Syllabus

IB Mathematics AA HL/SL Syllabus Overview

This is based on the IB Mathematics: Analysis and Approaches syllabus (first assessment 2021).

| Topic | Subtopic | Level | Key Understanding | Relevant Equations & Identities | | :-------------------------------- | :------------------------------------------- | :------ | :--------------------------------------------------------------------------------------------------------------------------- | :--------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------- | ------------------------------------------------------------------------------- | -------------------------------- | ------------------------------ | ------------- | --- | | 1. Number and Algebra | Arithmetic Sequences | SL & HL | Sequences with a constant common difference; finding the nth term and the sum of the first n terms. | u_n = u_1 + (n-1)d | | | | | | | | | Arithmetic Series | SL & HL | Sum formulae for arithmetic series in terms of first term, last term, common difference, and number of terms. | S_n = (n/2)(2u_1 + (n-1)d) = (n/2)(u_1 + u_n) | | | | | | | | | Geometric Sequences | SL & HL | Sequences with a constant common ratio; finding the nth term. | u_n = u_1 r^{n-1} | | | | | | | | | Geometric Series | SL & HL | Finite and infinite geometric series; convergence conditions for infinite series. | S_n = u_1(1 - r^n)/(1 - r); S_inf = u_1/(1 - r) for | r | < 1 | | | | | | | Sigma Notation | SL & HL | Using summation notation to express and evaluate series; rules for manipulating sums. | \sum_{r=1}^n u_r; \sum_{r=1}^n k = kn; \sum_{r=1}^n (a_r + b_r) = \sum a_r + \sum b_r | | | | | | | | | The Binomial Theorem | SL & HL | Expansion of (a + b)^n using binomial coefficients; Pascal's triangle and the relationship to combinations. | (a+b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r; \binom{n}{r} = n! / (r!(n-r)!) | | | | | | | | | Applications of the Binomial Theorem | SL & HL | Finding specific terms in a binomial expansion without fully expanding. | General term: T_{r+1} = \binom{n}{r} a^{n-r} b^r | | | | | | | | | Compound Interest | SL & HL | Calculating the future value of an investment or loan with compounding. | FV = PV(1 + r)^n where r is the rate per period and n is the number of periods | | | | | | | | | Compound Depreciation | SL & HL | Calculating the depreciated value of an asset over time. | FV = PV(1 - r)^n | | | | | | | | | Permutations | HL Only | Counting ordered arrangements of objects; counting with and without repetition. | ^nP_r = n!/(n-r)! | | | | | | | | | Combinations | HL Only | Counting unordered selections; applications to probability and combinatorial problems. | ^nC_r = n!/(r!(n-r)!) | | | | | | | | | Complex Numbers: Introduction | HL Only | The imaginary unit i where i^2 = -1; complex numbers in Cartesian form a + bi. | z = a + bi; i^2 = -1; i^3 = -i; i^4 = 1 | | | | | | | | | Complex Numbers: Operations | HL Only | Addition, subtraction, multiplication, and division of complex numbers; equating real and imaginary parts. | (a+bi)(c+di) = (ac-bd) + (ad+bc)i; division by multiplying by the conjugate | | | | | | | | | Modulus of a Complex Number | HL Only | The modulus is the distance from the origin to the point (a, b) on the Argand diagram. | | z | = \sqrt{a^2 + b^2} | | | | | | | Argument of a Complex Number | HL Only | The argument is the angle from the positive real axis to the line joining the origin to the point (a, b). | \arg(z) = \arctan(b/a) with quadrant adjustment; principal argument -\pi < \theta \le \pi | | | | | | | | | Polar Form | HL Only | Representing complex numbers in modulus-argument (polar) form on the Argand diagram. | z = r(\cos\theta + i\sin\theta) = r\,\mathrm{cis}\,\theta | | | | | | | | | Euler Form | HL Only | Euler's formula and the exponential representation of complex numbers. | e^{i\theta} = \cos\theta + i\sin\theta; z = re^{i\theta}; e^{i\pi} + 1 = 0 | | | | | | | | | De Moivre's Theorem | HL Only | Raising complex numbers to integer powers using polar form. | [r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta) | | | | | | | | | Roots of Complex Numbers | HL Only | Finding the nth roots of a complex number; roots lie equally spaced on a circle of radius r^{1/n}. | z_k = r^{1/n}[\cos((\theta + 2k\pi)/n) + i\sin((\theta + 2k\pi)/n)] for k = 0, 1, \ldots, n-1 | | | | | | | | | Complex Conjugate Root Theorem | HL Only | If a polynomial with real coefficients has a complex root a + bi, then a - bi is also a root. | If f(a+bi) = 0 then f(a-bi) = 0 for real coefficients | | | | | | | | | Sum and Product of Roots | HL Only | Relationship between coefficients and sums/products of roots for polynomial equations. | Quadratic: sum = -b/a, product = c/a; Cubic: sum = -b/a, sum of pairwise = c/a, product = -d/a | | | | | | | | | Proof by Contradiction | HL Only | Assuming the negation of a statement and arriving at a contradiction to prove the original statement. | N/A | | | | | | | | | Proof by Mathematical Induction | HL Only | Proving statements for all positive integers using the induction principle: base case, inductive hypothesis, inductive step. | N/A | | | | | | | | | Counterexamples | HL Only | Using a single counterexample to disprove a universal statement. | N/A | | | | | | | | | Direct Variation | HL Only | Modelling directly proportional relationships between two variables. | y = kx; y_1/x_1 = y_2/x_2 | | | | | | | | | Inverse Variation | HL Only | Modelling inversely proportional relationships between two variables. | y = k/x; x_1 y_1 = x_2 y_2 | | | | | | | | 2. Functions | Linear Functions | SL & HL | Equations and graphs of straight lines; gradient, intercepts, parallel and perpendicular lines. | y = mx + c; m = (y_2 - y_1)/(x_2 - x_1); perpendicular gradients: m_1 m_2 = -1 | | | | | | | | | Quadratic Functions: Standard Form | SL & HL | f(x) = ax^2 + bx + c; identifying the y-intercept and direction of opening. | y-intercept at (0, c); opens up if a > 0, down if a < 0 | | | | | | | | | Quadratic Functions: Vertex Form | SL & HL | f(x) = a(x - h)^2 + k; identifying the vertex, axis of symmetry, and optimal value. | Vertex at (h, k); axis of symmetry x = h | | | | | | | | | Quadratic Functions: Discriminant | SL & HL | The discriminant determines the number and nature of roots. | D = b^2 - 4ac; D > 0: two distinct real roots; D = 0: one repeated root; D < 0: no real roots | | | | | | | | | Exponential Functions | SL & HL | Properties and graphs; exponential growth and decay; the natural exponential function e^x. | y = a^x; y = e^x; horizontal asymptote at y = 0 | | | | | | | | | Logarithmic Functions | SL & HL | Definition as the inverse of exponentiation; domain and range; graphs of \log_a x. | y = \log_a x iff a^y = x; domain x > 0; passes through (1, 0) | | | | | | | | | Laws of Logarithms | SL & HL | Product, quotient, and power rules for logarithms; change of base formula. | \log_a(xy) = \log_a x + \log_a y; \log_a(x/y) = \log_a x - \log_a y; \log_a(x^n) = n\log_a x | | | | | | | | | Solving Equations | SL & HL | Solving equations analytically and graphically; intersection of graphs as solutions to f(x) = g(x). | N/A | | | | | | | | | Domain and Range | SL & HL | Identifying the domain (set of inputs) and range (set of outputs) of functions from their equations and graphs. | N/A | | | | | | | | | Even and Odd Functions | SL & HL | Symmetry properties: even functions are symmetric about the y-axis; odd functions have rotational symmetry about the origin. | Even: f(-x) = f(x); Odd: f(-x) = -f(x) | | | | | | | | | Horizontal Translations | SL & HL | Shifting graphs left or right by a constant. | y = f(x - b) shifts right by b units; y = f(x + b) shifts left by b units | | | | | | | | | Vertical Translations | SL & HL | Shifting graphs up or down by a constant. | y = f(x) + c shifts up by c units; y = f(x) - c shifts down by c units | | | | | | | | | Reflections | SL & HL | Reflecting graphs in the x-axis, y-axis, and the line y = x. | y = -f(x) (x-axis); y = f(-x) (y-axis); y = f^{-1}(x) (line y = x) | | | | | | | | | Stretches | SL & HL | Vertical and horizontal stretches and compressions of graphs. | y = af(x) (vertical stretch by factor a); y = f(bx) (horizontal stretch by factor 1/b) | | | | | | | | | Composite Functions | SL & HL | Combining two or more functions; finding the domain of a composite function. | (f \circ g)(x) = f(g(x)); domain of f \circ g is the set of x in domain of g where g(x) is in domain of f | | | | | | | | | Inverse Functions | SL & HL | Finding the inverse function algebraically; reflecting the graph of f in y = x to obtain f^{-1}. | f^{-1}(f(x)) = x; domain of f^{-1} = range of f; range of f^{-1} = domain of f | | | | | | | | | Polynomial Functions | SL & HL | Identifying polynomials by degree; determining end behaviour, intercepts, and turning points; sketching graphs. | f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 | | | | | | | | | Linear Inequalities | SL & HL | Solving and representing solutions to linear inequalities on a number line. | ax + b > 0; multiply/divide by negative reverses the inequality | | | | | | | | | Quadratic Inequalities | SL & HL | Solving quadratic inequalities using the discriminant and sign analysis. | Sign analysis of (x - \alpha)(x - \beta) > 0 or < 0 | | | | | | | | | Rational Inequalities | SL & HL | Solving inequalities involving rational expressions; critical values and test intervals. | Find zeros of numerator and denominator; test intervals; exclude values making denominator zero | | | | | | | | | Factor Theorem | HL Only | If (x - a) is a factor of f(x), then f(a) = 0; used to factorise polynomials. | f(a) = 0 implies (x - a) divides f(x); find integer roots among factors of constant term | | | | | | | | | Remainder Theorem | HL Only | The remainder when f(x) is divided by (x - a) equals f(a). | f(x) = (x - a)Q(x) + f(a); remainder when dividing by (ax - b) is f(b/a) | | | | | | | | | Rational Functions | HL Only | Functions of the form f(x) = p(x)/q(x); identifying vertical, horizontal, and oblique asymptotes. | Vertical asymptotes where q(x) = 0; horizontal asymptote from p(x)/q(x) as x \to \pm\infty | | | | | | | | | Domain Restriction for Inverses | HL Only | Restricting the domain of a non-injective function so that its inverse exists and is also a function. | N/A | | | | | | | | | Self-Inverse Functions | HL Only | Functions for which f = f^{-1}; their graphs are symmetric about y = x. | f(f(x)) = x; examples: f(x) = a - x, f(x) = 1/x | | | | | | | | | Reciprocal Function Graphs | HL Only | Sketching y = 1/f(x) from the graph of y = f(x); new asymptotes at zeros of f(x). | N/A | | | | | | | | | Absolute Value Graphs | HL Only | Sketching y = | f(x) | (reflect negative parts above x-axis) andy = f( | x | ) (reflect right side to left). | N/A | | | | | Modulus Equations | HL Only | Solving equations of the form | f(x) | = g(x)by considering cases for the sign off(x). | Split into f(x) = g(x) where f(x) \ge 0 and -f(x) = g(x) where f(x) < 0 | | | | | | | Modulus Inequalities | HL Only | Solving inequalities involving absolute value. | | x | < aiff-a < x < a; | x | > aiffx < -aorx > a | | | | 3. Geometry and Trigonometry | Distance in 3D Space | SL & HL | Calculating the distance between two points in three-dimensional space. | d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} | | | | | | | | | Midpoint in 3D Space | SL & HL | Finding the midpoint of a line segment in 3D. | M = ((x_1+x_2)/2, (y_1+y_2)/2, (z_1+z_2)/2) | | | | | | | | | Volume and Surface Area | SL & HL | Volume and surface area of cuboid, cylinder, cone, sphere, hemisphere, and composite solids. | V_{\mathrm{cyl}} = \pi r^2 h; V_{\mathrm{cone}} = (1/3)\pi r^2 h; V_{\mathrm{sphere}} = (4/3)\pi r^3; SA_{\mathrm{sphere}} = 4\pi r^2 | | | | | | | | | Right-Angled Trigonometry | SL & HL | SOH-CAH-TOA; finding unknown sides and angles in right-angled triangles. | \sin\theta = \mathrm{opp}/\mathrm{hyp}; \cos\theta = \mathrm{adj}/\mathrm{hyp}; \tan\theta = \mathrm{opp}/\mathrm{adj} | | | | | | | | | Trigonometric Functions and Graphs | SL & HL | Graphs of sin, cos, tan; period, amplitude, phase shift, and vertical shift. | y = a\sin(b(x - c)) + d; period = 2\pi/ | b | ; amplitude = | a |; phase shift = c | | | | | Sine Rule | SL & HL | Finding unknown sides and angles in any triangle. | a/\sin A = b/\sin B = c/\sin C; ambiguous case for SSA | | | | | | | | | Cosine Rule | SL & HL | Finding unknown sides and angles; generalised Pythagorean theorem for any triangle. | a^2 = b^2 + c^2 - 2bc\cos A; A = \cos^{-1}((b^2 + c^2 - a^2)/(2bc)) | | | | | | | | | Area of a Triangle | SL & HL | Formula using two sides and the included angle. | A = (1/2)ab\sin C | | | | | | | | | Pythagorean Identity | SL & HL | Fundamental trigonometric identity relating sin and cos. | \sin^2\theta + \cos^2\theta = 1 | | | | | | | | | Double Angle Identities | SL & HL | Expressions for \sin 2\theta, \cos 2\theta, and \tan 2\theta in terms of \sin\theta and \cos\theta. | \sin 2\theta = 2\sin\theta\cos\theta; \cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta | | | | | | | | | Vectors as Column Vectors | SL & HL | Representing vectors in component form; adding, subtracting, and scalar multiplication. | \mathbf{'\{'}v{'\}'} = \begin\{pmatrix\} a \\ b \end\{pmatrix\}; `\mathbf{'\{'}v{'\}'} + \mathbf{'\{'}w{'\}'} = \begin`\{pmatrix}` a+c \\ b+d \end`\{pmatrix} | | | | | | | | | Magnitude of a Vector | SL & HL | Calculating the length (magnitude) of a vector in 2D and 3D. | | \mathbf{'\{'}v{'\}'} | = \sqrt{a^2 + b^2}(2D); | \mathbf{'\{'}v{'\}'} | = \sqrt{a^2 + b^2 + c^2} (3D) | | | | | Unit Vectors | SL & HL | Vectors with magnitude 1; expressing a vector as a scalar multiple of a unit vector. | \hat{\mathbf{'\{'}v{'\}'}} = \mathbf{'\{'}v{'\}'}/ | \mathbf{'\{'}v{'\}'} | ; | \hat{\mathbf{'\{'}v{'\}'}} | = 1 | | | | | Scalar (Dot) Product | SL & HL | Definition in terms of components and in terms of magnitudes and angle; geometric interpretation as projection. | \mathbf{'\{'}v{'\}'} \cdot \mathbf{'\{'}w{'\}'} = a_1 b_1 + a_2 b_2 (+ a_3 b_3); \mathbf{'\{'}v{'\}'} \cdot \mathbf{'\{'}w{'\}'} = | \mathbf{'\{'}v{'\}'} | | \mathbf{'\{'}w{'\}'} | \cos\theta | | | | | Angle Between Two Vectors | SL & HL | Using the scalar product to determine the angle between two vectors. | \cos\theta = (\mathbf{'\{'}v{'\}'} \cdot \mathbf{'\{'}w{'\}'}) / ( | \mathbf{'\{'}v{'\}'} | | \mathbf{'\{'}w{'\}'} | ) | | | | | Perpendicularity of Vectors | SL & HL | Two vectors are perpendicular if and only if their scalar product is zero. | \mathbf{'\{'}v{'\}'} \perp \mathbf{'\{'}w{'\}'} iff \mathbf{'\{'}v{'\}'} \cdot \mathbf{'\{'}w{'\}'} = 0 | | | | | | | | | Vector Equation of a Line | SL & HL | Lines in two and three dimensions expressed in vector form with a position vector and direction vector. | \mathbf{'\{'}r{'\}'} = \mathbf{'\{'}a{'\}'} + \lambda\mathbf{'\{'}b{'\}'} where \mathbf{'\{'}a{'\}'} is on the line, \mathbf{'\{'}b{'\}'} is the direction vector | | | | | | | | | Parametric Line Equations | SL & HL | Expressing the coordinates of points on a line in terms of a parameter. | x = x_0 + \lambda a, y = y_0 + \lambda b, z = z_0 + \lambda c | | | | | | | | | Cartesian (Symmetric) Line Equations | SL & HL | Eliminating the parameter to obtain the symmetric form of a line equation. | (x - x_0)/a = (y - y_0)/b = (z - z_0)/c | | | | | | | | | Cross Product (Vector Product) | HL Only | The vector product of two vectors in 3D; result is a vector perpendicular to both inputs. | \mathbf{'\{'}v{'\}'} \times \mathbf{'\{'}w{'\}'} computed via determinant of a 3x3 matrix | | | | | | | | | Properties of the Cross Product | HL Only | Anti-commutativity; magnitude gives area of parallelogram; scalar triple product gives volume of parallelepiped. | \mathbf{'\{'}v{'\}'} \times \mathbf{'\{'}w{'\}'} = -(\mathbf{'\{'}w{'\}'} \times \mathbf{'\{'}v{'\}'}); | \mathbf{'\{'}v{'\}'} \times \mathbf{'\{'}w{'\}'} | = area of parallelogram | | | | | | | Equation of a Plane | HL Only | Cartesian, vector, and parametric forms of a plane in 3D. | \mathbf{'\{'}r{'\}'} \cdot \mathbf{'\{'}n{'\}'} = d; ax + by + cz = d; \mathbf{'\{'}r{'\}'} = \mathbf{'\{'}a{'\}'} + \lambda\mathbf{'\{'}b{'\}'} + \mu\mathbf{'\{'}c{'\}'} (parametric) | | | | | | | | | Intersection of Line and Plane | HL Only | Finding the point of intersection; determining whether a line is parallel to or lies in a plane. | Substitute parametric line equations into plane equation; solve for \lambda | | | | | | | | | Angle Between Line and Plane | HL Only | Finding the angle between a line and a plane using the direction vector and the normal vector. | \sin\theta = | \mathbf{'\{'}d{'\}'} \cdot \mathbf{'\{'}n{'\}'} | / ( | \mathbf{'\{'}d{'\}'} | | \mathbf{'\{'}n{'\}'} | ) | | | Angle Between Two Planes | HL Only | Finding the angle between two planes using their normal vectors. | \cos\theta = | \mathbf{'\{'}n{'\}'}\_1 \cdot \mathbf{'\{'}n{'\}'}\_2 | / ( | \mathbf{'\{'}n{'\}'}\_1 | | \mathbf{'\{'}n{'\}'}\_2 | ) | | | Complex Numbers on the Argand Diagram | HL Only | Representing complex numbers geometrically; the Argand plane with real and imaginary axes. | z = a + bi plotted at (a, b); multiplication: add arguments, multiply moduli | | | | | | | | | Compound Angle Identities | HL Only | Formulae for sine, cosine, and tangent of the sum and difference of two angles. | \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta; \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta | | | | | | | | | Reciprocal Trigonometric Functions | HL Only | Definitions and identities for sec, cosec, and cot. | \sec\theta = 1/\cos\theta; \cosec\theta = 1/\sin\theta; \cot\theta = \cos\theta/\sin\theta | | | | | | | | | Matrix Addition and Scalar Multiplication | HL Only | Adding and subtracting matrices of the same order; multiplying a matrix by a scalar. | (A + B)_\{ij\} = A_\{ij\} + B_\{ij\}; `(kA)_`\{ij}` = k \cdot A_`\{ij} | | | | | | | | | Matrix Multiplication | HL Only | Multiplying matrices; conditions for multiplication; identity matrix. | (AB)_\{ij\} = \sum_k A_\{ik\}B_\{kj\}; `AI = IA = A`; matrix multiplication is not commutative | | | | | | | | | Determinants (2x2 and 3x3) | HL Only | Calculating determinants; geometric interpretation as area/volume scale factor. | `\det\begin`\{pmatrix}` a & b \\ c & d \end`\{pmatrix}` = ad - bc`; 3x3 by cofactor expansion | | | | | | | | | Inverse of a 2x2 Matrix | HL Only | Formula for the inverse of a 2x2 matrix; verifying by multiplication. | `\begin`\{pmatrix}` a & b \\ c & d \end`\{pmatrix}`^{-1} = \frac{1}{ad-bc}\begin`\{pmatrix}` d & -b \\ -c & a \end`\{pmatrix} | | | | | | | | | Inverse of a 3x3 Matrix | HL Only | Finding the inverse using the adjugate (adjoint) matrix method. | A^{-1} = (1/\det A) \cdot \mathrm{adj}(A); A^{-1} exists iff \det A \ne 0 | | | | | | | | | Solving Systems with Inverse Matrices | HL Only | Solving A\mathbf{'\{'}x{'\}'} = \mathbf{'\{'}b{'\}'} by pre-multiplying by A^{-1}. | \mathbf{'\{'}x{'\}'} = A^{-1}\mathbf{'\{'}b{'\}'}; requires \det A \ne 0 (unique solution) | | | | | | | | | Augmented Matrices and Row Reduction | HL Only | Solving systems by Gaussian elimination; row operations on augmented matrices. | Row operations: swap rows, multiply row by scalar, add multiple of one row to another | | | | | | | | | Eigenvalues | HL Only | Scalar values \lambda satisfying A\mathbf{'\{'}v{'\}'} = \lambda\mathbf{'\{'}v{'\}'}; found by solving the characteristic equation. | \det(A - \lambda I) = 0; for 2x2: \lambda^2 - \mathrm{tr}(A)\lambda + \det(A) = 0 | | | | | | | | | Eigenvectors | HL Only | Non-zero vectors \mathbf{'\{'}v{'\}'} satisfying A\mathbf{'\{'}v{'\}'} = \lambda\mathbf{'\{'}v{'\}'}; finding for each eigenvalue. | Solve (A - \lambda I)\mathbf{'\{'}v{'\}'} = \mathbf{'\{'}0{'\}'} for each \lambda; eigenvectors are unique up to scalar multiples | | | | | | | | 4. Statistics and Probability | Populations and Samples | SL & HL | Distinguishing between populations (entire group) and samples (subset); representative sampling. | N/A | | | | | | | | | Sampling Methods | SL & HL | Simple random, systematic, stratified, and quota sampling; convenience, judgement, and voluntary response. | N/A | | | | | | | | | Reliability and Bias | SL & HL | Identifying sources of bias; understanding the effect of sample size on reliability. | N/A | | | | | | | | | Measures of Central Tendency | SL & HL | Mean, median, and mode; their relative merits for different data