(D) Let $f(x) = x^{2}-3x+k$. For the roots to lie in the interval $(0,1)$, the vertex of the parabola must lie within $(0,1)$. The $x$-coordinate of t

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no value of $k$ satisfies the requirement

(D) Let $f(x) = x^{2}-3x+k$. For the roots to lie in the interval $(0,1)$, the vertex of the parabola must lie within $(0,1)$. The $x$-coordinate of the vertex is given by $x = -\frac{b}{2a} = -\frac{-3}{2(1)} = \frac{3}{2} = 1.5$. Since $1.5 \notin (0,1)$, it is impossible for both roots to lie in the interval $(0,1)$ simultaneously. Therefore, there is no value of $k$ that satisfies the given condition.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Quadratic expressions and Position of roots

Medium

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

WBJEE 2016 All WBJEE

A
three
B
two
C
infinitely many
D
no value of $k$ satisfies the requirement

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

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Column-$I$	Column-$II$
$A$. Imaginary roots	$P$. $a \in (-1, 4)$
$B$. One root less than $3$ and other greater than $3$	$Q$. $a \in (-\infty, -1)$
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