Algebra Cheat Sheet - IIM CAT 2026

Mastering Algebra is pivotal for achieving a top percentile in the Quantitative Aptitude (QA) section of the IIM CAT 2026 exam. Algebra consistently carries high weightage in CAT, accounting for roughly 25% to 35% of the total Quantitative Aptitude questions. This comprehensive, LibreTexts-style chapter breakdown compiles core theoretical concepts, identities, formulas, and structural patterns derived from past year questions (PYQs) to streamline your revision.

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1. Exponents & Logarithms

Exponent and logarithm rules form the foundation for solving non-linear equations, interest rate problems, and series approximations in CAT QA. A strong grasp of base conversion and power manipulation is essential for speed and accuracy.

Laws of Exponents

For any non-zero real bases \(a\) and \(b\), and integers or real numbers \(m\) and \(n\):

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

Properties of Logarithms

Logarithmic transformations simplify exponential relations into linear operations. For base \(b > 0\) (\(b eq 1\)) and positive arguments \(x, y\):

• Product Property: \(\log_b(xy) = \log_b x + \log_b y\)
• Quotient Property: \(\log_b\bigl(\frac{x}{y}\bigr) = \log_b x - \log_b y\)
• Power Property: \(\log_b(x^k) = k \log_b x\)
• Change of Base Theorem: \(\log_b x = \frac{\log_c x}{\log_c b}\)

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2. Polynomials & Factoring

Quadratic equations and high-degree polynomials are central to CAT Algebra. Expect questions testing root identities, graphical behavior, maximum/minimum values, and discriminant conditions.

Special Algebraic Products & Factorization

• Square Identities: \((a \pm b)^2 = a^2 \pm 2ab + b^2\)
• Difference of 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: \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)

Quadratic Formula & Discriminant Analysis

For a general quadratic equation \(ax^2 + bx + c = 0\) with \(a eq 0\), the roots are given by:

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

The nature of the roots depends on the discriminant \(\Delta = b^2 - 4ac\):

• \(\Delta > 0\): Two distinct real roots.
• \(\Delta = 0\): Exactly one real repeated root (equal roots).
• \(\Delta < 0\): Two complex conjugate roots (no real roots).

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3. Sequences & Series

Questions involving progressions often test pattern recognition, formula application, and telescoping series summation.

Arithmetic Progression (AP)

In an AP with first term \(a_1\) and common difference \(d\):

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

Geometric Progression (GP)

In a GP with first term \(a_1\) and common ratio \(r\):

• \(n\)-th Term: \(a_n = a_1 r^{n-1}\)
• Sum of Finite GP (\(r eq 1\)): \(S_n = \frac{a_1(1-r^n)}{1-r}\)
• Sum to Infinite GP (when \(|r| < 1\)): \(S_\infty = \frac{a_1}{1-r}\)

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4. Complex Numbers

While basic, understanding imaginary units and standard forms aids in solving advanced polynomial root systems and functional equations in competitive exams.

Fundamentals of Complex Operations

• Powers of \(i\): \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\)
• Standard Form: \(z = a + bi\), where \(a, b \in \mathbb{R}\)
• Complex Conjugate: \(\bar{z} = a - bi\)
• Modulus (Magnitude): \(|z| = \sqrt{a^2 + b^2}\)

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5. Binomial Theorem

The Binomial Theorem facilitates polynomial expansions, remainder problems, and term identification in higher powers.

Binomial Expansion & General Term

For any positive integer \(n\):

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

Where the binomial coefficient is calculated as:

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

The general term \((k+1)\)-th term in the expansion is given by:

\[T_{k+1} = \binom{n}{k} x^{n-k} y^k\]

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6. Matrices & Determinants

Linear algebra principles like determinants and inverse calculations provide quick shortcuts for linear systems of equations.

2x2 Matrix Inverse & Determinant

For a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\):

• Determinant: \(\det(A) = ad - bc\)
• Inverse Matrix (when \(\det(A) eq 0\)):

\[A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\]

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7. CAT PYQ Analysis & Exam Pattern Insights

Chapter Weightage & Strategic Importance

Analysis of past CAT papers reveals the relative frequency and question distributions across Algebra topics:

• Quadratic & Higher Degree Equations: High Priority (~3-5 Questions/Slot). Questions emphasize range of functions, roots, and sign charts.
• Logarithms & Exponents: High Priority (~2-3 Questions/Slot). Focus on multi-base simplification and inequality constraints.
• Sequences & Series: Moderate-High Priority (~2 Questions/Slot). Strong emphasis on telescoping sums and AGP.
• Binomial Theorem & Matrices: Moderate-Low Priority (~0-1 Questions/Slot). Useful for remainder techniques and modular arithmetic applications.

NTA Pattern & Trends

In recent testing patterns, Indian Institutes of Management (IIMs) have favored non-routine algebra problems that combine multiple concepts—such as incorporating logarithms inside arithmetic progressions or using quadratic constraints to find maximum/minimum values without calculus. Mastering standard formulas along with rapid mental evaluation is crucial.

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8. Frequently Asked Questions (FAQ)

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