Coefficient of xⁿ — Finding specific coefficients
Binomial Theorem
9
JEE Qs
8%
Hard
75
min
Always write down the general term first, equate the total power of 'x' to the desired power, and critically check if 'r' is a valid non-negative integer before calculating the coefficient.
🧮 Key Formulas
✅ Key Points for JEE
- 1To find the coefficient of x^k in (ax^p + bx^q)^n, first write the general term T_(r+1) = nCr * (ax^p)^(n-r) * (bx^q)^r.
- 2Equate the total power of x in the general term to 'k' (i.e., p(n-r) + qr = k) and solve for 'r'. Ensure 'r' is a valid non-negative integer where 0 <= r <= n.
- 3Substitute the valid integer value(s) of 'r' back into the general term, excluding x^k, to obtain the desired coefficient.
- 4For expressions that are products of binomials or polynomials, e.g., (1+x)^a * (1+x^2)^b, identify all possible combinations of terms from each factor whose product results in x^k and sum their coefficients.
- 5The 'independent term' is the coefficient of x^0.
⚠️ Common Mistakes
- ✕Algebraic errors when solving for 'r' or simplifying the coefficient, especially with signs or fractional/negative powers.
- ✕Failing to check if the value of 'r' obtained is a valid non-negative integer within the range [0, n].
- ✕Missing some combinations of terms when dealing with products of two or more expansions, leading to an incomplete sum of coefficients.
- ✕Incorrectly identifying 'a' and 'b' (or their powers of x) in the general term formula T_(r+1).
📝 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 Maths Ch 8: Binomial Theorem