Home
Class 12
MATHS
The smallest value of k for which both t...

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 atleast 4 is

A

6

B

4

C

2

D

0

Text Solution

Verified by Experts

The correct Answer is:
D
Let `f(x)=x^(2)-8kx+16(k^(2)-k+1)` brgt
`:.Dgt0`
`implies64k^(2)-4.16(k^(2)-k+1)gt0`
`implieskgt1` …….i
`implies(-b)/(2a)gt4implies(8k)/2gt4`
`implieskgt1`………….ii
and `f(4)ge0`
`implies16-32k+16(k^(2)-k+1)ge0`
`impliesk^(2)-3k+2ge0`
`implies(k-1)(k-2)ge0`
`kle1` or `kge2`...........iii
From Eqs i, ii and iii we get
`kge2`
`k_("min")=2`
Promotional Banner

Similar Questions

Explore conceptually related problems

The positive value of K for which both the equations x^(2)+Kx+64=0 and x^(2)-8x+K=0 have real roots, is

If both the roots of the quadratic equation x^(2)-2 k x+(k^(2)+k-5)=0 are less than 5, then k lies in the interval

If the equation x^(2) + 2 (k+3) x +12k=0 has equal roots, then k=

One root of the equation x^(2) - 5x+ k = 0 is 2. Then k is :

Find the value of k for which the quadratic equations 9x^(2) - 24x+ k=0 has equal roots.

The product of the roots of the equations x^(2) + 5x+ (k+ 4) =0 is zero , then k is equal to

The equation (x^(2))/(12-k)+(y^(2))/(8-k)=1 represents

The number of values of 'k' for which the equation x^(2) - 3x + k = 0 has two distinct roots lying in the interval (0,1) is :

If 2 is the root of the quadratic equation 3x^(2)+px-8=0 and 4x^(2)-2px+K=0 has equal roots, find the value of K.

The number of values of k for which the system of equations : (k+1)x+8y=4k kx+(k+3)y=3k-1 has no solution is: