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!

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Answer here!   

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!

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Alternate:

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

$log(P) = k, \quad \dfrac{1}{2}log(Q) = k , \quad \dfrac{1}{3}log(R) = k$

$log(P) = k, \quad log(Q) = 2k , \quad log(R) = 3k$

$\boxed{P = 10^{k}, \quad Q = 10^{2k} , \quad R = 10^{3k}}$

by looking at the values of $P, Q, R$ we can easily guess that option B is True.

$Q^2 = PR$

$(10^{2t})^{2} = 10^{t} \cdot 10^{3t}$

$10^{2 \times 2t} = 10^{t +3t}$

$10^{4t} = 10^{4t}$

$L.H.S. = R.H.S$

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

Comments

Hi @JashanArora,

 

Thanks for posting the solution!

Had a doubt, how do we call Q^1/2 as P R^1/3

Thanks!!

Answer 3

B is correct, try to understand the solution 

Answer 4

Answer is Option B.

Answer 5

Answer: B