Module 1: The Real Number System

• Sets of Numbers Natural (1, 2, 3...), Integers (...-1, 0, 1...), Rational (fractions), Irrational (non-repeating decimals like pi), Real (all rational and irrational).
• Absolute Value The distance of a number from zero. |x| is x if x is positive, -x if x is negative.

Module 2: Elementary Set Theory

• Set Operations Union (A or B), Intersection (A and B), Complement (Not in A).
• Cardinality Principle n(A U B) = n(A) + n(B) - n(A ∩ B)

Module 3: Quadratic Equations (ax² + bx + c = 0)

• Quadratic Formula x = [-b ± sqrt(b² - 4ac)] / 2a
• Discriminant (D) D = b² - 4ac
  If D > 0, two distinct real roots.
  If D = 0, one repeated real root.
  If D < 0, two complex roots.
• Sum and Product of Roots (alpha, beta) Sum: alpha + beta = -b/a
  Product: alpha * beta = c/a

Module 4: Sequences and Series

• Arithmetic Progression (A.P.) nth Term: a_n = a + (n-1)d
  Sum of n terms: S_n = (n/2) * [2a + (n-1)d]
• Geometric Progression (G.P.) nth Term: a_n = a * r^(n-1)
  Sum of n terms: S_n = a * (1 - r^n) / (1 - r)
  Sum to Infinity: S_inf = a / (1 - r) (where |r| < 1)
• Harmonic Progression (H.P.) A sequence whose reciprocals form an A.P.

Module 5: Mathematical Induction

• Principle of Proof 1. Base Case: Prove the statement is true for n=1.
  2. Inductive Hypothesis: Assume the statement is true for n=k.
  3. Inductive Step: Prove the statement is true for n=k+1, using the assumption.

Module 6: The Binomial Theorem

• Expansion of (a+b)ⁿ The terms are found using combinations. The (r+1)th term is: nCr * a^(n-r) * b^r, where nCr is "n choose r".
• First few terms of (1+x)ⁿ 1 + nx + [n(n-1)/2]x² + [n(n-1)(n-2)/6]x³ + ...

Module 7: Trigonometry

• Angle Conversion 180 degrees = pi radians
• Pythagorean Identities sin²(x) + cos²(x) = 1
1 + tan²(x) = sec²(x)
1 + cot²(x) = csc²(x)
• Sum and Difference Formulas sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
• Double Angle Formulas sin(2A) = 2sin(A)cos(A)
cos(2A) = cos²(A) - sin²(A)

Module 8: Complex Numbers (z = a + bi)

• Modulus (r or |z|) |z| = sqrt(a² + b²)
• Polar Form z = r * (cos(theta) + i*sin(theta))
• De Moivre's Theorem [r*(cos(theta)+i*sin(theta))]^n = r^n*(cos(n*theta)+i*sin(n*theta))