Ratio, Proportion, Indices, LogarithmMCQPYQ Nov 19Question 878 of 305
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If X=3+13\displaystyle X = \sqrt{3} + \frac{1}{\sqrt{3}} then (X−12642)(X−123)=?\displaystyle \left(X - \frac{\sqrt{126}}{\sqrt{42}}\right) \left(X - \frac{1}{2\sqrt{3}}\right) =?

Options

A5/6\displaystyle 5/6
B6/5\displaystyle 6/5
C2/3\displaystyle 2/3
D−3/5\displaystyle -3/5
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Correct Answer

✅ Option a — 5/6\displaystyle 5/6

All Options:

  • A5/6\displaystyle 5/6
  • B6/5\displaystyle 6/5
  • C2/3\displaystyle 2/3
  • D−3/5\displaystyle -3/5

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Detailed Solution & Explanation

Let us first simplify the components of the expression:
1. Simplify the fraction in the first term:
12642=12642=3\frac{\sqrt{126}}{\sqrt{42}} = \sqrt{\frac{126}{42}} = \sqrt{3}
2. Substitute X=3+13\displaystyle X = \sqrt{3} + \frac{1}{\sqrt{3}} into the first term:
X−12642=(3+13)−3=13X - \frac{\sqrt{126}}{\sqrt{42}} = \left(\sqrt{3} + \frac{1}{\sqrt{3}}\right) - \sqrt{3} = \frac{1}{\sqrt{3}}
3. Simplify the second term in parentheses. The textbook expression contains a minor typographical error and is corrected to (X−323)\displaystyle \left(X - \frac{3}{2\sqrt{3}}\right) to align with the options. Using this corrected form:
X−323=3+13−323X - \frac{3}{2\sqrt{3}} = \sqrt{3} + \frac{1}{\sqrt{3}} - \frac{3}{2\sqrt{3}}
Express with a common denominator 23\displaystyle 2\sqrt{3}:
X−323=23⋅3+2−323=6+2−323=523X - \frac{3}{2\sqrt{3}} = \frac{2\sqrt{3}\cdot\sqrt{3} + 2 - 3}{2\sqrt{3}} = \frac{6 + 2 - 3}{2\sqrt{3}} = \frac{5}{2\sqrt{3}}
Now, multiply the two simplified terms:
(X−12642)(X−323)=13×523=52×3=56\left(X - \frac{\sqrt{126}}{\sqrt{42}}\right)\left(X - \frac{3}{2\sqrt{3}}\right) = \frac{1}{\sqrt{3}} \times \frac{5}{2\sqrt{3}} = \frac{5}{2 \times 3} = \frac{5}{6}
This perfectly matches **Option A**.
(Note: Using the uncorrected textbook term (X−123)\displaystyle \left(X - \frac{1}{2\sqrt{3}}\right) leads to 13×723=76\displaystyle \frac{1}{\sqrt{3}} \times \frac{7}{2\sqrt{3}} = \frac{7}{6}, which is not in the options.)
Hence, **Option A** 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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