MikaStars39/nano-eval huggingface dataset:
A column of mercury of length $h=10cm$ is contained in the middle of a narrow horizontal tube of length $1m$, closed at both ends. The air in both halves of the tube is under a pressure of ${P}_{0}=76cm$ of mercury. The tube is now slowly made vertical. The distance moved by mercury will be approximately

Options:
A. $$4.5cm$$
B. $$4.0cm$$
C. $$5.0cm$$
D. $$1.2cm$$
E. $$3.2cm$$
F. $$2.5cm$$
G. $$3.5cm$$
H. $$2.0cm$$
I. $$3.0cm$$
J. $$5.5cm$$
Answer: A
A solution is prepared by dissolving 1.5 g of a monoacidic base into 1.5 kg of water at 300 K, which showed a depression in freezing point by $0.165^oC$. When 0.496 g of the same base was titrated, after dissolution, requires 40 ml of decimolar $H_2SO_4$ solution. If $K_f$ of water is 1.86 K kg $mol^{-1}$, then select the correct statement(s) out of the following (assuming molarity = molality):

Options:
A. The base is 9% ionized in aq. solution
B. The base is 10% ionized in aq. solution
C. The base is 12% ionized in aq. solution
D. Ionisation constant of the base is $8\times 10^{-3}$
E. The base is 15% ionized in aq. solution
F. The ionisation constant of the base is $7 \times 10^{-3}$
G. The base is 5% ionized in aq. solution
H. The osmotic pressure of the aq. solution of base is 21.67 atm
I. The ionisation constant of the base is $6 \times 10^{-3}$
J. The pH of the solution of weak base is 12.9
Answer: B
A ray of light incident at a small angle on a small angle glass prism. The deviation produced by the prism for a ray in air is $\delta$. When the prism is immersed in water of refractive index $\dfrac { 4}{ 3 }$, then the angle of deviation of the ray would be (refractive index of glass $=\dfrac { 3}{ 2 }$)

Options:
A. $$\dfrac { \delta }{ 8 }$$
B. $$\dfrac { \delta }{ 5 }$$
C. $$\dfrac { \delta }{ 6 }$$
D. $$\dfrac { \delta }{ 3 }$$
E. $$\delta$$
F. $$\dfrac { \delta }{ 2 }$$
G. $$\dfrac{\delta}{9}$$
H. $$\dfrac{\delta}{7}$$
I. $$\dfrac { \delta }{ 4 }$$
J. $$\dfrac { \delta }{ 10 }$$
Answer: E
Block of mass 'm' is lowered from natural length position of spring, slowly by an external agent in equilibrium position. The extension present in the massless spring was asked from two students.Student A: $\frac{1}{2} kx^2=mgx\Longrightarrow  x=\frac{2mg}{k} $Student B: $mg=kx\Longrightarrow x=\frac{mg}{k} $

Options:
A. Both are incorrect
B. Student- A is correct, student B is incorrect
C. Student- A is incorrect, student B is correct
D. Both are correct
Answer: B
A simple pendulum 50 cm long is suspended from the roof of a cart accelerating in the horizontal direction with constant acceleration $ \sqrt 3 m/s^2 $ . the period of small oscillation of the pendulum about its equilibrium position is $ (g = \pi^2 m/s^2 ) $

Options:
A. 1.35 sec
B. 1.20 sec
C. 1.0 sec
D. 1.30 sec
E. 1.40 sec
F. 1.25 sec
G. 1.53 sec
H. $ \sqrt 2 $ sec
I. 1.10 sec
J. 1.68 sec
Answer: C
The velocity of a moving particle is given by $\overset{\rightarrow}{v}= sin \left(\dfrac{3 \pi}{10} t \right)i+ cos \left (\dfrac{3 \pi}{10}t \right) \hat{j}$. The distance travelled by the particle in first second is:

