Suppose \(x = √2+√3,\ y = √5+1,\ z = 2√2+1\) ; then which of the following is true?

Question

Suppose \(x = √2+√3,\ y = √5+1,\ z = 2√2+1\) ; then which of the following is true?

1. A. \(x < y < z\) (Correct answer)
2. B. \(z < y < x\)
3. C. \(y < x < z\)
4. D. \(x < z < y\)

Correct Answer

Option A — \(x < y < z\)

Detailed Solution & Explanation

The correct answer is x < y < z.

Key Points

• Approximate each expression to two decimal places:
  • x = √2 + √3 ≈ 1.414 + 1.732 = 3.146
  • y = √5 + 1 ≈ 2.236 + 1 = 3.236
  • z = 2√2 + 1 ≈ 2.828 + 1 = 3.828
• Ordering them gives x < y < z.

Additional Information

• Square roots worth memorising to three decimals: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449, √7 ≈ 2.646, √10 ≈ 3.162.
• Where two values sit very close, squaring both sides is the exact method — but note that squaring a sum of surds leaves a cross term, so x² = 5 + 2√6 rather than 5.
• Approximation is safe here because the three values differ in the first decimal place. Reach for exact comparison only when two options agree to two decimals.

प्रश्न (हिन्दी में)

मान लीजिए \(x = √2+√3,\ y = √5+1,\ z = 2√2+1\) ; तो निम्न में से कौन सा सत्य है?

1. A. \(x < y < z\)
2. B. \(z < y < x\)
3. C. \(y < x < z\)
4. D. \(x < z < y\)