GATE CSE 2011 | Question: 57

Question

If $\log (\text{P}) = (1/2)\log (\text{Q}) = (1/3)\log (\text{R})$, then which of the following options is TRUE?

• A. $\text{P}^2 = \text{Q}^3\text{R}^2$
• B. $\text{Q}^2=\text{P}\text{R}$
• C. $\text{Q}^2 = \text{R}^3\text{P}$
• D. $\text{R}=\text{P}^2\text{Q}^2$

Comments

Put P=$2^{10}$
      Q=$2^{20}$
      R= $2^{30}$

Put in options only b) option satisfies!

Answer 1

$B$. is the answer.

Following logarithm formula, we get:
$P=Q^{\frac{1}{2}}=R^{\frac{1}{3}}$
So, $Q^{2}= P^{4}= P\times P^{3}=PR.$

Comments

Simplest Approach!

Answer 2

$log(P)=\frac{1}{2}log(Q)=\frac{1}{3}log(R)$

$=>P=Q^{\frac{1}{2}}=R^{\frac{1}{3}}$

Now

$Q^{2}=Q^{\frac{1}{2}}Q^{\frac{1}{2}}Q^{\frac{1}{2}}Q^{\frac{1}{2}}$

$=>Q^{2}=PR^{\frac{1}{3}}R^{\frac{1}{3}}R^{\frac{1}{3}}$

$=>Q^{2}=PR$

 

Option B

Answer 3

Answer is Option B.