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NDA Mathematics: High-Yield Algebra Summary

Algebra contributes a major chunk of questions in the NDA Mathematics paper. Mastering core definitions, identities, and algebraic properties is essential for rapid problem-solving.

1. Complex Numbers

A complex number is expressed as \(z = a + ib\), where \(i = \sqrt{-1}\). Key identities and formulas include:

• Modulus: \(|z| = \sqrt{a^2 + b^2}\)
• Conjugate: \(\bar{z} = a - ib\), with property \(z \cdot \bar{z} = |z|^2\)
• Euler Formula: \(e^{i\theta} = \cos\theta + i\sin\theta\)
• De Moivre Theorem: \((\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)\)

2. Quadratic Equations

For the standard quadratic equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\):

• Discriminant: \(D = b^2 - 4ac\). Real and distinct roots occur when \(D > 0\); real and equal roots when \(D = 0\); complex roots when \(D < 0\).
• Sum of Roots: \(\alpha + \beta = -b/a\)
• Product of Roots: \(\alpha\beta = c/a\)

3. Sequences and Series

Arithmetic, Geometric, and Harmonic Progressions form standard series questions:

• Arithmetic Progression (AP): \(n\)-th term \(a_n = a + (n-1)d\); Sum \(S_n = (n/2)[2a + (n-1)d]\).
• Geometric Progression (GP): \(n\)-th term \(a_n = a \cdot r^{n-1}\); Sum of infinite GP \(S_{\infty} = a/(1-r)\) for \(|r| < 1\).
• AM-GM-HM Relation: \(AM \ge GM \ge HM\).

4. Permutations and Combinations

• Permutations (Arrangements): \(^nP_r = n! / (n-r)!\)
• Combinations (Selections): \(^nC_r = n! / [r!(n-r)!]\)
• Identity: \(^nC_r + ^nC_{r-1} = ^{n+1}C_r\)

5. Binomial Theorem

The expansion of \((x + y)^n\) is given by:

\[(x + y)^n = \sum_{k=0}^{n} ^nC_k x^{n-k} y^k\]

The general term is \(T_{k+1} = ^nC_k x^{n-k} y^k\). Total number of terms in expansion is \(n + 1\).