Ratio, Proportion, Indices, LogarithmMCQPYQ Sep 24Question 894 of 305
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What is the value of (xbxc)(b+c−a)×(xcxa)(c+a−b)×(xaxb)(a+b−c)\displaystyle \left(\frac{x^b}{x^c}\right)^{(b+c-a)} \times \left(\frac{x^c}{x^a}\right)^{(c+a-b)} \times \left(\frac{x^a}{x^b}\right)^{(a+b-c)}

Options

Ax(a+b+c)\displaystyle x^{(a+b+c)}
Bxabc\displaystyle x^{abc}
C-1
D1
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Correct Answer

✅ Option d — 1

All Options:

  • Ax(a+b+c)\displaystyle x^{(a+b+c)}
  • Bxabc\displaystyle x^{abc}
  • C-1
  • D1

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Detailed Solution & Explanation

We want to simplify the expression:
(xbxc)b+c−a×(xcxa)c+a−b×(xaxb)a+b−c\left(\frac{x^b}{x^c}\right)^{b+c-a} \times \left(\frac{x^c}{x^a}\right)^{c+a-b} \times \left(\frac{x^a}{x^b}\right)^{a+b-c}
Let us simplify each factor using exponent laws (xmxn=xm−n\displaystyle \frac{x^m}{x^n} = x^{m-n} and (xm)n=xmn\displaystyle (x^m)^n = x^{mn}):
1. **First factor:**
(xb−c)b+c−a=x(b−c)(b+c−a)=xb2−c2−ab+ac\left(x^{b-c}\right)^{b+c-a} = x^{(b-c)(b+c-a)} = x^{b^2 - c^2 - ab + ac}
2. **Second factor:**
(xc−a)c+a−b=x(c−a)(c+a−b)=xc2−a2−bc+ab\left(x^{c-a}\right)^{c+a-b} = x^{(c-a)(c+a-b)} = x^{c^2 - a^2 - bc + ab}
3. **Third factor:**
(xa−b)a+b−c=x(a−b)(a+b−c)=xa2−b2−ac+bc\left(x^{a-b}\right)^{a+b-c} = x^{(a-b)(a+b-c)} = x^{a^2 - b^2 - ac + bc}

Now, multiply the three factors by adding their exponents (xm⋅xn=xm+n\displaystyle x^m \cdot x^n = x^{m+n}):
Product=x(b2−c2−ab+ac)+(c2−a2−bc+ab)+(a2−b2−ac+bc)\text{Product} = x^{(b^2 - c^2 - ab + ac) + (c^2 - a^2 - bc + ab) + (a^2 - b^2 - ac + bc)}
Let us combine and simplify the exponent terms:
(b2−b2)+(c2−c2)+(a2−a2)+(ab−ab)+(bc−bc)+(ac−ac)=0(b^2 - b^2) + (c^2 - c^2) + (a^2 - a^2) + (ab - ab) + (bc - bc) + (ac - ac) = 0
Thus, the expression simplifies to:
x0=1x^0 = 1
This matches **Option D**.
Note: The textbook answer key incorrectly specifies **Option A** (x(a+b+c)\displaystyle x^{(a+b+c)}) as correct. However, our rigorous mathematical simplification clearly proves that all exponents cancel out to zero, yielding exactly 1\displaystyle 1 (Option D).
Hence, **Option D** 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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