Match the List - I with List - II

List - I (Isothermal process for ideal gas system)	List - II Work done \(\left(V_f>V_i\right)\)
A. Reversible expansion	I. w = 0
B. Free expansion	II. \(w =-n R T \ln \frac{V_{ f }}{ V _{ i }}\)
C. Irreversible expansion	III. \(w =- p _{ ex }\left( V _{ f }- V _{ i }\right)\)
D. Irreversible compression	IV. \(w =- p _{ ex }\left( V _{ i }- V _{ i }\right)\)

Choose the correct answer from the options given below :

The concentration of dissolved oxygen \((DO)\) is cold water can go upto ......... \(ppm\)

Consider the given reaction. The total number of oxygen atoms present per molecule of the product \((P)\) is _______.

Match \(List-I\) with \(List-II\) 

\(List-I\)	\(List-II\)
\((A)\)Glucose \(+ HI\)	\((I)\) Gluconic acid
\((B)\)Glucose \(+ Br _{2}\) water	\((II)\) Glucose pentacetate
\((C)\)Glucose \(+\) acetic anhydride	\((III)\) Saccharic acid
\((D)\)Glucose \(+ HNO _{3}\)	\((IV)\) Hexane

Choose the correct answer from the options given below.

Number of metal ions characterized by flame test among the following is _______.
\(Sr ^{2+}, Ba ^{2+}, Ca ^{2+}, Cu ^{2+}, Zn ^{2+}, Co ^{2+}, Fe ^{2+}\)

Which of the following are isostructural pairs ? \(A.\) \(SO _{4}^{2-}\) and \(CrO _{4}^{2-}\) \(B.\) \(SiCl _{4}\) and \(TiCl _{4}\) \(C.\) \(NH _{3}\) and \(NO _{3}^{-}\) \(D.\) \(BCl _{3}\) and \(BrCl _{3}\) \(BCl _{3}\) and \(BrCl _{3}\)

The heat of atomization of methane and ethane are \(360\,kJ/mol\) and \(620\,kJ/mol,\) respectively. The longest wavelength of light capable of breaking the \(C - C\) bond is (Avogadro number \(= 6.02 \times  10^{23},\) \(h = 6.62 \times 10^{-34}\,Js\) )

Let the foot of perpendicular from a point \(P(1,2,-1)\) to the straight line  \(L: \frac{x}{1}=\frac{y}{0}=\frac{z}{-1}\) be \(N\). Let a line be drawn from \(P\) parallel to the plane \(x+y+2 z=0\) which meets \(L\) at point \(Q\). If \(\alpha\) is the acute angle between the lines \(\mathrm{PN}\) and \(\mathrm{PQ}\), then \(\cos \alpha\) is equal to \(.....\)

Consider the function \(\mathrm{f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathrm{R}\) defined by \(f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1\). Consider the statements \((I)\) The curve \(y=f(x)\) intersects the \(x\)-axis exactly at one point \((II)\) The curve \(y=f(x)\) intersects the \(x\)-axis at \(\mathrm{x}=\cos \frac{\pi}{12}\) Then

If \({E}\) and \({H}\) represents the intensity of electric field and magnetising field respectively, then the unit of \(E/H\) will be :

For an electrochemical cell \(\mathrm{Sn}(\mathrm{s})\left|\mathrm{Sn}^{2+}(\mathrm{aq}, 1 \mathrm{M}) \| \mathrm{Pb}^{2+}(\mathrm{aq}, 1 \mathrm{M})\right| \mathrm{Pb}(\mathrm{s})\) the ratio \(\frac{\left[\mathrm{Sn}^{2+}\right]}{\left[\mathrm{Pb}^{2+}\right]}\) when this cell attains equilibrium is (Given \(\mathrm{E}_{\mathrm{Sn}^{2+} / \mathrm{Sn}}^{0}=-0.14 \mathrm{\;V}\) \(\left.\mathrm{E}_{\mathrm{Pb}^{+2}/{\mathrm{Pb}}}^{0}=-0.13 \;\mathrm{V}, \frac{2.303 \mathrm{RT}}{\mathrm{F}}=0.06\right)\)

\(\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}}(\sin \sqrt{t}) dt }{x^{3}}\) is equal to

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........