\(\mathrm{N}_{2} \mathrm{O}_{5(9)} \rightarrow 2 \mathrm{NO}_{2(9)}+\frac{1}{2} \mathrm{O}_{2(\mathrm{~g})}\) In the above first order reaction the initial concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is \(2.40 \times 10^{-2}\, \mathrm{~mol} \,\mathrm{~L}^{-1}\) at \(318 \,K.\) The concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) after \(1\, hour\) was \(1.60 \times 10^{-2}\, \mathrm{~mol} \,\mathrm{~L}^{-1}\), The rate constant of the reaction at \(318\, \mathrm{~K}\) is \(.....\,\times 10^{-3} \mathrm{~min}^{-1}\). (Nearest integer) [Given: \(\log 3=0.477, \log 5=0.699\) ]

Given below are two statements: Statement \(I:\) According to Bohr's model of hydrogen atom, the angular momentum of an electron in a given stationary state is quantised. Statement \(II:\) The concept of electron in Bohr's orbit, violates the Heisenberg uncertainty principle. In the light of the above statements, choose the most appropriate answer from the options given below:

Which of the following statements is false ?

`The vapour pressure vs. temperature curve for a solution solvent system is shown below.The boiling point of the solvent is \(........\,{ }^{\circ} C\)

For combustion of one mole of magnesium in an open container at \(300 \,K\) and \(1\, bar\) pressure, \(\Delta_{ C } H ^{\ominus}\) \(=-601.70 \,kJ\, mol ^{-1}\), the magnitude of change in internal energy for the reaction is \(....\,kJ\). (Nearest integer) (Given : \(R =8.3 \,J \,K ^{-1}\, mol ^{-1}\) )

The temperature at which the rate constants of the given below two gaseous reactions become equal is _______ K. (Nearest integer).
\(X \longrightarrow Y \quad k _1=10^6 e ^{\frac{-30000}{T}}\)
\(P \longrightarrow Q \quad k _2=10^4 e ^{\frac{-24000}{T}}\)
Given : ln 10 = 2.303

The reaction rate for the reaction \(\left[\mathrm{PtCl}_{4}\right]^{2-}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons\left[\mathrm{Pt}\left(\mathrm{H}_{2} \mathrm{O}\right) \mathrm{Cl}_{3}\right]^{-}+\mathrm{Cl}^{-}\) was measured as a function of concentrations of different species. It was observed that \(\frac{-\mathrm{d}\left[\left[\mathrm{PtCl}_{4}\right]^{2-}\right]}{\mathrm{dt}}=4.8 \times 10^{-5}\left[\left[\mathrm{PtCl}_{4}\right]^{2-}\right]-2.4 \times10^{-3}\)
\(\left[\left[\mathrm{Pt}\left(\mathrm{H}_{2} \mathrm{O}\right) \mathrm{Cl}_{3}\right]^{-}\right]\left[\mathrm{Cl}^{-}\right]\) where square brackets are used to denote molar concentrations. The equilibrium constant \(\mathrm{K}_{\mathrm{c}}=....\). (Nearest integer)

Two vertices of a triangle \(\mathrm{ABC}\) are \(\mathrm{A}(3,-1)\) and \(\mathrm{B}(-2,3)\), and its orthocentre is \(\mathrm{P}(1,1)\). If the coordinates of the point \(\mathrm{C}\) are \((\alpha, \beta)\) and the centre of the circle circumscribing the triangle \(\mathrm{PAB}\) is \((h, k)\), then the value of \((\alpha+\beta)+2(h+k)\) equals :

\(\text { If } \int \frac{2 e^{x}+3 e^{-x}}{4 e^{x}+7 e^{-x}} d x=\frac{1}{14}\left(u x+v \log _{c}\left(4 e^{x}+7 e^{-x}\right)\right)+C\) where \(\mathrm{C}\) is a constant of integration, then \(\mathrm{u}+\mathrm{v}\) is equal to .... .

Given below are two statements:
Statement (I) : for \(\mathrm{ClF}_3\), all three possible structures may be drawn as follows.

Statement (II) : Structure III is most stable, as the orbitals having the lone pairs are axial, where the \(\ell \mathrm{p}-\mathrm{bp}\) repulsion is minimum.
In the light of the above statements, choose the most appropriate answer from the options given below:

Electron beam used in an electron microscope, when accelerated by a voltage of \(20\,kV\). has a de-Broglie wavelength of \(\lambda_0\). If the voltage is increased to \(40\,kV\). then the de-Broglie wavelength associated with the electron beam would be:

The sum of the infinite series \(1+\frac{2}{3}+\frac{7}{3^{2}}+\frac{12}{3^{3}}+\frac{17}{3^{4}}+\frac{22}{3^{5}}+\ldots\) is equal to

The least positive integral value of \(\alpha\), for which the angle between the vectors \(\alpha \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \mathrm{k}\) and \(\alpha \hat{\mathrm{i}}+2 \alpha \hat{\mathrm{j}}-2 \mathrm{k}\) is acute, is