Question

Find the smallest positive integer value of k for which quadratic equation \(\rm \sqrt{3}x^2-\sqrt{2}kx +2 \sqrt{3}=0\), will have distinct real roots ?
1. 3
2. 4
3. 1
4. 2
5. None of these

Answer 1

Correct Answer - Option 2 : 4

Concept:

Discriminant of quadratic equation ax2 + bx + c = 0, is given by D = b² - 4ac.

If a quadratic equation ax2 + bx + c = 0 has distinct real roots then D > 0.

Calculation:

Given: Quadratic equation \(\rm \sqrt {3}x^2-\sqrt {2}kx +2 \sqrt {3}=0\), has distinct real roots.

As we know that, if a quadratic equation ax2 + bx + c = 0 has distinct real roots then Δ > 0 where Δ = b2 - 4ac.

Here, a = √3, b = - √2k and c = 2√3

⇒ \(\rm (-\sqrt {2}k)^2-4\times \sqrt {3}\times 2 \sqrt {3}>0\)   

⇒ 2k² - 24 > 0  ⇒ k² > 12      

∵ k is smallest positive integer, so k = 4

​Hence, option 4 is correct.