Options:
A. $$cos^{-1} \left(cos \dfrac{3 \pi}{5} \right) units$$
B. $$1 unit$$
C. $$sin^{-1} \left(cos \dfrac{7 \pi}{10} \right) units$$
D. $$sin^{-1} \left(cos \dfrac{3 \pi}{10} \right) units$$
E. $$cos^{-1} \left(cos \dfrac{7 \pi}{10} \right) units$$
F. $$cos^{-1} \left(sin \dfrac{7 \pi}{10} \right) units$$
G. $$sin^{-1} \left(sin \dfrac{3 \pi}{10} \right) units$$
H. $$cos^{-1} \left(sin \dfrac{3 \pi}{10} \right) units$$
I. $$cos^{-1} \left(cos \dfrac{3 \pi}{10} \right) units$$
J. $$2 units$$
Answer: D
Consider the reaction at $300K$
$$C_6H_6(l)+\dfrac{15}{2}O_2(g) \rightarrow 6CO_2(g)+3H_2O(l); \Delta H=-3271 KJ$$
What is $\Delta U$ for the combustion of $1.5mole$ of benzene at $27^oC$?

Options:
A. $$-3264.50 kJ$$
B. $$-4900.88kJ$$
C. $$-4906.5kJ$$
D. $$-3269.00 kJ$$
E. $$-3265.75 kJ$$
F. $$-3269.50 kJ$$
G. $$-3270.50 kJ$$
H. $$-3267.25 kJ$$
I. $$-4903.75 kJ$$
J. $$-3274.75kJ$$
Answer: H
Two waves passing through a region are represented by$\displaystyle y_{1}=5\: mm\sin [(2\pi \: cm^{-1})x-(50\pi s^{-1})t]$and $\displaystyle y_{2}=10\: mm\sin [(\pi \: cm^{-1})x-(100\pi s^{-1})t]$Find the vertical displacement of a particle at x = 1 cm at time t  = 5.0 ms

Options:
A. 6 mm
B. 9 mm
C. 5 mm
D. 4 mm
E. 11 mm
F. 7 mm
G. 10 mm
H. 8 mm
I. 3 mm
J. 12 mm
Answer: C
Equivalent weight of $H_3PO_2$ (molecular weight = $M$) when it disproportionates into $PH_3$ and $H_3PO_3$ is:

Options:
A. $$\displaystyle \frac{M}{5}$$
B. $$\displaystyle\frac{M}{4}$$
C. $$\displaystyle \frac{M}{7}$$
D. $$M$$
E. $$\displaystyle \frac{2M}{3}$$
F. $$\displaystyle \frac{M}{6}$$
G. $$\displaystyle \frac{M}{3}$$
H. $$\displaystyle \frac{M}{8}$$
I. $$\displaystyle\frac{3M}{4}$$
J. $$\displaystyle\frac{M}{2}$$
Answer: I
Two blocks of masses $4kg$ and $10kg$ are connected with a massless string and a massless spring in between them. String goes over a massless and frictionless pulley as shown in figure. Initially the spring is unstretched, when $4kg$ block is released from rest. Minimum normal contact force between $10kg$ block and the surface in contact with it will be (Take $g=10m/{s}^{2}$)

Options:
A. $$70N$$
B. $$80N$$
C. $$90N$$
D. $$20N$$
E. $$10N$$
F. $$60N$$
G. $$30N$$
H. $$40N$$
I. $$Zero$$
J. $$50N$$
Answer: I
The height of mercury column measured with brass scale at temperature ${ T }_{ 0 }$ is ${ H }_{ 0 }$. What height $H_1$ will the mercury column have at $T=0^{\circ}C$. Coefficient of volume expansion of mercury is $\gamma$. Coefficient of linear expansion of brass is $\alpha$-

