NDA Algebra Cheat Sheet & Revision Notes

Mastering Algebra is pivotal for clearing the Mathematics paper in the Union Public Service Commission (UPSC) National Defence Academy (NDA) examination. The NDA Mathematics paper consists of 120 questions, out of which Algebra accounts for roughly 25-30% of the total weightage. This comprehensive, LibreTexts-style revision guide covers all fundamental rules, special expansions, roots of equations, logarithms, complex numbers, and progression formulas derived directly from standard exam patterns.

1. Exponents and Radicals

Exponent rules and radical simplifications form the foundation of algebraic manipulation in NDA problems, especially when solving logarithmic equations or indices-based simplifications.

Laws of Exponents

For any real bases \(a\) and \(b\) with positive integers \(m\) and \(n\):

• Product Rule: \[a^m \cdot a^n = a^{m+n}\]
• Quotient Rule: \[\frac{a^m}{a^n} = a^{m-n}\]
• Power Rule: \[(a^m)^n = a^{mn}\]
• Power of a Product: \[(ab)^n = a^n b^n\]
• Negative Exponent: \[a^{-n} = \frac{1}{a^n}\]
• Zero Exponent: \[a^0 = 1 \quad (a eq 0)\]

Properties of Radicals

Radical operations streamline expressions involving roots:

• Product of Radicals: \[\sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}\]
• Quotient of Radicals: \[\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\]
• Nested Radicals: \[\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}\]
• Rational Exponent Representation: \[a^{m/n} = \sqrt[n]{a^m}\]

2. Factoring, Products, and Binomial Theorem

Special Algebraic Products

Memorizing algebraic identities speeds up factorizations in complex NDA questions:

• Square Identities: \[(a \pm b)^2 = a^2 \pm 2ab + b^2\]
• Difference of Two Squares: \[a^2 - b^2 = (a-b)(a+b)\]
• Cube Identities: \[(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3\]
• Sum and Difference of Cubes: \[a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\]

The Binomial Theorem

For any positive integer \(n\), the expansion of \((a+b)^n\) is given by:

\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\]

Where the combination formula for the binomial coefficient \(\binom{n}{k}\) is:

\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

3. Quadratic Equations and Discriminant Analysis

Quadratic equations regularly feature in multiple statement-based questions in the UPSC NDA examination.

A standard quadratic equation is expressed as \(ax^2 + bx + c = 0\), where \(a eq 0\). The roots \(x\) are calculated using the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Discriminant Analysis (\(\Delta\))

The nature of the roots is governed by the discriminant \(\Delta = b^2 - 4ac\):

• \(\Delta > 0\): Two distinct real roots.
• \(\Delta = 0\): One repeated (real) root.
• \(\Delta < 0\): Two complex conjugate roots.

4. Logarithmic Functions and Properties

Logarithmic identities are essential for simplifying non-linear relations and exponent equations.

Definition

\[\log_b(x) = y \iff b^y = x \quad (b > 0, b eq 1, x > 0)\]

Fundamental Logarithmic Rules

• Product Rule: \[\log_b(xy) = \log_b(x) + \log_b(y)\]
• Quotient Rule: \[\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)\]
• Power Rule: \[\log_b(x^k) = k \log_b(x)\]
• Identity & Base Rules: \[\log_b(1) = 0, \quad \log_b(b) = 1\]
• Change of Base Formula: \[\log_b(x) = \frac{\log_c(x)}{\log_c(b)}\]

5. Complex Numbers and Euler's Identity

A complex number \(z\) is written in standard form as \(z = a + bi\), where \(a, b \in \mathbb{R}\) and \(i = \sqrt{-1}\) (i.e., \(i^2 = -1\)).

Key Complex Number Concepts

• Complex Conjugate: \(\bar{z} = a - bi\)
• Modulus (Magnitude): \(|z| = \sqrt{a^2 + b^2}\)
• Euler's Formula: \[e^{i\theta} = \cos\theta + i\sin\theta\]

6. Sequences and Series (AP and GP)

Arithmetic Progression (AP)

A sequence where consecutive terms differ by a constant value \(d\).

• \(n\)-th Term Formula: \[a_n = a_1 + (n-1)d\]
• Sum of First \(n\) Terms: \[S_n = \frac{n}{2}\bigl(2a_1 + (n-1)d\bigr)\]

Geometric Progression (GP)

A sequence where consecutive terms share a constant ratio \(r\).

• \(n\)-th Term Formula: \[a_n = a_1 r^{n-1}\]
• Sum of First \(n\) Terms (\(r eq 1\)): \[S_n = \frac{a_1(1-r^n)}{1-r}\]
• Sum of an Infinite GP (\(|r| < 1\)): \[S_\infty = \frac{a_1}{1-r}\]

7. NDA Exam Analysis: PYQ Patterns & Weightage

Analysis of previous year questions (PYQs) from the UPSC NDA exam reveals consistent patterns across algebra topics:

• Quadratic Equations & Complex Numbers: Account for 8 to 12 questions per exam. Frequently feature statement evaluation (Statement I & II format).
• Sequences & Series: Contributes 4 to 6 questions, mainly focusing on properties of AP, GP, and AM-GM inequalities.
• Logarithms & Exponents: Generally integrated into direct simplifications or functional equations (2 to 4 questions).
• Binomial Theorem: 2 to 4 questions testing the general term, middle term, or sum of coefficients.

8. Frequently Asked Questions (FAQ)