For any quadratic expression fx-ax2-bx-c-a-0- having zeros as - - and if k is any real number such that k- then select the correct statements-

Given quadratic expression as: f(x)=ax^2+bx+c with a lt;0 α, β are roots of ax^2+bx+c=0 and k is any real number such that: α gt;β gt;k Now, using the giv ...

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Byju's Answer

Standard XII

Mathematics

Location of Roots when Compared with a constant 'k'

For any quadr...

Question

For any quadratic expression $f (x) = a x^{2} + b x + c; a < 0$ , having zeros as $α, β$ and if $k$ is any real number such that $k < β < α,$ then select the correct statement(s).

- \frac{b}{2 a} > k

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- \frac{b}{2 a} < k

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D > 0

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f (k) < 0

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Solution

The correct option is D $f (k) < 0$
Given quadratic expression as: $f (x) = a x^{2} + b x + c$ with $a < 0$
$α, β$ are roots of $a x^{2} + b x + c = 0$ and
$k$ is any real number such that: $α > β > k$
Now, using the given conditions we can plot the graph as:

From the graph, we can observe the following:
$(a) D > 0,$ as $f (x) = 0$ has two distinct real roots.
$(b) f (k) < 0$ , from the graph.
$(c)$ Abscissa of Vertex $= - \frac{b}{2 a} > k$ .

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For any quadratic expression f(x)=ax2+bx+c;a<0, having zeros as α,β and if k is any real number such that k<β<α, then select the correct statement(s).

For any quadratic expression $f (x) = a x^{2} + b x + c; a < 0$ , having zeros as $α, β$ and if $k$ is any real number such that $k < β < α,$ then select the correct statement(s).