Question

If both roots of the equation (x-1)(x-5) + k = 0 lies in the
interval (2, 4), then the range of k is
(1) [3,4]
(2) [3,4)
(3) (3,4
(4) (3, 4)​

Answer 1

Both the roots are less than 5

⇒22k±4k2−4(k2+k−5)<5

⇒k±5−k<5

⇒±5−k<5−k

⇒±1<5−k

⇒1<5−k

⇒k<4⇒k∈(−∞,4)