PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions

Which term of the G.P. :

\[2, 2\sqrt{2}, 4, . . .\text {  is }128 ?\]

Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?

Which term of the G.P. :

\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]

Prove that:
\[\frac{\tan \left( A + B \right)}{\cot \left( A - B \right)} = \frac{\tan^2 A - \tan^2 B}{1 - \tan^2 A \tan^2 B}\]

Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?

Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x

Prove that:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]

Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1

Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x

Find the 4th term from the end of the G.P.

\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]

Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]

If tan A = x tan B, prove that
\[\frac{\sin \left( A - B \right)}{\sin \left( A + B \right)} = \frac{x - 1}{x + 1}\]

If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) \[\frac{1}{a} - \frac{1}{b}\].