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

In the expansion of ${\left( {3x - \frac{1}{{{x^2}}}} \right)^{10}}$,the $5^{th}$ term from the end is:

If the three consecutive coefficients in the expansion of $(1 + x)^n$ are $28, 56$ and $70,$ then the value of $n$ is

If $n$ is the number of irrational terms in the expansion of $(3^{1/4} + 5^{1/8})^{60}$,then $(n - 1)$ is divisible by

If $k$ is the coefficient of $x^5$ in the expansion of $(2x^2 - \frac{1}{3x^3})^5$,then $\frac{3k}{2} = $

If for positive integers $r > 1$ and $n > 2$,the coefficients of the $(3r)^{th}$ and $(r + 2)^{th}$ powers of $x$ in the expansion of $(1 + x)^{2n}$ are equal,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module