begin{aligned} & text{If } frac{x^3}{(2x-1)(x | vedclass

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

Question

(C) We have,$\frac{x^3}{(2x-1)(x-1)^2} = \frac{x^3}{2x^3-5x^2+4x-1}$.Performing long division,we get:$\frac{x^3}{2x^3-5x^2+4x-1} = \frac{1}{2} + \frac

Vedclass · Accepted Answer

$\frac{17}{2}$

Vedclass · Answer

(C) We have,$\frac{x^3}{(2x-1)(x-1)^2} = \frac{x^3}{2x^3-5x^2+4x-1}$.Performing long division,we get:$\frac{x^3}{2x^3-5x^2+4x-1} = \frac{1}{2} + \frac{1}{2} \left( \frac{5x^2-4x+1}{(2x-1)(x-1)^2} \right) \dots (i)$.Now,let $\frac{5x^2-4x+1}{(2x-1)(x-1)^2} = \frac{B}{2x-1} + \frac{C}{x-1} + \frac{D}{(x-1)^2}$.$5x^2-4x+1 = B(x-1)^2 + C(x-1)(2x-1) + D(2x-1)$.Putting $x=1$,we get $5-4+1 = D(2-1) \Rightarrow D=2$.Putting $x=1/2$,we get $5(1/4)-4(1/2)+1 = B(1/2-1)^2$ $\Rightarrow 5/4-2+1 = B(1/4)$ $\Rightarrow 1/4 = B/4$ $\Rightarrow B=1$.Comparing coefficients of $x^2$: $5 = B + 2C$ $\Rightarrow 5 = 1 + 2C$ $\Rightarrow 2C = 4$ $\Rightarrow C=2$.Substituting these into $(i)$:$\frac{x^3}{(2x-1)(x-1)^2} = \frac{1}{2} + \frac{1}{2} \left( \frac{1}{2x-1} + \frac{2}{x-1} + \frac{2}{(x-1)^2} \right) = \frac{1}{2} + \frac{1/2}{2x-1} + \frac{1}{x-1} + \frac{1}{(x-1)^2}$.Comparing with the given expression,$A=1/2, B=1/2, C=1, D=1$.Then,$2A - 3B + 4C + 5D = 2(1/2) - 3(1/2) + 4(1) + 5(1) = 1 - 1.5 + 4 + 5 = 8.5 = \frac{17}{2}$.

$\begin{aligned} & \text{If } \frac{x^3}{(2x-1)(x-1)^2} = A + \frac{B}{2x-1} + \frac{C}{x-1} \\ & + \frac{D}{(x-1)^2}, \text{ then } 2A - 3B + 4C + 5D = \end{aligned}$

Similar Questions

Evaluate the partial fraction decomposition of $\frac{2x}{x^4 + x^2 + 1}$.

If $\frac{3 x^4-2 x^2+1}{(x-2)^4}=A+\frac{B}{x-2}+\frac{C}{(x-2)^2}+\frac{D}{(x-2)^3}+\frac{E}{(x-2)^4}$, then $2 A+3 B-C-D+E=$

What is the coefficient of $x^4$ in the expansion of $\frac{3x}{(x - 2)(x + 1)}$?

If $\frac{1}{x^4+x^2+1}=\frac{Ax+B}{x^2+ax+1}+\frac{Cx+D}{x^2-ax+1}$ then $A+B-C+D=$

If $\frac{x-4}{x^2-5x+6}$ can be expanded in the ascending powers of $x$,then the coefficient of $x^3$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module