The positive value of lambda for which the co | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The given expression is $x^2 \left( x^{1/2} + \lambda x^{-2} \right)^{10}$.The general term in the expansion of $\left( x^{1/2} + \lambda x^{-2} \

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) The given expression is $x^2 \left( x^{1/2} + \lambda x^{-2} ight)^{10}$.The general term in the expansion of $\left( x^{1/2} + \lambda x^{-2} ight)^{10}$ is given by $T_{r+1} = {}^{10}C_r (x^{1/2})^{10-r} (\lambda x^{-2})^r$.$T_{r+1} = {}^{10}C_r \lambda^r x^{(10-r)/2} x^{-2r} = {}^{10}C_r \lambda^r x^{(10-5r)/2}$.Multiplying by the external $x^2$,the general term of the full expression is $x^2 \cdot T_{r+1} = {}^{10}C_r \lambda^r x^{(10-5r)/2 + 2}$.To find the coefficient of $x^2$,we set the exponent equal to $2$:$\frac{10-5r}{2} + 2 = 2$ $\Rightarrow 10-5r = 0$ $\Rightarrow r = 2$.Substituting $r=2$ into the coefficient expression:${}^{10}C_2 \lambda^2 = 720$.Since ${}^{10}C_2 = \frac{10 	imes 9}{2} = 45$,we have $45 \lambda^2 = 720$.$\lambda^2 = \frac{720}{45} = 16$.Since $\lambda$ is positive,$\lambda = 4$.

The positive value of $\lambda$ for which the coefficient of $x^2$ in the expression $x^2 \left( \sqrt{x} + \frac{\lambda}{x^2} \right)^{10}$ is $720$ is

Similar Questions

The term independent of $x$ in the binomial expansion of $\left( 1 - \frac{1}{x} + 3x^5 \right) \left( 2x^2 - \frac{1}{x} \right)^8$ is

The coefficient of $x^{32}$ in the expansion of $(x^4 - \frac{1}{x^3})^{15}$ is

In the expansion of $(1+x)(1-x^2)(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3})^5, x \neq 0$,the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to

If the coefficients of the three successive terms in the binomial expansion of $(1 + x)^n$ are in the ratio $1 : 7 : 42,$ then the first of these terms in the expansion is (in $^{th}$)

The $16^{th}$ term in the expansion of $(\sqrt{x} - \sqrt{y})^{17}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module