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Mathematics - furthe

Differentiation

Derivatives and gradient functions (5) The product rule and quotient rule (4) Derivatives of exponential and natural logarithmic functions (6) Stationary points and second derivatives (17) Maximum and minimum points (8) Implicit differentiation (13) Second order differential equations(homogeneous and inhomogeneous case) (1) Calculating Higher-Order Derivatives (2) Derivatives of trigonometric functions (7) Second derivatives (2) Parametric differentiation (6) Tangents and normals (1) Increasing and decreasing functions (1) Definite integration (1) Area problems (1) Asymptotes (vertical,horizontal,oblique) (1) Maclaurin series (1)

Intergration

Intergration using reduction formulae (1) Integration by parts (1)

Rational Functions and graphs

Asymptotes (vertical,horizontal,oblique) (40) Skethcing curves related functions (29) Inequatities (3)

Coordinate geometry

The equation of a circle (4) Problems involving intersections of lines and circles (8) The equation of a straight line (1)

Polar Coordinate

Convert between polar and Cartesian form (4) Sketching polar curves (14) The area of polar curve (9) Application of polar coordinates (3)

Rotation of a rigid body-Old Syl

Moment of Inertia (7) Parallel and perpendicular axes theorems (6) Conservation of Energy for Rotational Rigid Body (4) Equation of Angular Motion (2)

Bivariate Data

Product Moment Correlation Coefficient (31) Equations of Regression Line (27) Estimate by Regression Line (7) Hypothesis Test Based on PMCC (15) Scatter Diagram and PMCC (5) Least Square Estimation (12) Mean Center (x bar, b bar), and Relation between PMCC and b1,b2 that r2=b1b2 (5)

Chi-square test

Goodness of fit for discrete distribution (10) Goodness of fit for continuous distribution (5) Hypothesis of chi-square tests (8) Contingency table (4)

Inferential statistics

Tests about difference in means by using normal or t-distribution (11) Confidence interval (9) Tests about mean by using normal or t-distribution (3) Paired t-tests (4)

Quadratics

Solving quadratic equations (by factorisation, completing the square and the quadratic formula) (3) Solving simultaneous equations and more complex quadratic euqations (2) The number of roots of a quadratic equation (4) Intersection of a line and a quadratic curve (3) Solving quadratic inequalities (1)

Vector

Vector Equation of Line and Plane (9) The vector product (5) Find the distance from a point to a plane (4) Find angle between two planes (6) Intersection of two planes (4) Find the distance between two skew lines (4) Find the distance from a point to a line (1) Find the distance between two parallel lines (1)

Roots of polynomial equations

Cubics (38) Substitution (15) Quartics (9) Quadratics (12)

Summation of Series

Converging Series (21) The Summation Formula ∑r, ∑r 2 , ∑r 3 (14)

Continuous random variables

Probability density function (17) Cumulative distribution function (13) Expectation of function of continuous random variables (10) Finding pdf and cdf of Y=g(x) (13)

Equilibrium of rigid body

Moment (9) Objects in equilibrium (10) Toppling and sliding (6)

Integration

Integration by parts (5) Integration methods choosing (2) Integration using reduction formulae (1) Volumes of revolution (4) Integration by substitution (5) Definite integration (16) Arc lengths and surface areas (5) Area problems (6) Integration of sinx, cosx and sec2x and further trigonometric functions (2) Improper integrals (1) Concept of integration and indefinite integration (1) Integration formulae (1) Integration of $1/(x^2+a^2)$ (1)

Vectors

Displacement vectors (1) Position vectors (1) The scalar product (4) The angle between two lines (4) The vector euqation of a line (4) Find angle between two planes (1) Intersection of two lines (3) The vector product (1) Vector Equation of Line and Plane (1) The vector equation of a line (1)

The binomial and geometric distributions

The geometric distribution (3)

Differential equations

Second order differential equations(homogeneous and inhomogeneous case) (9) Use of substitutions to transform differential equations (1) Solving differential equations (3) Forming a differential equation from a problem (2) Integrating factor and first order differential equations (1)

Matrix

Eigenvalues and eigenvectors (10) Matrix algebra (2) Diagonalise matrices (4)

Simple Harmonic Motion-Old Syl

Definition of Simple Harmonic Motion and Period and Amplitude (7) Standard Simple Harmonic Motion Formula (4) Differential Equation and Simple Harmonic Motion (7) Approximation by Simple Harmonic Motion (2)

Circular motion

Vertical circle (2) Angular speed and central force (1) Radial and Transverse Components of Acceleration-Old Syl (4)

Momentum

Collision and convservation of momentum (1) Conservation of momentum (1)

Trigonometry

Trigonometric identities (1) Further trigonometric identities and equations (2) Trigonometric equations (6) Inverse trigonometric functions (2) Trigonometric ratios of general angles (2) Graphs of trigonometric functions and transformations (1)

