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Further MathsQ&A by dot point
A short Q&A bank for every Singapore Further Maths syllabus dot point. Each question and answer is drawn directly from our worked dot-point page, so you can scan key concepts before opening the long-form answer.
Complex Numbers and Polynomials
- Represent complex numbers in Cartesian, polar and exponential form, perform arithmetic, and interpret them on the Argand diagram4Q&A pairs
- State and apply de Moivre's theorem to find powers of complex numbers and to derive multiple-angle and power-reduction trigonometric identities3Q&A pairs
- Sketch loci and regions in the Argand diagram defined by conditions on the modulus and argument of a complex number8Q&A pairs
- Use the relationships between the roots and coefficients of a polynomial and apply the conjugate root theorem for real polynomials7Q&A pairs
- Find the nth roots of unity and the nth roots of a general complex number, and describe their geometric arrangement on the Argand diagram7Q&A pairs
Differential Equations
- Solve first-order differential equations by separation of variables and by the integrating factor method, applying initial conditions5Q&A pairs
- Formulate differential equations from descriptions of rates of change and interpret the solutions in context, including long-term behaviour6Q&A pairs
- Solve non-homogeneous second-order linear differential equations by finding the complementary function and a particular integral for standard forcing terms5Q&A pairs
- Solve homogeneous second-order linear differential equations with constant coefficients using the auxiliary equation, covering real, repeated and complex roots8Q&A pairs
- Solve coupled systems of first-order linear differential equations by reduction to a single second-order equation4Q&A pairs
Further Calculus
- Calculate the arc length of a curve and the area of a surface of revolution for curves given in Cartesian or parametric form9Q&A pairs
- Integrate using trigonometric and hyperbolic substitutions and recognise standard integrals giving inverse trigonometric and logarithmic forms8Q&A pairs
- Evaluate improper integrals with infinite limits or integrands with a singularity, determining convergence by means of limits5Q&A pairs
- Derive Maclaurin series including by repeated implicit differentiation and use series to evaluate limits and approximations7Q&A pairs
- Derive reduction formulae using integration by parts and apply them to evaluate families of integrals7Q&A pairs
Further Probability and Statistics
- Work with continuous random variables defined by a probability density function, finding probabilities, the cumulative distribution function, expectation, variance and median7Q&A pairs
- Work with discrete random variables, their probability distributions, expectation, variance, and the expectation and variance of linear functions6Q&A pairs
- Compute unbiased estimates of a population mean and variance and construct and interpret confidence intervals for a population mean7Q&A pairs
- Carry out hypothesis tests and analyse Type I and Type II errors and the power of a test6Q&A pairs
- Apply non-parametric tests including the sign test and the Wilcoxon signed-rank test, and know when they are appropriate6Q&A pairs
- Recognise and apply the geometric and negative binomial distributions, including their probabilities, expectations and variances9Q&A pairs
Mathematical Induction, Inequalities and Recurrences
- Prove and apply inequalities including the use of the discriminant, completing the square, and standard results such as the AM-GM inequality5Q&A pairs
- Construct rigorous mathematical arguments using direct proof, proof by contradiction, proof by contrapositive, and disproof by counterexample9Q&A pairs
- Prove statements about sums, divisibility and inequalities for all positive integers using the principle of mathematical induction5Q&A pairs
- Solve first- and second-order linear recurrence relations with constant coefficients and find closed-form expressions for the nth term6Q&A pairs
- Sum finite series using the method of differences, standard results for powers of integers, and partial fractions6Q&A pairs
Matrices and Linear Spaces
- Diagonalise a matrix using its eigenvalues and eigenvectors and use the diagonal form to compute powers of the matrix4Q&A pairs
- Find the eigenvalues and eigenvectors of 2x2 and 3x3 matrices using the characteristic equation4Q&A pairs
- Find the inverse of a non-singular matrix and use matrices to solve systems of linear equations, recognising consistent, inconsistent and dependent cases5Q&A pairs
- Use the concepts of vector spaces and subspaces, linear independence, spanning sets, basis, dimension and the rank of a matrix4Q&A pairs
- Carry out matrix addition and multiplication and evaluate the determinant of 2x2 and 3x3 matrices, interpreting its geometric meaning6Q&A pairs
Numerical Methods
- Solve an equation by fixed-point iteration of the form x = g(x), and use the derivative condition to decide convergence3Q&A pairs
- Apply the Newton-Raphson method to find a root of an equation numerically and discuss its convergence and failure cases4Q&A pairs
- Approximate a definite integral using the trapezium rule and Simpson's rule and comment on the accuracy of the estimate5Q&A pairs
- Use Euler's method and the improved Euler (midpoint) method to obtain a numerical solution of a first-order differential equation6Q&A pairs
Vectors and the Geometry of Three Dimensions
- Find the intersection of lines and planes and compute shortest distances from a point to a line or plane and between two skew lines9Q&A pairs
- Write the vector, parametric and Cartesian equations of a line in three dimensions and classify the relationship between two lines7Q&A pairs
- Write the vector, scalar product and Cartesian equations of a plane using a normal vector and find the angle between planes and between a line and a plane8Q&A pairs
- Use the scalar and vector products and the scalar triple product to find angles, areas and volumes in three dimensions6Q&A pairs
- Apply vector methods to geometric problems including the foot of the perpendicular, reflections of points, and proofs of geometric properties9Q&A pairs