| Direction: For the following questions. Choose the correct answer to the codes [a], [b], [c] and [d] defined as follows. |
| Let us consider the any \[\Delta \,ABC,\] whose sum of all angles is \[180{}^\circ \]. |
| Statement I If \[\angle A\] is obtuse, then \[\tan \,B\,\tan C<1\]. |
| Statement II \[\tan \,A+\tan \,B+\tan \,C=\tan \,A\,\tan \,B\,\tan \,C\]. |
A) Statement I is true. Statement II is also true and Statement II is the correct explanation of Statement I.
B) Statement I is true/Statement II is also true and Statement II is not the correct explanation of Statement I.
C) Statement I is true. Statement II is false.
D) Statement I is false, Statement II is true.
Correct Answer: A
Solution :
I \[\tan \,(B+C)=\tan \,(\pi -A)\] \[=\tan \,A\,=\frac{\tan \,B+\tan \,C}{\tan \,B\,\tan \,C-1}<0\] \[(\because \,\,\,\text{A}\,\,\text{is}\,\text{obtuse})\] \[\therefore \] B and C are acute, \[\tan \,B\,+\tan \,C>0\] \[\because \] \[\tan \,B\,\tan \,C-1<0\] \[\Rightarrow \] \[\tan \,B\,\tan \,C<1\] II. \[\tan \,(A+B+C)\] \[=\frac{\tan \,A+\tan \,B+\tan \,C-\tan \,A\,\tan \,B\,\tan \,C}{1-\tan \,A\,\tan \,B-\tan \,B\,\tan \,C-\tan \,C\,\tan \,A}\] \[=\tan \,{{180}^{o}}=0\] \[\Rightarrow \] \[\tan \,A+\tan \,B+\tan \,C=\tan \,A\,\tan \,B\,\tan C\]
You need to login to perform this action.
You will be redirected in
3 sec
