Question

Assertion :In any $\Delta ABC$, minimum value of $\frac{r_1{r}_2{r}_3}{r^3}=27$ Reason: $x_1+x_2+x_3+....+x_n=\lambda$ (constant) then $x_1,x_2,x_3,....x_n$ is minimum when $x_1=x_2=x_3=....x_n$

• 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 correct but Reason is incorrect
• D. Assertion false but Reason is true

Answer 1

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

$(r_1.r_2.r_3{)}^{\frac{1}{3}}\ge \frac{3}{\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}}$
$(r_1.r_2.r_3{)}^{\frac{1}{3}}\ge \frac{3}{\frac{1}{r}}$ since $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r}$.

Hence
$(r_1.r_2.r_3{)}^{\frac{1}{3}}\ge 3r$

$(r_1.r_2.r_3)\ge 27r^3$

$\frac{r_1.r_2.r_3}{r^3}\ge 27$.

Now
$x_1+x_2+..x_n=\lambda$ then $\lambda$ is maximum when $x_1=x_2=x_3=..x_n\ne 0$.

Then $\lambda =nX$

Hence Reason is false but Assertion is true.