(A) Let $f(x) = x^2 + 2(a - 3)x + 9$.For a quadratic $f(x) = x^2 + bx + c$ to have roots $\alpha, \beta$ such that $\alpha < k < \beta$,the condition

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Statement-$I$ is true,Statement-$II$ is true,Statement-$II$ is a correct explanation for Statement-$I$.

(A) Let $f(x) = x^2 + 2(a - 3)x + 9$.For a quadratic $f(x) = x^2 + bx + c$ to have roots $\alpha, \beta$ such that $\alpha Here,$k = 6$,so we require $f(6) $f(6) = (6)^2 + 2(a - 3)(6) + 9 $36 + 12(a - 3) + 9 $45 + 12a - 36 $12a + 9 $12a $a Thus,Statement-$II$ is the condition for Statement-$I$ to hold,and both are true.

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

Medium

Statement-$I$: If the roots $\alpha, \beta$ of the equation $x^2 + 2(a - 3)x + 9 = 0$,$a \in R$ satisfy $\alpha < 6 < \beta$,then $a < -3/4$.
Statement-$II$: If $f(x) = x^2 + 2(a - 3)x + 9$,then $f(6) < 0 \implies a < -3/4$.

A
Statement-$I$ is true,Statement-$II$ is true,Statement-$II$ is a correct explanation for Statement-$I$.
B
Statement-$I$ is true,Statement-$II$ is true,Statement-$II$ is not a correct explanation for Statement-$I$.
C
Statement-$I$ is true,Statement-$II$ is false.
D
Statement-$I$ is false,Statement-$II$ is true.

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

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Statement-$I$: If the roots $\alpha, \beta$ of the equation $x^2 + 2(a - 3)x + 9 = 0$,$a \in R$ satisfy $\alpha < 6 < \beta$,then $a < -3/4$.Statement-$II$: If $f(x) = x^2 + 2(a - 3)x + 9$,then $f(6) < 0 \implies a < -3/4$.

Similar Questions

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 set of all real values of $\lambda$ for which the quadratic equation $(\lambda^{2}+1)x^{2}-4\lambda x+2=0$ has exactly one root in the interval $(0,1)$ is

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

All the values of $k$ such that the quadratic expression $2kx^2 - (4k+1)x + 2$ is negative for exactly three integral values of $x$,lie in the interval

Consider the quadratic equation $(c - 5)x^2 - 2cx + (c - 4) = 0$,where $c \ne 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and the other root lies in the interval $(2, 3)$. Then the number of elements in $S$ is

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Statement-$I$: If the roots $\alpha, \beta$ of the equation $x^2 + 2(a - 3)x + 9 = 0$,$a \in R$ satisfy $\alpha < 6 < \beta$,then $a < -3/4$.
Statement-$II$: If $f(x) = x^2 + 2(a - 3)x + 9$,then $f(6) < 0 \implies a < -3/4$.