Algebra Cheat Sheet - NDA Mathematics Complete Study Guide

Chapter 1: Exponents and Logarithms

Logarithms and exponents form the foundational backbone for algebraic simplification and calculus in the NDA written exam. Questions often combine logarithmic properties with equations or inequalities.

1.1 Fundamental Rules of Exponents

For any base \(a > 0\) and real exponents \(m\) 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: \[(ab)^n = a^n b^n\]

1.2 Fundamental Logarithm Identities

For base \(b > 0\) (where \(b eq 1\)) and positive arguments \(x, y\):

• Product Identity: \Builder[\log_b(xy) = \log_b x + \log_b y\]
• Quotient Identity: \Builder[\log_b(\frac{x}{y}) = \log_b x - \log_b y\]
• Power Identity: \[\log_b(x^k) = k \log_b x\]
• Change of Base Formula: \[\log_b x = \frac{\log_a x}{\log_a b}\]

Chapter 2: Quadratic Equations and Discriminant Analysis

Quadratic equations appear consistently in the NDA mathematics paper, testing nature of roots, relations between roots and coefficients, and graphical representations.

2.1 Standard Form and the Quadratic Formula

A second-degree polynomial equation given by \(ax^2 + bx + c = 0\) (where \(a eq 0\)) has roots given by:

2.2 The Discriminant (\(\Delta\)) and Nature of Roots

The term under the radical sign, \(\Delta = b^2 - 4ac\), dictates the nature of the solutions:

• Case 1 (\(\Delta > 0\)): Two distinct real roots.
• Case 2 (\(\Delta = 0\)): One repeated real root (equal roots).
• Case 3 (\(\Delta < 0\)): Two complex conjugate roots.

Chapter 3: Polynomials, Factoring, and Binomial Theorem

Factoring identities assist in rapid algebraic expansion and simplification, while the Binomial Theorem is directly tested in middle-term and coefficient problems.

3.1 Important Algebraic Factoring Identities

• Difference of Squares: \[a^2 - b^2 = (a-b)(a+b)\]
• Difference of Cubes: \[a^3 - b^3 = (a-b)(a^2 + ab + b^2)\]
• Sum of Cubes: \[a^3 + b^3 = (a+b)(a^2 - ab + b^2)\]

3.2 The Binomial Theorem

For any non-negative integer \(n\), the expansion of \((x + y)^n\) is given by:

where the binomial coefficient is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Chapter 4: Sequences and Series (AP & GP)

Progressions are highly scoring in NDA. Mastery over general terms and sum formulas yields direct marks.

4.1 Arithmetic Progression (AP)

An AP has a constant common difference \(d = a_n - a_{n-1}\).

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

4.2 Geometric Progression (GP)

A GP has a constant common ratio \(r = \frac{a_n}{a_{n-1}}\).

• \(n\)-th Term: \[a_n = a_1 r^{n-1}\]
• Sum of First \(n\) Terms (\(r eq 1\)): \[S_n = \frac{a_1(1 - r^n)}{1 - r}\]

Chapter 5: Inequalities and Absolute Value Functions

Absolute value properties govern domain, range, and inequality questions in NDA Algebra.

5.1 Solving Absolute Value Inequalities

For a positive real constant \(c > 0\):

• \[|x| < c \iff -c < x < c\]
• \[|x| > c \iff x > c \text{ or } x < -c\]

5.2 Triangle Inequality

For any real numbers \(x\) and \(y\):

Chapter 6: Complex Numbers and Euler's Identity

Complex numbers extend the real line into a 2D plane (Argand Plane), crucial for solving quadratic equations with negative discriminants and rotation geometry.

6.1 Standard Form, Modulus, and Conjugate

Let \(z = a + bi\), where \(i = \sqrt{-1}\) (or \(i^2 = -1\)):

• Modulus (Magnitude): \[|z| = \sqrt{a^2 + b^2}\]
• Complex Conjugate: \[\bar{z} = a - bi\]

6.2 Euler's Formula

Euler's formula connects complex exponentials with trigonometric functions:

NDA Exam Analysis & Topic Weightage

PYQ Analysis & Pattern: Algebra accounts for approximately 25–30 questions out of 120 in the NDA Mathematics Paper (approx. 60–75 marks out of 300). Questions frequently test mixed concepts—such as combining complex numbers with geometric progressions or logarithms with quadratic roots.

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