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MathsMediumClass 11

Middle Term — Even/odd n

Binomial Theorem

9

JEE Qs

8%

Hard

45

min

Always carefully identify 'n' as even or odd first, determine the correct term position(s), and then meticulously apply the general term formula for calculation, paying attention to signs and exponents.

🧮 Key Formulas

T_(r+1) = nCr * (first_term)^(n-r) * (second_term)^r
If n is even, total terms = n+1 (odd). The middle term is T_((n/2)+1)
If n is odd, total terms = n+1 (even). The two middle terms are T_((n+1)/2) and T_((n+3)/2)

✅ Key Points for JEE

  • 1The total number of terms in the expansion of (a+b)^n is (n+1). This determines whether there's one or two middle terms.
  • 2If 'n' (the power) is even, there is a single middle term. Its position is (n/2) + 1.
  • 3If 'n' (the power) is odd, there are two middle terms. Their positions are (n+1)/2 and (n+3)/2.
  • 4Once the position (k) of the middle term(s) is identified, use T_(r+1) where r = k-1 to find the actual term(s) using the general term formula.
  • 5Pay close attention to the signs and exponents of the individual terms 'a' and 'b' when substituting into the general term formula.

⚠️ Common Mistakes

  • ✕Confusing the term number (k) with the 'r' in the general term formula T_(r+1). (e.g., for the 5th term, r=4).
  • ✕Incorrectly identifying 'n' as even or odd, leading to the wrong formula for term position.
  • ✕Errors in calculating the binomial coefficient nCr or simplifying the powers of 'a' and 'b'.
  • ✕Neglecting negative signs within the 'a' or 'b' terms (e.g., in (x - 1/x)^n, 'b' is -1/x).

📝 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