 # Singapore Additional Mathematics Curriculum (Scope And Sequence) For 9th Grade And 10th Grade / Secondary 3 And Secondary 4 / GCE O Level

Our Singapore Additional Math books for 9th Grade / Secondary 3 and Singapore Additional Math books for 10th Grade / Secondary 4 are written in English and based on the Singapore Additional Math curriculum for 9th Grade And 10th Grade / Secondary 3 And Secondary 4, which covers the following topics.

If your child uses our Singapore Additional Math books for 9th Grade / Secondary 3 and Singapore Additional Math books for 10th Grade / Secondary 4, he will be able to:

(Some of the following symbols may not display properly.)

## Set language and notation

• use set language and notation, and Venn diagrams to describe sets and represent relationships between sets as follows:
• A = {x: x is a natural number}
• B = {(x, y): y = mx + c}
• C = {x: a < x < b }
• D = {a, b, c,. . . }
• understand and use the notation for the following :
• Union of A and B
• Intersection of A and B
• Number of elements in set A
• ". . . is an element of . . ."
• ". . . is not an element of . . ."
• Complement of set A
• The empty set
• Universal set
• A is a subset of B
• A is a proper subset of B
• A is not a subset of B
• A is not a proper subset of B

## Functions

• understand the terms function, domain, range (image set), one-one function, inverse function and composition of functions
• use the notation f(x) = sin x, f: x --> lg x, (x > 0), f -1(x) and f2 (x) [=f(f(x))]
• understand the relationship between y = f(x) and y = | f(x) |, where f(x) may be linear, quadratic or trigonometric
• explain in words why a given function is a function or why it does not have an inverse
• find the inverse of a one-one function and form composite functions
• use sketch graphs to show the relationship between a function and its inverse

• find the maximum or minimum value of the quadratic function f : x --> ax2 + bx + c by any method
• use the maximum or minimum value of f(x) to sketch the graph or determine the range for a given domain
• know the conditions for f(x) = 0 to have (i) two real roots, (ii) two equal roots, (iii) no real roots; and the related conditions for a given line to (i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve
• solve quadratic equations for real roots and find the solution set for quadratic inequalities

## Indices and surds

• perform simple operations with indices and with surds, including rationalising the denominator

## Factors of polynomials

• know and use the remainder and factor theorems
• find factors of polynomials
• solve cubic equations

## Simultaneous equations

• solve simultaneous equations in two unknowns with at least one linear equation

## Logarithmic and exponential functions

• know simple properties and graphs of the logarithmic and exponential functions including lnx and ex (series expansions are not required)
• know and use the laws of logarithms (including change of base of logarithms)
• solve equations of the form ax= b

## Straight line graphs

• interpret the equation of a straight line graph in the form y = m x + c
• transform given relationships, including y = axn and y = Abx, to straight line form and hence determine unknown constants by calculating the gradient or intercept of the transformed graph
• solve questions involving mid-point and length of line
• know and use the condition for two lines to be parallel or perpendicular

## Circular measure

• solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure

## Trigonometry

• know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent)
• understand amplitude and periodicity and the relationship between graphs of e.g. sin x and sin 2x
• draw and use the graphs of y = a sin(bx) + c, y = a cos(bx) + c, y = a tan(bx) + c, where a, b are positive integers and c is an integer
• use the relationships sin A / cos A = tan A, cos A / sin A = cot A, sin2 A + cos2 A = 1, sec2 A = 1 + tan2 A, cosec2 A = 1 + cot2 A, and solve simple trigonometric equations involving the six trigonometric functions and the above relationships (not including general solution of trigonometric equations)
• prove simple trigonometric identities

## Permutations and combinations

• recognise and distinguish between a permutation case and a combination case
• know and use the notation n!, (with 0! = 1), and the expressions for permutations and combinations of n items taken r at a time
• answer simple problems on arrangement and selection (cases with repetition of objects, or with objects arranged in a circle or involving both permutations and combinations, are excluded)

## Binomial expansions

• use the Binomial Theorem for expansion of (a + b)n for positive integral n
• use the general term (nr)an - r br, 0 < r < n (knowledge of the greatest term and properties of the coefficients is not required)

## Vectors in 2 dimensions

• use vectors in any form, e.g. (ab), p, ai - bj
• know and use position vectors and unit vectors
• find the magnitude of a vector. Add and subtract vectors and multiply vectors by scalars
• compose and resolve velocities
• use relative velocity including solving problems on interception (but not closest approach)

## Matrices

• display information in the form of a matrix of any order and interpret the data in a given matrix
• solve problems involving the calculation of the sum and product (where appropriate) of two matrices and interpret the results
• calculate the product of a scalar quantity and a matrix
• use the algebra of 2 x 2 matrices (including the zero and identity matrix)
• calculate the determinant and inverse of a non-singular 2 x 2 matrix and solve simultaneous line equations

## Differentiation and integration

• understand the idea of a derived function
• use the notations f '(x), f "(x), dy/dx, d2y/dx2 [=d/dx(dy/dx)]
• use the derivatives of the standard functions xn (for any rational n), sin x, cos x, tan x, ex, lnx, together with constant multiples, sums and composite functions of these
• differentiate products and quotients of functions
• apply differentiation to gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems
• discriminate between maxima and minima by any method
• understand integration as the reverse process of differentiation
• integrate sums of terms in powers of x excluding 1/x
• integrate functions of the form (ax + b)n (excluding n = -1), eax+b, sin (ax + b), cos (ax + b)
• evaluate definite integrals and apply integration to the evaluation of plane areas
• apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration, and the use of x-t and v-t graphs

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