If a = √7 − √3, and b = √5 − √2, then which of the following is true?

Correct — C. Replace each root by its value to three decimals.

a = √7 − √3 ≈ 2.646 − 1.732 = 0.914
b = √5 − √2 ≈ 2.236 − 1.414 = 0.822

Compare: 0.914 > 0.822, a gap of about 0.09
Rounding each root to three decimals moves each difference by at most 0.001, far less than that gap.

So a > b → option (c)

Why the others are wrong

• (a)
  a=b — 0.914 and 0.822 differ in the first decimal place, a gap far larger than any rounding error, so the two values cannot be equal.
• (b)
  a<b — It reverses the working: a ≈ 0.914, b ≈ 0.822. For a fixed gap, larger radicands shrink a difference of roots, but a's radicands are further apart (7 − 3 = 4 against 5 − 2 = 3), and that wins here.
• (d)
  Cannot be determined — Both a and b are fixed numbers with no variable in them, so exactly one of =, < or > holds, and the decimals settle it: 0.914 > 0.822.

To compare two differences of square roots, approximate each root to three decimals and subtract. Knowing √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 and √7 ≈ 2.646 makes this quick.

The exact route is rationalising: √x − √y = (x − y)⁄(√x + √y). Here a = 4⁄(√7 + √3) and b = 3⁄(√5 + √2).

That form explains the result: a has the larger numerator (4 against 3), and its larger denominator (≈ 4.378 against ≈ 3.650) is not enough to pull it below b.

Key facts

• √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 and √7 ≈ 2.646, to three decimals.
• √x − √y = (x − y)⁄(√x + √y) for positive x and y.
• For a fixed gap between x and y, √x − √y shrinks as x and y grow: √2 − √1 ≈ 0.414 but √5 − √4 ≈ 0.236.

Common traps

• Deciding by the radicand gaps alone (4 against 3): it agrees here, but √100 − √91 ≈ 0.46 is smaller than √5 − √1 ≈ 1.24 despite the bigger gap.
• Choosing 'Cannot be determined' because the values look close: an expression with no variable always has a definite size.

No directly related past PYQ was found.