When \(20 \mathrm{~g}\) of naphthoic acid \(\left(\mathrm{C}_{11} \mathrm{H}_8 \mathrm{O}_2\right)\) is dissolved in \(50 \mathrm{~g}\) of benzene \(\left(K_f=1.72 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\), a freezing point depression of \(2 \mathrm{~K}\) is observed. The van't Hoff factor \((i)\) is

A sample \((5.6 \mathrm{~g})\) containing iron is completely dissolved in cold dilute \(\mathrm{HCl}\) to prepare a \(250 \mathrm{~mL}\) of solution. Titration of \(25.0 \mathrm{~mL}\) of this solution requires \(12.5 \mathrm{~mL}\) of \(0.03 \mathrm{M} \mathrm{KMnO}_{4}\) solution to reach the end point. Number of moles of \(\mathrm{Fe}^{2+}\) present in \(250 \mathrm{~mL}\) solution is \(\mathbf{x} \times 10^{-2}\) (consider complete dissolution of \(\mathrm{FeCl}_{2}\) ). The amount of iron present in the sample is \(\mathbf{y} \%\) by weight.
(Assume: \(\mathrm{KMnO}_{4}\) reacts only with \(\mathrm{Fe}^{2+}\) in the solution Use: Molar mass of iron as \(56 \mathrm{~g} \mathrm{~mol}^{-1}\) )
The value of $\mathrm{y}$ is________ .

Statement I Boron always forms covalent bond
Statement II The small size of \(\mathrm{B}^{3+}\) favours formation of covalent bond.

Among the triatomic molecules/ions, $\mathrm{B}\mathrm{e}\mathrm{C}{\mathrm{l}}_2,\mathrm{ }{\mathrm{N}}_3^{-},\mathrm{ }{\mathrm{N}}_2\mathrm{O},\mathrm{ }{\mathrm{N}\mathrm{O}}_2^{+},\mathrm{ }{\mathrm{O}}_3,\mathrm{S}\mathrm{C}{\mathrm{l}}_2,\mathrm{ }{\mathrm{I}\mathrm{C}\mathrm{l}}_2^{-},\mathrm{ }{\mathrm{I}}_3^{-}$ and $\mathrm{X}\mathrm{e}{\mathrm{F}}_2$ , the total number of linear molecule(s)/ion(s) where the hybridization of the central atom does not have contribution from the d-orbital(s) is
[Atomic number: S = 16, Cl = 17, I = 53 and Xe = 54]

In the reaction,
$\text{P}+\text{Q}\to \text{R}+\text{S}$
the time taken for $75\%$ reaction of $\mathrm{P}$is twice the time taken for$50\%$ reaction of $\mathrm{P}$. The concentration of $\mathrm{Q}$varies with reaction time as shown in the figure.The overall order of the reaction is:

Among cellulose, polyvinyl chloride, nylon and natural rubber, the polymer in which the intermolecular force of attraction is weakest in

Reaction of \(\mathbf{x}\) g of \(\mathrm{Sn}\) with \(\mathrm{HCl}\) quantitatively produced a salt. Entire amount of the salt reacted with \(\mathbf{y} g\) of nitrobenzene in the presence of required amount of \(\mathrm{HCl}\) to produce \(1.29 \mathrm{~g}\) of an organic salt (quantitatively).
(Use Molar masses (in \(\mathrm{g} \mathrm{mol}^{-1}\) ) of \(\mathrm{H}, \mathrm{C}, \mathrm{N}, \mathrm{O}, \mathrm{Cl}\) and \(\mathrm{Sn}\) as \(1,12,14,16,35\) and 119 , respectively).
The value of $\mathrm{x}$ is _______ .

A ray of light travelling in water is incident on its surface open to air. The angle of incidence is \(\theta\), which is less than the critical angle. Then there will be

A length - scale $\left(\mathcal{l}\right)$ depends on the permittivity $\left(\mathrm{\epsilon }\right)$ of a dielectric material, Boltzmann constant $k_B$ , the absolute temperature T, the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles, Which of the following expressions (s) for $\mathcal{l}$ is (are) dimensionally correct?

For how many values of $p$, the circle $x^2+y^2+2x+4y-p=0$ and the coordinate axes have exactly three common points?

A loop carrying current \(I\) lies in the \(x-y\) plane as shown in the figure. The unit vector \(\hat{k}\) is coming out of the plane of the paper. The magnetic moment of the current loop is

Let $\mathrm{\Delta }\mathrm{P}\mathrm{Q}\mathrm{R}$ be a triangle. Let $\overrightarrow{\mathrm{a}}=\mathrm{ }\overrightarrow{\mathrm{Q}\mathrm{R}},\mathrm{ }\mathrm{ }\overrightarrow{\mathrm{b}}=\mathrm{ }\overrightarrow{\mathrm{R}\mathrm{P}}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\overrightarrow{\mathrm{c}}=\mathrm{ }\overrightarrow{\mathrm{P}\mathrm{Q}}.$ If $\left|\overrightarrow{\mathrm{a}}\right|=12,\mathrm{ }\left|\overrightarrow{\mathrm{b}}\right|=4\sqrt{3}$ and $\overrightarrow{\mathrm{b}}\mathrm{ }\cdot \mathrm{ }\overrightarrow{\mathrm{c}}=24,$ then which of the following is (are) true ?

Let $f_1: \mathbb{R}\to \mathbb{R}, f_2\left(-\frac{\pi }{2},\frac{\pi }{2} \right)\to \mathbb{R}, f_3:\left(-1, e^{\frac{\pi }{2}}-2\right)\to \mathbb{R}$ and $f_4: \mathrm{ℝ}\to \mathrm{ℝ}$ be functions defined by
(i) $f_1\left(x\right)=\sin \left(\sqrt{1-e^{-x^2}}\right)$
(ii) $f_2\left(x\right)=\left\{\begin{array}{ll}\frac{\left|\sin x\right|}{{\tan }^{-1}\left(x\right)}& , x\ne 0\\ 1& , x=0\end{array}\right.$ , where the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
(iii) $f_3\left(x\right)=\left[\sin \left({\log }_{\text{e}}\left(x+2\right)\right)\right]$ , where, for $t\in R, \left[t\right]$ denotes the greatest integer less than or equal to $t$ ,
(iv) $f_4\left(x\right)=\left\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right) & , x\ne 0\\ 0 & , x=0\end{array}\right.$

LIST-I	LIST-II
A. the function $f_1$ is	P. NOT continuous at $x=0$
B. The function $f_2$ is	Q. continuous at $x = 0$ and NOT differentiable at $x=0$
C. The function $f_3$ is	R. differentiable at $x=0$ and its derivative is NOT continuous at $x=0$
D. The function $f_4$ is	S. differentiable at $x=0$ and its derivative is continuous at $x=0$

The correct option is :