In a certain test, $${a_i}$$ students gave wrong answers to atleast i questions, where i = 1, 2,..., k. No student gave more than k wrong answers. The total number of wrong answers given is.....................................

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

The inequality |z-4| < |z-2| represents the region given by

Show that the equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has no real solution.