List of top Mathematics Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

For the Fourier series of the following function of period \( 2\pi \): 

The ratio (to the nearest integer) of the Fourier coefficients of the first and the third harmonic is: 
 

Which one of the following graphs represents the derivative \( f'(x) = \frac{df}{dx} \) of the function \( f(x) = \frac{1}{1+x^2} \) most closely (graphs are schematic and not drawn to scale)? 

The integral of the vector \( \mathbf{A}(\rho, \varphi, z) = \frac{40}{\rho} \cos \varphi \hat{\rho} \) (standard notation for cylindrical coordinates is used) over the volume of a cylinder of height \( L \) and radius \( R_0 \), is:
Consider two particles moving along the x-axis. In terms of their coordinates \( x_1 \) and \( x_2 \), their velocities are given as \( \frac{dx_1}{dt} = x_2 - x_1 \) and \( \frac{dx_2}{dt} = x_1 - x_2 \), respectively. When they start moving from their initial locations of \( x_1(0) = 1 \) and \( x_2(0) = -1 \), the time dependence of both \( x_1 \) and \( x_2 \) contains a term of the form \( e^{at} \), where \( a \) is a constant. The value of \( a \) (an integer) is:
Consider the differential equation \( y'' + 2y' + y = 0 \). If \( y(0) = 0 \) and \( y'(0) = 1 \), then the value of \( y(2) \) is:
The volume integral of the function \( f(r, \theta, \phi) = r^2 \cos\theta \) over the region \( 0 \leq r \leq 2, 0 \leq \theta \leq \frac{\pi}{3} \) and \( 0 \leq \phi \leq 2\pi \) is:
If the Boolean function \( Z = PQ + PQR + PQRS + PQRST + PQRSTU \), then \( Z \) is:

Which one of the following graphs represents the derivative \( f'(x) = \frac{df}{dx} \) of the function \( f(x) = \frac{1}{1+x^2} \) most closely (graphs are schematic and not drawn to scale)? 

The integral of the vector \( \mathbf{A}(\rho, \varphi, z) = \frac{40}{\rho} \cos \varphi \hat{\rho} \) (standard notation for cylindrical coordinates is used) over the volume of a cylinder of height \( L \) and radius \( R_0 \), is:
Consider the differential equation \( y'' + 2y' + y = 0 \). If \( y(0) = 0 \) and \( y'(0) = 1 \), then the value of \( y(2) \) is:
Consider two particles moving along the x-axis. In terms of their coordinates \( x_1 \) and \( x_2 \), their velocities are given as \( \frac{dx_1}{dt} = x_2 - x_1 \) and \( \frac{dx_2}{dt} = x_1 - x_2 \), respectively. When they start moving from their initial locations of \( x_1(0) = 1 \) and \( x_2(0) = -1 \), the time dependence of both \( x_1 \) and \( x_2 \) contains a term of the form \( e^{at} \), where \( a \) is a constant. The value of \( a \) (an integer) is:
The volume integral of the function \( f(r, \theta, \phi) = r^2 \cos\theta \) over the region \( 0 \leq r \leq 2, 0 \leq \theta \leq \frac{\pi}{3} \) and \( 0 \leq \phi \leq 2\pi \) is:
Consider a system of \( N \) particles obeying classical statistics, each of which can have an energy 0 or \( E \). The system is in thermal contact with a reservoir maintained at a temperature \( T \). Let \( k \) denote the Boltzmann constant. Which one of the following statements regarding the total energy \( U \) and the heat capacity \( C \) of the system is correct?