International Baccalaureate Diploma (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 (IB), not a summary of it.

1. 1

   Number and Algebra

   4 topics

   • Sequences and Series
     • Arithmetic and geometric sequences
     • Sigma notation and the binomial theorem
   • Exponents and Logarithms
     • Laws of exponents and logarithms
     • Solving exponential equations
   • Proof and Counting (HL)
     • Proof by induction and contradiction
     • Permutations and combinations
   • Complex Numbers (HL)
2. 2

   Functions

   3 topics

   • Function Concepts and Transformations
     • Domain, range, and composite functions
     • Inverse functions and graph transformations
   • Polynomial and Rational Functions
   • Exponential and Logarithmic Functions
3. 3

   Geometry and Trigonometry

   3 topics

   • Trigonometric Functions and Identities
     • Unit circle and radian measure
     • Trig identities and equations
   • Triangle Geometry
     • Sine and cosine rules
     • Area and 3D applications
   • Vectors (HL)
4. 4

   Statistics and Probability

   3 topics

   • Descriptive Statistics and Correlation
     • Measures of central tendency and spread
     • Regression and correlation
   • Probability
     • Conditional probability and Bayes (HL)
     • Discrete random variables
   • Probability Distributions
     • Binomial distribution
     • Normal distribution
5. 5

   Calculus

   3 topics

   • Differentiation
     • Limits and derivative rules
     • Chain, product, and quotient rules
     • Optimization and kinematics
   • Integration
     • Indefinite and definite integrals
     • Area, volume, and applications
   • Differential Equations and Maclaurin Series (HL)
6. 6

   Toolkit and Internal Assessment

   3 topics

   • Mathematical Modelling and Inquiry
   • Use of Technology (GDC)
   • Mathematical Exploration (IA)

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

19 of 51 cards from the Group 5: Mathematics (Analysis and Approaches) deck — real questions with worked answers.

1. What is the formula for the nth term of an arithmetic sequence?

   u_n = u_1 + (n - 1)d, where u_1 is the first term and d is the common difference.

2. What is the formula for the sum of the first n terms of an arithmetic series?

   S_n = (n/2)(2u_1 + (n - 1)d) = (n/2)(u_1 + u_n).

3. What is the formula for the nth term and the sum of n terms of a geometric sequence?

   nth term: u_n = u_1 r^(n-1). Sum: S_n = u_1(r^n - 1)/(r - 1) = u_1(1 - r^n)/(1 - r), r not equal to 1.

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

   It converges when |r| < 1, and the sum to infinity is S_infinity = u_1/(1 - r).

5. State the Binomial Theorem for (a + b)^n where n is a positive integer.

   (a + b)^n = sum from k=0 to n of C(n,k) a^(n-k) b^k, where C(n,k) = n!/(k!(n-k)!).

6. State the three main laws of exponents (logarithm-related index laws).

   a^m * a^n = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(mn). Also a^0 = 1 and a^(-n) = 1/a^n.

7. State the three core laws of logarithms (product, quotient, power).

   log(xy) = log x + log y; log(x/y) = log x - log y; log(x^n) = n log x.

8. What is the change of base formula for logarithms?

   log_a x = (log_b x)/(log_b a), commonly log_a x = (ln x)/(ln a).

9. How do you solve the equation a^x = b (where a is not equal to 1)?

   Take logarithms of both sides: x = log_a b = (ln b)/(ln a).

10. (HL) What is the difference between proof by induction, proof by contradiction, and proof by contrapositive?

    Induction: prove a base case then assume P(k) and prove P(k+1). Contradiction: assume the statement is false and derive a logical impossibility. Contrapositive: to prove 'if P then Q', instead prove 'if not Q then not P'.

11. (HL) What are the two steps of a proof by mathematical induction?

    1) Base case: show the statement holds for n = 1 (or the smallest value). 2) Inductive step: assume true for n = k, then prove it holds for n = k + 1; conclude it holds for all integers n >= 1.

12. (HL) State the formula for the number of permutations and combinations of r objects from n.

    Permutations (order matters): nPr = n!/(n-r)!. Combinations (order does not matter): nCr = n!/(r!(n-r)!).

13. (HL) What are the modulus and argument of a complex number z = a + bi?

    Modulus: |z| = sqrt(a^2 + b^2). Argument: arg(z) = arctan(b/a) (adjusted for the correct quadrant).

14. (HL) State Euler's form and the modulus-argument (polar) form of a complex number.

    Polar form: z = r(cos theta + i sin theta) = r cis theta. Euler's form: z = r e^(i theta), where r = |z| and theta = arg(z).

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

    [r(cos theta + i sin theta)]^n = r^n (cos n theta + i sin n theta), or (r e^(i theta))^n = r^n e^(i n theta).

16. (HL) How do you find the nth roots of a complex number z = r e^(i theta)?

    The n roots are r^(1/n) e^(i(theta + 2 pi k)/n) for k = 0, 1, ..., n-1; they are equally spaced on a circle of radius r^(1/n).

17. What is the difference between the domain and the range of a function?

    Domain is the set of all permitted input (x) values; range is the set of all resulting output (y) values.

18. What is an inverse function f^(-1), and what is its graphical relationship to f?

    f^(-1) reverses f, so f^(-1)(f(x)) = x. Its graph is the reflection of y = f(x) in the line y = x; the inverse exists only if f is one-to-one.

19. Describe the effect of the transformations y = f(x) + a, y = f(x + a), y = a f(x), and y = f(ax).

    f(x) + a: translation a units up. f(x + a): translation a units left. a f(x): vertical stretch by factor a. f(ax): horizontal stretch by factor 1/a.

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