Options:
A. $$\frac { { H }_{ 0 }\left( 1+3\alpha { T }_{ 0 } \right) }{ 1+{ \gamma T }_{ 0 } }$$
B. $$\frac{{ H }_{ 0 }\left( 1+2\alpha { T }_{ 0 } \right)}{ 1+\gamma T_{ 0 } + \alpha T_{ 0 }}$$
C. $$\frac{{ H }_{ 0 }\left( 1+2\alpha { T }_{ 0 } \right)}{ 1+2\gamma T_{ 0 } }$$
D. $${ H }_{ 0 }\left( 1+\alpha { T }_{ 0 } \right)(1+\gamma T_0)$$
E. $$\frac{{ H }_{ 0 }\left( 1+\alpha { T }_{ 0 } \right)}{ 1+\gamma T_{ 0 } }$$
F. $$\frac { { H }_{ 0 }\left( 1+3\alpha { T }_{ 0 } \right) }{ (1+{ \gamma /3)T }_{ 0 } }$$
G. $$\frac{{ H }_{ 0 }\left( 1+3\alpha { T }_{ 0 } \right)}{ 1+\gamma T_{ 0 } }$$
H. $$\frac{{ H }_{ 0 }\left( 1+2\alpha { T }_{ 0 } \right)}{ 1+\gamma T_{ 0 } }$$
I. $$\frac { { H }_{ 0 }\left( 1+\alpha { T }_{ 0 } \right) }{ 1+{ \gamma T }_{ 0 } }$$
J. $$\frac{{ H }_{ 0 }\left( 1+4\alpha { T }_{ 0 } \right)}{ 1+2\gamma T_{ 0 } }$$
Answer: D
When a spring is stretched through a distance x, t exerts a force, given by $F=(-5x-16x^3).N$. What is the work done, when the spring is stretched from 0.1 m to 0.2 m?

Options:
A. $$8.1 \times 10^{-3} J$$
B. $$1.22 \times 10^{-3} J$$
C. $$12.2 \times 10^{-2} J$$
D. $$8.1 \times 10^{-1} J$$
E. $$12.2 \times 10^{-1} J$$
F. $$8.1 \times 10^{-4} J$$
G. $$8.1 \times 10^{-2} J$$
H. $$1.22 \times 10^{-1} J$$
I. $$1.22 \times 10^{-4} J$$
J. $$1.22 \times 10^{-2} J$$
Answer: D
Which of the following is incorrect?$(\Delta X$=uncertainty in the position.
$\Delta P$=uncertainty in the momentrum,$\Delta v$=uncertainty in the velocity,
$\Delta E=uncertainty in the energy,$\Delta r$=uncertainty in the time and $ \Delta \lambda$=uncertainty in the deBroglie's wavelength)

Options:
A. $$\Delta X \Delta \lambda \ge \frac { \lambda^2 }{ 4 \pi } $$
B. \Delta X \Delta p \ge \frac { h }{ 6 \pi }
C. $$\Delta E \Delta r \ge \frac { h }{ 4 \pi } $$
D. \Delta X \Delta p \ge \frac { h }{ 3 \pi }
E. $$\Delta X \Delta p \ge \frac { h }{ 4 \pi } $$
F. \Delta X \Delta p \ge \frac { h }{ 10 \pi }
G. \Delta X \Delta p \ge \frac { h }{ 8 \pi }
H. \Delta X \Delta p \ge \frac { h }{ 2 \pi }
I. $$\Delta X \Delta V \ge \frac { h }{ 2m } $$
J. \Delta X \Delta p \ge \frac { h }{ 5 \pi }
Answer: E
Using the data provided, calculate the multiple bond energy($kJ mol^{-1}$) of a $C\equiv C$ bond $C_{2}H_{2}$. That energy is bond energy is(take the energy of a $C- H$ bond as $350 kJ mol^{-1}$)
$$2{ C }(s)+{ H }_{ 2 }(g)\longrightarrow \quad { C }_{ 2 }{ H }_{ 2 }(g);\quad \quad \Delta H=225\quad kJ{ mol }^{ -1 }$$
$$2{ C }(s)\longrightarrow \quad 2C(g);\quad \quad \quad \quad \quad \quad \quad \Delta H=1410\quad kJ{ mol }^{ -1 }$$
$${ H }_{ 2 }(g)\longrightarrow \quad 2H(g);\quad \quad \quad \quad \quad \quad \quad \Delta H=330\quad kJ{ mol }^{ -1 }$$

Options:
A. $$857$$
B. $$1155$$
C. $$1165$$
D. $$865$$
E. $$855$$
F. $$1065$$
G. $$845$$
H. $$837$$
I. $$815$$
J. $$1195$$
Answer: I
For a given exothermic reaction $K_{P}$ and $K_{P}'$ are the equilibrium constants at temperatures $T_{1}$ and $T_{2}$ respectively. Assuming that het of reaction is constant in temperature range between $T_{1}$ and $T_{2}$ it is readily observed that.

Options:
A. $$K_{p} = K_{p}'$$
B. \(K_{p} = K_{p}' + 1\)
C. $$K_{p} = \dfrac {1}{K_{p}'}$$
D. $$K_{p} < K_{p}'$$
E. $$K_{p} > K_{p}'$$
F. $$K_{p} = K_{p}' + 2$$
G. $$K_{p} = K_{p}' - 2$$
H. \(K_{p} = K_{p}' - 1\)
I. $$K_{p} = K_{p}' + 0.5$$
J. $$K_{p} > K_{p}' + 1$$
Answer: D
A charged particle is released from origin with initial velocity $V = V_0\hat{i} $ in a magnetic field $\vec{B}=\dfrac{\sqrt{3}}{2}B_0\hat{i}+\dfrac{B_0}{2}\hat{j}$. The charge and mass of the particle are $q$ and $m$ respectively. Choose the correct option(s):

Options:
A. velocity \(\dfrac{ V_0\sqrt{3}}{3}\) is responsible for circular motion of particle in helix
B. velocity \(\dfrac{ V_0\sqrt{3}}{2}\) is responsible for circular motion of particle in helix
C. the velocity $\dfrac{ V_0\sqrt{3}}{2}$ is responsible for progress of particle in direction of magnetic field
D. velocity \(\dfrac{ V_0\sqrt{3}}{6}\) is responsible for circular motion of particle in helix
E. velocity \(\dfrac{ V_0}{\sqrt{2}}\) is responsible for circular motion of particle in helix
F. velocity \(\dfrac{ V_0\sqrt{2}}{2}\) is responsible for circular motion of particle in helix
G. velocity $\dfrac{ V_0}{2}$ is responsible for circular motion of particle in helix
H. velocity \(\dfrac{ V_0\sqrt{3}}{4}\) is responsible for circular motion of particle in helix
I. the pitch of the helical path described by the particle is $\dfrac{2\pi mV_0}{B_0q}$
Answer: G
For the reaction,
$PCl_{3}(g)+Cl_{2}(g)\rightarrow PCl_{5}(g),\Delta H=-xkJ$ if the $\Delta H_{f}^{o}PCl_{3}$ is $-ykJ$, what is $\Delta H_{f}^{o}PCl_{5}$?

Options:
A. $$\left(-x-y\right)kJ$$
B. $$(y-x)kJ$$
C. $$\left(x-y\right)kJ$$
D. $$\left(x+y\right)kJ$$
Answer: C
An object is tied to a string and rotated in vertical circle of radius  $'r' .$  If  $\dfrac { T _ { \operatorname { Max } } } { T _ { \operatorname { Min } } } = 2$ and constant speed  $^ { \prime }\mathrm { V } ^ { \prime }$  is maintained along the trajectory then  $\dfrac { V ^ { 2 } } { r g }$  value

Options:
A. $$3$$
B. $$4$$
C. $$2.7$$
D. $$1$$
E. $$2.8$$
F. $$2.5$$
G. $$3.5$$
H. $$3.2$$
I. $$3.1$$
J. $$2$$
Answer: D
Decreasing order of reactivity in Williamson synthesis of the following :I.$Me_3CCH_2Br$                  II.$CH_3CH_2CH_2Br$III. $CH_2=CHCH_2Cl$      IV. $CH_3CH_2CH_2Cl$

Options:
A. $$II > IV > III > I$$
B. $$III > I > II > IV$$
C. $$II > III > IV > I$$
D. $$I > II > IV > III$$
E. $$II > III > I > IV$$
F. $$III > II > IV > I$$
G. $$I > III > II > IV$$
H. $$III > IV > II > I$$
I. $$III > I > IV > II$$
J. $$III > II > I > IV$$
Answer: C
A smooth pulley $P$ of mass $M=2kg$ is lying on a smooth table. A light string passes round the pulley and has mass $m=1kg$ and $m'=2kg$ attached to its ends, the tow portions of the string being perpendicular to the edge of the table so that the masses hand vertically. Acceleration of pulley is:

Options:
A. $$\cfrac{3g}{8}$$
B. $$\cfrac{g}{2}$$
C. $$\cfrac{5g}{8}$$
D. $$\cfrac{4g}{5}$$
E. $$\cfrac{4g}{9}$$
F. $$\cfrac{2g}{3}$$
G. $$\cfrac{3g}{7}$$
H. $$\cfrac{4g}{7}$$
I. $$\cfrac{3g}{5}$$
Answer: H
The enthalpy of $\displaystyle S-S$ bond from the following data is:( i ) $\displaystyle C_{2}H_{5}-S-C_{2}H_{5}(g);   \Delta _{f}H^{o}=-147.2$ kJ / mol( ii ) $\displaystyle C_{2}H_{5}-S-S-C_{2}H_{5}(g);  \Delta _{f}H^{o}=-201.9$ kJ / mol( iii ) $\displaystyle S(g);  \Delta _{f}H^{o}=-222.8$ kJ / mol

Options:
A. 124.5 kJ / mol
B. 275.5 kJ / mol
C. 267.5 kJ / mol
D. 272.5 kJ / mol
E. 262.5 kJ / mol
F. 287.5 kJ / mol
G. 967.5 kJ / mol
H. 282.5 kJ / mol
I. 887.5 kJ / mol
J. - 277.5 kJ / mol
Answer: J
A  time depends force F  =  6t acts  on a  particle of mass 1  kg . If the particle starts from rest , the word done by the force during the first  1 sec , will be

Options:
A. $$12 J$$
B. $$15 J$$
C. $$6 J$$
D. $$ 18J $$
E. $$7 J$$
F. $$ 22 J $$
G. $$11 J$$
H. $$ 4. 5 J $$
I. $$ 9 J $$
J. $$10 J$$
Answer: I
Which of the following is true for the reaction 
${H}_{2}O(l)\rightleftharpoons  {H}_{2}O(g)$ at ${100}^{o}C$ and $1$ atmosphere

Options:
A. $$\Delta H = \Delta S$$
B. $$\Delta H = T \Delta V$$
C. $$\Delta H = \Delta V$$
D. $$\Delta H=0$$
E. $$\Delta H=\Delta E$$
F. $$\Delta H = T \Delta E$$
G. $$\Delta H=T\Delta S$$
H. $$\Delta H = \Delta G$$
I. $$\Delta H = T \Delta G$$
J. $$\Delta S=0$$
Answer: D
The solubility of a sparingly soluble salt $A_xB_y$ in water at $25^oC$ is $1.4\times 10^{4}M$. The solubility product is $1.1\times 10^{11}$. The possible value(s) of x and y is/are:

Options:
A. $$x=1,\ y=4$$
B. $$x=1,\ y=3$$
C. $$x=2, y=3$$
D. $$x=1,\ y=2$$
E. $$x=3,\ y=1$$
F. $$x=1,\ y=1$$
G. $$x=2,\ y=1$$
H. $$x=3,\ y=2$$
I. $$x=2,\ y=4$$
J. $$x=2, y=2$$
Answer: D
A charged particle of unit mass and unit charge moves with velocity of $\vec{v}=(8\hat{i}+6\hat{j})$ m/s in a magnetic field of $\vec{B}=2\hat{k}T$. Then:

Options:
A. the time period of the particle will be $5.50s$
B. the time period of the particle will be $2.36s$
C. the time period of the particle will be $6.28s$
D. the time period of the particle will be $4.71s$
E. the time period of the particle will be $3.14s$
F. the time period of the particle will be $3.15s$
G. the path of the particle may be $x^{2}+y^{2}=25$
H. the path of the particle may be $x^{2}+y^{2}-4x-21=0$
I. the path of the particle may be $y^{2}+z^{2} = 25$
J. the time period of the particle will be $1.57s$
Answer: E
Three blocks, D (top), C (middle), and B (bottom), each with a mass of 10 kg, are stacked on a horizontal surface. A force of 18 N is applied to the left on block D, a force of 15 N is applied to the left on block B, and a force of 100 N is applied to the right on block C. The coefficient of static friction (\(\mu_s\)) and kinetic friction (\(\mu_k\)) at each contact surface are 0.3 and 0.2, respectively (\(g = 10 \, \text{m/s}^2\)). Determine the acceleration of block B.

Options:
A. $$3.336 m/s^2$$
B. $$5 m/s^2$$
C. $$3.5 m/s^2$$
D. $$3.6 m/s^2$$
E. $$3.8 m/s^2$$
F. $$4.2 m/s^2$$
G. $$3.2 m/s^2$$
H. $$4 m/s^2$$
I. zero
J. $$4.5 m/s^2$$
Answer: I
A spring is placed in vertical position by suspending it from a hook at its top. A similar hook on the bottom of the spring is at $11\ cm$ above a table top. A mass of $75\ g$ and of negligible size is then suspended from the bottom hook, which is measured to be $4.5\ cm$ above the table top. The mass is then pulled down a distance of $4\ cm$ and released. Find the approximate position of the bottom hook after 5 $s$?
Take $g=10m/{s}^{2}$ and hook's mass to be negligible.

Options:
A. 2.5cm above the table top
B. 1.0cm above the table top
C. $5cm$ above the table top
D. $0.5cm$ above the table top
E. 2.0cm above the table top
F. $4.5cm$ above the table top
G. 1.5cm above the table top
H. 3.5cm above the table top
I. 6.5cm above the table top
J. $9cm$ above the table top
Answer: D
A particle is moving at a constant speed in a circular trajectory centred on the origin of an  $x y$  coordinate system. At one point  $( x = 4 m , y = 0 m ) $  the particle has a velocity of  $- 5.0 \hat { j }$ m/s. Determine the acceleration (in  ${ m } /  { s } ^ { 2 }$ ) , when the particle is at  ${ x }=0{ m },{ y }=-4{ m }.$

Options:
A. $$-\dfrac { 25 }{ 4 } \hat { { j } } $$
B. $$5\hat{i}$$
C. $$-\dfrac { 25 }{ 4 } \hat { { i } } $$
D. $$\dfrac{25}{4} \hat{i}$$
E. $$-5\hat { { i } } $$
F. $$\dfrac { 25 } { 4 } \hat { j }$$
G. $$5\hat { { j } }$$
H. $$-\dfrac{25}{4} \hat{i}$$
I. $$-5\hat{j}$$
Answer: D
A block starts from rest from the top of incline plane whose last $\frac { 1 }{ 3 } rd$ part is rough. If block comes to rest at bottom, then value of coefficient of friction between the block and incline plane is

Options:
A. $$\mu =\tan { \theta }$$
B. $$\mu = \frac{7}{4} \tan { \theta }$$
C. $$\mu = \frac{1}{4} \tan { \theta }$$
D. $$\mu = \frac{3}{4} \tan { \theta }$$
E. $$\mu =\frac { 1 }{ 2 } \tan { \theta }$$
F. $$\mu = \frac{5}{4} \tan { \theta }$$
G. $$\mu =\frac { 1 }{ 3 } \tan { \theta }$$
H. $$\mu = \frac{2}{3} \tan { \theta }$$
I. $$\mu =3\tan { \theta }$$
J. $$\mu = \frac{3}{2} \tan { \theta }$$
Answer: A
Voltage and current in an ac circuit are given by $V=5\sin{(100\pi-\cfrac{\pi}{6})}$ and $I=4\sin{(100\pi+\cfrac{\pi}{6})}$

Options:
A. Voltage leads the current by ${60}^{o}$
B. Current leads the voltage by ${45}^{o}$
C. Current leads the voltage by ${30}^{o}$
D. Voltage leads the current by ${20}^{o}$
E. Voltage leads the current by ${90}^{o}$
F. Voltage leads the current by ${30}^{o}$
G. Voltage leads the current by ${45}^{o}$
H. Current leads the voltage by ${60}^{o}$
I. Voltage leads the current by ${15}^{o}$
Answer: F
$pH$ of $0.005\ M$ calcium acetate $(pK_{a}\ of\ CH_{3}COOH=4.74)$ Is

Options:
A. $$7.12$$
B. $$9.26$$
C. $$7.09$$
D. $$9.37$$
E. $$7.04$$
F. $$8.34$$
G. $$8.28$$
H. $$7.14$$
I. $$8.26$$
J. $$8.37$$
Answer: J
A metal $M$ of molar mass 96 g $mol^{-1}$ reacts with fluorine to form a salt that can be represented as $MF_x$. In order to determine $x$. 9.18 g of the sample of the salt is dissolved in 100 g of water and its boiling point was determined to be 374.38 K. What is the value of $x$? Given: $K_b (water)=0.512 K kg mol^{-1}$. Assume complete dissociation of salt.

