Maths IIA — Formula Sheet
Key formulas for all 10 chapters of TS Intermediate 2nd Year Mathematics IIA as per the TGBIE Annual Plan 2025-26. Paper: 75 marks · 3 hours.
1.
Complex Numbers
- z = a + ib where a = Re(z), b = Im(z)
- |z| = √(a² + b²) — modulus of z
- arg(z) = θ = tan⁻¹(b/a) — amplitude (principal value: −π < θ ≤ π)
- Polar form: z = r(cos θ + i sin θ) = r cis θ
- Conjugate: z̄ = a − ib; |z|² = z · z̄
- z₁z₂: |z₁z₂| = |z₁||z₂|, arg(z₁z₂) = arg(z₁) + arg(z₂)
2.
De Moivre's Theorem
- (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ (for integer n)
- (cos θ + i sin θ)^(p/q) = cos(pθ/q) + i sin(pθ/q) (rational p/q)
- nth roots of unity: ωᵏ = cos(2kπ/n) + i sin(2kπ/n), k = 0,1,...,n−1
- Sum of all nth roots of unity = 0
- Product of all nth roots of unity = (−1)^(n−1)
3.
Quadratic Expressions
- ax² + bx + c = 0; roots α, β: α + β = −b/a, αβ = c/a
- Discriminant: Δ = b² − 4ac; Δ > 0: real distinct; Δ = 0: equal; Δ < 0: complex
- ax² + bx + c > 0 for all x iff a > 0 and Δ < 0
- ax² + bx + c < 0 for all x iff a < 0 and Δ < 0
- Maximum value of ax² + bx + c (a < 0): −Δ/(4a) at x = −b/(2a)
- Minimum value (a > 0): −Δ/(4a) at x = −b/(2a)
4.
Theory of Equations
- nth degree equation: aₙxⁿ + … + a₀ = 0; sum of roots = −aₙ₋₁/aₙ
- Sum of products of roots taken r at a time = (−1)ʳ · aₙ₋ᵣ / aₙ
- Reciprocal equation: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ = 0 with aₖ = aₙ₋ₖ
- If α + iβ is a root (a,b real), then α − iβ is also a root
- If α + √β is a root (rational coefficients), then α − √β is also a root
5.
Permutations and Combinations
- ⁿPᵣ = n!/(n−r)! — number of permutations of n things taken r at a time
- ⁿCᵣ = n!/[r!(n−r)!] — combinations; ⁿCᵣ = ⁿCₙ₋ᵣ
- Circular permutations of n distinct things: (n−1)!
- Permutations with repetitions: if p, q, r are alike → n!/(p!q!r!)
- ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal's identity)
6.
Binomial Theorem
- (x + a)ⁿ = Σ ⁿCᵣ xⁿ⁻ʳ aʳ (r = 0 to n), n ∈ ℕ
- General term: Tᵣ₊₁ = ⁿCᵣ xⁿ⁻ʳ aʳ
- Middle term(s): T_(n/2+1) if n even; T_((n+1)/2) and T_((n+3)/2) if n odd
- Binomial for rational index (|x| < 1): (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + …
- (1+x)⁻¹ = 1 − x + x² − x³ + …
- (1−x)⁻¹ = 1 + x + x² + x³ + …
7.
Partial Fractions
- Non-repeated linear factor (ax+b): A/(ax+b)
- Repeated linear factor (ax+b)²: A/(ax+b) + B/(ax+b)²
- Irreducible quadratic (ax²+bx+c): (Ax+B)/(ax²+bx+c)
- Method: clear denominators, equate coefficients or substitute roots
8.
Measures of Dispersion
- Range = Maximum value − Minimum value
- Mean Deviation (from mean x̄): MD = (1/n) Σ|xᵢ − x̄|
- Variance: σ² = (1/n) Σ(xᵢ − x̄)²
- Short-cut formula: σ² = (1/n) Σxᵢ² − x̄²
- Standard Deviation: σ = √(variance)
- Coefficient of Variation: CV = (σ/x̄) × 100%
9.
Probability
- P(A) = n(A)/n(S); 0 ≤ P(A) ≤ 1; P(S) = 1
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- If A, B mutually exclusive: P(A ∪ B) = P(A) + P(B)
- Conditional probability: P(A|B) = P(A ∩ B)/P(B)
- Multiplication theorem: P(A ∩ B) = P(A) · P(B|A)
- If independent: P(A ∩ B) = P(A) · P(B)
- Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ)
10.
Random Variables & Probability Distributions
- Mean (Expectation): E(X) = μ = Σ xᵢ P(xᵢ)
- Variance: Var(X) = E(X²) − [E(X)]² = Σ xᵢ² P(xᵢ) − μ²
- Binomial Distribution: P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ, where q = 1 − p
- Binomial mean = np; Variance = npq
- Poisson Distribution: P(X=k) = e⁻ᵘ · uᵏ / k!, where u = np
- Poisson mean = variance = u