types (discrete vs continuous, skewed data). | \bar{x} = \sum x_i / n; median = middle value; mode = most frequent value | | | | | | | | | Grouped Data | SL & HL | Estimating the mean from grouped data using mid-interval values. | \bar{x} \approx \sum(f_i x_i) / \sum f_i where x_i is the midpoint of class i | | | | | | | | | Variance and Standard Deviation | SL & HL | Measuring the spread of data around the mean; using the unbiased estimator for sample variance. | s^2 = \sum(x_i - \bar{x})^2 / (n-1); s = \sqrt{s^2}; IQR = Q_3 - Q_1 | | | | | | | | | Histograms | SL & HL | Frequency histograms with equal and unequal class widths; frequency density. | Frequency density = frequency / class width; area of bar = frequency | | | | | | | | | Box-and-Whisker Plots | SL & HL | Five-number summary (min, Q1, median, Q3, max); identifying outliers and comparing distributions. | Outlier: value < Q_1 - 1.5 \times IQR or > Q_3 + 1.5 \times IQR | | | | | | | | | Cumulative Frequency Graphs | SL & HL | Constructing ogives; estimating median, quartiles, and percentiles from the graph. | N/A | | | | | | | | | Set Theory and Notation | SL & HL | Universal set, subsets, union, intersection, complement; two-set and three-set Venn diagrams. | P(A \cup B) = P(A) + P(B) - P(A \cap B); P(A') = 1 - P(A) | | | | | | | | | Conditional Probability | SL & HL | The probability of an event given that another event has occurred; restricted sample space. | P(A | B) = P(A \cap B) / P(B); P(A \cap B) = P(A | B)P(B) | | | | | | | Independence of Events | SL & HL | Two events are independent if the occurrence of one does not affect the probability of the other. | A \perp B iff P(A \cap B) = P(A)P(B) iff P(A | B) = P(A) | | | | | | | | Tree Diagrams and Sample Spaces | SL & HL | Using tree diagrams for sequential events; exhaustive sample spaces for equally likely outcomes. | N/A | | | | | | | | | Discrete Random Variables | SL & HL | Random variables that take countable values; probability mass functions (PMF). | Conditions: 0 \le p(x) \le 1 for all x; \sum p(x) = 1 | | | | | | | | | Expected Value | SL & HL | The long-run average value of a discrete random variable. | E(X) = \sum x_i p(x_i); linearity: E(aX + b) = aE(X) + b | | | | | | | | | Variance of a Discrete Random Variable | SL & HL | Measuring the spread of a discrete random variable about its mean. | Var(X) = E(X^2) - [E(X)]^2; Var(aX + b) = a^2 Var(X) | | | | | | | | | Binomial Distribution | SL & HL | Modelling the number of successes in n independent Bernoulli trials with probability p. | X \sim B(n, p); P(X = x) = \binom{n}{x}p^x(1-p)^{n-x}; E(X) = np; Var(X) = np(1-p) | | | | | | | | | Normal Distribution | SL & HL | Properties of the normal curve; symmetric about \mu; approximately 68-95-99.7 rule; standardisation. | X \sim N(\mu, \sigma^2); Z = (X - \mu)/\sigma where Z \sim N(0, 1) | | | | | | | | | Standard Normal Distribution | SL & HL | The standard normal Z; using tables or technology to find probabilities. | P(Z < z) from tables; P(a < Z < b) = P(Z < b) - P(Z < a) | | | | | | | | | Linear Regression: Equation of Line | SL & HL | Equation of the least squares regression line of y on x; interpreting slope and y-intercept. | y - \bar{y} = r(s_y/s_x)(x - \bar{x}); y = a + bx where b = r(s_y/s_x) | | | | | | | | | Pearson's Correlation Coefficient | SL & HL | Measuring the strength and direction of linear association; interpreting r and r^2. | r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i-\bar{x})^2 \sum(y_i-\bar{y})^2}}; -1 \le r \le 1 | | | | | | | | | Bayes' Theorem | HL Only | Reversing conditional probabilities; partitioning the sample space using the law of total probability. | P(A | B) = P(B | A)P(A) / P(B); P(B) = \sum_i P(B | A_i)P(A_i) | | | | | | Poisson Distribution | HL Only | Modelling the number of rare events in a fixed interval; Poisson as a limit of the binomial. | X \sim Po(\lambda); P(X = x) = e^{-\lambda}\lambda^x / x!; E(X) = Var(X) = \lambda | | | | | | | | | Probability Density Functions | HL Only | Continuous analogue of a PMF; f(x) \ge 0 and total area under the curve equals 1. | P(a \le X \le b) = \int_a^b f(x)\,dx; \int_{-\infty}^{\infty} f(x)\,dx = 1 | | | | | | | | | Cumulative Distribution Functions | HL Only | F(x) = P(X \le x); the CDF is the integral of the PDF; F'(x) = f(x). | F(x) = \int_{-\infty}^x f(t)\,dt; P(a < X < b) = F(b) - F(a) | | | | | | | | | Expected Value (Continuous) | HL Only | The mean of a continuous random variable; calculated by integration. | E(X) = \int_{-\infty}^{\infty} xf(x)\,dx; E(X^2) = \int_{-\infty}^{\infty} x^2 f(x)\,dx | | | | | | | | | Variance (Continuous) | HL Only | The variance of a continuous random variable. | Var(X) = E(X^2) - [E(X)]^2 | | | | | | | | | Median, Mode, Quartiles (Continuous) | HL Only | Finding the median, mode, and quartiles for continuous distributions. | Median: solve F(m) = 0.5; Q_1: solve F(q) = 0.25; Q_3: solve F(q) = 0.75 | | | | | | | | | Linear Transformation of a Variable | HL Only | Effect of Y = aX + b on mean, variance, and standard deviation. | E(Y) = aE(X) + b; Var(Y) = a^2 Var(X); \sigma_Y = | a | \sigma_X | | | | | | | Linear Combinations of Independent Variables | HL Only | Mean and variance of sums and differences of independent random variables. | E(aX + bY) = aE(X) + bE(Y); Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) for independent X, Y | | | | | | | | | Inverse Normal Distribution | HL Only | Given a probability, finding the corresponding value of X. | X = \mu + Z\sigma where Z is found from the inverse normal function | | | | | | | | | Student's t-Distribution | HL Only | Used when population standard deviation is unknown and sample size is small; degrees of freedom = n - 1. | CI for \mu: \bar{x} \pm t_{n-1,\alpha/2} \cdot s/\sqrt{n} | | | | | | | | | Hypothesis Testing: Concepts | HL Only | Null and alternative hypotheses; significance level \alpha; p-values; Type I and Type II errors. | H_0 vs H_1; reject H_0 if p-value < \alpha | | | | | | | | | Chi-Squared Test for Independence | HL Only | Testing whether two categorical variables are independent; observed vs expected frequencies. | \chi^2 = \sum (O_i - E_i)^2 / E_i; E_i = (\mathrm{row total} \times \mathrm{col total}) / \mathrm{grand total} | | | | | | | | | Chi-Squared Goodness of Fit | HL Only | Testing whether observed data fits a specified theoretical distribution. | Same formula; df = (k - 1 - p) where k = categories, p = estimated parameters | | | | | | | | 5. Calculus | Informal Understanding of Limits | SL & HL | The concept of a limit; understanding behaviour as x approaches a value or infinity; one-sided limits. | \lim_{x \to a} f(x) = L; \lim_{x \to a^+} vs \lim_{x \to a^-} | | | | | | | | | Continuity | SL & HL | A function is continuous at a point if the limit equals the function