Electrolysis of 600 mL aqueous solution of NaCl for 5 min changes the pH of the solution to 12 . The current in Amperes used for the given electrolysis is ____. (Nearest integer).

Which of the following elements have half-filled \(f\)-orbitals in their ground state ? (Given : atomic number \(Sm =62 ; Eu =63 ; Tb =65 ; Gd =64, Pm =61 \text { ) }\) \(A.\) \(Sm\)   \(B.\) \(Eu\)   \(C.\) \(Tb\)   \(D.\) \(Gd\)   \(E.\) \(Pm\) Choose the correct answer from the options given below:

The observed osmotic pressure for a \(0.10\, M\) solution of \(Fe(NH_4)_2(SO_4)_2\) at \(25\,^oC\) is \(10.8\, atm\). The expected and experimental ( observed) values of van't Hoff factor \((i)\) will be respectively: \((R\,= 0.082\, L\, atm\, k^{-1}\, mol^{-1})\)

A metal chloride contains \(55.0 \%\) of chlorine by weight.\(100\,mL\) vapours of the metal chloride at \(STP\) weigh \(0.57\,g\). The molecular formula of the metal chloride is \(...\).(Given : Atomic mass of chlorine is \(35.5\,u\) )

In Carius tube, an organic compound ' \(X\) ' is treated with sodium peroxide to form a mineral acid ' \(Y\) '. The solution of \(BaCl _2\) is added to ' \(Y\) ' to form a precipitate ' \(Z\) '. ' \(Z\) ' is used for the quantitative estimation of an extra element. ' \(X\) ' could be:

\(1\, \mathrm{~kg}\) of \(0.75\) molal aqueous solution of sucrose can be cooled up to \(-4^{\circ} \mathrm{C}\) before freezing. The amount of ice (in \(g\)) that will be separated out is .... . (Nearest integer) \(\left[\right.\) Given \(\left.: \mathrm{K}_{\mathrm{f}}\left(\mathrm{H}_{2} \mathrm{O}\right)=1.86\, \mathrm{~K}\, \mathrm{~kg}\, \mathrm{~mol}^{-1}\right]\)

The results given in the below table were obtained during kinetic studies of the following reaction: \(2 A + B \longrightarrow C + D\)

Experiment	\([ A ] / molL ^{-1}\)	\([ B ] / molL ^{-1}\)	Initial \(rate/molL\) \(^{-1}\) \(\min ^{-1}\)
\(I\)	\(0.1\)	\(0.1\)	\(6.00 \times 10^{-3}\)
\(II\)	\(0.1\)	\(0.2\)	\(2.40 \times 10^{-2}\)
\(III\)	\(0.2\)	\(0.1\)	\(1.20 \times 10^{-2}\)
\(IV\)	\(X\)	\(0.2\)	\(7.20 \times 10^{-2}\)
\(V\)	\(0.3\)	\(Y\)	\(2.88 \times 10^{-1}\)

\(X\) and \(Y\) in the given table are respectively :  

Let \(f(x) = \lim_{y \to 0} \dfrac{(1 - \cos(xy)) \tan(xy)}{y^3}\). Then the number of solutions of the equation \(f(x) = \sin x\), \(x \in \mathbf{R}\) is :

At \(300\, K\), the vapour pressure of a solution containing \(1\) mole of \(n-\)hexane and \(3\) moles of \(n-\)heptane is \(550\, mm\) of \(Hg\). At the same temperature, if one more mole of n-heptane is added to this solution, the vapour pressure of the solution increases by \(10\, mm\) of \(Hg\). What is the vapour pressure in \(mm Hg\) of \(n-\)heptane in its pure state..........\(?\)

Let \(P (-2,-1,1)\) and \(Q \left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)\) be the vertices of the rhombus PRQS. If the direction ratios of the diagonal \(RS\) are \(\alpha,-1, \beta\), where both \(\alpha\) and \(\beta\) are integers of minimum absolute values, then \(\alpha^{2}+\beta^{2}\) is equal to \(.....\)

If \( x=-1 \) and \( x=2 \) are extreme point of \(f\left( x \right) = \alpha \log \left| x \right| + \beta {x^2} + x\) then \(\left( {\alpha ,\beta } \right)\) 

Which of the following are correct expression for torque acting on a body?
A. \(\vec{\tau}=\vec{r}\times\vec{L}\)
B. \(\vec{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}})\)
C. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \mathrm{p}}{\mathrm{dt}}\)
D. \(\vec{\tau}=\mathrm{I} \vec{\alpha}\)
E. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
( \(\overrightarrow{\mathrm{r}}=\) position vector; \(\overrightarrow{\mathrm{p}}=\) linear momentum;
\(\overrightarrow{\mathrm{L}}=\) angular momentum; \(\vec{\alpha}=\) angular acceleration;
\(I=\) moment of inertia; \(\vec{F}=\) force; \(t=\) time)
Choose the correct answer from the options given below :

Let \(y=y(x)\) be the solution of the differential equation \(\left(3 y^2-5 x^2\right) y d x+2 x\left(x^2-y^2\right) d y=0\) such that \(y(1)=1\). then \(\left|(y(2))^3-12 y(2)\right|\) is equal to: