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Singapore · SEABSyllabus
Additional Mathematics syllabus, dot point by dot point
Every dot point in the Singapore Additional Mathematicssyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.8, Anthropic's latest AI.
Algebra: Surds, Indices and Polynomials
Module overview →- How do the laws of indices let us simplify expressions with positive, negative, zero and fractional powers?Apply the laws of indices to simplify expressions involving positive, negative, zero and fractional powers and to solve simple exponential equations8 min answer →
- How do the remainder and factor theorems let us find remainders and factors of a polynomial without long division?Use the remainder theorem to find the remainder on division by a linear factor and the factor theorem to identify and extract factors of a polynomial9 min answer →
- How do we solve a cubic equation by combining the factor theorem with factorisation of the resulting quadratic?Solve cubic and higher polynomial equations by factorising fully and applying the zero-product principle to find all real roots8 min answer →
- How do we simplify expressions containing square roots and remove surds from a denominator?Simplify surds, perform the four operations on surds, and rationalise denominators including those of the form a plus root b8 min answer →
Binomial Theorem and Partial Fractions
Module overview →- How does the binomial theorem let us expand a power of a two-term expression without multiplying it out by hand?Expand expressions of the form a plus b to the power n for a positive integer n using the binomial theorem and binomial coefficients8 min answer →
- How do we pick out a single term, such as the coefficient of x cubed or the constant term, from a binomial expansion?Use the general term of a binomial expansion to find a specified term, the coefficient of a given power, or the term independent of x8 min answer →
- How do we split a proper rational fraction with distinct linear factors into simpler partial fractions?Express a proper rational fraction with distinct linear factors in the denominator as a sum of partial fractions8 min answer →
- How does the partial-fraction form change when the denominator has a repeated factor or an irreducible quadratic factor?Express proper fractions with repeated linear factors or an irreducible quadratic factor as partial fractions, choosing the correct numerator forms9 min answer →
Coordinate Geometry and Circles
Module overview →- How can we find the area of a polygon directly from the coordinates of its vertices?Find the area of a triangle or polygon from the coordinates of its vertices using the shoelace determinant method8 min answer →
- How do gradients, distances and midpoints let us describe and relate straight lines in the plane?Find the gradient, length and midpoint of a line segment, write the equation of a line, and use the conditions for parallel and perpendicular lines9 min answer →
- How do we write the equation of a circle and recover its centre and radius from either standard or general form?Write the equation of a circle in standard and general form and find the centre and radius by completing the square9 min answer →
- How do we find where a line meets a circle and the equation of a tangent to a circle?Find the intersection of a line and a circle, determine tangency using the discriminant or perpendicular radius, and find tangent equations9 min answer →
Differentiation and Its Applications
Module overview →- What does the derivative measure, and how do we differentiate powers and the standard functions?Interpret the derivative as a gradient and rate of change, and differentiate powers of x and the standard exponential, logarithmic and trigonometric functions9 min answer →
- How do the product, quotient and chain rules let us differentiate products, quotients and composite functions?Apply the product, quotient and chain rules, individually and in combination, to differentiate products, quotients and composite functions9 min answer →
- How does the chain rule connect the rates at which two related quantities change with time?Use the chain rule to relate connected rates of change, finding one rate from another for two quantities linked by an equation9 min answer →
- How do we locate the stationary points of a curve and decide whether each is a maximum, a minimum or a point of inflexion?Find stationary points by setting the first derivative to zero and determine their nature using the first or second derivative test9 min answer →
- How do we use the derivative to find the equation of a tangent or a normal to a curve at a point?Use the derivative as the gradient to find the equations of the tangent and the normal to a curve at a given point8 min answer →
Integration and Its Applications
Module overview →- How do we find the area of a region enclosed between two curves, or between a curve and a line?Find the area enclosed between two curves, or a curve and a line, by integrating the difference of the upper and lower functions between their intersection points9 min answer →
