Binomial Theorem Important Solved Problems for JEE Mains Exam free pdf Download

📚 Binomial Theorem Solved Problems for JEE Exam – Class 11 Math

📝 Author: Neeraj Anand
🏛 Published by: ANAND TECHNICAL PUBLISHERS
📖 Available at: ANAND CLASSES

🔍 For CBSE Board & JEE Mains/Advanced Aspirants

🔢 Mastering Binomial Theorem with Solved Problems

The Binomial Theorem is an essential topic in Class 11 Mathematics, particularly for students preparing for CBSE Board Exams, JEE Mains, and JEE Advanced. It provides a powerful method for expanding algebraic expressions of the form (a + b)ⁿ and is widely used in problem-solving.

In this article, we will cover:
✔️ Basic & Advanced Problems on Binomial Expansion
✔️ Application-Based Problems for JEE
✔️ Shortcuts & Tricks for Fast Calculation
✔️ Important Questions with Solutions

📥 Download the PDF by Neeraj Anand (ANAND CLASSES) for detailed explanations and extra practice questions!

Solved Examples

Example.1 :: Find the positive value of λ for which the coefficient of x² in the expression x²[√x + (λ/x²)]¹⁰ is 720.

Solution:

⇒ x² [¹⁰C_r . (√x)^10-r . (λ/x²)^r] = x² [¹⁰C_r . λ^r . x^(10-r)/2 . x^-2r]

= x² [¹⁰C_r . λ^r . x^(10-5r)/2]

Therefore, r = 2

Hence, ¹⁰C₂ . λ² = 720

⇒ λ² = 16

⇒ λ = ±4.

Example.2: Let (x + 10)⁵⁰ + (x – 10)⁵⁰ = a₀ +a₁x + a₂ x₂ + . . . . . + a₅₀ x⁵⁰ for all x ∈R, then a₂/a₀ is equal to?

Solution:

⇒ (x + 10)⁵⁰ + (x – 10)⁵⁰:

a₂ = 2 × ⁵⁰C₂ × 10⁴⁸

a₀ = 2 × 10⁵⁰

⇒ a₂/a₀ = ⁵⁰C₂/10² = 12.25.

Example.3 : Find the coefficient of x⁹ in the expansion of (1 + x) (1 + x² ) (1 + x³) . . . . . . (1 + x¹⁰⁰).

Solution:

x⁹ can be formed in 8 ways.

i.e., x⁹ x¹⁺⁸ x²⁺⁷ x³⁺⁶ x⁴⁺⁵, x¹⁺³⁺⁵, x²⁺³⁺⁴

∴ The coefficient of x⁹ = 1 + 1 + 1 + . . . . + 8 times = 8.

Example.4 : Find the total number of terms in the expansion of (x + a)¹⁰⁰ + (x – a)¹⁰⁰.

Solution:

⇒ (x + a)¹⁰⁰ + (x – a)¹⁰⁰ = 2[¹⁰⁰C₀ x¹⁰⁰. ¹⁰⁰C₂  x⁹⁸ . a² + . . . . . . +  ¹⁰⁰C₁₀₀ a¹⁰⁰]

∴ Total Terms = 51

Example.5 : Find the coefficient of t⁴ in the expansion of [(1-t⁶)/(1 – t)].

Solution:

⇒ [(1-t⁶)/(1 – t)] = (1 – t¹⁸ – 3t⁶ + 3t¹²) (1 – t)^-3

Coefficient of t in (1 – t)^-3 = 3 + 4 – 1

C₄ = ⁶C₂ =15

The coefficient of x^r in (1 – x)^-n = (r + n – 1) C_r

Example.6: Find the coefficient of x⁴ in the expansion of (1 + x + x² +x³)¹¹.

Solution:

By expanding the given equation using the expansion formula, we can get the coefficient x⁴

i.e. 1 + x + x² + x³ = (1 + x) + x² (1 + x) = (1 + x) (1 + x²)

⇒ (1 + x + x² + x³) x¹¹ = (1+x)¹¹ (1+x²)¹¹

= 1+ ¹¹C₁ x² + ¹¹C₂ x² + ¹¹C₃ x³ + ¹¹C₄ x⁴ . . . . . . .

= 1 + ¹¹C₁ x² + ¹¹C₂ x⁴ + . . . . . .

To find the term in from the product of two brackets on the right-hand-side, consider the following products terms as

= 1 × ¹¹C₂ x⁴ + ¹¹C₂ x² × ¹¹C₁ x² + ¹¹C₄ x⁴

= ¹¹C₂ + ¹¹C₂ × ¹¹C₁ + ¹¹C₄ ] x⁴

⇒ [55 + 605 + 330] x⁴ = 990x⁴

∴ The coefficient of x⁴ is 990.

Example.7 : Find the number of terms free from the radical sign in the expansion of (√5 + ⁴√n)¹⁰⁰.

Solution:

T_r+1 = ¹⁰⁰C_r . 5^{(100 – r)/2} n^r/4

Where r = 0, 1, 2, . . . . . . , 100

r must be 0, 4, 8, … 100

The number of rational terms = 26

Example.8 : Find the degree of the polynomial [x + {√(3^(3-1))}^1/2]⁵ + [x + {√(3^(3-1))}^1/2]⁵.

Solution:

[x + { √(3^(3-1)) }^1/2 ]⁵:

= 2 [⁵C₀ x⁵ + ⁵C₂ x⁵ (x³ – 1) + ⁵C₄ . x . (x³ – 1)²]

Therefore, the highest power = 7.

Example.9 : Find the last three digits of 27²⁶.

Solution:

By reducing 27²⁶ into the form (730 – 1)ⁿ and using simple binomial expansion, we will get the required digits.

We have 27² = 729

Now 27²⁶ = (729)¹³ = (730 – 1)¹³

= ¹³C₀ (730)¹³ – ¹³C₁ (730)¹² + ¹³C₂ (730)¹¹ – . . . . . – ¹³C₁₀ (730)³ + ¹³C₁₁(730)² – ¹³C₁₂ (730) + 1

= 1000m + [(13 × 12)]/2] × (14)² – (13) × (730) + 1

Where ‘m’ is a positive integer

= 1000m + 15288 – 9490 = 1000m + 5799

Thus, the last three digits of 17²⁵⁶ are 799.

📚 Why Choose Neeraj Anand’s Book?

✔️ Comprehensive Explanation – Step-by-step solutions
✔️ JEE-Focused Approach – Covers JEE Mains & Advanced syllabus
✔️ Practice Questions – Includes previous year JEE questions
✔️ Tricks & Shortcuts – Solve problems faster and efficiently

📥 Download the PDF & Start Practicing Today!

🚀 Master Binomial Theorem & Score High in JEE!