(B) Given $f(x) = ax^2 + 2bx + c$. Since $f(x) = 0$ has imaginary roots,the discriminant $D = (2b)^2 - 4ac < 0$,which implies $4b^2 - 4ac < 0$,or $b^2

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(B) Given $f(x) = ax^2 + 2bx + c$. Since $f(x) = 0$ has imaginary roots,the discriminant $D = (2b)^2 - 4ac Since $a \neq 0$,if $a > 0$,then $ac > b^2 \geq 0$,so $c > 0$. However,the graph of $f(x)$ would be an upward parabola,and since $D Therefore,we must have $a Since $a Thus,$f(x) Specifically,at $x = 0$,$f(0) = c$. Since the entire graph is below the $x$-axis,$f(0) < 0$,which means $c < 0$.

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

Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

A
$c > 0$
B
$c < 0$
C
$c = 0$
D
Data insufficient

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

The range of $a$ for which the roots of $x^2 - 2x - a^2 + 1 = 0$ lie between the roots (exclusive) of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ is:

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The number of values of $k$ for which the equation $x^2 - 3x + k = 0$ has two real and distinct roots lying in the interval $(0, 1)$ is

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The set of values of $a$ for which the inequality $x^2 - (a + 2)x - (a + 3) < 0$ is satisfied by at least one positive real $x$ is:

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If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2 + bx + c = 0$ and $k$ is a real number,then the condition for $\alpha < k < \beta$ is given by:

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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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Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

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The range of $a$ for which the roots of $x^2 - 2x - a^2 + 1 = 0$ lie between the roots (exclusive) of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ is:

The number of values of $k$ for which the equation $x^2 - 3x + k = 0$ has two real and distinct roots lying in the interval $(0, 1)$ is

The set of values of $a$ for which the inequality $x^2 - (a + 2)x - (a + 3) < 0$ is satisfied by at least one positive real $x$ is:

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