value; identifying discontinuities. | f is continuous at c if \lim_{x \to c} f(x) = f(c); types: removable, jump, infinite | | | | | | | | | Differentiation: Power Rule | SL & HL | The derivative of x^n; differentiating sums, differences, and constant multiples. | f(x) = x^n \implies f'(x) = nx^{n-1}; derivative of a constant is 0 | | | | | | | | | Derivatives of Exponential Functions | SL & HL | Differentiating e^x and a^x. | d/dx(e^x) = e^x; d/dx(a^x) = a^x \ln a; d/dx(e^{f(x)}) = f'(x)e^{f(x)} | | | | | | | | | Derivatives of Logarithmic Functions | SL & HL | Differentiating \ln x and \log_a x. | d/dx(\ln x) = 1/x; d/dx(\ln f(x)) = f'(x)/f(x) | | | | | | | | | Derivative of Trigonometric Functions | SL & HL | Differentiating \sin x, \cos x, and \tan x (angles in radians). | d/dx(\sin x) = \cos x; d/dx(\cos x) = -\sin x; d/dx(\tan x) = \sec^2 x | | | | | | | | | Product Rule | SL & HL | Differentiating the product of two functions. | (uv)' = u'v + uv' | | | | | | | | | Quotient Rule | SL & HL | Differentiating the quotient of two functions. | (u/v)' = (u'v - uv') / v^2 | | | | | | | | | Chain Rule | SL & HL | Differentiating composite functions. | dy/dx = (dy/du)(du/dx); d/dx[f(g(x))] = f'(g(x)) \cdot g'(x) | | | | | | | | | First Derivative Test | SL & HL | Using the sign of f'(x) to classify stationary points as local maxima or minima. | f'(x) changes + to -: local max; - to +: local min; no change: point of inflection | | | | | | | | | Second Derivative | SL & HL | Finding f''(x); using it to determine concavity and points of inflection. | f''(x) = d/dx[f'(x)]; f''(x) > 0: concave up; f''(x) < 0: concave down | | | | | | | | | Equation of a Tangent | SL & HL | Finding the gradient at a point and writing the equation of the tangent line. | y - y_0 = f'(x_0)(x - x_0); gradient = f'(x_0) | | | | | | | | | Equation of a Normal | SL & HL | The normal is perpendicular to the tangent; gradient is the negative reciprocal. | y - y_0 = -(1/f'(x_0))(x - x_0); condition: f'(x_0) \ne 0 | | | | | | | | | Points of Inflection | SL & HL | Points where the concavity changes; second derivative changes sign. | f''(x_0) = 0 and f''(x) changes sign at x_0 | | | | | | | | | Optimisation | SL & HL | Formulating and solving real-world problems involving maxima and minima. | Find f'(x) = 0, classify critical points, check endpoints if domain is restricted | | | | | | | | | Indefinite Integration | SL & HL | Antidifferentiation; the constant of integration C. | \int x^n\,dx = x^{n+1}/(n+1) + C for n \ne -1; \int 1/x\,dx = \ln | x | + C; \int e^x\,dx = e^x + C | | | | | | | Definite Integration | SL & HL | Evaluating the definite integral using the Fundamental Theorem of Calculus. | \int_a^b f(x)\,dx = [F(x)]_a^b = F(b) - F(a) | | | | | | | | | Area Under a Curve | SL & HL | Using definite integration to find the area between a curve and the x-axis. | A = \int_a^b f(x)\,dx when f(x) \ge 0; split at roots where f(x) changes sign | | | | | | | | | Area Between Two Curves | SL & HL | Finding the enclosed area between two functions over a given interval. | A = \int_a^b [f(x) - g(x)]\,dx where f(x) \ge g(x) on [a, b] | | | | | | | | | Trapezoidal Rule | SL & HL | Numerical approximation of a definite integral; accuracy increases with more trapezia. | \int_a^b f(x)\,dx \approx (h/2)[y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n] where h = (b-a)/n | | | | | | | | | Kinematics: Displacement and Velocity | SL & HL | Displacement, velocity, and acceleration as functions of time. | v = ds/dt; a = dv/dt = d^2s/dt^2; s = \int v\,dt; v = \int a\,dt | | | | | | | | | Kinematics: Total Distance Travelled | SL & HL | Finding total distance travelled from a velocity-time graph. | Total distance = \int_a^b | v(t) | \,dt; split where v(t) changes sign | | | | | | | Implicit Differentiation | HL Only | Differentiating equations where y is defined implicitly as a function of x. | d/dx[y^n] = ny^{n-1}\,dy/dx; d/dx[xy] = y + x\,dy/dx | | | | | | | | | Related Rates of Change | HL Only | Connecting the rates of change of related quantities using the chain rule. | dy/dt = (dy/dx)(dx/dt); differentiate both sides of a relation with respect to t | | | | | | | | | Integration by Substitution | HL Only | Using a substitution to simplify integrals; changing limits for definite integrals. | \int f(g(x))g'(x)\,dx = \int f(u)\,du where u = g(x), du = g'(x)\,dx | | | | | | | | | Integration by Parts | HL Only | Reversing the product rule; choosing u and dv strategically using LIATE. | \int u\,dv = uv - \int v\,du; repeated application when necessary | | | | | | | | | Definite Integrals Using Technology | HL Only | Evaluating definite integrals numerically using a graphing calculator (GDC). | N/A | | | | | | | | | Volumes of Revolution about the x-Axis | HL Only | Volume generated by rotating a region about the x-axis. | V_x = \pi\int_a^b [f(x)]^2\,dx | | | | | | | | | Volumes of Revolution about the y-Axis | HL Only | Volume generated by rotating a region about the y-axis. | V_y = \pi\int_c^d [g(y)]^2\,dy | | | | | | | | | Separable Differential Equations | HL Only | First-order ODEs of the form dy/dx = f(x)g(y); separating variables and integrating. | \int dy/g(y) = \int f(x)\,dx + C; verify solution by substitution | | | | | | | | | Maclaurin Series: Concept | HL Only | Expressing f(x) as an infinite polynomial series about x = 0; interval of convergence. | f(x) = \sum_{n=0}^{\infty} f^{(n)}(0)\,x^n / n! | | | | | | | | | Common Maclaurin Series | HL Only | Standard Maclaurin series for e^x, \sin x, \cos x, \ln(1+x), (1+x)^n. | e^x = 1 + x + x^2/2! + x^3/3! + \cdots; \sin x = x - x^3/3! + x^5/5! - \cdots | | | | | | | | | Integration by Partial Fractions | HL Only | Decomposing proper rational functions into partial fractions before integrating. | P(x)/Q(x) = A/(x-a) + B/(x-b) + \cdots; each term integrates to a log or power | | | | | | | | | L'Hopital's Rule | HL Only | Evaluating limits of indeterminate forms by differentiating numerator and denominator. | \lim_{x \to a} f(x)/g(x) = \lim_{x \to a} f'(x)/g'(x) for 0/0 or \infty/\infty forms | | | | | | | | | Improper Integrals: Infinite Limits | HL Only | Integrals where one or both limits are infinite; determining convergence or divergence. | \int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx; converges if the limit is finite | | | | | | | | | Improper Integrals: Unbounded Integrands | HL Only | Integrals where the integrand has a vertical asymptote within the interval. | \int_0^1 (1/\sqrt{x})\,dx = \lim_{a \to 0^+} \int_a^1 (1/\sqrt{x})\,dx = 2; split at the discontinuity | | | | | | | | | Convergence of Improper Integrals | HL Only | Determining whether an improper integral converges to a finite value or diverges. | \int_1^{\infty} (1/x^p)\,dx converges for p > 1, diverges for p \le 1 | | | | | | |

