Mathematics — Standard (041) — CBSE Class 10 board question

For rational roots, discriminant $b^2 - 4ac$ must be a perfect square (≥ 0).

Here $a = 2$, $b = k$, $c = -4$.

$D = k^2 - 4(2)(-4) = k^2 + 32$

For rational roots, $k^2 + 32$ must be a perfect square. The least positive value of $k$ that satisfies this is $k = \mathbf{2}$.

Answer: (B) 2

(Check: $k=2 \Rightarrow D = 4+32=36=6^2$ ✓, roots rational)

Source: Chapter 4, Section 4.4 Nature of Roots

• Rational roots require the discriminant to be a perfect square (not just ≥ 0).
• Here $D = k^2 + 32$; since $k^2 + 32 > 0$ always, roots are always real, but for rationality $k^2 + 32$ must be a perfect square.
• Testing small positive integers: $k=1 \Rightarrow 33$ (not perfect square); $k=2 \Rightarrow 36 = 6^2$ ✓.
• The question asks for the least positive value, so the answer is 2, not $\pm2$.