IAT IISER 2022

2 g of naphthoic acid (molecular weight $=172 \mathrm{~g} \mathrm{~mol}^{-} 1$ ) dissolved in 20 mL of benzene shows a freezing point depression of 2 K . For benzene, the freezing point depression constant, $K_f=5$ $\mathrm{K} \mathrm{kg} \mathrm{mol}{ }^{-1}$ and the density is $0.88 \mathrm{~g} \mathrm{~mL}^{-1}$. What is the magnitude of the van't Hoff factor?

A randomly chosen card from a deck of 52 cards is given to be a black card (Spade or Club). What is the probability that it is either a face card (King, Queen or Jack) or a Spade?

Choose the correct statement about the structure of $\mathrm{C}_{60}$ fullerene

Consider a very long cylinder of radius $a$ having a uniform positive charge density $\rho$. A sphere of radius $a$ has been carved out (see figure), leaving no charge in that region. The distance radially outward to the cylinder, as measured from the center of the sphere is $x$. At what value of $x$ will the electric field be maximum?

Which of the following expressions represents the hydrogen atom wave function $\psi(r)$ shown in the figure below? ( $r$ is the distance of the electron from the nucleus and $a_0$ is a constant)

An aqueous solution contains 1.0 M of $\mathrm{X}^{2+}$ and 0.001 M of $\mathrm{Y}^2+$ ions at $25^{\circ} \mathrm{C} . \mathrm{X}^{2+}$ and $\mathrm{Y}^{2+}$ ions do not interact with each other. This solution is put in an electrolytic cell and the voltage is gradually increased till a current begins to flow through the cell. The voltage is maintained at this point and a deposit is observed on the cathode. What is the composition of the material deposited on the cathode?

(Given: Atomic weight of $X$ is 63 and $Y$ is 200.)

$$ \begin{aligned} & X^{2+}+2 e^{-} \rightarrow X, E^0=0.35 \\ & Y^{2+}+2 e^{-} \rightarrow Y, E^0=0.40 \mathrm{~V} \end{aligned} $$

( $E^0$ is the standard reduction potential.)

$$ \text { The first to fifth ionization energies (IE) of two p-block elements } X \text { and } Y \text { are given below. } $$

$$ \begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{IE}_1(\mathrm{eV}) & \mathrm{IE}_2(\mathrm{eV}) & \mathrm{IE}_3(\mathrm{eV}) & \mathrm{IE}_4(\mathrm{eV}) & \mathrm{IE}_5(\mathrm{eV}) \\ \hline \mathbf{X} & 6.0 & 18.8 & 28.4 & 120.0 & 153.7 \\ \hline \mathbf{Y} & 8.2 & 16.3 & 33.5 & 45.1 & 166.7 \\ \hline \end{array} $$

$$ \text { The number of valence electrons in } X \text { and } Y \text { respectively are } $$

The van der Waals equation for a real gas is given by $\left(P+\frac{a}{V^2}\right)(V-b)=R T$. What is the dimension of $\left(\frac{a}{b}\right)$ ?

$$ \text { Which of the following molecules are aromatic? } $$

$$ \text { The major product } P \text { of the following reaction is } $$

$$ \text {In the following reaction sequence, the major products } M, N \text {, and } O \text { respectively are } $$

Let $\omega$ be a complex root of the quadratic polynomial $x^2+x+1$. The value of

$$ \left(\omega+\frac{1}{\omega}\right) \cdot\left(\omega^2+\frac{1}{\omega^2}\right) \cdots\left(\omega^{100}+\frac{1}{\omega^{100}}\right) $$

is

Let $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_1 x+a_0$ be a polynomial. Suppose that $f(0)=0$,

$$ \left.\left.\frac{d f}{d x}\right]_{x=0}=1, \frac{d^2 f}{d x^2}\right]_{x=0}=4 $$

and

$$ \frac{d^3 f}{d x^3}=\frac{d^5 f}{d x^5} $$

Then $f(5)=$

The value of the integral

$$ \int_1^{100} \frac{[x]}{x} d x $$

where $[x]$ is the greatest integer less than or equal to $x$ for any real number $x$, is

Let ' $X$ ' be the perpendicular distance from the centromere of each chromosome to the equatorial plane of a human cell. Which of the following stages of the cell cycle is most likely to have the HIGHEST average value of $X^{\prime}$ ?

Shown below are some of the reactions that occur in the metabolic pathway leading to complete oxidation of glucose during aerobic respiration.

i. Pyruvate $\rightarrow$ Acetyl CoA

ii. Dihydroxy acetone phosphate $\rightarrow$ Glyceraldehyde-3-phosphate

iii. Oxaloacetate $\rightarrow$ Citrate

iv. Fumarate $\rightarrow$ Malate

Choose the CORRECT sequence of reactions during the complete oxidation of glucose.