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Worked Examples

Worked Example: Arithmetic and Geometric Series (Topic 1)

An arithmetic sequence has first term 5 and common difference 3. A geometric sequence has first term 2 and common ratio 2. Find the smallest value of $n$ such that the sum of the first $n$ terms of the geometric sequence exceeds the sum of the first $n$ terms of the arithmetic sequence.

Solution

• Arithmetic sum: $S_n = \dfrac{n}{2}(2(5) + (n-1)(3)) = \dfrac{n}{2}(10 + 3n - 3) = \dfrac{n(3n + 7)}{2}$
• Geometric sum: $T_n = \dfrac{2(2^n - 1)}{2 - 1} = 2(2^n - 1) = 2^{n+1} - 2$
• We need $2^{n+1} - 2 \gt \dfrac{n(3n + 7)}{2}$.
• Testing values:
  • $n = 4$: geometric $= 30$, arithmetic $= \dfrac{4(19)}{2} = 38$. Geometric is smaller.
  • $n = 5$: geometric $= 62$, arithmetic $= \dfrac{5(22)}{2} = 55$. Geometric exceeds arithmetic.
• Answer: $n = 5$

Worked Example: Binomial Expansion (Topic 1)

Expand $(1 + 2x)^5$ in ascending powers of $x$ up to and including the term in $x^3$.

