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4.2 Quadratic equations and inequations — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS4.2 Quadratic equations and inequations1 Mark
MCQ
The polynomial equation $x^3-3 a x^2+\left(27 a^2+9\right) x+2016=0$ has
  • ✓
    exactly one real root for any real $a$
  • B
    three real roots for any real $a$
  • C
    three real roots for any $a \geq 0$, and exactly one real root for any $a < 0$
  • D
    three real roots for any $a \leq 0$, and exactly one real root for any $a > 0$
✓

Answer

Correct option: A.
exactly one real root for any real $a$
a
(a)

We have,

$x^3-3 a x^2+\left(27 a^2+9\right) x+2016=0$

Let

$f(x) =x^3-3 a x^2+\left(27 a^2+9\right) x+2016$

$f^{\prime}(x) =3 x^2-6 a x+27 a^2+9$

$f^{\prime}(x) =3\left(x^2-2 a x+9 a^2+3\right)$

$f^{\prime}(x) =3\left((x-a)^2+8 a^2+3\right)$

$f^{\prime}(x) > 0, \forall x \in R$

$\therefore(x) \text { is increasing function, } \forall a \in R$.

$f(x)$ has exactly one real root for any real $\varepsilon$.

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