IIT-JEE MAINS 2027 formula sheet for Algebra! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

High-Yield Algebra Formula Sheet: Fundamental Concepts

Mastering Algebra for the IIT-JEE Mains requires a solid grasp of core algebraic structures, identities, and analytical techniques. This chapter-wise summary outlines the essential equations and theories necessary for rapid revision and accurate problem-solving.

1. Quadratic Equations & Expressions

For a standard quadratic equation \(ax^2+bx+c=0\) with roots \(\alpha\) and \(\beta\):

• Sum of roots: \(\alpha+\beta=-\frac{b}{a}\)
• Product of roots: \(\alpha\beta=\frac{c}{a}\)
• Discriminant analysis: \(D=b^2-4ac\). Roots are real and distinct if \(D>0\), real and equal if \(D=0\), and complex conjugate pairs if \(D<0\).
• Condition for common roots: Two equations \(a_1x^2+b_1x+c_1=0\) and \(a_2x^2+b_2x+c_2=0\) share both roots if \[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]

2. Sequences and Series

Fundamental formulas for progression models:

• Arithmetic Progression (AP): \(n\)-th term \(T_n=a+(n-1)d\); Sum of \(n\) terms \(S_n=\frac{n}{2}\bigl(2a+(n-1)d\bigr)\).
• Geometric Progression (GP): \(n\)-th term \(T_n=ar^{n-1}\); Sum of infinite terms \(S_{\infty}=\frac{a}{1-r}\) for \(|r|<1\).
• AM-GM Inequality: For non-negative real numbers, \(AM \ge GM\), represented as \[\frac{a+b}{2} \ge \sqrt{ab}\]

3. Binomial Theorem

Expansion of \((x+y)^n\) for a positive integer exponent \(n\):

• General term: \(T_{r+1} = {}^n C_r x^{n-r} y^r\)
• Sum of binomial coefficients: \({}^n C_0 + {}^n C_1 + \dots + {}^n C_n = 2^n\)

4. Complex Numbers

For a complex number \(z=x+iy\):

• Modulus and Argument: \(|z|=\sqrt{x^2+y^2}\); Polar form \(z=|z|(\cos\theta + i\sin\theta)\).
• Euler's Formula: \(e^{i\theta}=\cos\theta + i\sin\theta\).
• De Moivre's Theorem: \((\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)\).