IIM CAT 2026 cheat sheet for Algebra! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Core Concepts & Formulas in Algebra for CAT 2026

Algebra constitutes a significant portion of the Quantitative Aptitude section in the IIM CAT exam. Mastering foundational operations, algebraic identities, quadratic equations, and progressions is crucial for tackling high-level problem-solving scenarios efficiently.

1. Algebraic Identities & Polynomials

Fundamental expansions form the basis for simplifying complex equations and expressions rapidly during timed tests:

• \( (a+b)^2 = a^2 + 2ab + b^2 \)
• \( (a-b)^2 = a^2 - 2ab + b^2 \)
• \( a^2 - b^2 = (a-b)(a+b) \)
• \( (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \)
• \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \)
• \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)
• Special Identity: If \( a+b+c = 0 \), then \( a^3 + b^3 + c^3 = 3abc \).

2. Quadratic Equations & Roots

For a standard quadratic equation \( ax^2 + bx + c = 0 \) where \( a \neq 0 \):

• Roots are given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Sum of roots: \( \alpha + \beta = -\frac{b}{a} \)
• Product of roots: \( \alpha\beta = \frac{c}{a} \)
• Discriminant: \( D = b^2 - 4ac \). Real roots exist if \( D \ge 0 \); equal roots occur if \( D = 0 \).

3. Sequences & Progressions

Arithmetic and Geometric Progressions frequently appear in CAT algebraic sequences:

• Arithmetic Progression (AP): \( n^{\text{th}} \) term is \( T_n = a + (n-1)d \). Sum of \( n \) terms is \[ S_n = \frac{n}{2}\bigl(2a + (n-1)d\bigr) \]
• Geometric Progression (GP): \( n^{\text{th}} \) term is \( T_n = a r^{n-1} \). Sum of infinite terms for \( |r| < 1 \) is \( S_{\infty} = \frac{a}{1-r} \).