- How does a definite integral give the area under a curve, and how do we evaluate it?Evaluate definite integrals using limits and use them to find the area of a region bounded by a curve and the x-axis9 min answer →
- How is integration the reverse of differentiation, and how do we integrate powers and standard functions?Integrate powers of x and standard functions as the reverse of differentiation, including the constant of integration, and integrate linear composites9 min answer →
- How do we integrate exponential, reciprocal and trigonometric functions, including those of a linear expression?Integrate the exponential, reciprocal and trigonometric functions and their linear composites as the reverse of the corresponding derivatives9 min answer →
Kinematics
Module overview →- How do we solve practical motion problems involving maximum displacement, total distance and changes of direction?Solve kinematics problems involving maximum or minimum displacement and velocity, total distance travelled, and changes of direction9 min answer →
- How are displacement, velocity and acceleration related for a particle moving in a straight line?Define displacement, velocity and acceleration for motion in a straight line and interpret their signs and the graphs that connect them8 min answer →
- How do differentiation and integration move between displacement, velocity and acceleration?Differentiate to pass from displacement to velocity to acceleration, and integrate to reverse the process, fixing constants from initial conditions9 min answer →
Logarithmic and Exponential Functions
Module overview →- How do we solve equations where the unknown is in an exponent or inside a logarithm?Solve exponential equations by taking logarithms and logarithmic equations by converting to index form, rejecting invalid solutions8 min answer →
- What do the graphs of exponential and logarithmic functions look like, and how are they related?Sketch the graphs of exponential and logarithmic functions, identify their key features, and recognise them as reflections of each other8 min answer →
- How do the laws of logarithms let us combine, split and simplify logarithmic expressions?State and apply the product, quotient and power laws of logarithms and the change-of-base relationship to simplify and evaluate expressions8 min answer →
- How can we transform a non-linear relationship into a straight line to find its unknown constants?Transform a non-linear relationship into the form Y equals mX plus c and use the gradient and intercept of the straight-line graph to find unknown constants9 min answer →
Quadratic Functions and Equations
Module overview →- What does the discriminant tell us about the number and type of roots of a quadratic equation?Use the discriminant b squared minus 4ac to determine whether a quadratic has two, one or no real roots and to solve related problems8 min answer →
- How can a substitution turn a non-quadratic equation into a quadratic we already know how to solve?Solve equations reducible to quadratic form by a suitable substitution, including equations in powers, surds and exponentials8 min answer →
- How does completing the square reveal the maximum or minimum value and the line of symmetry of a quadratic?Express a quadratic in completed-square form and use it to find the vertex, the maximum or minimum value, and the line of symmetry9 min answer →
- How do we solve a quadratic inequality and express the solution as a range of values?Solve quadratic inequalities by factorising and reasoning about the sign of the quadratic between and beyond its roots8 min answer →
Trigonometry and Identities
Module overview →- How do the addition and double angle formulae let us expand and simplify trigonometric expressions of combined angles?Apply the addition formulae for sine, cosine and tangent and the double angle formulae to expand, simplify and evaluate trigonometric expressions9 min answer →
- How does the R-formula combine a sine and a cosine term into one, and what does it tell us about maximum and minimum values?Express a sine plus cosine as a single R sine or R cosine function and use it to find maximum and minimum values and to solve equations9 min answer →
- How do we find all solutions of a trigonometric equation within a given range?Solve trigonometric equations within a stated interval, finding the basic angle and using symmetry to obtain every solution9 min answer →
- How do we use the Pythagorean and reciprocal identities to simplify expressions and prove trigonometric statements?State and use the Pythagorean, reciprocal and quotient identities to simplify expressions and prove trigonometric identities9 min answer →
- How do the unit circle and reference angles extend the trigonometric ratios to any angle?Define sine, cosine and tangent for any angle using the unit circle, determine signs by quadrant, and use reference angles and special angles9 min answer →