Ratio, Proportion, Indices, LogarithmMCQMTP March 21Question 959 of 305
All Questions

If x2+y2=7xy\displaystyle x^2 + y^2 = 7xy, then log⁡13(x+y)\displaystyle \log \frac{1}{3} (x + y) then x\displaystyle x is

Options

A(log⁡x+log⁡y)\displaystyle (\log x + \log y)
B12(log⁡x+log⁡y)\displaystyle \frac{1}{2} (\log x + \log y)
C13(log⁡x+log⁡y)\displaystyle \frac{1}{3} (\log x + \log y)
D3(log⁡x+log⁡y)\displaystyle 3 (\log x + \log y)
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Correct Answer

✅ Option b — 12(log⁡x+log⁡y)\displaystyle \frac{1}{2} (\log x + \log y)

All Options:

  • A(log⁡x+log⁡y)\displaystyle (\log x + \log y)
  • B12(log⁡x+log⁡y)\displaystyle \frac{1}{2} (\log x + \log y)
  • C13(log⁡x+log⁡y)\displaystyle \frac{1}{3} (\log x + \log y)
  • D3(log⁡x+log⁡y)\displaystyle 3 (\log x + \log y)

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Detailed Solution & Explanation

Given x2+y2=7xy\displaystyle x^2 + y^2 = 7xy.

Add 2xy\displaystyle 2xy to both sides:

x2+2xy+y2=9xyx^2 + 2xy + y^2 = 9xy

(x+y)2=9xy(x + y)^2 = 9xy

x+y=3xyx + y = 3\sqrt{xy}

x+y3=xy\frac{x + y}{3} = \sqrt{xy}

Taking logarithm of both sides:

log⁡(x+y3)=log⁡(xy)\log\left(\frac{x + y}{3}\right) = \log(\sqrt{xy})

log⁡(x+y3)=12log⁡(xy)=12(log⁡x+log⁡y)\log\left(\frac{x+y}{3}\right) = \frac{1}{2}\log(xy) = \frac{1}{2}(\log x + \log y)

**The answer is (b) 12(log⁡x+log⁡y)\displaystyle \frac{1}{2}(\log x + \log y).**

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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