Solution

• Using the binomial theorem with $a = 1$, $b = 2x$, $n = 5$:
• $(1 + 2x)^5 = 1 + 5(2x) + 10(2x)^2 + 10(2x)^3 + \cdots$
• $= 1 + 10x + 40x^2 + 80x^3 + \cdots$

Worked Example: Transformations of Functions (Topic 2)

The graph of $y = f(x)$ passes through the point $(3, 5)$. Find the coordinates of the corresponding point on the graph of $y = -2f(x - 1) + 3$.

Solution

• Apply transformations step by step:
  1. $f(x) \to f(x - 1)$: translate right by 1: $(3, 5) \to (4, 5)$
  2. $\to 2f(x - 1)$: vertical stretch by factor 2: $(4, 5) \to (4, 10)$
  3. $\to -2f(x - 1)$: reflect in $x$-axis: $(4, 10) \to (4, -10)$
  4. $\to -2f(x - 1) + 3$: translate up by 3: $(4, -10) \to (4, -7)$
• Answer: $(4, -7)$

Worked Example: Quadratic Inequalities (Topic 2)

Solve the inequality $x^2 - 5x + 6 \lt 0$.

Solution

• Factorise: $x^2 - 5x + 6 = (x - 2)(x - 3)$
• The parabola opens upward (positive $x^2$ coefficient) with roots at $x = 2$ and $x = 3$.
• The expression is negative between the roots: $2 \lt x \lt 3$

Worked Example: Trigonometric Equations (Topic 3)

Solve $\cos 2x = \cos x$ for $0 \le x \le \pi$.

Solution

• Using $\cos 2x = 2\cos^2 x - 1$:
• $2\cos^2 x - 1 = \cos x \implies 2\cos^2 x - \cos x - 1 = 0$
• Let $u = \cos x$: $2u^2 - u - 1 = 0 \implies (2u + 1)(u - 1) = 0$
• $u = -1/2$ or $u = 1$
• $\cos x = -1/2 \implies x = 2\pi/3$ (only solution in $[0, \pi]$)
• $\cos x = 1 \implies x = 0$
• Solutions: $x = 0, 2\pi/3$

Worked Example: Vector Line Intersection (Topic 3, HL)

Find the point of intersection of the lines $\mathbf{'\{'}r{'\}'}_1 = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \lambda\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{'\{'}r{'\}'}_2 = \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix} + \mu\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}$.

Solution

• Equate components:
  • $1 + 2\lambda = 3 + \mu$ ... (i)
  • $2 - \lambda = \mu$ ... (ii)
  • $3\lambda = 1 - \mu$ ... (iii)
• From (i) and (ii): $1 + 2\lambda = 3 + 2 - \lambda \implies 3\lambda = 4 \implies \lambda = 4/3$
• From (ii): $\mu = 2 - 4/3 = 2/3$
• Verify with (iii): $3(4/3) = 4$ and $1 - 2/3 = 1/3$. These do not match, so the lines are skew (they do not intersect).

If you get this wrong, revise: Vector equations of lines and intersection conditions (Topic 3).

Worked Example: Probability with Venn Diagrams (Topic 4)

In a survey, 60 people were asked whether they liked tea ($T$) and coffee ($C$). 35 liked tea, 40 liked coffee, and 15 liked both. Find the probability that a randomly chosen person liked neither.

Solution

• $n(T \cup C) = n(T) + n(C) - n(T \cap C) = 35 + 40 - 15 = 60$
• Everyone liked at least one, so $n((T \cup C)') = 0$.
• $P(\mathrm{neither}) = 0/60 = 0$

Worked Example: Normal Distribution (Topic 4)

Heights of a population are normally distributed with mean $170$ cm and standard deviation $8$ cm. Find the probability that a randomly selected person is between $162$ cm and $178$ cm tall.

Solution

• Standardise:
  • $z_1 = (162 - 170)/8 = -1.00$
  • $z_2 = (178 - 170)/8 = 1.00$
• $P(162 \lt X \lt 178) = P(-1 \lt Z \lt 1) = \Phi(1) - \Phi(-1) = 0.8413 - 0.1587 = 0.6826$

Worked Example: Differentiation -- Quotient Rule (Topic 5)

Find the equation of the normal to the curve $y = \dfrac{x^2 + 1}{x - 1}$ at the point where $x = 2$.

Solution

• Quotient rule: $u = x^2 + 1$, $v = x - 1$. Then $u' = 2x$, $v' = 1$.
• $\dfrac{dy}{dx} = \dfrac{2x(x - 1) - (x^2 + 1)}{(x - 1)^2} = \dfrac{2x^2 - 2x - x^2 - 1}{(x - 1)^2} = \dfrac{x^2 - 2x - 1}{(x - 1)^2}$
• At $x = 2$: $\dfrac{dy}{dx} = \dfrac{4 - 4 - 1}{1} = -1$
• Point: $y(2) = (4 + 1)/(2 - 1) = 5$, so the point is $(2, 5)$.
• Normal gradient: $m_{\mathrm{normal}} = 1$ (negative reciprocal of $-1$).
• Normal equation: $y - 5 = 1(x - 2) \implies y = x + 3$

Worked Example: Volume of Revolution (Topic 5, HL)

Find the volume generated when the region bounded by $y = \sqrt{x}$, the $x$-axis, and $x = 4$ is rotated $360$ degrees about the $x$-axis.

Solution

• $V = \pi\displaystyle\int_0^4 (\sqrt{x})^2\,dx = \pi\displaystyle\int_0^4 x\,dx = \pi\left[\dfrac{x^2}{2}\right]_0^4 = \pi \times 8 = 8\pi$

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Notes by Level

Standard Level (SL)

The SL course covers Topics 1-5 at a foundational level. Students are expected to understand and apply core concepts in number and algebra, functions, geometry and trigonometry, statistics and probability, and calculus. Internal assessment (IA) counts for 20% of the final grade, and the remaining 80% is assessed through two written examination papers (Paper 1: non-calculator, Paper 2: calculator).

Higher Level (HL)

The HL course extends the SL content with additional subtopics in all five areas. HL students sit two additional papers (Paper 3: problem-solving with extended response questions) and cover more advanced techniques including complex numbers, matrices, further calculus, and hypothesis testing. The HL course demands deeper conceptual understanding and greater fluency with algebraic manipulation.

