First slide

Theory of equations

Question

$The possible value of p for which graph of the function f (x) = 2 p^{2} - 3 p \tan x + \tan^{2} x + 1 does not lie below x -axis$
$for all x \in (\frac{- π}{2}, \frac{π}{2}) is$

Moderate

A

0
B

4
C

3
D

8

Solution

$\begin{array}{l} Let \tan x = t \in R \\ y = t^{2} - 3 p t + 2 p^{2} + 1, t \in R \end{array}$
Graph does not lie below x- axis,
$\begin{array}{l} ∴ D \leq 0 \\ ∴ 9 p^{2} - 4 (2 p^{2} + 1) \leq 0 \\ \Rightarrow p^{2} - 4 \leq 0 \\ \Rightarrow p \in [- 2, 2] \end{array}$

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