Ratio, Proportion, Indices, LogarithmMCQPYQ Dec 23Question 892 of 305
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Given x=5+35−3\displaystyle x = \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} and y=5−35+3\displaystyle y = \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}. Then find the value of 1x2+1y2\displaystyle \frac{1}{x^2} + \frac{1}{y^2}.

Options

A63
B61
C62
D60
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Correct Answer

✅ Option c — 62

All Options:

  • A63
  • B61
  • C62
  • D60

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Detailed Solution & Explanation

We are given:
x=5+35−3andy=5−35+3x = \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} \quad \text{and} \quad y = \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}
Let us first rationalize the denominator of x\displaystyle x:
x=(5+3)2(5−3)(5+3)=5+3+2155−3=8+2152=4+15x = \frac{(\sqrt{5}+\sqrt{3})^2}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})} = \frac{5 + 3 + 2\sqrt{15}}{5 - 3} = \frac{8 + 2\sqrt{15}}{2} = 4 + \sqrt{15}
Similarly, since y\displaystyle y is the reciprocal of x\displaystyle x:
y=4−15y = 4 - \sqrt{15}
Now, calculate the sum (x+y)\displaystyle (x+y) and product (xy)\displaystyle (xy):
- Sum: x+y=(4+15)+(4−15)=8\displaystyle x + y = (4 + \sqrt{15}) + (4 - \sqrt{15}) = 8
- Product: xy=(4+15)(4−15)=42−(15)2=16−15=1\displaystyle xy = (4 + \sqrt{15})(4 - \sqrt{15}) = 4^2 - (\sqrt{15})^2 = 16 - 15 = 1
We want to find the value of 1x2+1y2\displaystyle \frac{1}{x^2} + \frac{1}{y^2}:
1x2+1y2=y2+x2x2y2=x2+y2(xy)2\frac{1}{x^2} + \frac{1}{y^2} = \frac{y^2 + x^2}{x^2 y^2} = \frac{x^2 + y^2}{(xy)^2}
Since xy=1\displaystyle xy = 1:
1x2+1y2=x2+y2\frac{1}{x^2} + \frac{1}{y^2} = x^2 + y^2
Using the algebraic identity x2+y2=(x+y)2−2xy\displaystyle x^2 + y^2 = (x+y)^2 - 2xy:
x2+y2=82−2(1)=64−2=62x^2 + y^2 = 8^2 - 2(1) = 64 - 2 = 62
Thus, the value of the expression is exactly 62\displaystyle 62. This corresponds to **Option C**.
Note: The textbook answer key incorrectly specifies **Option B** (61\displaystyle 61) as correct. However, our mathematical proof clearly shows that the value is 62\displaystyle 62 (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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