JEE ADVANCED 2018 PAPER 2 OFFLINE

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass \(m=0.4\) \(kg\) is at rest on this surface. An impulse of \(1.0\) \(Ns\) is applied to the block at time \(t=0\) so that it starts moving along the \(x\)-axis with a velocity \(v\left( t \right) = {v_0}{e^{ - t/\tau }},\) where \({v_0}\) is a constant and \(\tau = 4s.\) The displacement of the block, in metres, at \(t = \tau\) is ______________ Take \({e^{ - 1}} = 0.37.\)

The surface of copper gets tarnished by the formation of copper oxide. \({N_2}\) gas was passed to prevent the oxide formation during heating of copper at \(1250\) \(K.\) However, the \({N_2}\) gas contains \(1\) mole % of water vapor as impurity. The water vapor oxidises copper as per the reaction given below : \(2Cu\left( s \right) + {H_2}O\left( g \right) \to C{u_2}O\left( s \right) + {H_2}\left( g \right)\)

\({P_{H2}}\) is the minimum partial pressure of \({H_2}\) (in bar) needed to prevent the oxidation at \(1250\) \(K.\) The value of \(\ln \left( {{P_{H2}}} \right)\) is ________.

Given: total pressure \(=1\) bar, \(R\) (universal gas constant ) \(=\) \(8J{K^{ - 1}}\,\,mo{l^{ - 1}},\) \(\ln \left( {10} \right) = 2.3.\,\) \(Cu(s)\) and \(C{u_2}O\left( s \right)\) are naturally immiscible.

At \(1250\) \(K:2Cu(s)\) \(+ {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}{O_2}\left( g \right) \to C{u_2}O\left( s \right);\) \(\Delta {G^ \circ } = - 78,000J\,mo{l^{ - 1}}\)

$${H_2}\left( g \right) + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}{O_2}\left( g \right) \to {H_2}O\left( g \right);$$

\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\) \(\Delta {G^ \circ } = - 1,78,000J\,mo{l^{ - 1}};\) (\(G\) is the Gibbs energy)

Let \(X = {({}^{10}{C_1})^2} + 2{({}^{10}{C_2})^2} + 3{({}^{10}{C_3})^2} + ... + 10{({}^{10}{C_{10}})^2}\),

where \({}^{10}{C_r}\), r \(\in\){1, 2, ..., 10} denote binomial coefficients. Then, the value of \({1 \over {1430}}X\) is ..........

One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is \(100\,K\) and the universal gas constant \(R=8.0\) \(J\,mo{l^{ - 1}}{K^{ - 1}},\) the decrease in its internal energy, in Joule, is ____________.

To measure the quantity of \(MnC{l_2}\) dissolved in an aqueous solution, it was completely converted to \(KMn{O_4}\) using the reaction,

\(MnC{l_2} + {K_2}{S_2}{O_8} + {H_2}O \to KMn{O_4} + {H_2}S{O_4} + HCl\) (equation not balanced).

Few drops of concentrated \(HCl\) were added to this solution and gently warmed. Further, oxalic acid (\(225\) \(mg\)) was added in portions till the colour of the permanganate ion disappeared. The quantity of \(MnC{l_2}\) (in mg) present in the initial solution is ____________.

(Atomic weights in \(g\,\,mo{l^{ - 1}}:Mn = 55,Cl = 35.5\) )

Let P be a matrix of order 3 \(\times\) 3 such that all the entries in P are from the set {\(-\)1, 0, 1}. Then, the maximum possible value of the determinant of P is ............ .