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

If \(\left(\frac{1-i}{1+i}\right)^{n/2} = -1\), where \(i = \sqrt{-1}\), one possible value of n is
In Cartesian coordinates, consider the functions \(u(x, y) = \frac{1{2}(x^2 - y^2)\) and \(v(x,y) = xy\). If \((r, \theta)\) are the polar coordinates, the Jacobian determinant \(|\frac{\partial(u,v)}{\partial(r,\theta)}|\) is}
If the system of linear equations
x + my + az = 0
2x + ay + mz = 0
ax + 2y - z = 0
with m and a as non-zero constants, admits a non-trivial solution, then which one of the following conditions is correct?
Given a function \(f(x, y) = \frac{x}{a}e^y + \frac{y}{b}e^x\), where \(x = at\) and \(y = bt\) (a and b are non-zero constants), the value of \(\frac{df}{dt}\) at \(t = 0\) is
Consider a volume V enclosed by a closed surface S having unit surface normal \(\hat{n}\). For \(\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}\), the value of the surface integral \(\frac{1}{9} \oint_{S} \mathbf{r} \cdot \hat{n} \,dS\) is
Which one of the following figures represents the vector field \(\mathbf{A} = y\hat{i}\)?
(\(\hat{i}\) is the unit vector along the x-direction)
Two point-particles having masses \(m_1\) and \(m_2\) approach each other in perpendicular directions with speeds \(v_1\) and \(v_2\), respectively, as shown in the figure below. After an elastic collision, they move away from each other in perpendicular directions with speeds \(v'_1\) and \(v'_2\), respectively.
The ratio \(\frac{v'_1}{v_1}\) 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?