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Links to Existing Content

• Number and Algebra -- arithmetic and geometric sequences, sigma notation, binomial theorem
• Complex Numbers -- complex numbers in Cartesian, polar, and Euler form; De Moivre's theorem; roots of polynomials
• Trigonometry -- trigonometric functions, identities, sine and cosine rules
• Vectors -- vector operations, scalar and cross products, lines and planes
• Integration -- antiderivatives, definite integration, area, integration techniques
• Logic (Discrete Maths) -- propositional logic, truth tables, proof
• Maths Index -- overview of all mathematics notes

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Problem Set

Number and Algebra

1. Find the sum of the infinite geometric series $8 + 4 + 2 + 1 + \cdots$

Solution

• $u_1 = 8$, $r = 4/8 = 0.5$. Since $|r| \lt 1$, the sum converges.
• $S_\infty = \dfrac{8}{1 - 0.5} = 16$

If you get this wrong, revise: Sum to infinity of geometric series (Topic 1).

2. Find the coefficient of $x^4$ in the expansion of $(1 - 3x)^8$.

Solution

• General term: $\dbinom{8}{r}(-3x)^r$. For $x^4$, need $r = 4$.
• $\dbinom{8}{4}(-3)^4 = 70 \times 81 = 5670$

If you get this wrong, revise: Binomial theorem, finding specific coefficients (Topic 1).

Functions

3. The function $f(x) = 2x^3 - 3x^2 - 12x + 5$ has a local maximum and a local minimum. Find the coordinates of both.

Solution

• $f'(x) = 6x^2 - 6x - 12 = 6(x^2 - x - 2) = 6(x - 2)(x + 1)$
• Stationary points at $x = 2$ and $x = -1$.
• $f''(x) = 12x - 6$
• At $x = -1$: $f''(-1) = -18 \lt 0$ (local max). $f(-1) = -2 - 3 + 12 + 5 = 12$. Point: $(-1, 12)$.
• At $x = 2$: $f''(2) = 18 \gt 0$ (local min). $f(2) = 16 - 12 - 24 + 5 = -15$. Point: $(2, -15)$.

If you get this wrong, revise: First and second derivative tests (Topic 5).

4. Find the inverse of $f(x) = \dfrac{3x + 2}{x - 1}$ and state its domain.

Solution

• Let $y = \dfrac{3x + 2}{x - 1}$. Swap: $x = \dfrac{3y + 2}{y - 1}$.
• $x(y - 1) = 3y + 2 \implies xy - x = 3y + 2 \implies xy - 3y = x + 2$
• $y(x - 3) = x + 2 \implies f^{-1}(x) = \dfrac{x + 2}{x - 3}$
• Domain of $f^{-1}$: $x \neq 3$ (i.e. $\mathbb{'\{'}R{'\}'} \setminus \\{3\\}$).

If you get this wrong, revise: Finding inverse functions and domain restrictions (Topic 2).

Geometry and Trigonometry

5. Triangle $ABC$ has sides $a = 9$, $b = 7$, $c = 5$. Find the largest angle.

Solution

• The largest angle is opposite the longest side, so angle $A$:
• $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc} = \dfrac{49 + 25 - 81}{70} = \dfrac{-7}{70} = -0.1$
• $A = \arccos(-0.1) = 95.7$ degrees

If you get this wrong, revise: Cosine rule for finding an angle (Topic 3).

6. Find the area of a triangle with sides $6$ cm and $8$ cm and an included angle of $50$ degrees.

Solution

• $A = \dfrac{1}{2}ab\sin C = \dfrac{1}{2}(6)(8)\sin 50^\circ = 24 \times 0.7660 = 18.4$ cm$^2$

If you get this wrong, revise: Area of a triangle formula $A = \frac{1}{2}ab\sin C$ (Topic 3).

Probability and Statistics

7. A bag contains 4 red and 6 blue marbles. Three marbles are drawn without replacement. Find the probability that exactly two are red.

Solution

• Total ways to choose 3 from 10: $\dbinom{10}{3} = 120$
• Ways to choose 2 red from 4 and 1 blue from 6: $\dbinom{4}{2} \times \dbinom{6}{1} = 6 \times 6 = 36$
• $P = 36/120 = 3/10 = 0.3$

If you get this wrong, revise: Combinations and probability (Topics 1, 4).

8. Scores on a test are normally distributed with $\mu = 65$ and $\sigma = 10$. The top 10% of students receive an A grade. Find the minimum score for an A.

Solution

• We need the 90th percentile: $P(X \lt c) = 0.90$
• $P(Z \lt z) = 0.90 \implies z = 1.282$
• $c = \mu + z\sigma = 65 + 1.282 \times 10 = 65 + 12.82 = 77.8$
• Minimum score for an A: $78$ (rounding up).

If you get this wrong, revise: Inverse normal distribution and percentiles (Topic 4).

Calculus

9. Find $\displaystyle\int_0^{\pi/2} \sin x \cos x\,dx$.

Solution

• Let $u = \sin x$, $du = \cos x\,dx$:
• $\displaystyle\int_0^{\pi/2} u\,du = \left[\dfrac{u^2}{2}\right]_0^{\pi/2} = \left[\dfrac{\sin^2 x}{2}\right]_0^{\pi/2} = \dfrac{1}{2} - 0 = \dfrac{1}{2}$

If you get this wrong, revise: Integration by substitution (Topic 5).

10. Find the equation of the tangent to $y = \ln(x^2 + 1)$ at $x = 1$.

Solution

• $\dfrac{dy}{dx} = \dfrac{2x}{x^2 + 1}$ (chain rule)
• At $x = 1$: gradient $= 2/2 = 1$; $y = \ln 2$
• Tangent: $y - \ln 2 = 1(x - 1) \implies y = x - 1 + \ln 2$

If you get this wrong, revise: Chain rule and equation of a tangent (Topic 5).

11. Evaluate $\displaystyle\int_1^e \dfrac{\ln x}{x}\,dx$.

Solution

• Substitution: let $u = \ln x$, $du = dx/x$.
• When $x = 1$, $u = 0$; when $x = e$, $u = 1$.
• $\displaystyle\int_0^1 u\,du = \left[\dfrac{u^2}{2}\right]_0^1 = \dfrac{1}{2}$

If you get this wrong, revise: Integration by substitution (Topic 5).

12. Solve the differential equation $\dfrac{dy}{dx} = \dfrac{x}{y}$ given that $y = 2$ when $x = 1$.

Solution

• Separate variables: $y\,dy = x\,dx$
• Integrate: $\dfrac{y^2}{2} = \dfrac{x^2}{2} + C$
• Apply initial condition: $4/2 = 1/2 + C \implies C = 3/2$
• $y^2 = x^2 + 3$
• Since $y = 2 \gt 0$ at $x = 1$: $y = \sqrt{x^2 + 3}$

If you get this wrong, revise: Separable differential equations (HL, Topic 5).