# Singapore Additional Mathematics Curriculum (Scope And Sequence) For 9th Grade And 10th Grade / Secondary 3 And Secondary 4 / GCE O LevelOur 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 f^{2} (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
## Quadratic functions - find the maximum or minimum value of the quadratic function f : x --> ax
^{2} + 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 e
^{x} (series expansions are not required) - know and use the laws of logarithms (including change of base of logarithms)
- solve equations of the form a
^{x}= b ## Straight line graphs - interpret the equation of a straight line graph in the form y = m x + c
- transform given relationships, including y = ax
^{n} and y = Ab^{x}, 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, sin
^{2} A + cos^{2} A = 1, sec^{2} A = 1 + tan^{2} A, cosec^{2} A = 1 + cot^{2} 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 (
^{n}_{r})a^{n - r} b^{r}, 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. (
^{a}_{b}), 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, d
^{2}y/dx^{2} [=d/dx(dy/dx)] - use the derivatives of the standard functions x
^{n} (for any rational n), sin x, cos x, tan x, e^{x}, 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), e^{ax+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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