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BITSAT 2023Paper 2
Exam
BITSAT
Year
2023
Paper
Paper 2
Subject
Mathematics
Chapter
Binomial Theorem

BITSAT 2023 · Paper 2 · Mathematics · Binomial Theorem

If x^ 18 occurs in the rth term in the expansion of...

If x18x^{18} occurs in the rth term in the expansion of \left(x^4+\frac{1}{x^3}\right)^{15},thenwhatisthevalueof, then what is the value ofr$ ?

Options

  1. A
    3
  2. B
    5
  3. C
    7
  4. D
    9

Correct answer

C.
7

Step-by-step solution

In the expansion of (x4+1x3)15\left(x^4+\frac{1}{x^3}\right)^{15}, let TrT_r is the rth r^{\text{th }} term
Tr=(15r−1)(x4)15−r+1(1x3)r−1=(15r−1)x64−4r−3r+3=15Cr−1x67−7r\begin{aligned} \\ T_r & =\binom{15}{r-1}\left(x^4\right)^{15-r+1}\left(\frac{1}{x^3}\right)^{r-1} \\ \\ & =\binom{15}{r-1} x^{64-4 r-3 r+3}=15_{C_{r-1}} x^{67-7 r} \\ \end{aligned}
\mathrmx18\mathrmx^{18} occurs in this term
⇒18=67−7r⇒7r=49⇒r=7.\Rightarrow 18=67-7 r \Rightarrow 7 r=49 \Rightarrow r=7 .
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