Question

If the roots of the equation x² - 4x - log₁₀N = 0 are real, then what is the minimum value of N ?
1. 0.1
2. 0.01
3. 0.001
4. 0.0001

Answer 1

Correct Answer - Option 4 : 0.0001

Given:

Roots of x2 - 4x - log10N = 0 are real

Formula used:

Roots of quadratic equation ax² + bx + c = 0 are real 

if D ≥ 0  i.e b² - 4ac ≥ 0 

And lf logaN = x

Then N = a^x

Calculation:

We have the quadratic equation x2 - 4x - log10N = 0

On comparing this equation with ax2 + bx + c = 0, we get 

a = 1, b = -4 and c = - log10N

Now, According to the question 

D = b2 - 4ac  ≥ 0

⇒ (- 4)² - 4 × 1 × (- log10N) ≥ 0

⇒ 16 + 4log10N ≥ 0

⇒ 4log10N ≥ -16

⇒ log10N ≥ - 4

⇒ N ≥ 10^-4

⇒ N ≥ 0.0001

∴ The minimum value of N is 0.0001.