MCQ 11 Mark
Assertion (A): If the product of positive real numbers $a, b, c$ is 27 , then minimum value of $a+b+c$ is 9 .
Reason (R): For positive real numbers, A.M. $\leq$ G.M.
Reason (R): For positive real numbers, A.M. $\leq$ G.M.
- ABoth A and R are true and R is the correct explanation of A .
- BBoth $A$ and $R$ are true but $R$ is not the correct explanation of A.
- ✓A is true but R is false.
- D$A$ is false but $R$ is true.
Answer
View full question & answer→Correct option: C.
A is true but R is false.
(c) A is true but R is false.
Explanation:
We know that for positive numbers A.M. $\geq$ G.M.
$\Rightarrow \frac{a+b+c}{3} \geq \sqrt[3]{a \cdot b \cdot c}$
$\Rightarrow \frac{a+b+c}{3} \geq \sqrt[3]{27} \Rightarrow a+b+c \geq 3 \times 3$
$\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c} \geq 9$
$\therefore$ Minimum value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is 9 .
$\therefore \mathrm{A}$ is true. R is false.
Explanation:
We know that for positive numbers A.M. $\geq$ G.M.
$\Rightarrow \frac{a+b+c}{3} \geq \sqrt[3]{a \cdot b \cdot c}$
$\Rightarrow \frac{a+b+c}{3} \geq \sqrt[3]{27} \Rightarrow a+b+c \geq 3 \times 3$
$\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c} \geq 9$
$\therefore$ Minimum value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is 9 .
$\therefore \mathrm{A}$ is true. R is false.