Projectile

Motion in the vertical plane (1) Trajectory (1)

Distribution-Old Syl

Geometric Distribution and Negative Exponential Distribution (2) Means and Variance of Geometric Distribution and Negative Exponential Distribution (2)

Matrices

Determinants (13) Inverse matrix (10) System of equations (3) Matrix operations (9) Matrix transformations (6)

Complex numbers

Complex numbers and arithmetic operations (1) Solving equations with complex roots (6) The modulus-argument form and exponential form (6) Geometry of complex arithmetic operations (1) The nth roots of a complex number (2) De Moivre’s theorem (2)

Proof by induction

Proof by induction (2)

Series

Arthmetic progressions (2) Geometric progressions (3)

Complex Number

De Moivre’s theorem (1) Powers of sine and cosine (1)

Hookes law

Hookes law (1) Elastic potential energy (1) Work-energy principle (1)

Probability distributions

Discrete random variables (1) Probability distributions (1) Expectation and variance of a discrete random variable (1)

Circular measure

Arc length and area of a sector (3) Radians and degrees (1)

Functions

Definition of functions (1)

Logarithmic and exponential functions

Solving logarithmic equations, exponential equations and inequalities (2) Definition and the rules of logarithms (1)

Probability generating function

Probability generating function (6) Mean and variance by using pgf (6) Sum of independent random variables (3)

Measures of central tendency

The mean (1) The median (1)

Nonparametric tests

Wilcoxon matched-paire signed-rank test (2) Wilcoxon signed-rank (3) Paired sample signed test (1) Single sample sign test (1)

Algebra

The modulus function and its graph (1) Solving modulus inequalities (1)

Infinite Sequences and Series (BC Only)

Defining Convergent and Divergent Infinite Series (1) The nth Term Test for Divergence (1)

Mathematics

Complex numbers

Complex numbers and arithmetic operations (9) The complex plane and Argand diagrams (7) Geometry of complex arithmetic operations (6) Imaginary numbers (2) Solving equations with complex roots (10) The modulus-argument form and exponential form (3) De Moivre’s theorem (1)

Series

Arthmetic progressions (11) Geometric progressions (9) Binomial expansions (3) Arithmetic progressions (1)

Algebra

The factor theorem and remainder theorem (9) Division of polynomials (7) The modulus function and its graph (2) Solving modulus inequalities (2) General binomial expansions (for n that are not integers) (1)

Quadratics

Solving quadratic equations (by factorisation, completing the square and the quadratic formula) (8) The number of roots of a quadratic equation (7) The graphs of quadratic functions (3) Intersection of a line and a quadratic curve (3) Solving quadratic inequalities (2)

Roots of polynomial equations

Cubics (10) Quartics (6) Substitution (3) Quadratics (2)

Logarithmic and exponential functions

Definition and the rules of logarithms (14) Solving logarithmic equations, exponential equations and inequalities (17)

Trigonometry

Trigonometric equations (9) Compound angle formulae (6) Trigonometric ratios of general angles (6) Trigonometric identities (5) Graphs of trigonometric functions and transformations (3) The Sine and Cosine Rules (2) Double angle formulae (1) Inverse trigonometric functions (1) The equation of a circle (1)

Probability

Basic concepts in probability (18) Independent events and the multiplication law (11) Mutually exclusive events and the addition law (2) Conditional probability (6) The geometric distribution (1) Problem solving with permutations and combinations (2)

Permutations and combinations

Problem solving with permutations and combinations (34) The factorial function (10) Permutations (9) Combinations (11)

Coordinate geometry

The equation of a circle (12) Problems involving intersections of lines and circles (14) Parallel and perpendicular lines (3) Middle point, gradient and length of line segment (4) The equation of a straight line (1)

Functions

Inverse functions (3) Definition of functions (2) Transformations of functions and combining transformations (1) Composite functions (1)

Circular measure

Arc length and area of a sector (2) Radians and degrees (1)

Discrete Mathematics

Permutations (1) Combinations (1) Problem solving with permutations and combinations (1)

Summation of Series

The Summation Formula ∑r, ∑r 2 , ∑r 3 (1) Converging Series (1)

Vectors

The scalar product (2) The angle between two lines (1) Position vectors (1)

Integration

Definite integration (1) Area problems (1)

Trigonometric equations

Trigonometric equations (1) Graphs of trigonometric functions and transformations (1)

The modulus function and its graph

Solving modulus inequalities (1)

Number Theory

Primes, GCD and LCM (1)

Equilibrium of rigid body

Objects in equilibrium (6) Toppling and sliding (5) Moment (1)

Kinematics of motion in a straight line

Acceleration as the derivative of velocity with respect to time (1)

Differentiation

Connected rates of change and its application (1)
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