IAT IISER 2025

Consider two waves, which are given by $y_1(x, t)=A \sin (k x-\omega t)$ and $y_2(x, t)=\sqrt{3} A \cos (k x-\omega t)$, where $k$ is the wave number and $\omega$ is the angular frequency. The amplitude of the resultant waveform obtained by the superposition of the two waves is $A_s$ and its phase difference with $y_1$ is $\phi_s$. What are $A_s$ and $\phi_s$ ?

The work done when one mole of an ideal gas expands at constant temperature $T$ from volume $V$ to $2 V$ (in two equal steps of volume in a linear fashion) is $\frac{7}{12} R T$. How much more work would be done by the gas if it expands in three equal steps?

[ $R$ is the universal gas constant]

Let $A$ be a $3 \times 3$ matrix with real entries such that

$$A=\left[\begin{array}{ccc} 4 & -1 & \cos x \\ -1 & 5 x & 25 \\ x^2+1 & 25 & 7 \end{array}\right]$$

For how many values of $x$, the matrix $A$ is symmetric?

Which one of the following options describes a triglyceride?

A block of mass $M$ lies at rest connected to a massless spring of spring constant $k$ on a frictionless surface. A bullet of mass $m$ hits the block horizontally with speed $v$ as shown in the figure and is completely stuck to the block. What is the maximum compression in the spring resulting from this impact (assuming that at this point the spring is still not fully compressed)?

What is the time period of revolution of an electron in the fourth Bohr orbit of $\mathrm{He}^{+}$?

[Bohr radius $=52.9$ picometers, mass of an electron $=9.11 \times 10^{-31} \mathrm{~kg}$, Planck's constant $=6.626 \times 10^{-34} \mathrm{Js}$ ]