The smallest value of k, for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1)$ = 0 are real, distinct and have values at least 4,is

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B
2

$α a n d β \geq 4$

$x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$

(i) D > 0

64 $k^{2} - 64 (k^{2} - k + 1)$ > 0

k - 1 > 0 $\Rightarrow$ k > 1

(ii) $\frac{8 k}{2}$ > 4 $\Rightarrow$ k > 1

(iii) f(4) $\geq$ 0

16 - 32k + 16 $(k^{2} - k + 1) \geq 0$

$k^{2} - 3 k + 2 \geq 0 \Rightarrow k \leq 1$ or k $\geq$ 2

$∴$ k $\in$ [2, $\infty$ )

$∴$ least value of k = 2

Suggest Corrections

Similar questions

Q. The smallest value of k, for which both the roots of the equation

x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0

are real, distinct and have values at least 4, is

Q. The smallest integer value of

k

for which roots of the equation

x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0

are real is

If $k_{1}, k_{2}, k_{3}$ are real numbers and the equation $k_{1} x^{2} + k_{2} x + k_{3} = 0$ have three roots, then value of $k_{1} + k_{2} + k_{3}$ is

Q. If the roots of the equation

x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0

are real, distinct and have values at least

4

, then which of the following is (are) true?

The positive value of k, for which both equations: x² + kx + 64 =0 and x² - 8x + k =0 have real roots is___________

Join BYJU'S Learning Program

The smallest value of k, for which both the roots of the equation x2−8kx+16(k2−k+1) = 0 are real, distinct and have values at least 4,is

The correct option is B 2 α and β ≥4 x2−8kx+16(k2−k+1)=0 (i) D > 0 64 k2−64(k2−k+1) > 0 k - 1 > 0 ⇒ k > 1 (ii) 8k2 > 4 ⇒ k > 1 (iii) f(4) ≥ 0 16 - 32k + 16 (k2−k+1)≥0 k2−3k+2≥0⇒k≤1 or k ≥ 2 ∴ k ∈ [2,∞) ∴ least value of k = 2

The smallest value of k, for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1)$ = 0 are real, distinct and have values at least 4,is