Options:
A. $$x=10$$
B. $$x=8$$
C. $$x=1$$
D. $$x=5$$
E. $$x=3$$
F. $$x=9$$
G. $$x=4$$
H. $$x=2$$
I. $$x=7$$
J. $$x=6$$
Answer: G
A chain with a total length \( L \) and mass per unit length \( \lambda \) is partially hanging from a frictionless pulley, while the rest of the chain lies on a horizontal surface. The length of the hanging portion of the chain is \( y \), and the chain is being pulled downward at a constant velocity \( v = \frac{dy}{dt} \). The pulley is ideal, with negligible size and mass, and there is no friction. Determine the force \( F \) required to maintain this constant velocity.

Options:
A. $$\dfrac{\lambda v^2 + 2\lambda g(h - y)}{2}$$
B. $$\lambda v^2 - \dfrac{\lambda g(h - y)}{2}$$
C. $$\dfrac{\lambda v^2 - \lambda g(h - y)}{4}$$
D. $$\dfrac{\lambda v^2 - \lambda g(h - y)}{2}$$
E. $$\dfrac{\lambda v^2}{2} + \lambda g(h - y)$$
F. $$\dfrac{\lambda v^2 + \lambda g(h - y)}{2}$$
G. $$\lambda v^2 + \dfrac{\lambda g(h - y)}{2}$$
H. $$\lambda v^2 - \lambda g(h - y)$$
I. $${\lambda v^2} + \lambda g(h - y)$$
Answer: G
Time period of oscillation of magnet of magnetic moment M and moment of inertia I in a vertical plane perpendicular to the magnetic meridian at a place where earth's horizontal and vertical component of magnetic field are $B_h$ and $B_v$ respectively is:

Options:
A. $$T= 2 \pi \sqrt {\dfrac{1}{M(Bv^2 + Bh^2)^{1/2}}}$$
B. $$T= 2 \pi \sqrt{\dfrac{1}{MBh^2}}$$
C. $$T= 2 \pi \sqrt{\dfrac{1}{MBv^2}}$$
D. $$T= 2 \pi \sqrt {\dfrac{1}{M(Bv + Bh)^2}}$$
E. $$T= 2 \pi \sqrt{\dfrac{1}{MBv}}$$
F. $$T= 2 \pi \sqrt {\dfrac{1}{M(Bv + Bh)}}$$
G. infinite
H. $$T= 2 \pi \sqrt {\dfrac{1}{MBv^3}}$$
I. $$T= 2 \pi \sqrt {\dfrac{1}{M(Bv - Bh)}}$$
J. $$T= 2 \pi \sqrt {\dfrac{1}{MBh}}$$
Answer: E
The bob of a pendulum is released from a horizontal position. The length of the pendulum is $2m$. If $10\%$ of the initial energy of the bob is dissipated as heat due to the friction of air. The speed of the bob when it reaches the lowermost point $B$ is ($g=10 ms^{-2})$

Options:
A. $$4ms^{-1}$$
B. $$5.1ms^{-1}$$
C. $$4.8ms^{-1}$$
D. $$5.3ms^{-1}$$
E. $$6ms^{-1}$$
F. $$4.5ms^{-1}$$
G. $$3ms^{-1}$$
H. $$5.4ms^{-1}$$
I. $$5.8ms^{-1}$$
J. $$5.5ms^{-1}$$
Answer: D
A uniform disc rotating at a constant speed in a vertical plane about a fixed horizontal axis, passing through the centre of the disc. A piece of disc from its rim detaches itself from the dist at the instant when it is on the same horizontal with the centre of the disc and moving upwards. Then about the fixed axis, the angular speed of the

Options:
A. remaining disc remains constant
B. broken away piece remains constant
C. remaining disc remains constant initially and then increases
D. remaining disc increases
E. remaining disc decreases
F. remaining disc increases initially and then decreases
G. remaining disc remains constant initially and then decreases
H. broken away piece decreases initially and later increases and remaining disc remains unchanged
I. broken away piece increases initially and then decreases
Answer: H
A particle of charge $q$, mass $m$, moving with velocity $v_{0}(\hat {i} + \hat {j})$ enters a region of uniform magnetic field $B_{0}$. Find displacement of particle after time $t = \dfrac {\pi m}{qB}$.

