Singapore · SEABQ&A
MathsQ&A by dot point
A short Q&A bank for every Singapore 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.
Calculus
- Find and classify stationary points, determine increasing and decreasing intervals and concavity, and solve optimisation problems in context4Q&A pairs
- Evaluate definite integrals, use them to find the area under a curve and between curves, and apply the fundamental theorem of calculus6Q&A pairs
- Solve first-order differential equations by direct integration and by separating variables, find particular solutions from conditions, and interpret solutions in context8Q&A pairs
- Differentiate standard functions and use the product, quotient and chain rules to differentiate products, quotients and composite functions6Q&A pairs
- Differentiate relations defined implicitly and curves defined parametrically, and find gradients, tangents and second derivatives in each case6Q&A pairs
- Integrate standard functions and use substitution, integration by parts and partial fractions to evaluate a wide range of integrals9Q&A pairs
- Derive and use the Maclaurin series of a function, apply the standard series for common functions, and use series to obtain approximations6Q&A pairs
- Find equations of tangents and normals to curves, and solve connected rates of change problems using the chain rule6Q&A pairs
- Find volumes of revolution generated by rotating a region about the x-axis or y-axis, including the volume between two curves9Q&A pairs
Functions and Graphs
- Identify and use the key features of a curve - intercepts, turning points, asymptotes, symmetry and behaviour at infinity - to produce and interpret graph sketches5Q&A pairs
- Form and find the domain of composite functions, determine when a composite is defined, find inverse functions and their domains, and use the graphical relationship between a function and its inverse5Q&A pairs
- Recognise and sketch the standard conics (circle, ellipse, parabola, hyperbola) from their equations, and sketch and analyse curves defined parametrically4Q&A pairs
- Define a function and its domain and range, decide whether a relation is a function or one-to-one, and find the range of a given function over a stated domain5Q&A pairs
- Sketch graphs of rational functions of the form a linear over linear and a quadratic over linear, finding intercepts, asymptotes, stationary points and the regions where the curve lies4Q&A pairs
- Solve quadratic, polynomial and rational inequalities algebraically and graphically, using a sign analysis and respecting the sign of any denominator8Q&A pairs
- Define the modulus function, sketch graphs involving the modulus of a function, and solve equations and inequalities involving the modulus4Q&A pairs
- Relate the graph of y equals a f(b(x + c)) + d to the graph of y equals f(x) through translations, stretches and reflections, and apply combined transformations in the correct order5Q&A pairs
Probability and Statistics
- Model situations with the binomial and Poisson distributions, state the conditions for each, and compute probabilities, means and variances6Q&A pairs
- Calculate conditional probabilities, test for independence, and apply the conditional probability formula and the law of total probability4Q&A pairs
- Compute and interpret the product moment correlation coefficient, find the least squares regression line, and use it for prediction within the data range5Q&A pairs
- Construct probability distributions for discrete random variables and compute the expectation and variance, including for functions of the variable6Q&A pairs
- Carry out a hypothesis test for a population mean, stating hypotheses, computing a test statistic or p-value, and interpreting the conclusion in context3Q&A pairs
- Approximate the binomial by the Poisson or the normal, and the Poisson by the normal, under stated conditions, applying a continuity correction where appropriate5Q&A pairs
- Model continuous data with the normal distribution, standardise to the Z-distribution to find probabilities, and find values from given probabilities5Q&A pairs
- Use the addition and multiplication principles, permutations and combinations to count arrangements and selections, including cases with restrictions6Q&A pairs
- Use the probability rules for the complement, union and intersection of events, and apply Venn diagrams and tree diagrams to combined events5Q&A pairs
- Describe the distribution of the sample mean, use the Central Limit Theorem, and find unbiased estimates of the population mean and variance from a sample7Q&A pairs
Sequences and Series
- Use the formulae for the nth term and the sum of the first n terms of an arithmetic progression, and solve problems involving arithmetic sequences and series6Q&A pairs
- Expand (1 + x) to the power n for rational n as a series, state the range of validity, and use the expansion to obtain approximations5Q&A pairs
- Describe the behaviour of a sequence as n tends to infinity, determine the convergence of a geometric series, and interpret the limit of a sequence or partial sum6Q&A pairs
- Use the formulae for the nth term and the sum of a geometric progression, determine convergence, and find the sum to infinity of a convergent geometric series4Q&A pairs
- Use the principle of mathematical induction to prove statements about sums of series, divisibility and other results indexed by the positive integers5Q&A pairs
- Use the method of differences, including the use of partial fractions, to find the sum of a series whose terms telescope, and deduce the sum to infinity where it exists6Q&A pairs
- Use recurrence relations to generate sequences, find and verify a conjectured formula for the nth term, and analyse long-term behaviour5Q&A pairs
- Use sigma notation and the standard results for the sums of integers, squares and cubes, and the linearity of summation, to evaluate finite series5Q&A pairs
Vectors and Complex Numbers
- Represent complex numbers on an Argand diagram and identify and sketch loci defined by conditions on the modulus and argument5Q&A pairs
- Perform arithmetic with complex numbers in Cartesian form, use the conjugate, and solve polynomial equations including the use of conjugate root pairs3Q&A pairs
- Express complex numbers in modulus-argument and exponential form, convert between forms, and use them to multiply, divide and take powers via de Moivre's theorem7Q&A pairs
- Write the vector and Cartesian equations of a line in three dimensions, find the intersection of two lines, and classify lines as parallel, intersecting or skew7Q&A pairs
- Write the scalar product and Cartesian equations of a plane, find the intersection of a line with a plane and of two planes, and compute distances and angles involving planes9Q&A pairs
- Find the nth roots of a complex number using de Moivre's theorem, and solve polynomial equations with complex roots, interpreting the roots geometrically6Q&A pairs
- Define and compute the scalar (dot) product, use it to find angles between vectors, test for perpendicularity, and find the projection of one vector onto another5Q&A pairs
- Define and compute the vector (cross) product, use it to find a vector perpendicular to two given vectors, the area of a triangle or parallelogram, and the sine of the angle between vectors5Q&A pairs
- Represent vectors in component and position form, add and scale them, find magnitudes and unit vectors, and use the ratio theorem for points dividing a line segment4Q&A pairs