Physics after Engineering

Around 110 students were shortlisted for the interview round of department of Physical Sciences at IISER Kolkata, based on the all-India ranks of the candidates in JAM and JEST exams for a total of 5 seats in their Integrated PhD program. In addition to that, every candidate had to send two letters of recommendation to the department. My interview was conducted on 29th June. There were five professors in my panel. They first asked me my favourite topics to which I replied, 'Quantum Mechanics' and 'Lagrangian and Hamiltonian Mechanics'. They directly started asking the questions. Following questions were asked by the panel.  Prof-1: Topic: Graphs 1-i) Plot sin(x^2). 1-ii) What will be the asymptotic behaviour of the nodal distance i.e. the distance between two consecutive nodes? Prof-2: Topic: Lagrangian Mechanics 2-i) Write the Lagrangian of a charged particle in an EM field. 2-ii) Apply a Gauge transformation and re-write the Lagrangian. 2-iii) Are the equations of mo