The number of integral values of k , for which one root of the equation 2 x 2 - 8 x + k = 0 lies in the interval (1, 2) and its other root lies in the interval (2, 3), is [2023]

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Solution

Q.
The number of integral values of $k$ , for which one root of the equation $2 x^{2} - 8 x + k = 0$ lies in the interval (1, 2) and its other root lies in the interval (2, 3), is [2023]

1 2

2 0

3 1

4 3

Ans.
(3)

$2 x^{2} - 8 x + k = 0$ represents an upward parabola.

$f (1) \cdot f (2) < 0 and f (2) \cdot f (3) < 0$

$(2 {(1)}^{2} - 8 (1) + k) \cdot (2 {(2)}^{2} - 8 (2) + k) < 0$

$and (2 {(2)}^{2} - 8 (2) + k) (2 {(3)}^{2} - 8 (3) + k) < 0$

$(k - 6) (k - 8) < 0 and (k - 8) (k - 6) < 0$

$k \in (6, 8) and k \in (6, 8)$

$Integral value of k = 7$

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