Ratio, Proportion, Indices, LogarithmMCQPYQ July 21Question 881 of 305
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If xy+yz+zx=−1\displaystyle xy + yz + zx = -1 then the value of (x+y1+xy+z+y1+zy+x+z1+zx)\displaystyle \left(\frac{x+y}{1+xy} + \frac{z+y}{1+zy} + \frac{x+z}{1+zx}\right) is:

Options

Axyz\displaystyle xyz
B−1xyz\displaystyle \frac{-1}{xyz}
C1xyz\displaystyle \frac{1}{xyz}
D1x+y+z\displaystyle \frac{1}{x+y+z}
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Correct Answer

✅ Option c — 1xyz\displaystyle \frac{1}{xyz}

All Options:

  • Axyz\displaystyle xyz
  • B−1xyz\displaystyle \frac{-1}{xyz}
  • C1xyz\displaystyle \frac{1}{xyz}
  • D1x+y+z\displaystyle \frac{1}{x+y+z}

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Detailed Solution & Explanation

We are given that:
xy+yz+zx=−1xy + yz + zx = -1
We want to find the value of:
x+y1+xy+z+y1+zy+x+z1+zx\frac{x+y}{1+xy} + \frac{z+y}{1+zy} + \frac{x+z}{1+zx}
Let us rewrite the denominator terms using the given equation:
1. For 1+xy\displaystyle 1+xy:
1+xy=−(yz+zx)=−z(y+x)=−z(x+y)1+xy = -(yz+zx) = -z(y+x) = -z(x+y)
2. For 1+zy\displaystyle 1+zy:
1+zy=−(xy+zx)=−x(y+z)=−x(z+y)1+zy = -(xy+zx) = -x(y+z) = -x(z+y)
3. For 1+zx\displaystyle 1+zx:
1+zx=−(xy+yz)=−y(x+z)1+zx = -(xy+yz) = -y(x+z)
Substitute these expressions back into the original fraction:
x+y−z(x+y)+z+y−x(z+y)+x+z−y(x+z)\frac{x+y}{-z(x+y)} + \frac{z+y}{-x(z+y)} + \frac{x+z}{-y(x+z)}
Cancel out the common binomial terms in the numerator and denominator:
=1−z+1−x+1−y= \frac{1}{-z} + \frac{1}{-x} + \frac{1}{-y}
=−(1x+1y+1z)= -\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)
To combine these terms, find a common denominator xyz\displaystyle xyz:
=−(yz+zx+xyxyz)= -\left(\frac{yz + zx + xy}{xyz}\right)
Substitute the given value xy+yz+zx=−1\displaystyle xy + yz + zx = -1 into the numerator:
=−(−1xyz)=1xyz= -\left(\frac{-1}{xyz}\right) = \frac{1}{xyz}
Thus, the value of the expression is 1xyz\displaystyle \frac{1}{xyz}. This corresponds to **Option C**.
Note: The textbook answer key incorrectly specifies **Option B** (−1xyz\displaystyle -\frac{1}{xyz}) as the correct option. However, as mathematically proven above, the signs cancel out to yield a positive value (Option C).
Hence, **Option C** is the correct answer.

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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