GCE Advanced Level (A-Levels) Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in GCE Advanced Level (A-Levels), not a summary of it.

1. 1

   Pure Mathematics: Algebra and Functions

   6 topics

   • Indices, Surds and Quadratics
     • Laws of indices and fractional/negative exponents
     • Manipulating and rationalising surds
     • Completing the square and the quadratic formula
     • The discriminant and nature of roots
   • Polynomials and Algebraic Division
     • Factor and remainder theorems
     • Algebraic long division
     • Partial fractions
   • Functions and Graphs
     • Domain, range and composite functions
     • Inverse functions and their graphs
     • Modulus function and transformations
   • Simultaneous Equations and Inequalities
     • Linear and quadratic simultaneous equations
     • Solving linear and quadratic inequalities
     • Set and interval notation
   • Binomial Expansion
     • Expansion for positive integer powers
     • Expansion for fractional and negative powers
     • Range of validity
   • Sequences and Series
     • Arithmetic sequences and series
     • Geometric sequences, series and sum to infinity
     • Recurrence relations and sigma notation
2. 2

   Pure Mathematics: Calculus

   6 topics

   • Differentiation from First Principles and Rules
     • Limit definition of the derivative
     • Power, product, quotient and chain rules
     • Differentiating trigonometric, exponential and logarithmic functions
   • Applications of Differentiation
     • Tangents, normals and rates of change
     • Stationary points and the second derivative test
     • Increasing/decreasing functions and optimisation
     • Connected rates of change
   • Integration Techniques
     • Integration as the reverse of differentiation
     • Integration by substitution and by parts
     • Integrating using partial fractions
   • Definite Integrals and Areas
     • Area under a curve and between curves
     • The trapezium rule for approximation
   • Differential Equations
     • Separation of variables
     • Forming and solving first-order models
     • Modelling with exponential growth and decay
   • Parametric Equations
     • Converting between parametric and Cartesian forms
     • Parametric differentiation and integration
3. 3

   Pure Mathematics: Trigonometry and Coordinate Geometry

   6 topics

   • Trigonometric Functions and Identities
     • Radian measure, arc length and sector area
     • Sine and cosine rules and area of a triangle
     • Reciprocal and inverse trigonometric functions
   • Trigonometric Identities and Equations
     • Pythagorean and addition formulae
     • Double angle and the R-formula (a cos x + b sin x)
     • Solving trigonometric equations
   • Exponentials and Logarithms
     • The exponential function and natural logarithm
     • Laws of logarithms and solving equations
     • Linearising data using logarithms
   • Coordinate Geometry of Lines and Circles
     • Equations of straight lines and gradients
     • Equation of a circle and tangent properties
   • Vectors
     • Vectors in two and three dimensions
     • Magnitude, unit vectors and position vectors
     • Geometric problems with vectors
   • Numerical Methods
     • Locating roots by sign change
     • Iteration and the Newton-Raphson method
4. 4

   Statistics

   5 topics

   • Data Presentation and Interpretation
     • Measures of location and spread
     • Histograms, box plots and cumulative frequency
     • Outliers and cleaning data
   • Probability
     • Venn diagrams, tree diagrams and conditional probability
     • Independent and mutually exclusive events
   • Statistical Distributions
     • The binomial distribution
     • The normal distribution and standardisation
     • Normal approximation to the binomial
   • Hypothesis Testing
     • Null and alternative hypotheses
     • Testing a binomial proportion
     • Testing a correlation coefficient and a normal mean
   • The Large Data Set
     • Sampling methods and bias
     • Interpreting real contextual data
5. 5

   Mechanics

   4 topics

   • Kinematics
     • Displacement-time and velocity-time graphs
     • Constant acceleration (SUVAT) equations
     • Variable acceleration using calculus
   • Forces and Newton's Laws
     • Newton's three laws of motion
     • Resolving forces and friction
     • Connected particles and pulleys
   • Moments
     • Moment of a force about a point
     • Equilibrium of rigid bodies
   • Projectiles
     • Horizontal and vertical components
     • Range, time of flight and maximum height

Mathematics flashcards for GCE Advanced Level (A-Levels)

21 of 61 cards from the Mathematics deck — real questions with worked answers.

1. How do you simplify a surd of the form $\frac{1}{\sqrt{a}}$ by rationalising the denominator?

   Multiply numerator and denominator by $\sqrt{a}$: $\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$. For $\frac{1}{a+\sqrt{b}}$, multiply by the conjugate $a-\sqrt{b}$.

2. State the laws of indices for $a^{m}\times a^{n}$, $\frac{a^{m}}{a^{n}}$, $(a^{m})^{n}$, $a^{0}$ and $a^{-n}$.

   $a^{m}\times a^{n}=a^{m+n}$; $\frac{a^{m}}{a^{n}}=a^{m-n}$; $(a^{m})^{n}=a^{mn}$; $a^{0}=1$; $a^{-n}=\frac{1}{a^{n}}$; and $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$.

