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📅 Year 2024
Q1CBSE Board Exam 20243 Marks
Find the term independent of x in the expansion of (x − 1/x)¹⁰.
▶ Show Detailed Solution
General term: T(r+1) = ¹⁰C(r) x10−r (−1/x)r = ¹⁰C(r) (−1)r x10−r−r = ¹⁰C(r)(−1)r x10−2r.
Independent of x ⇒ 10 − 2r = 0 ⇒ r = 5.
T₆ = ¹⁰C₅ (−1)⁵ = −252.
Term independent of x = −252.
Q2CBSE Board Exam 20242 Marks
Expand (2x − 3)⁴ using the Binomial Theorem.
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(2x − 3)⁴ = Σ ⁴C(r)(2x)4−r(−3)r
= (2x)⁴ + 4(2x)³(−3) + 6(2x)²(9) + 4(2x)(−27) + 81
= 16x⁴ − 96x³ + 216x² − 216x + 81.
(2x − 3)⁴ = 16x⁴ − 96x³ + 216x² − 216x + 81.
📅 Year 2023
Q1CBSE Board Exam 20233 Marks
Find the coefficient of x⁵ in the expansion of (x + 3)⁸.
▶ Show Detailed Solution
General term: T(r+1) = ⁸C(r) x8−r 3r.
Coefficient of x⁵ ⇒ 8 − r = 5 ⇒ r = 3.
Coefficient = ⁸C₃ · 3³ = 56 × 27 = 1512.