Q107 / 160 · TS EAMCET 2022 (19 Jul Shift 2) · Mathematics

$\frac{d}{d x}\left[\left(x^{\frac{5}{2}}-x^{\frac{3}{2}}+1\right)\left(x^2-3 x+5\right)\right]=$

  • Correct answer
    A
    $\frac{9}{2} x^{\frac{7}{2}}-14 x^{\frac{5}{2}}+20 x^{\frac{3}{2}}-\frac{15}{2} x^{\frac{1}{2}}+2 x-3$
  • B
    $\frac{9}{2} x^{\frac{7}{2}}-7 x^{\frac{5}{2}}+5 x^{\frac{3}{2}}-\frac{3}{2} x^{\frac{1}{2}}+2 x-3$
  • C
    $9 x^{\frac{7}{2}}-14 x^{\frac{5}{2}}+20 x^{\frac{3}{2}}-15 x^{\frac{1}{2}}+2 x-3$
  • D
    $\frac{9}{2} x^{\frac{7}{2}}-\frac{7}{2} x^{\frac{5}{2}}+\frac{5}{2} x^{\frac{3}{2}}-\frac{15}{2} x^{\frac{1}{2}}+2 x-3$

Solution

We have $\frac{d}{d x}\left[\left(x^{\frac{5}{2}}-x^{\frac{3}{2}}+1\right)\left(x^2-3 x+5\right)\right]$ $$ \begin{aligned} & \Rightarrow\left(\frac{5}{2} x^{\frac{3}{2}}-\frac{3}{2} x^{\frac{1}{2}}\right)\left(x^2-3 x+5\right)+\left(x^{\frac{5}{2}}-x^{\frac{3}{2}}+1\right)(2 x-3) \\ & \Rightarrow \frac{5}{2} x^{\frac{7}{2}}-9 x^{\frac{5}{2}}+17 x^{\frac{3}{2}}-\frac{15}{2} x^{\frac{1}{2}}+2 x^{\frac{7}{2}}-5 x^{\frac{5}{2}}+3 x^{\frac{3}{2}}+2 x-3 \\ & \Rightarrow \frac{9}{2} x^{\frac{7}{2}}-14 x^{\frac{5}{2}}+20 x^{\frac{3}{2}}-\frac{15}{2} x^{\frac{1}{2}}+2 x-3 \end{aligned} $$

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

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