(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 expression	The minimum value
i) $x^2 + 4x + 6$	a) $1$
ii) $x^2 - 2x + 5$	b) $2$
iii) $x^2 + 6x + 18$	c) $4$
iv) $x^2 - 4x + 5$	d) $9$

DifficultAP EAMCET 2017

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Medium

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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

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Match the following quadratic expressions with their minimum values:
Quadratic expression The minimum value
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