First slide

Theory of equations

Question

The set of values of $' a'$ for which $x^{2} + a x + \sin^{- 1} (x^{2} - 4 x + 5) + \cos^{- 1} (x^{2} - 4 x + 5) = 0$ has at least one real root is given by

Moderate

A

$(- \infty, - \sqrt{2} π] \cup [\sqrt{2 π}, \infty)$
B

$(- \infty, - \sqrt{2 π}) \cup (\sqrt{2 π}, \infty)$
C

$R$
D

none of these

Solution

We have,
$\begin{array}{l} x^{2} + a x + \sin^{- 1} (x^{2} - 4 x + 5) + \cos^{- 1} (x^{2} - 4 x + 5) = 0 \\ \Rightarrow x^{2} + a x + \frac{π}{2} = 0 \end{array}$
This equation will have real roots, if
$\begin{array}{l} a^{2} - 2 π \geq 0 \\ \Rightarrow (a - \sqrt{2 π}) (a + \sqrt{2 π}) \geq 0 \\ \Rightarrow a \in (- \infty, - \sqrt{2 π}] \cup [\sqrt{2 π}, \infty) \end{array}$

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