First slide

Properties of ITF

Question

The set of values of $λ$ for which $x^{2} - λ x + \sin^{- 1} (\sin 4) > 0$ for all $x \in R$ , is

Moderate

A

$ϕ$
B

(-2, 2)
C

$R$
D

none of these

Solution

We have,
$\begin{array}{l} \sin^{- 1} (\sin 4) = \sin^{- 1} (\sin (π - 4)) = π - 4 . \\ ∴ x^{2} - λ x + \sin^{- 1} (\sin 4) > 0 for all x \in R \\ \Rightarrow x^{2} - λ x + (π - 4) > 0 for all x \in R \\ \Rightarrow λ^{2} - 4 (π - 4) < 0 \Rightarrow λ^{2} + 16 - 4 π < 0 \end{array}$
But, $λ^{2} + 16 - 4 π > 0$ for all $λ \in R$ .
So, there is no value of $λ$ for which the given in equation holds true.

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