If - - are the two roots of the quadratic polynomial fx-x2-bax-ca- a 0- And if exactly one of the roots lies between k1- k2- Then select the correct statements where - - k1- k2 -

Given the quadratic polynomial f(x)=x^2+bax+ca; a 0 Now, given two numbers k_1 & k_2 such that exactly one root lies between k_1 & k_2 The condition can ...

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Byju's Answer

Standard XII

Mathematics

Location of Roots when Compared to two constants 'k1' & 'k2'

If α, β are t...

Question

If $α, β$ are the two roots of the quadratic polynomial $f (x) = x^{2} + \frac{b}{a} x + \frac{c}{a}; a \neq 0.$ And if exactly one of the roots lies between $k_{1}, k_{2} .$ Then select the correct statements where $α, β \neq k_{1}, k_{2}$ .

f (k_{1}) f (k_{2}) > 0

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f (k_{1}) f (k_{2}) = 0

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f (k_{1}) f (k_{2}) < 0

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D > 0

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Solution

The correct option is D $D > 0$
Given the quadratic polynomial $f (x) = x^{2} + \frac{b}{a} x + \frac{c}{a}; a \neq 0$
Now, given two numbers $k_{1} & k_{2}$ such that exactly one root lies between $k_{1} & k_{2}$
The condition can be portrayed as:

Now, from observing we get:
$A . D > 0$
Also, from the case $k_{1} < α < k_{2}$
$f (k_{1}) > 0 & f (k_{2}) < 0$
And from the case $k_{1} < β < k_{2}$
$f (k_{1}) < 0 & f (k_{2}) > 0$
Thus, combining both of them, we get:
$B . f (k_{1}) f (k_{2}) < 0$

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If α,β are the two roots of the quadratic polynomial f(x)=x2+bax+ca;a≠0. And if exactly one of the roots lies between k1,k2. Then select the correct statements where α,β≠k1,k2 .

If $α, β$ are the two roots of the quadratic polynomial $f (x) = x^{2} + \frac{b}{a} x + \frac{c}{a}; a \neq 0.$ And if exactly one of the roots lies between $k_{1}, k_{2} .$ Then select the correct statements where $α, β \neq k_{1}, k_{2}$ .