IIT-JEE 1982

The third term of a geometric progression is 4. The product of the first five terms is

Does there exist a geometric progression containing \(27, 8\) and \(12\) as three of its terms? If it exits, how many such progressions are possible ?

The set of lines \(ax + by + c = 0,\) where \(3a + 2b + 4c = 0\) is concurrent at the point ..........

\(y = {10^x}\) is the reflection of \({\log _{10}}\,x\) in the line whose equation is ...........

Without using tables, prove that \(\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {\sin \,{{54}^ \circ }} \right) = {1 \over 8}.\)

Find all values of \(\lambda\) such that \(x, y, z,\)\(\, \ne\)\((0,0,0)\) and
$$\left( {\overrightarrow i + \overrightarrow j + 3\overrightarrow k } \right)x + \left( {3\overrightarrow i - 3\overrightarrow j + \overrightarrow k } \r...

For non-zero vectors \({\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },\) $$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right).\overrightarrow c } \right| = \left| {\overrightarrow a } \right|\left| {\overrig...

\({A_1},{A_2},.................{A_n}\) are the vertices of a regular plane polygon with \(n\) sides and \(O\) is its centre. Show that
$$\sum\limits_{i = 1}^{n - 1} {\left( {\overrightarrow {O{A_i}} \times {{\overrightarrow {OA} }_{i + 1}...

In the arrangement shown in Fig. the ends P and Q of an unstretchable string move downwards with uniform speed U. Pulleys A and B are fixed.
Mass M moves upwards with a speed

A particle is moving eastwards with a velocity of 5 m/s. In 10s the velocity changes to 5 m/s northwards. The average acceleration in this time is

Write the dimensions of the followings in terms of mass, time, length and charge
(i) magnetic flux
(ii) rigidity modulus