At \(T(\mathrm{~K})\), copper (atomic mass \(=635 \mathrm{u}\) ) has fcc unit cell structure with edge length of \(x Å\). What is the approximate density of \(\mathrm{Cu}\) in \(\mathrm{g} \mathrm{cm}^{-3}\) at that temperature?
\(\left(N_A=6.0 \times 10^{23} \mathrm{~mol}^{-1}\right)\)

Identify the conditions at which van der Waals equation of state changes to ideal gas equation.

\(\mathrm{CH}_3 \mathrm{COOH} \stackrel{\mathrm{LiAlH}_4}{\longrightarrow} A\)
\(\mathrm{A}+\mathrm{CH}_3 \mathrm{COOH} \stackrel{\mathrm{H}_3 \mathrm{O}^{+}}{\longrightarrow} \mathrm{B}+\mathrm{H}_2 \mathrm{O}\)
In the above reactions ' \(A\) ' and ' \(B\) ' respectively are

0.05 mole of a non-volatile solute is dissolved in \(500 \mathrm{~g}\) of water. What is the depression in freezing point of resultant solution?
\(\left(K_f\left(\mathrm{H}_2 \mathrm{O}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\)

(i) \(\mathrm{H}_3 \mathrm{PO}_4(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{H}_2 \mathrm{PO}_4^{-}(a q)\)
(ii) \(\mathrm{H}_2 \mathrm{PO}_4^{-}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{HPO}_4^{2-}(a q)\)
(iii) \(\mathrm{HPO}_4^{2-}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{PO}_4^{3-}(a q)\)
The equilibrium constants for the above reactions at a certain temperature are \(K_1, K_2\) and \(K_3\) respectively. The equilibrium constant for the reaction \(\mathrm{H}_3 \mathrm{PO}_4(a q) \rightleftharpoons 3 \mathrm{H}^{+}(a q)+\mathrm{PO}_4^{3-}(a q)\) in terms of \(K_1, K_2\) and \(K_3\) is

Gaseous state is chracterised by which of the following physical properties?
1. Gases mix evenly and completely on all proportions without any mechanical aid.
II. Gases exert pressure on all directions evenly.
MI. Gases are highly compressible and have lower density than liquids and solids.
IV. The volume and shape of the gas is fixed.

Match the following
\(\begin{array}{ll}
\text{List - I} & \text{List - II} \\
\text{A) } \mathrm{Al}_2 \mathrm{O}_3+2 \mathrm{NaOH}+3 \mathrm{H}_2 \mathrm{O} \rightarrow 2 \mathrm{Na}\left[\mathrm{Al}(\mathrm{OH})_4\right] & \text{I) Roasting} \\
\text{B) } \mathrm{Ni}(\mathrm{CO})_4 \xrightarrow{230^{\circ} \mathrm{C}} \mathrm{Ni}+4 \mathrm{CO} & \text{II) Calcination} \\
\text{C) } \mathrm{Fe}_2 \mathrm{O}_3 \cdot 3 \mathrm{H}_2 \mathrm{O} \xrightarrow{\Delta} \mathrm{Fe}_2 \mathrm{O}_3+3 \mathrm{H}_2 \mathrm{O} & \text{II) Vapour phase refining} \\
\text{D) } 2 \mathrm{PbS}+3 \mathrm{O}_2 \rightarrow 2 \mathrm{PbO}+2 \mathrm{SO}_2 & \text{IV) Electrolysis} \\
& \text{V) Leaching}
\end{array}\)
The correct answer is

\(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors such that \(\mathbf{a}+2 \mathbf{b}\) is also a unit vector. If \(\theta\) is the angle between a and \(\mathbf{b}\), then \(\sin \theta+\cos ^3 \theta+\tan ^5 \theta\) is equal to

\(\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=p \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+q \hat{\mathbf{k}}\) and \(\mathbf{d}=p \hat{\mathbf{i}}+q \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) are given vectors. If the projection of \(\mathbf{c}\) on \(\mathbf{a}\) is \(5 \sqrt{3}\) units and if a, \(\mathbf{b}\) and \(\mathbf{c}\) form a parallelopiped of volume 5 cubic units, then \(\tan ^{-1}(\mathbf{b}\). d \()=\)

The work function of a metal is \(2 \mathrm{eV}\). If a radiation of wavelength \(3000 Ã…\) is incident on it, the maximum kinetic energy of the emitted photoelectrons is (Planck's constant \(h=6.6 \times 10^{-34} \mathrm{Js}\); velocity of light \(\left.c=3 \times 10^8 \mathrm{~m} / \mathrm{s} ; 1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}\right)\)

If \(\bar{a}=\bar{i}+\bar{j}, \bar{b}=2 \bar{j}-\bar{k}\) are two vectors such that \(\bar{r} \times \bar{a}=\bar{b} \times \bar{a}, \bar{r} \times \bar{b}=\bar{a} \times \bar{b}\), then the unit vector in the direction of \(\overline{\mathrm{r}}\) is

One mole of a gas expands such that its volume ' \(V\) ' changes with absolute temperature ' \(T\) ' in accordance with the relation \(V=K T^2\) where ' \(K\) ' is a constant. If the temperature of the gas changes by \(60^{\circ} \mathrm{C}\), then work done by the gas is ( \(R\) is universal gas constant).

If \(2 \cdot 4^{2 k+1}+3^{3 k+1}=11 t\) and \(2 \cdot 4^{2 k+3}+3^{3 k+4}=11\) \(\left(p t+3^q\right)\), where \(k, t \in Z^{+}\), then \((p, q)=\)