## Math AA HL Spring Group Course

7 Max. Students

NY: 08:00-12:30, UK: 13:00-17:30, Dubai: 16:00-20:30 (4.5 Hours a day incl. 1/2 an hour break) Join our expert tutor, Susovan as he provides you with Math AA HL Spring Revision Tuition. See our detailed Curriculum breakdown below.

For problems using several concepts (e.g. integral of trig functions), they can be covered in either of the three days

**Important:** The lesson will solve a lot of actual IB problems!!!

**Day 1:**

- Introduction: Revision of prerequisites: number systems, integers, positive and negative integers, fractions and their decimal representations, rational and irrational numbers, and their symbols wherever needed. Modulus of a number. Examples.
- Number and Algebra I:

* Laws of exponents with integer exponents, rational exponents.

* Laws of logarithms.

* Solving exponential equations, including using logarithms.

*Operations with numbers in the form a × 10^k where 1 ≤ a < 10 and k is an integer. - Number and Algebra II:

*Arithmetic and Geometric sequences and series with finite numbers of terms: Use of the formulae for the n-th term and the sum of the first n terms of the sequence. Use of sigma notation for sums of arithmetic sequences. Applications - compound interests.

* Sum of infinite convergent geometric sequences - Number and Algebra III:

Permutations, Combinations, and the Binomial Theorem with integer indices: expansion of (a + b)^n , n ∈ ℕ. Counting principles, including permutations and combinations. Extension of the binomial theorem to fractional and negative indices, ie (a + b)^n , n ∈ ℚ. - Number and Algebra IV: Familiarity with the existence of equation x² + 1= 0 and its solution as the complex number i or -i. The complex plane. Representation of complex numbers as z=x + iy and re^{iθ}. Addition, subtraction, multiplication, division of complex numbers. Modulus and arguments. DeMoivre’s theorem.
- Number and Algebra V:

*Simple deductive proof, numerical and algebraic; how to layout a left-hand side to right-hand side (LHS to RHS) proof. The symbols and notation for equality and identity. Proof by mathematical induction, proof by contradiction. Use of a counterexample to show that a statement is not always true. - Number and Algebra VI: Basic familiarity with the fact that there are systems of linear equations. Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions, or no solution.

**Day 2:**

- Functions I:

*Concept of a function, domain, range and graph. Function notation,

*The graph of a function, Determine key features of graphs, transformations of graphs.

*Composite functions, identity functions, inverse functions

*Solutions of g(x) ≥ f(x), both graphically and analytically. The graphs of the functions, y = |f(x)| and y = f(|x|), y = 1/f(x) , y = f(ax + b), y = [f(x)]^2 . - Functions II:

*Linear functions and straight lines: Different forms of the equation of a straight line. Gradient; intercept.

*The quadratic function f(x) = ax² + bx + c: its graph, y-intercept (0, c). Axis of symmetry. The form f(x) = a(x − p)(x − q), x-intercepts (p, 0) and (q, 0). The form f(x) = a (x − h)²+ k, vertex (h, k). Solution of quadratic equations and inequalities. The quadratic formula. The discriminant Δ = b²− 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots. - Functions III: The reciprocal function f(x) = 1/x , x ≠ 0: its graph and self-inverse nature. Rational functions and their graphs. Equations of vertical and horizontal asymptotes.

Polynomial functions, their graphs and equations; zeros, roots, and factors. The factor and remainder theorems. Rational functions. Odd and even functions. - Geometry and trigonometry I:

*Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere, and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane. - Geometry and trigonometry II:

*Use of sine, cosine, and tangent ratios to find the sides and angles of right-angled triangles. The sine rule, cosine rule, area of a triangle.

*Applications of right and non-right angled trigonometry, including Pythagoras’s theorem. Angles of elevation and depression.

*The circle: radian measure of angles; length of an arc; area of a sector.

*Definition of cosθ, sinθ in terms of the unit circle. Definition of tanθ as sinθ/cosθ. Exact values of trigonometric ratios of 0, π/6 , π/4 , π/3 , π/2 and their multiples. Extension of the sine rule to the ambiguous case.

*The Pythagorean identity cos² θ + sin² θ = 1. Double angle identities for sine and cosine.

*Definition of the reciprocal trigonometric ratios secθ, cosecθ, and cotθ. Pythagorean identities: 1 + tan²θ = sec²θ, 1 + cot² θ = cosec² θ The inverse trigonometric functions f(x) = arcsinx, f(x) = arccosx, f(x) = arctanx; their domains and ranges; their graphs. Compound angle identities. Double angle identity for tan.Relationships between trigonometric functions and the symmetry properties of their graphs. - Geometry and trigonometry III:

*Basic concept of what a vector is; vectors in depth. position vectors; displacement vectors.

Representation of vectors using directed line segments. Basis vectors. Dot/inner and cross/vector product of two vectors. Perpendicular and parallel vectors.

*Vector equation of a line in two and three dimensions: r = a + λb. The angle between lines. Application to kinematics. Vector equation of a plane. Angles between lines and planes.

**Day 3:**

- Statistics and Probability:

*Concepts of population, sample, random sample, discrete and continuous data. Outliers.

*Presentation of data (discrete and continuous): frequency distributions (tables). Histograms, cumulative frequencies.Median, quartile, percentiles etc. Box and whisker plots.

*Measures of central tendency (mean, median, and mode).

*Measures of dispersion (interquartile range, standard deviation, and variance).

*Linear correlation of bivariate data. Pearson’s product-moment correlation coefficient.

*Scatter diagrams; lines of best fit, by eye, passing through the mean point.

*Regression, regression lines, and their use for prediction purposes. Interpretation of parameters of regression lines.

* Probability, random samples, conditional probability, independent events, discrete random variables, binomial distribution, its mean, and variance.

*Use of Bayes’ theorem for a maximum of three events.

*Normal distribution, its curve, normal and inverse normal calculations. Bayes’ theorem. The variance of a discrete random variable. Continuous random variables X and their probability density functions. Mean median mode etc. Effect of linear transformations on X. - Calculus I:

*Limits and continuity. Informal understanding of continuity and differentiability of a function at a point.Understanding of limits (convergence and divergence). Definition of derivative from first principles.

*Derivatives, derivatives of power functions, and polynomials. Product, quotient, chain rules.The second derivative. Higher derivatives. Implicit differentiation.

*Increasing and decreasing functions. Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0.

*Tangents and normals at a given point, and their equations.

* Local maximum and minimum. - Calculus II:

*Indefinite integrals. The indefinite integral of x^n (n ∈ ℚ), sin x, cos x, 1/x, and e^x. Integration by substitution, integration by parts.

*Definite integrals, including analytical approach.

*Areas of a region enclosed by a curve. Areas between curves. - Calculus III: Basic knowledge of what a differential equation is.

*Differential equations in-depth: First order differential equations. Numerical solution of dy/dx = f(x, y) using Euler’s method.

* Separation of variables, homogeneous differential equations. Integrating factor.

*Maclaurin series to obtain expansions for e^x , sinx, cosx, ln(1 + x), (1 + x) p , p ∈ ℚ. Maclaurin series developed from differential equations.

**"Full past paper practice"**