If the equation $2 t^{3} - 9 t^{2} + 30 - a = 0$ has three real and distinct roots, then

a > 30

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a < 3

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3 < a < 30

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a < 3 o r a > 30

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Solution

The correct option is C $3 < a < 30$
$2 t^{3} - 9 t^{2} + 30 - a = 0$
$f (t) = 2 t^{3} - 9 t^{3} + 30 - a = 0$
$f^{'} (t) = 6 t^{2} - 18 t$
$= 6 t (t - 3)$
$t = 0, t = 3$
For three real distinct roots
$f (0) \cdot f (3) < 0$
$(30 - a) (3 - a) < 0$
$(a - 3) (a - 30) < 0$
$3 < a < 30$

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Interval in which $‘ a^{'}$ lies so that $x^{3} - 3 x + a = 0$ has three real and distinct roots.

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a = \sqrt{3}, c = 3

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