(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 2 Complex Numbers and Quadratic Equations: Quadratic Equations 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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2 Complex Numbers and Quadratic Equations: Quadratic Equations

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Quadratic expressions and Position of roots

The value of $p$ such that both the roots of the equation $(p - 5)x^2 - 2px + (p - 4) = 0$ are positive,one is less than $2$ and the other lies between $2$ and $3$,lies in the interval:

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If the roots of the equation $x^2+x+a=0$ exceed $a$,then

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If $\alpha, \beta$ are the roots of $x^2 - 3x + a = 0, a \in R$ and $\alpha < 1 < \beta$,then :-

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If the maximum value of $2x - 7 - ax^2$ cannot exceed $20$,then the minimum value of $a$ is

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For what set of values of $K$ do both roots of the equation $4x^2 - 20Kx + (25K^2 + 15K - 66) = 0$ lie below $2$?

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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 value of $p$ such that both the roots of the equation $(p - 5)x^2 - 2px + (p - 4) = 0$ are positive,one is less than $2$ and the other lies between $2$ and $3$,lies in the interval:

If the roots of the equation $x^2+x+a=0$ exceed $a$,then

If $\alpha, \beta$ are the roots of $x^2 - 3x + a = 0, a \in R$ and $\alpha < 1 < \beta$,then :-

If the maximum value of $2x - 7 - ax^2$ cannot exceed $20$,then the minimum value of $a$ is

For what set of values of $K$ do both roots of the equation $4x^2 - 20Kx + (25K^2 + 15K - 66) = 0$ lie below $2$?

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