If a12a-5b-2c - 0 and the equation ax2-bx-c-0 has the roots - and - then

The correct options are B α D 3 a^2(12+5b/a+2c/a) gt;0 12 5(α+β)+2αβ gt;0 αβ 3α 2β+6+αβ 2α 3β+6 gt;0 ⇒ (α 2)(β 3)+(α 3)(β 2) gt ...

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Byju's Answer

Standard XII

Mathematics

Algebra of Limits

If a12a+5b+...

Question

If $a (12 a + 5 b + 2 c) > 0$ and the equation $a x^{2} + b x + c = 0$ has the roots $α$ and $β$ , then

2 < α < β < 3

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α < β < 2

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3 < α < β

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2 < α < 3 < β

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Solution

The correct options are
B $α < β < 2$
D $3 < α < β$
$a^{2} (12 + \frac{5 b}{a} + \frac{2 c}{a}) > 0$
$12 - 5 (α + β) + 2 α β > 0$
$α β - 3 α - 2 β + 6 + α β - 2 α - 3 β + 6 > 0$
$\Rightarrow (α - 2) (β - 3) + (α - 3) (β - 2) > 0$

Now, we will check by options
According to this figure, LHS $(+) (-) + (-) (+)$ . lt is not $> 0$
option (A) is not correct.
According to this Figure, LHS $(-) (-) + (-) (-)$ . lt is $> 0$
Option (B) is correct.
According to this Figure, LHS $(+) (+) + (+) (+)$ . lt is $> 0$
Option (C) is correct.
According to this Figure, LHS $(+) (-) + (-) (+)$ . lt is not $> 0$
Option (D) is not correct .

Hence, options 'B' and 'C' are correct.

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If a(12a+5b+2c)>0 and the equation ax2+bx+c=0 has the roots α and β, then

If $a (12 a + 5 b + 2 c) > 0$ and the equation $a x^{2} + b x + c = 0$ has the roots $α$ and $β$ , then