Q61 / 160 · TS EAMCET 2024 (11 May Shift 1) · Mathematics

If $\frac{x^2}{2 x^4+7 x^2+6}=\frac{A x+B}{x^2+a}+\frac{C x+D}{a x^2+3}$, then $A+B+C-2 D$

  • A
    $2 a$
  • B
    $-2 a$
  • C
    $-4 a$
  • Correct answer
    D
    $4 a$

Solution

$\begin{aligned} & \frac{x^2}{2 x^4+7 x^2+6}=\frac{x^2}{\left(x^2+2\right)\left(2 x^2+3\right)} \\ = & \frac{x^2+2-2}{\left(x^2+2\right)\left(2 x^2+3\right)}=\frac{1}{2 x^2+3}-\frac{2}{\left(x^2+2\right)\left(2 x^2+3\right)}\end{aligned}$ $\begin{aligned} & =\frac{1}{2 x^2+3}+2\left(\frac{1}{x^2+2}-\frac{2}{2 x^2+3}\right)=\frac{2}{x^2+2}-\frac{3}{2 x^2+3} \\ & a=2, A=0, C=0, B=2, D=-3 \\ & A+B+C-2 D=0+2+0-2 \times(-3)=8=4 a .\end{aligned}$

Hence, the correct answer is option: 'D'.

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