The value of k for which the number 3 lies between the roots of equation x^2 + (1 - 2k)x + (k^2 - k - 2) = 0 is given byA) 2 B) k > 2C) 2 D) k > 5[Full Explaination needed]

note that the graph of y= f(x)= x^2 + (1 - 2k)x + (k^2 - k - 2) would be an upward parabola. so, a necessary and sufficient condition for 3 to lie in between roots is f(3) should be less than 0. this is in effect the same as applying intermediate value theorem as  f is continuous.

here, f(3)= 9 + 3(1 –  2k) + k^2 - k - 2 must be less than 0.

or k^2 – 7k + 10 must be less than 0.

or (k – 2)(k – 5) must be less than 0.

so that Ans: k should lie in (2, 5).

your option A and C are wrongly typed so idk what they are.

but D is definitely wrong and B is partially correct.

anways, the ques has been answered.

Sir, I think there was a problem in placing the options
The Updated Options are
A) 2k5
B) k > 2
C) 2k3
D) k > 5

yeah so obviously option A is correct as it is the same as (2, 5).

if any other doubt, feel free to ask.

 kindly approve :)