Lesson 9: Mathematics: Algebra — Complex Numbers, Quadratics, Sequences
Algebra in JEE is about structure: a quadratic is determined by its roots, a sequence by its first term and its rule, and a complex number by its modulus and argument. Once the structure is written down, the question usually reduces to one clean calculation.
This lesson covers complex numbers, quadratic equations, logarithms, and arithmetic and geometric sequences, with the identities that save the most time.
Complex numbers and quadratics
A complex number z = a + bi has modulus |z| = √(a² + b²) and conjugate a − bi. Remember i² = −1, so (1 + i)² = 1 + 2i + i² = 2i. For a quadratic ax² + bx + c = 0, the sum of roots is −b/a, the product is c/a, and the discriminant b² − 4ac decides whether roots are real and distinct, equal, or non-real.
For z = 3 + 4i: |z| = √(9 + 16) = √25 = 5.
For (1 + i)²: expand to 1 + 2i + i² = 1 + 2i − 1 = 2i.
For x² − 5x + 6 = 0: discriminant = 25 − 24 = 1, so roots are real and distinct; by sum 5 and product 6 they are 2 and 3.
Sequences and logarithms
An arithmetic sequence has aₙ = a₁ + (n − 1)d. A geometric sequence has aₙ = a₁rⁿ⁻¹. Logarithms convert multiplication into addition: log(ab) = log a + log b, and log₂(64) = 6 because 2⁶ = 64.
Arithmetic sequence with a₁ = 2 and d = 3: a₁₀ = 2 + 9 × 3 = 2 + 27 = 29.
Geometric sum 3 + 6 + 12 + 24 = 45 (ratio 2, four terms; check by formula 3(2⁴ − 1)/(2 − 1) = 3 × 15 = 45).
Common pitfalls
Do not forget i² = −1 when expanding complex products: dropping the minus sign is the commonest complex-number error.
Do not confuse the sequence index: a₁₀ uses (n − 1) = 9 steps from a₁, not 10.
Further study
- NCERT Class 11 Mathematics (complex numbers, quadratic equations, sequences and series) and Class 12 Mathematics (revision of algebra).
📝 Quiz
1. What is |3 + 4i|?
Show answer
b) √(9 + 16) = √25 = 5.
2. What is (1 + i)²?
Show answer
b) 1 + 2i + i² = 1 + 2i − 1 = 2i.
3. An arithmetic sequence has a₁ = 2, d = 3. What is a₁₀?
Show answer
b) 2 + 9 × 3 = 29.
4. What is 3 + 6 + 12 + 24?
Show answer
b) Geometric with ratio 2: 3 + 6 + 12 + 24 = 45.