Advertisements
Advertisements
प्रश्न
If the sum of the binomial coefficients of the expansion \[\left( 2x + \frac{1}{x} \right)^n\] is equal to 256, then the term independent of x is
विकल्प
1120
1020
512
none of these
Advertisements
उत्तर
1120
\[\text{ Suppose (r + 1)th tem in the given expansion is independent of x . } \]
\[\text{ Then, we have } \]
\[ T_{r + 1} = ^{n}{}{C}_r (2x )^{n - r} \left( \frac{1}{x} \right)^r \]
\[ = ^{n}{}{C}_r 2^{n - r} x^{n - 2r} \]
\[\text{ For this term to be independent of x, we must have } \]
\[n - 2r = 0\]
\[ \Rightarrow r = n/2\]
\[ \therefore \text{ Required term } = ^{n}{}{C}_{n/2} 2^{n - n/2} = \frac{n!}{\left[ \left( n/2 \right)! \right]^2} 2^{n/2} \]
\[\text{ We know } : \]
\[\text{ Sum of the given expansion } = 256\]
\[\text{ Thus, we have } \]
\[ 2^n . 1^n = 256\]
\[ \Rightarrow n = 8\]
\[ \therefore \text{ Required term } = \frac{8!}{\left( 4 \right)! \left( 4 \right)!} 2^4 = 1120\]
APPEARS IN
संबंधित प्रश्न
Using binomial theorem, write down the expansions .
(i) \[\left( 2x + 3y \right)^5\]
Using binomial theorem, write down the expansions :
(ii) \[\left( 2x - 3y \right)^4\]
Using binomial theorem, write down the expansions :
(iv) \[\left( 1 - 3x \right)^7\]
Using binomial theorem, write down the expansions :
(vii) \[\left( \sqrt[3]{x} - \sqrt[3]{a} \right)^6\]
Evaluate the
(iii)\[\left( 1 + 2 \sqrt{x} \right)^5 + \left( 1 - 2 \sqrt{x} \right)^5\]
Evaluate the
(v) \[\left( 3 + \sqrt{2} \right)^5 - \left( 3 - \sqrt{2} \right)^5\]
Evaluate the
(vi) \[\left( 2 + \sqrt{3} \right)^7 + \left( 2 - \sqrt{3} \right)^7\]
Evaluate the
(vii) \[\left( \sqrt{3} + 1 \right)^5 - \left( \sqrt{3} - 1 \right)^5\]
Evaluate the
(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]
Find \[\left( a + b \right)^4 - \left( a - b \right)^4\] . Hence, evaluate \[\left( \sqrt{3} + \sqrt{2} \right)^4 - \left( \sqrt{3} - \sqrt{2} \right)^4\] .
Using binomial theorem evaluate .
(iii) (101)4
Using binomial theorem evaluate .
(iv) (98)5
Find the coefficient of:
(ii) x7 in the expansion of \[\left( x - \frac{1}{x^2} \right)^{40}\]
Find the coefficient of:
(iii) \[x^{- 15}\] in the expansion of \[\left( 3 x^2 - \frac{a}{3 x^3} \right)^{10}\]
Find the coefficient of:
(v) \[x^m\] in the expansion of \[\left( x + \frac{1}{x} \right)^n\]
Find the coefficient of:
(vi) x in the expansion of \[\left( 1 - 2 x^3 + 3 x^5 \right) \left( 1 + \frac{1}{x} \right)^8\]
Find the coefficient of:
(vii) \[a^5 b^7\] in the expansion of \[\left( a - 2b \right)^{12}\]
Which term in the expansion of \[\left\{ \left( \frac{x}{\sqrt{y}} \right)^{1/3} + \left( \frac{y}{x^{1/3}} \right)^{1/2} \right\}^{21}\] contains x and y to one and the same power?
Write the sum of the coefficients in the expansion of \[\left( 1 - 3x + x^2 \right)^{111}\]
If a and b are coefficients of xn in the expansions of \[\left( 1 + x \right)^{2n} \text{ and } \left( 1 + x \right)^{2n - 1}\] respectively, then write the relation between a and b.
If a and b denote the sum of the coefficients in the expansions of \[\left( 1 - 3x + 10 x^2 \right)^n\] and \[\left( 1 + x^2 \right)^n\] respectively, then write the relation between a and b.
The term without x in the expansion of \[\left( 2x - \frac{1}{2 x^2} \right)^{12}\] is
If the coefficient of x in \[\left( x^2 + \frac{\lambda}{x} \right)^5\] is 270, then \[\lambda =\]
The coefficient of x5 in the expansion of \[\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}\]
The coefficient of x8 y10 in the expansion of (x + y)18 is
If the coefficients of x2 and x3 in the expansion of (3 + ax)9 are the same, then the value of a is
