Advertisements
Advertisements
प्रश्न
If the coefficient of x in \[\left( x^2 + \frac{\lambda}{x} \right)^5\] is 270, then \[\lambda =\]
विकल्प
3
4
5
none of these
Advertisements
उत्तर
3
\[\text{ The coefficient of x in the given expansion where x occurs at the (r + 1)th term } . \]
\[\text{ We have } \]
\[ ^{5}{}{C}_r ( x^2 )^{5 - r} \left( \frac{\lambda}{x} \right)^r \]
\[ =^{5}{}{C}_r \lambda^r x^{10 - 2r - r} \]
\[\text{ For it to contain x, we must have: } \]
\[10 - 3r = 1\]
\[ \Rightarrow r = 3 \]
\[ \therefore \text{ Coefficient of x in the given expansion: } \]
\[ ^{5}{}{C}_3 \lambda^3 = 10 \lambda^3 \]
\[\text{ Now, we have } \]
\[10 \lambda^3 = 270\]
\[ \Rightarrow \lambda^3 = 27\]
\[ \Rightarrow \lambda = 3\]
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 .
(iii) \[\left( x - \frac{1}{x} \right)^6\]
Using binomial theorem, write down the expansions :
(v) \[\left( ax - \frac{b}{x} \right)^6\]
Using binomial theorem, write down the expansions :
(vi) \[\left( \frac{\sqrt{x}}{a} - \sqrt{\frac{a}{x}} \right)^6\]
Using binomial theorem, write down the expansions :
(vii) \[\left( \sqrt[3]{x} - \sqrt[3]{a} \right)^6\]
Using binomial theorem, write down the expansions :
(viii) \[\left( 1 + 2x - 3 x^2 \right)^5\]
Evaluate the
(ii) \[\left( x + \sqrt{x^2 - 1} \right)^6 + \left( x - \sqrt{x^2 - 1} \right)^6\]
Evaluate the
(vii) \[\left( \sqrt{3} + 1 \right)^5 - \left( \sqrt{3} - 1 \right)^5\]
Evaluate the
(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]
Evaluate the
(x) \[\left\{ a^2 + \sqrt{a^2 - 1} \right\}^4 + \left\{ a^2 - \sqrt{a^2 - 1} \right\}^4\]
Find \[\left( x + 1 \right)^6 + \left( x - 1 \right)^6\] . Hence, or otherwise evaluate \[\left( \sqrt{2} + 1 \right)^6 + \sqrt{2} - 1^6\] .
Using binomial theorem evaluate :
(i) (96)3
Using binomial theorem evaluate .
(ii) (102)5
Using binomial theorem evaluate .
(iii) (101)4
Using binomial theorem evaluate .
(iv) (98)5
Using binomial theorem, prove that \[2^{3n} - 7n - 1\] is divisible by 49, where \[n \in N\] .
Using binomial theorem, prove that \[3^{2n + 2} - 8n - 9\] is divisible by 64, \[n \in N\] .
Find the coefficient of:
(i) x10 in the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)^{20}\]
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:
(iv) \[x^9\] in the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\]
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 \[\left( a - 2b \right)^{12}\]
Find the coefficient of:
(viii) x in the expansion of \[\left( 1 - 3x + 7 x^2 \right) \left( 1 - x \right)^{16}\]
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?
Does the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)\] contain any term involving x9?
Write the sum of the coefficients in the expansion of \[\left( 1 - 3x + x^2 \right)^{111}\]
The coefficient of x4 in \[\left( \frac{x}{2} - \frac{3}{x^2} \right)^{10}\] is
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
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
