Ratio, Proportion, Indices, LogarithmMCQPYQ May 18Question 787 of 302
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If p:q\displaystyle p:q is the sub-duplicate ratio of p−x2:q−x2\displaystyle p-x^2:q-x^2, then x2\displaystyle x^2 is

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option d — None of these

All Options:

  • App+q\displaystyle \frac{p}{p+q}
  • Bqp+q\displaystyle \frac{q}{p+q}
  • Cqp\displaystyle qp
  • DNone of these

Detailed Solution & Explanation

Let the given ratio be $p : q$, which is the sub-duplicate ratio of (p−x2):(q−x2)\displaystyle (p - x^2) : (q - x^2).
By the definition of the sub-duplicate ratio, we have:
pq=p−x2q−x2\displaystyle \frac{p}{q} = \sqrt{p−x2q−x2\displaystyle \frac{p - x^2}{q - x^2}}
Squaring both sides of the equation:
p2q2=p−x2q−x2p2q2=p−x2q−x2\displaystyle \frac{p^2}{q^2} = \frac{p - x^2}{q - x^2}
Cross-multiplying to solve for x2\displaystyle x^2:
p2(q−x2)=q2(p−x2)p^2(q - x^2) = q^2(p - x^2)
p2q−p2x2=q2p−q2x2p^2q - p^2x^2 = q^2p - q^2x^2
Rearranging terms to group all x2\displaystyle x^2 terms on one side:
q2x2−p2x2=q2p−p2qq^2x^2 - p^2x^2 = q^2p - p^2q
Factoring out x2\displaystyle x^2 and pq\displaystyle pq respectively:
x2(q2−p2)=pq(q−p)x^2(q^2 - p^2) = pq(q - p)
Applying the difference of squares identity q2−p2=(q−p)(q+p)\displaystyle q^2 - p^2 = (q - p)(q + p):
x2(q−p)(q+p)=pq(q−p)x^2(q - p)(q + p) = pq(q - p)
Assuming p≠q\displaystyle p ≠\displaystyle \neq q, we divide both sides by (q−p)\displaystyle (q - p):
x2(p+q)=pq  ⟹  x2=pqp+qx^2(p + q) = pq \implies x^2 = pqp+q\displaystyle \frac{pq}{p + q}
Comparing with the given choices:
- Option A: pp+q\displaystyle pp+q\displaystyle \frac{p}{p+q}
- Option B: qp+q\displaystyle qp+q\displaystyle \frac{q}{p+q}
- Option C: qpp−q\displaystyle qpp−q\displaystyle \frac{qp}{p-q}
None of these represents the correct expression pqp+q\displaystyle pqp+q\displaystyle \frac{pq}{p+q}. Hence, the correct choice is **None of these**.

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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