The resistance of the meter bridge AB in given figure is $4 \Omega$. With a cell of emf $\varepsilon = 0.5 \mathrm {~V}$ and rheostat resistance $\mathrm { R } _ { \mathrm { h } } = 2 \Omega$ the null point is obtained at some point J. When the cell is replaced by another one of emf $\varepsilon = \varepsilon _ { 2 }$ the same null point J is found for $\mathrm { R } _ { \mathrm { h } } = 6 \Omega$. The emf $\varepsilon _ { 2 }$ is:
(1) 0.4 V
(2) 0.3 V
(3) 0.6 V
(4) 0.5 V

In a meter bridge, the wire of length $1 m$ has a non-uniform cross-section such that, the variation $\frac { d R } { d l }$ of its resistance $R$ with length $l$ is $\frac { d R } { d l } \propto \frac { 1 } { \sqrt { l } }$. Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point $P$. What is the length $AP$?
(1) 0.2 m
(2) 0.35 m
(3) 0.3 m
(4) $0.25 m$

A Sample of radioactive material $A$, that has an activity of $10 \mathrm{ mCi}$ ($1 \mathrm{ Ci} = 3.7 \times 10 ^ { 10 }$ decays $\mathrm { s } ^ { - 1 }$), has twice the number of nuclei as another sample of a different radioactive material $B$ which has an activity of $20 \mathrm{ mCi}$. The correct choices for half-lives of $A$ and $B$ would then be, respectively:
(1) 5 days and 10 days
(2) 10 days and 40 days
(3) 20 days and 10 days
(4) 20 days and 5 days

For emission line of atomic hydrogen from $n _ { i } = 8$ to $n _ { f } = n$, the plot of wave number $\bar { v }$ against $\frac { 1 } { n ^ { 2 } }$ will be: (The Rydberg constant, $\mathrm { R } _ { \mathrm { H } }$ is in wave number unit)
(1) Linear with slope $- R _ { H }$
(2) Non linear
(3) Linear with slope $R _ { H }$
(4) Linear with intercept $- \mathrm { R } _ { \mathrm { H } }$

The ground state energy of a hydrogen atom is $-13.6$ eV. The energy of second excited state of $\mathrm { He } ^ { + }$ ion in eV is:
(1) $-27.2$
(2) $-6.04$
(3) $-3.4$
(4) $-54.4$

The de Broglie wavelength $( \lambda )$ associated with a photoelectron varies with the frequency $( v )$ of the incident radiation as, [$v _ { 0 }$ is threshold frequency]:
(1) $\lambda \propto \frac { 1 } { \left( v - v _ { 0 } \right) }$
(2) $\lambda \propto \frac { 1 } { \left( v - v _ { 0 } \right) ^ { \frac { 1 } { 4 } } }$
(3) $\lambda \propto \frac { 1 } { \left( v - v _ { 0 } \right) ^ { \frac { 3 } { 2 } } }$
(4) $\lambda \propto \frac { 1 } { \left( v - v _ { 0 } \right) ^ { \frac { 1 } { 2 } } }$

The ratio of the shortest wavelength of two spectral series of hydrogen spectrum is found to be about 9. The spectral series are:
(1) Paschen and Pfund
(2) Balmer and Brackett
(3) Lyman and Paschen
(4) Brackett and Pfund

An ideal gas is allowed to expand from 1 L to 10 L against a constant external pressure of 1 bar. The work done in kJ is:
(1) $+10.0$
(2) $-2.0$
(3) $-0.9$
(4) $-9.0$

0.5 moles of gas $A$ and $x$ moles of gas $B$ exert a pressure of 200 Pa in a container of volume $10 \mathrm {~m} ^ { 3 }$ at 1000 K. Given, R is the gas constant in $\mathrm { JK } ^ { - 1 } \mathrm {~mol} ^ { - 1 }$, $x$ is:
(1) $\frac { 4 + R } { 2 R }$
(2) $\frac { 2 R } { 4 + 2R }$
(3) $\frac { 4 - R } { 2 R }$
(4) $\frac { 2 R } { 4 - R }$

At room temperature, a dilute solution of urea is prepared by dissolving 0.60 g of urea in 360 g of water. If the vapour pressure of pure water at this temperature is 35 mm Hg, lowering of vapour pressure will be: (molar mass of urea $= 60 \mathrm {~g} \mathrm {~mol} ^ { - 1 }$)
(1) 0.028 mm Hg
(2) 0.027 mm Hg
(3) 0.031 mm Hg
(4) 0.017 mm Hg

A solution of sodium sulphate contains $92 g$ of $\mathrm { Na } ^ { + }$ ions per kilogram of water. The molality of $\mathrm { Na } ^ { + }$ ions in that solution in $\mathrm { mol \, kg} ^ { - 1 }$ is:
(1) 16
(2) 4
(3) 12
(4) 8

An element has a face-centered cubic (fcc) structure with a cell edge of $a$. The distance between the centres of two nearest tetrahedral voids in the lattice is
(1) $\frac{a}{2}$
(2) $\frac{3}{2}a$
(3) $a$
(4) $\sqrt{2}\,a$

