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

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