If I have an elliptically polarized light with the major axis along 10 degrees to the x-axis. Now I place a quarter-wave plate with its fast axis along 45 degrees to the x-axis. What will be the final state of polarisation?
The final polarization state will be elliptical. A quarter-wave plate transforms circularly polarized light to linearly polarized light and vice versa only under certain circumstances. You can use Jones vectors and matrices to determine the output state of polarization altering systems.
Not always. To convert into linear polarization, you have to place the QWP at an appropriate angle. You can calculate this angle using either the Jones analysis or using the Muller matrix.