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MathsMediumClass 11

Greatest Term

Binomial Theorem

9

JEE Qs

8%

Hard

60

min

Master the inequality derived from |T_{r+1}/T_r| >= 1 and carefully handle the absolute values and integer/non-integer cases of the resulting 'K' value.

🧮 Key Formulas

T_{r+1} = nCr * a^(n-r) * b^r
|T_{r+1}/T_r| = |(n-r+1)/r * (b/a)|
Let K = (n+1) * |b| / (|a| + |b|). If K is an integer, the greatest terms are T_K and T_{K+1} (equal in magnitude).
If K is not an integer, the greatest term is T_{[K]+1} (where [K] is the greatest integer less than or equal to K).

✅ Key Points for JEE

  • 1The method for finding the numerically greatest term relies on comparing the ratio of consecutive terms, |T_{r+1}/T_r|, with 1.
  • 2If |T_{r+1}/T_r| > 1, then |T_{r+1}| is greater than |T_r|. The terms are increasing.
  • 3If |T_{r+1}/T_r| < 1, then |T_{r+1}| is less than |T_r|. The terms are decreasing.
  • 4Solving the inequality |(n-r+1)/r * (b/a)| >= 1 for 'r' provides the range of indices for which terms are increasing or equal, leading to the identification of the greatest term(s).
  • 5Always consider the absolute value of the ratio |b/a| to find the numerically greatest term, even if some terms in the expansion are negative.

⚠️ Common Mistakes

  • ✕Not taking the absolute value of the ratio |b/a| when dealing with terms that might be negative, leading to incorrect numerical comparison.
  • ✕Incorrectly solving the inequality |T_{r+1}/T_r| >= 1 for 'r', especially when 'r' appears in the denominator and inequalities need careful handling.
  • ✕Misinterpreting the value of K (often denoted as m or P) to find the index of the greatest term, particularly when K is an integer, leading to missing one of the two equal greatest terms.

📝 Practice Questions

See all

Q21.If ∑30r=1 r2(30Cr)230Cr−1

2025·NumericalMedium

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

2025·MCQMedium

Q22.If ∑5r=0 11C22r2r+2 = mn , gcd(m, n) = 1

2025·NumericalHard

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

2025·MCQHard

Q24.The sum of all rational terms in the expansion of (1 + 21/2 + 31/2) 6 is equal to

2025·NumericalMedium

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

2025·MCQMedium

NCERT Chapters

  • Class 11 Maths Ch 8: Binomial Theorem