First slide

Introduction to ITF

Question

If $t = \sec^{- 1} (x + \frac{1}{x}) + \sec^{- 1} (y + \frac{1}{y})$ where xy < 0, then the possible values of t is/are

Moderate

A

$\frac{4 π}{5}$
B

$\frac{7 π}{10}$
C

$\frac{9 π}{10}$
D

$\frac{21 π}{20}$

Solution

Since xy < 0 $\Rightarrow$ x <0, y> 0 or x> 0, y <0
$\begin{array}{l} x + \frac{1}{x} \geq 2, y + \frac{1}{y} \leq - 2 (or) \\ x + \frac{1}{x} \leq - 2, y + \frac{1}{y} \geq 2 \end{array}$
according to the case
for first case $\sec^{- 1} (x + \frac{1}{x}) \in [\frac{π}{3}, \frac{π}{2}), \sec^{- 1} (y + \frac{1}{y}) \in (\frac{π}{2}, \frac{2 π}{3})$
$\Rightarrow \sec^{- 1} (x + \frac{1}{\dot{x}}) + \sec^{- 1} (y + \frac{1}{y}) \in (\frac{5 π}{6}, \frac{7 π}{6})$
Similarly for second case also, $t \in (\frac{5 π}{6}, \frac{7 π}{6})$

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