How to Apply Binomial Theorem in Solving Problems with Examples
FAQs on Binomial Theorem and Its Simple Applications: Practice Paper
1. What is the Binomial Theorem?
The Binomial Theorem is a fundamental algebraic expansion for expressing the power of a binomial in terms of a sum involving binomial coefficients.
Key points include:
- The theorem states: (a + b)n = Σ [nCr × an–r × br], for r = 0 to n.
- It uses combinatorial coefficients (nCr) for expansion.
- It applies to any positive integer exponent n.
- Common in CBSE and school algebra syllabus for Class 11 and 12.
2. State the general formula for the Binomial Theorem for any positive integer n.
The general formula for the Binomial Theorem is:
(a + b)n = Σ [nCr × an–r × br], where r = 0 to n.
- nCr represents the binomial coefficient, calculated as n! / [r!(n–r)!].
- This formula allows expansion of any binomial raised to a positive integer power into a sum of terms.
- Each term contains powers of a and b decreasing and increasing, respectively.
3. What are binomial coefficients, and how are they calculated?
Binomial coefficients are the numerical coefficients in the binomial expansion, denoted by nCr or C(n, r).
- Formula: nCr = n! / [r!(n–r)!]
- They indicate the number of ways to choose r objects from n.
- Found in Pascals Triangle and algebraic expansions.
4. What are some simple applications of the Binomial Theorem in exams?
Simple applications of the Binomial Theorem in exams include:
- Expanding binomial expressions, e.g., (2x – 3)4.
- Finding specific terms, such as the middle term or the term independent of x.
- Calculating coefficients of given terms in the expansion.
5. How do you find the rth term in the expansion of (a + b)n?
The rth term in the expansion of (a + b)n is given by Tr = nC(r-1) × an-(r-1) × br-1.
- Replace r with the required term number.
- This formula helps directly find the value of any term without fully expanding the expression.
6. Give an example: Expand (x + 2)3 using the Binomial Theorem.
Expanding (x + 2)3 using the Binomial Theorem:
The general expansion is Σ [3Cr × x3–r × 2r], for r = 0 to 3.
- Term 1 (r=0): 1 × x3 × 1 = x3
- Term 2 (r=1): 3 × x2 × 2 = 6x2
- Term 3 (r=2): 3 × x × 4 = 12x
- Term 4 (r=3): 1 × 1 × 8 = 8
7. What is the middle term in the binomial expansion of (a + b)n?
The middle term in the expansion of (a + b)n depends on whether n is even or odd.
- If n is even, there is one middle term at position (n/2) + 1.
- If n is odd, there are two middle terms at positions (n+1)/2 and (n+3)/2.
8. What is Pascal’s Triangle and how is it related to the Binomial Theorem?
Pascal’s Triangle is a triangular array of numbers showing the binomial coefficients.
- Each row corresponds to the coefficients in the expansion of (a + b)n.
- Each entry is the sum of the two numbers directly above.
- Helps quickly find coefficients for binomial expansions.
9. When is the Binomial Theorem applicable for negative or fractional exponents?
The Binomial Theorem in its standard form applies to positive integers only; however, for negative or fractional exponents, an infinite series form known as the Binomial Series is used.
- Covered in higher mathematics and not usually in CBSE Class 11/12 syllabus.
- Involves infinite terms and convergence conditions.
10. What are the main points to remember while applying the Binomial Theorem in board exams?
Main points for applying the Binomial Theorem in board exams:
- Recall the binomial coefficients formula (nCr) accurately.
- Ensure correct placement of powers for a and b in each term.
- Study coefficient-based questions and term positions (like the 5th term).
- Practice both expansions and finding specific terms.
11. What is the expansion of (a + b)5?
The expansion of (a + b)5 is:
(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
This expansion uses binomial coefficients 1, 5, 10, 10, 5, 1 from Pascal’s Triangle and follows the pattern of decreasing powers of a and increasing of b.
12. How do you use the Binomial Theorem to find the coefficient of x3 in the expansion of (1 + x)6?
To find the coefficient of x3 in (1 + x)6:
- The general term is 6Cr × (1)6–r × xr.
- Set r = 3 for x3: 6C3 × 1 × x3 = 20 × x3.
