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Algebra

Complex numbers, quadratics, sequences and series, binomial theorem, and matrices

Key Concepts
  • Complex numbers: modulus, argument and the Argand plane
  • Quadratics: the discriminant decides the nature of the roots
  • Sequences and series: arithmetic and geometric progressions
  • Binomial theorem for any positive integral index
  • Matrices and determinants for solving linear systems
Important Formulae
Quadratic roots x = (−b ± √(b²−4ac))/2a
Sum & product of roots α+β = −b/a; αβ = c/a
AP sum S_n = n/2 [2a + (n−1)d]
GP sum S_n = a(rⁿ−1)/(r−1); S_∞ = a/(1−r), |r|<1
Binomial term T_{r+1} = ⁿC_r a^{n−r} b^r
Quick Tips
  • Discriminant: D>0 real & distinct, D=0 equal, D<0 complex conjugate roots.
  • |z|² = z·z̄; multiplying complex numbers adds their arguments.
  • The n nth-roots of unity sum to zero.
Sample Practice Questions

  1. The number of terms in the binomial expansion of (a + b)ⁿ is:

    • n
    • n + 1
    • n − 1
    • 2n
    Show answer

    Answer: n + 1

  2. The sum of the first 10 natural numbers is:

    • 45
    • 50
    • 55
    • 100
    Show answer

    Answer: 55

  3. The value of i² (where i is the imaginary unit) is:

    • 1
    • −1
    • i
    • 0
    Show answer

    Answer: −1

  4. The sum of the roots of x² − 7x + 12 = 0 is:

    • 7
    • 12
    • −7
    • 5
    Show answer

    Answer: 7

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Practice Questions

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