Module 1: The Real Number System
Estimated Study Time: 1-2 hours
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Number Systems Hierarchy:
- Natural Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers.
- Absolute Value: The absolute value of x is x if x is 0 or greater, and -x if x is less than 0.
Module 2: Elementary Set Theory
Estimated Study Time: 3-5 hours
- Set Operations: Intersection (A and B), Union (A or B), Complement (Not A).
- Cardinality Principle: The number of elements in the union of A and B equals the sum of elements in A and B, minus the elements in their intersection.
Module 3: Theory of Quadratic Equations
Estimated Study Time: 4-6 hours
- Standard Form: ax2 + bx + c = 0, where 'a' is not zero.
- Quadratic Formula: x equals [-b ± sqrt(b2 - 4ac)] / 2a.
- Discriminant (b2 - 4ac): Determines if the roots are real and distinct, real and equal, or complex.
- Sum and Product of Roots: The sum of roots is -b/a; the product is c/a.
Module 4: Sequences and Series
Estimated Study Time: 5-7 hours
- Arithmetic Progression (A.P.): The nth term is a + (n-1)d. The sum of n terms is (n/2) * [2a + (n-1)d].
- Geometric Progression (G.P.): The nth term is a*r(n-1). The sum to infinity is a / (1-r), for |r| < 1.
Module 5: Mathematical Induction
Estimated Study Time: 2-4 hours
A proof technique with three steps: Base Case, Inductive Hypothesis, and Inductive Step.
Module 6: The Binomial Theorem
Estimated Study Time: 3-5 hours
- For Positive Integer n: Formula for expanding expressions of the form (x+a)n.
- For Real n: Formula for expanding (1+x)n, valid for |x| < 1.
Module 7: Trigonometry
Estimated Study Time: 6-8 hours
- Key Identities: sin2(theta) + cos2(theta) = 1. Addition and double angle formulas.
- Formulas (theta in radians): Arc Length (l = r*theta), Sector Area (A = 0.5 * r2 * theta).
Module 8: Complex Numbers
Estimated Study Time: 6-8 hours
- Standard Form: z = a + bi, where i is the square root of -1.
- Polar Form: z = r * (cos(theta) + i*sin(theta)).
- De Moivre's Theorem: For finding powers and roots of complex numbers.