Advertisements
Advertisements
Question
The coefficient of the term independent of x in the expansion of \[\left( ax + \frac{b}{x} \right)^{14}\] is
Options
\[14! a^7 b^7\]
\[\frac{14!}{7!} a^7 b^7\]
\[\frac{14!}{\left( 7! \right)^2} a^7 b^7\]
\[\frac{14!}{\left( 7! \right)^3} a^7 b^7\]
Solution
\[\frac{14!}{\left( 7! \right)^2} a^7 b^7\]
\[\text { Suppose (r + 1)th term in the given expansion is independent of x } . \]
\[\text{ Then, we have} \]
\[ T_{r + 1} = ^{14}{}{C}_r (ax )^{14 - r} \left( \frac{b}{x} \right)^r \]
\[ = ^{14}{}{C}_r a^{14 - r} b^r x^{14 - 2r} \]
\[\text{ For this term to be independent of x, we must have } \]
\[14 - 2r = 0\]
\[ \Rightarrow r = 7\]
\[ \therefore \text{ Required term } = ^{14}{}{C}_7 a^{14 - 7} b^7 = \frac{14!}{(7! )^2} a^7 b^7\]
APPEARS IN
RELATED QUESTIONS
Find the 11th term from the beginning and the 11th term from the end in the expansion of \[\left( 2x - \frac{1}{x^2} \right)^{25}\] .
Find the 7th term in the expansion of \[\left( 3 x^2 - \frac{1}{x^3} \right)^{10}\] .
Find the 5th term from the end in the expansion of \[\left( 3x - \frac{1}{x^2} \right)^{10}\]
Find the 8th term in the expansion of \[\left( x^{3/2} y^{1/2} - x^{1/2} y^{3/2} \right)^{10}\]
Find the 7th term in the expansion of \[\left( \frac{4x}{5} + \frac{5}{2x} \right)^8\]
Find the 4th term from the beginning and 4th term from the end in the expansion of \[\left( x + \frac{2}{x} \right)^9\] .
Find the 4th term from the end in the expansion of \[\left( \frac{4x}{5} - \frac{5}{2x} \right)^8\] .
Find the 7th term from the end in the expansion of \[\left( 2 x^2 - \frac{3}{2x} \right)^8\] .
Find the sixth term in the expansion \[\left( y^\frac{1}{2} + x^\frac{1}{3} \right)^n\] , if the binomial coefficient of the third term from the end is 45.
Find n in the binomial \[\left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n\] , if the ratio of 7th term from the beginning to the 7th term from the end is \[\frac{1}{6}\]
if the seventh term from the beginning and end in the binomial expansion of \[\left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n\] are equal, find n.
Write the number of terms in the expansion of \[\left( 2 + \sqrt{3}x \right)^{10} + \left( 2 - \sqrt{3}x \right)^{10}\] .
Write the number of terms in the expansion of \[\left( 2 + \sqrt{3}x \right)^{10} + \left( 2 - \sqrt{3}x \right)^{10}\] .
Which term is independent of x, in the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9 ?\]
Write the number of terms in the expansion of \[\left[ \left( 2x + y^3 \right)^4 \right]^7\] .
Find the number of terms in the expansion of\[\left( a + b + c \right)^n\]
If rth term in the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)^{12}\] is without x, then r is equal to
If the fifth term of the expansion \[\left( a^{2/3} + a^{- 1} \right)^n\] does not contain 'a'. Then n is equal to
The coefficient of \[x^{- 3}\] in the expansion of \[\left( x - \frac{m}{x} \right)^{11}\] is
If the coefficients of the (n + 1)th term and the (n + 3)th term in the expansion of (1 + x)20are equal, then the value of n is
If the coefficients of 2nd, 3rd and 4th terms in the expansion of \[\left( 1 + x \right)^n , n \in N\] are in A.P., then n =
If the seventh terms from the beginning and the end in the expansion of `(root(3)(2) + 1/(root(3)(3)))^n` are equal, then n equals ______.
