Question

STATEMENT 1: lf two angles of a triangle satisfy the equation ${\tan }^2\theta +p\tan \theta -1=0$, then the triangle is obtuse angled for all values $p$.
STATEMENT 2: In a $\Delta ABC,\sum \tan A\tan B=1$

• A. Statement-1 is true, Statement-2 is true, Statement-2 is correct explanation of Statement- 1
• B. Statement-1 is true, Statement-2 is true,Statement-2 is not correct explanation for Statement-1
• C. Statement-1 is true, Statement-2 is false
• D. Statement-1 is false, Statement-2 is true

Answer 1

Answer: (C)

Statement-1 is true, Statement-2 is false

$\sum \tan A\tan B=1$ not possible for triangle ABC.
$\sum \tan A=\Pi \tan A$
$\tan A\tan B=-1$ (Product of roots $=-1$)
Only possibility of this is when one root is positive and other is negative, which can happen only if one of the angle is obtuse.
$\because \tan \theta$ is negative only when $\theta \in (\frac{\pi }{2},\pi )$ [For Triangle $ABC$]
That implies triangle is obtuse angled for all values of $p$.