1 g of a non-volatile non-electrolyte solute is dissolved in 100 g of two different solvents A and B whose ebullioscopic constants are in the ratio of $1 : 5$. The ratio of the elevation in their boiling points, $\frac { \Delta \mathrm { T } _ { \mathrm { b } } \mathrm { A } } { \Delta \mathrm { T } _ { \mathrm { b } } \mathrm { B } }$, is: (assuming they have the same molar mass)
(1) $10 : 1$
(2) $1 : 5$
(3) $1 : 0.2$
(4) $5 : 1$

The mole fraction of a solvent in aqueous solution of a solute is 0.8. The molality (in $\text{mol kg}^{-1}$) of the aqueous solution is:
(1) $13.88 \times 10^{-3}$
(2) $13.88 \times 10^{-1}$
(3) $13.88$
(4) $13.88 \times 10^{-2}$

The anodic half-cell of lead-acid battery is recharged using electricity of 0.05 Faraday. The amount of $\mathrm { PbSO } _ { 4 }$ electrolyzed in g during the process is: (Molar mass of $\mathrm { PbSO } _ { 4 } = 303 \mathrm {~g \, mol} ^ { - 1 }$)
(1) 22.8
(2) 15.2
(3) 11.4
(4) 7.6

The following results were obtained during kinetic studies of the reaction. $2 \mathrm {~A} + \mathrm { B } \rightarrow$ product

Experiment	A in mol L ${ } ^ { - 1 }$	B in mol L$^{-1}$	Initial rate of reaction in $\mathrm { mol } \mathrm { L } ^ { - 1 } \mathrm {~min} ^ { - 1 }$
I	0.10	0.20	$6.93 \times 10 ^ { - 3 }$
II	0.10	0.25	$6.93 \times 10 ^ { - 3 }$
III	0.20	0.30	$1.386 \times 10 ^ { - 2 }$

The time (in minutes) required to consume half of A is
(1) 100
(2) 10
(3) 5
(4) 1

For the reaction of $\mathrm { H } _ { 2 }$ with $\mathrm { I } _ { 2 }$, the rate constant is $2.5 \times 10 ^ { - 4 } \mathrm { dm } ^ { 3 } \mathrm {~mol} ^ { - 1 } \mathrm {~s} ^ { - 1 }$ at $327 ^ { \circ } \mathrm { C }$ and $1.0 \mathrm { dm } ^ { 3 } \mathrm {~mol} ^ { - 1 } \mathrm {~s} ^ { - 1 }$ at $527 ^ { \circ } \mathrm { C }$. The activation energy for the reaction, in kJ mol$^{-1}$ is: $\mathrm { R } = 8.314 \mathrm { JK } ^ { - 1 } \mathrm {~mol} ^ { - 1 }$
(1) 166
(2) 59
(3) 72
(4) 150

For any two statement $p$ and $q$, the negative of the expression $p \vee ( \sim p \wedge q )$ is
(1) $\sim p \vee \sim q$
(2) $p \wedge q$
(3) $\sim p \wedge \sim q$
(4) $p \leftrightarrow q$

The expression $\sim ( \sim p \rightarrow q )$ is logically equivalent to
(1) $p \wedge \sim q$
(2) $\sim p \wedge \sim q$
(3) $p \wedge q$
(4) $\sim p \wedge q$

Which one of the following statements is not a tautology?
(1) $p \vee q \rightarrow p \vee ( \sim q )$
(2) $p \wedge q \rightarrow ( \sim p \vee q )$
(3) $p \rightarrow p \vee q$
(4) $p \wedge q \rightarrow p$

If $p \Rightarrow ( q \vee r )$ is False, then the truth values of $p , q , r$ are respectively, (where T is True and F is False)
(1) $T , F , F$
(2) $F , T , T$
(3) $F , F , F$
(4) $T , T , F$

The logical statement $[\sim(\sim p \vee q) \vee (p \wedge r)] \wedge (\sim q \wedge r)$ is equivalent to
(1) $(\sim p \wedge \sim q) \wedge r$
(2) $(p \wedge r) \wedge \sim q$
(3) $(p \wedge \sim q) \vee r$
(4) $\sim p \vee r$

If the angle of elevation of a cloud from a point $P$ which is $25 m$ above a lake be $30 ^ { \circ }$ and the angle of depression of reflection of the cloud in the lake from $P$ be $60 ^ { \circ }$, then the height of the cloud (in meters) from the surface of the lake is :
(1) 50
(2) 60
(3) 45
(4) 42

Let $Z$ be the set of integers. If $A = \left\{ x \in Z : 2 ^ { ( x + 2 ) \left( x ^ { 2 } - 5 x + 6 \right) } = 1 \right\}$ and $B = \{ x \in Z : - 3 < 2 x - 1 < 9 \}$, then the number of subsets of the set $A \times B$, is :
(1) $2 ^ { 12 }$
(2) $2 ^ { 10 }$
(3) $2 ^ { 18 }$
(4) $2 ^ { 15 }$

If $A = \left[ \begin{array} { c c c } 1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1 \end{array} \right]$, then for all $\theta \in \left( \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } \right) , \operatorname { det } ( A )$ lies in the interval :
(1) $\left( 1 , \frac { 5 } { 2 } \right]$
(2) $\left[ \frac { 5 } { 2 } , 4 \right)$
(3) $\left( \frac { 3 } { 2 } , 3 \right]$
(4) $\left( 0 , \frac { 3 } { 2 } \right]$