Scottish Advanced Higher Mathematics Syllabus

Mathematics syllabus: all 5 chapters

Expand any chapter to see its topics and sub-topics.

1. 1

   Methods of Proof and Number Theory

   3 topics

   • Logic and methods of proof
     • Direct proof and proof by deduction
     • Proof by contradiction
     • Proof by contrapositive
     • Disproof by counterexample
     • Proof by induction
   • Number systems and notation
     • Natural numbers, integers, rationals, reals and complex numbers
     • Set notation and the algebra of sets
   • Euclidean algorithm and divisibility
     • Greatest common divisor
     • Expressing gcd as a linear combination
2. 2

   Algebra, Sequences and Series

   4 topics

   • Binomial theorem
     • Binomial coefficients and Pascal's triangle
     • General term and expansion
   • Partial fractions
     • Distinct linear factors
     • Repeated and irreducible quadratic factors
   • Sequences and series
     • Arithmetic and geometric sequences
     • Summation using the sigma notation and standard results
     • Maclaurin series expansions
   • Matrices and systems of equations
     • Matrix algebra, determinant and inverse
     • Gaussian elimination and consistency
3. 3

   Differential and Integral Calculus

   4 topics

   • Differentiation techniques
     • Chain, product and quotient rules
     • Implicit and parametric differentiation
     • Logarithmic differentiation
     • Higher derivatives
   • Applications of differentiation
     • Related rates of change
     • Curve sketching with asymptotes
   • Integration techniques
     • Integration by substitution
     • Integration by parts
     • Integration using partial fractions
   • Applications of integration
     • Volumes of revolution
     • Solving differential equations
4. 4

   Complex Numbers and Differential Equations

   3 topics

   • Complex numbers
     • Cartesian, polar and exponential forms
     • De Moivre's theorem and roots
     • Loci in the complex plane
     • Fundamental theorem of algebra
   • First order differential equations
     • Variables separable
     • Integrating factor method
   • Second order differential equations
     • Homogeneous equations and auxiliary equation
     • Particular integrals and general solutions
5. 5

   Vectors, Functions and Applications

   3 topics

   • Vectors in three dimensions
     • Vector, scalar and triple products
     • Equations of lines and planes
     • Intersection and angle problems
   • Properties of functions
     • Continuity and differentiability
     • Critical points and concavity
   • Number theory in applications
     • Modular arithmetic
     • Fibonacci and recurrence relations

Scottish Advanced Higher Mathematics flashcards — 51 questions with answers

1. What is the contrapositive of the statement "if $P$ then $Q$", and why is it useful in proof?

   The contrapositive is "if not $Q$ then not $P$" (i.e. $\neg Q \Rightarrow \neg P$). It is logically equivalent to the original statement, so proving the contrapositive proves the original.

2. Describe the structure of a proof by contradiction.

   Assume the negation of the statement you wish to prove is true, then derive a logical contradiction (something known to be false). Since the assumption leads to absurdity, the original statement must be true.

3. State the principle of proof by mathematical induction.

   To prove $P(n)$ for all integers $n \geq n_0$: (1) Base case — show $P(n_0)$ is true; (2) Inductive step — assume $P(k)$ true and show $P(k+1)$ follows. Then $P(n)$ holds for all $n \geq n_0$.

4. What is a counterexample, and what can it prove?

   A counterexample is a single specific case for which a statement fails. It disproves a universal ("for all") statement, but cannot prove one.

5. What is the difference between a necessary condition and a sufficient condition?

   $P$ is sufficient for $Q$ if $P \Rightarrow Q$ (having $P$ guarantees $Q$). $P$ is necessary for $Q$ if $Q \Rightarrow P$ ($Q$ cannot hold without $P$). If both, $P \Leftrightarrow Q$.

6. Name the standard number sets and their symbols from natural numbers up to complex numbers.

   Naturals $\mathbb{N}$, integers $\mathbb{Z}$, rationals $\mathbb{Q}$, reals $\mathbb{R}$, complex $\mathbb{C}$. They nest: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$.

7. Define a rational number and an irrational number.

   A rational number can be written as $\frac{p}{q}$ with $p, q \in \mathbb{Z}$ and $q \neq 0$. An irrational number cannot be expressed as such a ratio (e.g. $\sqrt{2}$, $\pi$); its decimal expansion neither terminates nor recurs.

8. What does it mean for an integer $a$ to divide an integer $b$, written $a \mid b$?

   $a \mid b$ means there exists an integer $k$ such that $b = ak$. Equivalently, $b$ divided by $a$ leaves no remainder.

9. State the Division Algorithm for integers.

   For integers $a$ and $b$ with $b > 0$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that $a = bq + r$ with $0 \leq r < b$.

10. Describe how the Euclidean algorithm finds $\gcd(a, b)$.

    Repeatedly apply $a = bq + r$, then replace $(a,b)$ with $(b, r)$, continuing until the remainder is $0$. The last non-zero remainder is $\gcd(a, b)$.

11. What is Bézout's identity, and how is it obtained?

    For integers $a, b$ there exist integers $x, y$ such that $ax + by = \gcd(a, b)$. The coefficients are found by back-substitution through the steps of the Euclidean algorithm.

12. State the Binomial Theorem for $(a+b)^{n}$ where $n$ is a positive integer.

    $$(a+b)^{n} = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r}$$ where $\binom{n}{r} = \dfrac{n!}{r!(n-r)!}$.

13. Write the general term in the expansion of $(a+b)^{n}$.

    The general term is $\binom{n}{r} a^{n-r} b^{r}$, corresponding to the $(r+1)$-th term of the expansion.

14. What is the value of $\binom{n}{0}$ and $\binom{n}{n}$, and state the symmetry property.

    $\binom{n}{0} = \binom{n}{n} = 1$. Symmetry: $\binom{n}{r} = \binom{n}{n-r}$. Pascal's rule: $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}$.

15. How do you decompose $\frac{f(x)}{(x-a)(x-b)}$ into partial fractions (distinct linear factors)?

    Write $\dfrac{f(x)}{(x-a)(x-b)} = \dfrac{A}{x-a} + \dfrac{B}{x-b}$, then solve for $A$ and $B$ by equating numerators (or by substituting $x=a$, $x=b$).

16. What is the partial fraction form when a denominator has a repeated linear factor $(x-a)^{2}$?

    $$\frac{f(x)}{(x-a)^{2}} = \frac{A}{x-a} + \frac{B}{(x-a)^{2}}$$ A separate term is needed for each power up to the multiplicity.

17. What partial fraction term corresponds to an irreducible quadratic factor $(x^{2}+px+q)$ in the denominator?

    A linear numerator over the quadratic: $\dfrac{Bx + C}{x^{2}+px+q}$.

18. State the formula for the $n$-th term and the sum of an arithmetic series with first term $a$ and common difference $d$.

    $n$-th term: $u_n = a + (n-1)d$. Sum: $S_n = \dfrac{n}{2}\big(2a + (n-1)d\big) = \dfrac{n}{2}(a + l)$ where $l$ is the last term.

19. State the formula for the $n$-th term and the sum of a geometric series with first term $a$ and common ratio $r$.

    $n$-th term: $u_n = ar^{n-1}$. Sum: $S_n = \dfrac{a(1 - r^{n})}{1 - r}$ for $r \neq 1$.

20. When does an infinite geometric series converge, and what is its sum?

    It converges when $|r| < 1$, with sum to infinity $$S_\infty = \frac{a}{1 - r}.$$

Scottish Advanced Higher Mathematics FAQ