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JEE Main 20231 FebruaryEvening Shift

Exam: JEE Main
Year: 2023
Date: 1 February
Shift: Evening Shift
Subject: Mathematics
Chapter: Quadratic Equation

JEE Main 2023 · 1 February · Evening Shift · Mathematics · Quadratic Equation

The number of integral values of k , for which one root of the...

The number of integral values of

k

, for which one root of the equation

2{x}^{2} - 8x + k = 0

lies in the interval \left(1, 2\right)

and its other root lies in the interval \left(2, 3\right)

, is :

The number of integral values of k , for which one root of the equation 2 x ^ 2 - 8x + k = 0 lies in the interval (1,

Options

A $2$
B $0$
C $1$
D $3$

Correct answer

C. $1$

Step-by-step solution

Let $f ( x ) = 2{x}^{2} - 8x + k = 0$

The number of integral values of k , for which one root of the equation 2 x ^ 2 - 8x + k = 0 lies in the interval (1,

$f ( 1 ) > 0$

$f ( 2 ) < 0$

$f ( 3 ) > 0$

Now, $f ( 1 ) > 0 ⇒ 2 - 8 + k > 0 ⇒ k > 6 . . . . . . ( 1 )$

$f ( 2 ) < 0 ⇒ 8 - 16 + k < 0 ⇒ k < 8 . . . . . . ( 2 )$

$f ( 3 ) > 0 ⇒ 18 - 24 + k > 0 ⇒ k > 6 . . . . . . ( 3 )$

From equations \left(1\right), \left(2\right) \text{ and $} \left(3\right)$ ,

$k \in ( 6, 8 )$

$∴$ Integral value of $k = 7$

Therefore number of integral value of $k$ is one

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