'normal component' and the 'frictional component') and use this representation in solving problems
 use the model of a 'smooth' contact and understand the limitations of this model
 understand the concept of limiting friction and limiting equilibrium, recall the definition of coefficient of friction; and use the relationship as appropriate (knowledge of angle of friction will not be required)
 recall and use Newton's third law

Kinematics of motion in a straight line  understand the concepts of distance and speed, as scalar quantities, and of displacement, velocity and acceleration, as vector quantities, and understand the relationships between them
 sketch and interpret xt and vt graphs, and in particular understand and use the facts that:
 the area under a vt graph represents displacement
 the gradient of an xt graph represents velocity
 the gradient of a vt graph represents acceleration
 use appropriate formulae for motion with constant acceleration in a straight line

Newton's laws of motion  recall and use Newton's first and second laws of motion
 apply Newton's laws to the linear motion of a particle of constant mass moving under the action of constant forces (including friction)
 solve problems on the motion of two particles, connected by a light inextensible string which may pass over a fixed smooth light pulley or peg

Energy, work and power  understand the concept of the work done by a force, and calculate the work done by a constant force when its point of application undergoes a displacement not necessarily parallel to the force (use of the scalar product is not required)
 understand the concepts of gravitational potential energy and kinetic energy, and recall and use appropriate formulae
 understand and use the relationship between the change in energy of a system and the work done by the external forces, and use where appropriate the principle of conservation of energy
 recall and use the definition of power as the rate at which a force does work, and use the relationship between power, force and velocity for a force acting in the direction of motion
 solve problems involving, for example, the instantaneous acceleration of a car moving on a hill with resistance

Linear motion under a variable force  solve simple problems on the linear motion of a particle of constant mass moving under the action of variable forces by setting up and solving an appropriate differential equation (use of for velocity and , as appropriate, for acceleration is expected, and any differential equations to be solved will be first order with separable variables)

Motion of a projectile  model the motion of a projectile as a particle moving with constant acceleration, and understand the limitations of this model
 use horizontal and vertical equations of motion to solve problems on the motion of projectiles (including finding the magnitude and direction of the velocity at a given time or position and finding the range on a horizontal plane)
 derive and use the cartesian equation of the trajectory of a projectile, including cases where the initial speed and/or angle of projection is unknown (knowledge of the range on an inclined plane is not required)

Hooke's law  recall and use Hooke's law as a model relating the force in an elastic string or spring to the extension or compression, and understand and use the term 'modulus of elasticity'
 understand the concept of elastic potential energy, and recall and use the appropriate formula for its calculation
 use considerations of work and energy to solve problems involving elastic strings and springs

Uniform circular motion  understand the concept of angular speed for a particle moving in a circle with constant speed, and recall and use the relation (no proof required)
 understand that the acceleration of particle moving in a circle with constant speed is directed towards the centre of the circle and has magnitude (no proof required)
 use Newton's second law to solve problems which can be modelled as the motion of a particle moving in a circle with constant speed
Probability and statistics 
Probability  use addition and multiplication of probabilities, as appropriate, in simple cases, and understand the representation of events by means of tree diagrams
 understand the meaning of mutually exclusive and independent events, and calculate and use conditional probabilities in simple cases
 understand and use the notations P(A), P(AB) and the equations and = P(A) P(BA) = P(B) P(AB) (the general form of Bayes' theorem is not required)

Discrete random variables  understand the concept of a discrete random variable
 construct a probability distribution table relating to a given situation, and calculate E(X) and Var(X)
 appreciate conditions under which a uniform distribution or a binomial distribution B(n,p) may be a suitable probability model, and recall and use formulae for the calculation of binomial probabilities
 understand conditions under which a Poisson distribution Po() may be a suitable probability model, and recall and use the formula for the calculation of Poisson probabilities
 recall and use the means and variances of binomial and Poisson distributions
 use a Poisson distribution as an approximation to a binomial distribution, where appropriate (candidates should know that the conditions n > 50 and np < 5, approximately, can generally be taken to be suitable)

The normal distribution  recall the general shape of a normal curve, and understand how the shape and location of the distribution are affected by the values of (in general terms only; no knowledge of mathematical properties of the normal density function is included)
 standardise a normal variable and use normal distribution tables
 use the normal distribution as a probability model, where appropriate, and solve problems concerning a variable X, where , including:
 finding the value of P(X < x_{1}) given the values of
 use of the symmetry of the normal distribution
 finding a relationship between given the value of P(X < x_{1})
 repeated application of the above
 recall conditions under which a normal distribution may be used to approximate a binomial distribution (n sufficiently large to ensure that np > 5 and nq > 5. approximately) or Poisson distribution (, approximately), and calculate such approximations, including the use of a continuity correction

Samples  understand the distinction between a sample and a population, and appreciate the necessity for randomness in choosing samples
 explain in simple terms why a given sampling method may be unsatisfactory (a detailed knowledge of sampling and survey methods is not required)
 recognise that the sample mean can be regarded as a random variable, and use the facts that
 use the fact that X is normally distributed if X is normally distributed
 use the Central Limit Theorem (without proof) to treat X as being normally distributed when the sample size is sufficiently large ('large' samples will usually be of size at least 50, but candidates should know that using the approximation of normality can sometimes be useful with samples that are smaller than this)
 calculate unbiased estimates of the population mean and population variance from a sample (only a simple understanding of the term 'unbiased' is required)
 determine, from a sample from a normal distribution of known variance or from a large sample, a confidence interval for the population mean
 determine, from a large sample, a confidence interval for a population proportion

Linear combinations of random variables  recall and use the results in the course of solving problems that, for either discrete or continuous random variables,
 E(aX + b) = aE(X) + b and Var(aX + b) = a^{2} Var(X)
 E(aX + bY) = aE(X) + bE(Y)
 Var(aX + bY) = a^{2} Var(X) + b^{2} for independent X and Y
 recall and use the results that:
 if X has a normal distribution then so does aX + b
 if X and Y have independent normal distributions then aX + bY has a normal distribution
 if X and Y have independent Poisson distributions then X + Y has a Poisson distribution

Continuous random variables  understand and use the concept of a probability density function, and recall and use the properties of a density function (which may be defined 'piecewise')
 use a given probability density function to calculate the mean, mode and variance of a distribution, and in general use the result in simple cases, where f(x) is the probability density function of X and g(X) is a function of X
 understand and use the relationship between the probability density function and the distribution function and use either to evaluate the median, quartiles and other percentiles
 use a probability density function or a distribution function in the context of a model, including in particular the continuous uniform (rectangular) distribution

Hypothesis testing  understand and use the concepts of hypothesis (null and alternative), test statistic, significance level, and hypothesis test (1tail and 2tail)
 formulate hypotheses and apply a hypothesis test concerning the population mean using:
 a sample drawn from a normal distribution of known variance
 a large sample drawn from any distribution of unknown variance
 formulate hypotheses concerning a population proportion, and apply a hypothesis test using a normal approximation to a binomial distribution
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