Algebra Cheat Sheet and Formula Notes - IIM CAT 2026

Welcome to the comprehensive LibreTexts-style study module for Algebra tailored specifically for the IIM CAT 2026 examination. Algebra forms the backbone of the Quantitative Aptitude section, accounting for a substantial weightage of questions every year. Mastering these core formulas, concepts, and previous years' question (PYQ) patterns is essential for securing a 99+ percentile.

Table of Contents

• 1. Exponents and Logarithms
• 2. Quadratic Equations
• 3. Polynomials and Factoring
• 4. Sequences and Series
• 5. Complex Numbers
• Frequently Asked Questions

---

1. Exponents and Logarithms

Chapter Weightage & PYQ Analysis: Exponents and logarithms frequently appear in CAT quantitative aptitude both as direct simplification problems and as underlying constraints in functions and equations. Expect 1-2 questions directly or indirectly linked to logarithmic properties.

Core algebraic rules for exponents and logarithms include:

\[a^m \cdot a^n = a^{m+n}, \quad \log_b(xy) = \log_b x + \log_b y\] \[\frac{a^m}{a^n} = a^{m-n}, \quad \log_b\Bigl(\frac{x}{y}\Bigr) = \log_b x - \log_b y\] \[(a^m)^n = a^{mn}, \quad \log_b(x^k) = k \log_b x\] \[a^{-n} = \frac{1}{a^n}, \quad \log_b x = \frac{\log_a x}{\log_a b}\]

2. Quadratic Equations

CAT Pattern Insight: Quadratic equations and roots properties are high-frequency scoring areas. CAT examiners often test properties of roots, common roots, and maximum/minimum values of quadratic expressions.

For a standard quadratic equation of the form \(ax^2 + bx + c = 0\):

Quadratic Formula

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

Discriminant (\(\Delta\))

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

Vieta's Formulas

\[\text{Sum} = x_1 + x_2 = -\frac{b}{a}, \quad \text{Product} = x_1 x_2 = \frac{c}{a}\]

3. Polynomials and Factoring

Exam Context: Algebraic identities and factorizations streamline time-consuming calculations in quantitative reasoning sections.

\[a^2 - b^2 = (a-b)(a+b)\] \[a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\] \[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]

Binomial Theorem

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

4. Sequences and Series

Chapter Weightage: Arithmetic progressions (AP) and Geometric progressions (GP) consistently contribute 2-3 questions in the CAT exam, often integrated with inequalities or functions.

Arithmetic Sequence

\[a_n = a_1 + (n-1)d\] \[S_n = \frac{n}{2}[2a_1 + (n-1)d]\]

Geometric Sequence

\[a_n = a_1 r^{n-1}\] \[S_n = \frac{a_1(1 - r^n)}{1 - r}, \quad r eq 1\]

5. Complex Numbers

Advanced NTA/CAT Pattern: While less frequent than arithmetic or quadratic equations, complex numbers occasionally feature in high-difficulty algebra sets focusing on magnitude, arguments, and roots of unity.

Let \(z = a + bi = r(\cos \theta + i \sin \theta) = re^{i\theta}\).

Euler's Formula

\[e^{i\theta} = \cos \theta + i \sin \theta\]

De Moivre's Theorem

\[(r(\cos \theta + i \sin \theta))^n = r^n(\cos(n\theta) + i \sin(n\theta))\]

---

People Also Ask (FAQ)

1. How many questions can I expect from Algebra in IIM CAT 2026?

Algebra typically accounts for 30% to 40% of the Quantitative Aptitude section in CAT, translating roughly to 6 to 8 questions.

2. Are complex numbers compulsory for CAT preparation?

Complex numbers are occasionally tested, primarily focusing on basic properties, moduli, and Euler's form, making them vital for aspirants aiming for a 99+ percentile.

3. What is the best way to revise algebra formulas before CAT?

Consistently practicing problem sets using a structured formula cheat sheet and solving past 5 years' PYQs is the most effective revision strategy.