List of top Physics Questions on Mathematical Methods asked in IIT JAM PH

A square laminar sheet with side a and mass M, has mass per unit area given by \(\sigma(x)=\sigma_0[1-\frac{x}{a}]\), (see figure). Moment of inertia of the sheet about y-axis is

Consider a binary number with m digits, where m is an even number. This binary number has alternating 1's and 0's, with digit 1 in the highest place value. The decimal equivalent of this binary number is

Consider the 2 X 2 matrix M= \(\begin{pmatrix} 0 & a \\ a & b \end{pmatrix}\), where a,b\(\gt\)0. Then,

The Boolean expression Y = \(\overline{PQ}\)R+ Q\(\bar{R}\) + \(\bar{P}\)QR + PQR simplifies to

The integral \(\iint(x^2+y^2)dxdy\) over the area of a disk of radius 2 in the xy plane is ___π.

Consider a unit circle C in the xy plane with center at the origin. The line integral of the vector field, $ \overrightarrow{F}(x, y, z) = -2y-3z\hat{y} + x\hat{z}$, taken anticlockwise over C is _________ π.

Consider the second order ordinary differential equation, y" +4y' +5y = 0. If y(0) = 0 and y'(0) = 1, then the value of у(π/2) is _________. (Round off to 3 decimal places).

Consider an electron with mass m and energy E moving along the x-axis towards a finite step potential of height $U_0$, as shown in the figure. In region $1 (x < 0)$, the momentum of the electron is $p_1 = \sqrt{2mE}$. The reflection coefficient at the barrier is given by $R = (\frac{p_1-p_2}{p_1+p_2})^2$, where $p_2$ is the momentum in region 2. If, in the limit $E>> U_0$, R, then the integer n is ________.

A current density for a fluid flow is given by,
$ \overrightarrow{j} (x,y,z,t) = \frac{8e^t}{ (1+x^2+y^2+z^2)} \hat{x}$
At time t = 0, the mass density p(x, y, z, 0) = 1.
Using the equation of continuity, $p(1,1,1,1)$ is found to be ___________ (Round off to 2 decimal places).

Consider the transformation to a new set of coordinates $(\xi,\eta)$ from rectangular coordinates $(x,y)$, where $\xi=2x+3y$ and $\eta=3x-2y$. In the $(\xi,\eta)$ coordinate system, the area element $dx\,dy$ is