Question

Assertion :If $x={\left(\frac{2}{3}\right)}^4÷{\left(\frac{2}{3}\right)}^2$, then value of $x^2+2x+3$ is 0 Reason: If ${(-\frac{1}{2})}^4\times (-2{)}^8=(-2{)}^{4x}$ then x=1.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is incorrect but Reason is correct

Statement 1-$x=(\frac{2}{3}{)}^4÷(\frac{2}{3}{)}^2$
$=(\frac{2}{3}{)}^{4-2}=(\frac{2}{3}{)}^2=\frac{4}{9}$
Now,$x^2+2x+3=(\frac{4}{9}{)}^2+2(\frac{4}{9})+3=4\frac{7}{81}\ne 0$
$\therefore$ Statement 1 is false.
Statement 2- We have,$(-\frac{1}{2}{)}^4\times (-2{)}^8=(-2{)}^{4x}$
$\Rightarrow (-1{)}^4\times (2{)}^{-4}\times (-1{)}^8\times (2{)}^9=(-2{)}^{4x}$
$\Rightarrow (-1{)}^{12}\times (2{)}^{8-4}=(-2{)}^{4x}\Rightarrow (2{)}^4=(-2{)}^{4x}$
$\Rightarrow (-1{)}^4\times (2{)}^4=(-2{)}^{4x}\Rightarrow (-2{)}^4=(-2{)}^{4x}$
On comparing, we get
$4=4x,\Rightarrow x=1$
$\therefore$ Statement-2 is true.