International Baccalaureate Diploma Programme (IB) Group 5: Mathematics (Analysis and Approaches) Syllabus

Group 5: Mathematics (Analysis and Approaches) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Group 5: Mathematics (Analysis and Approaches) in International Baccalaureate Diploma Programme (IB), not a summary of it.

1. 1

   Number and Algebra

   4 topics

   • Sequences, series and the binomial theorem
     • Arithmetic and geometric sequences
     • Sigma notation and binomial expansion
   • Exponents and logarithms
     • Laws of indices and logarithms
     • Solving exponential equations
   • Proof and reasoning (HL emphasis)
     • Proof by induction
     • Proof by contradiction and counterexample
   • Complex numbers (Higher Level)
2. 2

   Functions

   3 topics

   • Function concepts and notation
     • Domain, range and inverse functions
     • Composite functions
   • Families of functions
     • Quadratic, rational and exponential functions
     • Logarithmic and polynomial functions
   • Transformations and graphing
     • Translations, reflections and stretches
3. 3

   Geometry and Trigonometry

   3 topics

   • Trigonometric ratios and the unit circle
     • Radian measure and arc length
     • Trigonometric functions and graphs
   • Trigonometric identities and equations
   • Vectors (Higher Level)
     • Scalar and vector products
     • Lines and planes in space
4. 4

   Statistics and Probability

   3 topics

   • Descriptive statistics
     • Measures of central tendency and spread
     • Correlation and regression
   • Probability theory
     • Conditional probability and independence
     • Bayes' theorem (Higher Level)
   • Probability distributions
     • Binomial distribution
     • Normal distribution
5. 5

   Calculus

   3 topics

   • Differentiation
     • Rules of differentiation
     • Applications: tangents, optimisation and kinematics
   • Integration
     • Indefinite and definite integrals
     • Area under curves and volumes of revolution
   • Differential equations and Maclaurin series (HL)
6. 6

   Toolkit and Assessment

   4 topics

   • Mathematical inquiry and modelling
   • Use of technology (GDC)
   • Internal Assessment: Mathematical exploration
     • Choosing a topic and aim
     • Mathematical communication and reflection
   • Examination paper structure
     • Calculator and non-calculator papers
     • Paper 3 problem-solving (Higher Level)

Group 5: Mathematics (Analysis and Approaches) flashcards for International Baccalaureate Diploma Programme (IB)

19 of 56 cards from the Group 5: Mathematics (Analysis and Approaches) deck — real questions with worked answers.

1. State the formula for the $n$th term of an arithmetic sequence with first term $u_1$ and common difference $d$.

   $u_n = u_1 + (n-1)d$

2. What is the sum of the first $n$ terms of an arithmetic sequence?

   $S_n = \frac{n}{2}(2u_1 + (n-1)d) = \frac{n}{2}(u_1 + u_n)$

3. Give the $n$th term and the sum of the first $n$ terms of a geometric sequence with first term $u_1$ and common ratio $r$.

   $u_n = u_1 r^{n-1}$ and $S_n = \frac{u_1(r^n - 1)}{r-1} = \frac{u_1(1 - r^n)}{1-r}$, $r \neq 1$

4. Under what condition does an infinite geometric series converge, and what is its sum?

   It converges when $|r| < 1$, with sum $S_\infty = \frac{u_1}{1-r}$

5. State the binomial theorem for $(a+b)^n$ where $n \in \mathbb{Z}^{+}$.

   $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}$, where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

6. Simplify $\log_a x + \log_a y$ and $\log_a x - \log_a y$ using the laws of logarithms.

   $\log_a(xy)$ and $\log_a\!\left(\frac{x}{y}\right)$ respectively

7. State the change of base formula for logarithms.

   $\log_a x = \frac{\log_b x}{\log_b a}$

8. Express the relationship between exponential and logarithmic form: if $a^x = b$, then what is $x$?

   $x = \log_a b$ (with $a > 0$, $a \neq 1$, $b > 0$)

9. What is $\log_a(x^n)$ equal to, and what are $\log_a 1$ and $\log_a a$?

   $\log_a(x^n) = n\log_a x$; $\log_a 1 = 0$; $\log_a a = 1$

10. In a proof by contradiction, what is the general strategy?

    Assume the negation of the statement to be proved, then derive a logical contradiction, which establishes that the original statement must be true.

11. Outline the three steps of proof by mathematical induction (HL).

    1. Base case: prove the statement true for $n=1$ (or the first value). 2. Inductive step: assume true for $n=k$. 3. Prove it then holds for $n=k+1$, concluding it is true for all $n \geq 1$.

12. What does a counterexample prove, and what can it not prove?

    A single counterexample proves a universal statement is false; it cannot prove a statement true.

13. Define a complex number in Cartesian form and identify its real and imaginary parts (HL).

    $z = a + bi$ where $i^2 = -1$; real part $\operatorname{Re}(z) = a$, imaginary part $\operatorname{Im}(z) = b$.

14. Give the modulus-argument (polar) and Euler forms of a complex number (HL).

    $z = r(\cos\theta + i\sin\theta) = r\,\mathrm{e}^{i\theta}$, where $r = |z|$ and $\theta = \arg(z)$

15. State De Moivre's theorem (HL).

    $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$

16. For $z = a+bi$, give the modulus $|z|$ and the complex conjugate $z^{*}$ (HL).

    $|z| = \sqrt{a^2 + b^2}$ and $z^{*} = a - bi$

17. Define the domain and range of a function.

    The domain is the set of all permitted input values ($x$); the range is the set of all resulting output values ($f(x)$).

18. What condition must a function satisfy to have an inverse $f^{-1}$, and what does its graph look like?

    It must be one-to-one (injective); the graph of $f^{-1}$ is the reflection of $f$ in the line $y = x$.

19. Define the composite function $(f \circ g)(x)$.

    $(f \circ g)(x) = f(g(x))$: apply $g$ first, then $f$.

See more Group 5: Mathematics (Analysis and Approaches) flashcards →

Planning Group 5: Mathematics (Analysis and Approaches) for International Baccalaureate Diploma Programme (IB)

Group 5: Mathematics (Analysis and Approaches) (International Baccalaureate Diploma Programme (IB)) FAQ