CAT 2026 Algebra Formula Sheet & Topic-Wise Revision Notes

1. Exponents and Logarithms

Logarithms and indices are among the most scoring areas in CAT QA. They frequently appear as direct conceptual questions or merged with functions and series.

Core Exponent Laws

• Product Rule: \(a^m \cdot a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^n = a^n b^n\)
• Negative & Radical Exponents: \(a^{-n} = \frac{1}{a^n}\), \(a^{1/n} = \sqrt[n]{a}\)

Logarithm Properties

• 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: \(\log_b x = \frac{\log_c x}{\log_c b}\)

2. Polynomials and Quadratic Equations

Quadratic equations and algebraic identities form the backbone of polynomial questions in IIM entrance tests.

Algebraic Identities

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

Quadratic Formula & Nature of Roots

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

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

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

3. Sequences and Series (AP & GP)

Progressions are tested extensively in both CAT and non-CAT management exams (XAT, SNAP, NMAT).

Arithmetic Progression (AP)

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

Geometric Progression (GP)

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

• \(n\)-th term: \(a_n = a_1 r^{n-1}\)
• Sum of first \(n\) terms (\(r eq 1\)): \[S_n = \frac{a_1(1 - r^n)}{1 - r}\]
• Sum of infinite GP (when \(|r| < 1\)): \[S_\infty = \frac{a_1}{1 - r}\]

4. Complex Numbers Fundamentals

Complex numbers appear periodically in advanced algebra sets and higher-level aptitude exams.

• Definition: \(z = x + iy\) where \(i^2 = -1\).
• Magnitude: \(|z| = \sqrt{x^2 + y^2}\)
• Conjugate: \(\bar{z} = x - iy\)
• Polar Form: \(z = r(\cos\theta + i\sin\theta) = re^{i\theta}\)
• Euler's Formula: \(e^{i\theta} = \cos\theta + i\sin\theta\)

5. Binomial Theorem

The binomial theorem is vital for expansion, finding general terms, remainder problems, and approximations.

Where the binomial coefficient is calculated as:

6. CAT PYQ Analysis & Weightage

Analysis of past 10 years of CAT question papers reveals key patterns for CAT 2026 preparation:

• High-Yield Topics: Quadratic Equations, Logarithms, and Sequences & Series consistently yield 5–8 questions per slot.
• NTA / IIM Exam Pattern: Focus has shifted towards conceptual clarity, application-based word problems, and mixed concepts (e.g., combining AP/GP with Logarithms).
• Target Strategy: Memorize fundamental identities and practice speed-solving previous year questions (PYQs) to build pattern recognition.

7. People Also Ask (FAQ)