Q51 / 160 · AP EAMCET 2022 (06 Jul Shift 2) · Mathematics

The sum of squares of roots of the equation $x^{\frac{2}{3}}+x^{\frac{1}{3}}-2=0$ is

  • A
    82
  • Correct answer
    B
    65
  • C
    50
  • D
    37

Solution

Given, equation is $$ \begin{array}{rlrl} & & x^{2 / 3}+x^{1 / 3}-2 & =0 \\ \Rightarrow & & x^{2 / 3}+x^{1 / 3} & =2 \\ & \Rightarrow & \left(x^{2 / 3}+x^{1 / 3}\right)^3 & =2^3 \\ & & & \\ {\left[\therefore(a+b)^3\right.} & \left.=a^3+b^3+3 a b(a+b)\right] \end{array} $$ $\begin{aligned} & \Rightarrow \quad x^2+x+3 x \times 2=8 \Rightarrow \\ & x^2+7 x-8=0 \\ & \Rightarrow \quad(x+8)(x-1)=0 \Rightarrow x=1 ;-8 \\ & \text { Sum of square roots }=(1)^2+(-8)^2 \\ & \quad=1+64=65\end{aligned}$

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

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