Binomial Theorem — General Term + Coefficient
Binomial Theorem
48
JEE Qs
12%
Hard
60
min
Master the precise identification of 'a', 'b', and 'n' along with meticulous algebraic manipulation of exponents to accurately find 'r' for any specific term or coefficient.
🧮 Key Formulas
✅ Key Points for JEE
- 1Always correctly identify 'a', 'b' (including its sign), and 'n' from the given binomial expression (e.g., for (2x - 1/x^2)^10, a=2x, b=-1/x^2, n=10).
- 2The general term is denoted T_{r+1}, and 'r' represents the power of the second term 'b', ranging from 0 to n.
- 3To find the term independent of x (constant term), set the total power of x in T_{r+1} to 0 and solve for 'r'. Ensure 'r' is a non-negative integer within [0, n].
- 4To find the coefficient of x^k, set the total power of x in T_{r+1} to 'k' and solve for 'r'. If a valid 'r' is found, substitute it into the coefficient part of T_{r+1} (nCr * (coefficient of a)^(n-r) * (coefficient of b)^r).
- 5The sum of the powers of 'a' and 'b' in any term T_{r+1} (i.e., (n-r) + r) must always equal 'n'.
⚠️ Common Mistakes
- ✕Incorrectly identifying 'a' or 'b' from the binomial, especially mishandling negative signs or parts of variables (e.g., treating x as part of 'a' when 'a' should be just the numerical coefficient).
- ✕Algebraic errors when simplifying the powers of variables to find 'r' (e.g., errors in exponent rules like (x^p)^q = x^(pq) or x^p / x^q = x^(p-q)).
- ✕Forgetting to verify that the calculated value of 'r' is a non-negative integer and lies within the valid range [0, n]. If 'r' is not an integer or outside this range, the specific term/coefficient does not exist.
- ✕Confusing T_r with T_{r+1}, leading to an off-by-one error in determining the term number or the power of 'b'.
📝 Practice Questions
See allQ21.If ∑30r=1 r2(30Cr)230Cr−1
Q20.If the area of the region {(x, y) : −1 ≤x ≤1, 0 ≤y ≤a + e|x| −e−x, a > 0} is e2+8e+1e , then the value of is : (1) 8 (2) 7 (3) 5 (4) 6
Q22.If ∑5r=0 11C22r2r+2 = mn , gcd(m, n) = 1
Q3. Let α, β, γ and δ be the coefficients of x7, x5, x3 and x respectively in the expansion of 5 5 αu + βv = 18 + , x > 1. If u and v satisfy the equations then u + v equals : (x + √x3 −1) (x −√x3 −1) γu + δv = 20 (1) 5 (2) 3 (3) 4 (4) 8
Q24.The sum of all rational terms in the expansion of (1 + 21/2 + 31/2) 6 is equal to
Q6. The product of all the rational roots of the equation (x2 −9x + 11)2 −(x −4)(x −5) = 3, is equal to (1) 14 (2) 21 (3) 28 (4) 7
NCERT Chapters
- Class 11 Mathematics Ch 8: Binomial Theorem