3. What is the quadratic formula, and what does the discriminant tell you?

   $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. The discriminant $\Delta=b^{2}-4ac$: if $\Delta>0$ two real roots, $\Delta=0$ one repeated root, $\Delta<0$ no real roots.

4. How do you complete the square for $ax^{2}+bx+c$, and what does it reveal?

   Write $a\left(x+\frac{b}{2a}\right)^{2}+\left(c-\frac{b^{2}}{4a}\right)$. The vertex (turning point) is at $\left(-\frac{b}{2a},\,c-\frac{b^{2}}{4a}\right)$.

5. State the Factor Theorem and the Remainder Theorem for a polynomial $f(x)$.

   Remainder Theorem: dividing $f(x)$ by $(x-a)$ leaves remainder $f(a)$. Factor Theorem: $(x-a)$ is a factor of $f(x)$ if and only if $f(a)=0$.

6. What is a function, and how do you define its domain and range?

   A function maps each input to exactly one output. The domain is the set of permitted inputs $x$; the range is the set of resulting outputs $f(x)$.

7. How does the graph of $y=f(x)$ transform under $y=f(x)+a$, $y=f(x+a)$, $y=af(x)$ and $y=f(ax)$?

   $f(x)+a$: translate up by $a$. $f(x+a)$: translate left by $a$. $af(x)$: vertical stretch factor $a$. $f(ax)$: horizontal stretch factor $\frac{1}{a}$.

8. How do you find the inverse function $f^{-1}(x)$, and what is its relationship to $f$ graphically?

   Write $y=f(x)$, swap $x$ and $y$, then solve for $y$. The graph of $f^{-1}$ is the reflection of $f$ in the line $y=x$, with domain and range swapped.

9. What is the general method for solving simultaneous equations where one is linear and one is quadratic?

   Rearrange the linear equation for one variable, substitute into the quadratic to get a single quadratic equation, solve it, then back-substitute to find the other variable.

10. When solving a quadratic inequality such as $ax^{2}+bx+c>0$, what is the recommended method?

    Find the roots, sketch the parabola, then read off the regions. For $>0$ (upward parabola) take values outside the roots; for $<0$ take values between the roots.

11. State the binomial expansion of $(a+b)^{n}$ for positive integer $n$.

    $(a+b)^{n}=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^{r}$, where $\binom{n}{r}=\frac{n!}{r!(n-r)!}$ are the binomial coefficients (Pascal's triangle).

12. State the binomial series for $(1+x)^{n}$ where $n$ is any rational number, and its condition for validity.

    $(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+\frac{n(n-1)(n-2)}{3!}x^{3}+\cdots$, valid for $|x|<1$.

13. Give the formulas for the $n$th term and sum of an arithmetic series with first term $a$ and common difference $d$.

    $n$th term: $u_{n}=a+(n-1)d$. Sum: $S_{n}=\frac{n}{2}\left[2a+(n-1)d\right]=\frac{n}{2}(a+l)$ where $l$ is the last term.

14. Give the formulas for the $n$th term and sum of a geometric series with first term $a$ and common ratio $r$.

    $n$th term: $u_{n}=ar^{n-1}$. Sum: $S_{n}=\frac{a(1-r^{n})}{1-r}$. Sum to infinity $S_{\infty}=\frac{a}{1-r}$ exists only when $|r|<1$.

15. State the definition of the derivative from first principles.

    $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$, the limit of the gradient of a chord as the interval shrinks to zero.

16. State the product rule, quotient rule and chain rule for differentiation.

    Product: $(uv)'=u'v+uv'$. Quotient: $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^{2}}$. Chain: $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

17. Give the derivatives of $\sin x$, $\cos x$, $\tan x$, $e^{x}$ and $\ln x$.

    $\frac{d}{dx}\sin x=\cos x$; $\frac{d}{dx}\cos x=-\sin x$; $\frac{d}{dx}\tan x=\sec^{2}x$; $\frac{d}{dx}e^{x}=e^{x}$; $\frac{d}{dx}\ln x=\frac{1}{x}$.

18. How do you classify a stationary point using the second derivative test?

    At a stationary point ($f'(x)=0$): if $f''(x)>0$ it is a minimum; if $f''(x)<0$ it is a maximum; if $f''(x)=0$ the test is inconclusive (examine the sign of $f'$ either side).

19. What conditions identify an increasing function, a decreasing function and a point of inflection?

    Increasing where $f'(x)>0$; decreasing where $f'(x)<0$. A point of inflection is where the concavity changes, i.e. $f''(x)=0$ and $f''$ changes sign.

20. State the standard integration rule for $x^{n}$ and the rule for $\frac{1}{x}$.

    $\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c$ for $n\neq -1$; and $\int\frac{1}{x}\,dx=\ln|x|+c$. Always include the constant of integration $c$.

21. State the formula for integration by parts.

    $\int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx$. Choose $u$ as the part that simplifies on differentiating (use LIATE as a guide).

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