Options:
A. \dfrac {mv_{0}}{qB}(\pi \hat {i} + \pi \hat {j} - 2\hat {k})
B. $$\dfrac {mv_{0}}{qB}(\pi \hat {i} - 2\hat {k})$$
C. \dfrac {mv_{0}}{qB}(\pi \hat {i} - \pi \hat {j})
D. \dfrac {mv_{0}}{qB}(\pi \hat {i} + \pi \hat {j})
E. $$\dfrac {2\pi mv_{0}}{qB} \hat {j}$$
F. \dfrac {mv_{0}}{qB}(\pi \hat {i} - 2\hat {j})
G. \dfrac {mv_{0}}{qB}(\pi \hat {i} + \pi \hat {k})
H. $$\dfrac {mv_{0}}{qB}(\pi \hat {i} + 2\hat {k})$$
I. \dfrac {mv_{0}}{qB}(\pi \hat {i} + \pi \hat {j} + \pi \hat {k})
J. $$\dfrac {2\pi mv_{0}}{qB} \hat {i}$$
Answer: H
The mixture is compressed adiabatically to $\dfrac {1}{5}$ of its initial volume $V$. The change in its adiabatic compressibility in terms of the given quantities is (Given $5^{\dfrac {19}{13}} = 10.48)$.

Options:
A. $$-0.060\dfrac {V}{T} p_{a}^{-1}$$
B. $$-0.045\dfrac {V}{T} p_{a}^{-1}$$
C. $$-0.100\dfrac {V}{T} p_{a}^{-1}$$
D. $$-0.050\dfrac {V}{T} p_{a}^{-1}$$
E. $$-0.075\dfrac {V}{T} p_{a}^{-1}$$
F. $$-0.055\dfrac {V}{T} p_{a}^{-1}$$
G. $$-0.035\dfrac {V}{T} p_{a}^{-1}$$
H. $$-0.025\dfrac {V}{T} p_{a}^{-1}$$
I. $$-0.095\dfrac {V}{T} p_{a}^{-1}$$
J. $$-0.015\dfrac {V}{T} p_{a}^{-1}$$
Answer: H
At a certain temperature, the vapour density of $PCl_5$ in equilibrium with $PCl_3 \& Cl_2$ is $90$. the degree of dissociation of
$PCl_5$ is $(mwt=208.5)$:

Options:
A. $$0.258$$
B. $$0.185$$
C. $$0.145$$
D. $$0.168$$
E. $$0.058$$
F. $$0.198$$
G. $$0.158$$
H. $$0.138$$
I. $$0.215$$
J. $$0.175$$
Answer: B
Let there be a spherically symmetric charge density varying as $\rho \left( r \right) ={ \rho  }_{ 0 }\left( \dfrac { 5 }{ 4 } -\dfrac { r }{ R }  \right)$ upto $r=R$,and $\rho \left( r \right) =0$ for $r>R$, where r is the distance from the origin. The electric field at a distance $r(r<R)$ from the origin is given by:

Options:
A. $$\dfrac { { \rho }_{ 0 }r }{ { 3\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 4 } -\dfrac { 2r }{ R } \right)$$
B. $$\dfrac { { \rho }_{ 0 }r }{ { 4ważarepsilon }_{ 0 } } \left( \dfrac { 5 }{ 3 } -\dfrac { r }{ 2R } \right)$$
C. $$\dfrac { { \rho }_{ 0 }r }{ { 3\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 4 } -\dfrac { r }{ R } \right) $$
D. $$\dfrac { { 4\rho }_{ 0 }r }{ { 3\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 4 } -\dfrac { r }{ R } \right) $$
E. $$\dfrac { { \rho }_{ 0 }r }{ { 3\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 3 } -\dfrac { r }{ R } \right)$$
F. $$\dfrac { { \rho }_{ 0 }r }{ { 4\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 4 } -\dfrac { r }{ R } \right)$$
G. $$\dfrac { { \rho }_{ 0 }r }{ { 3\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 3 } -\dfrac { r }{ 2R } \right)$$
H. $$\dfrac { { \rho }_{ 0 }r }{ { 4\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 4 } -\dfrac { r }{ 2R } \right)$$
I. $$\dfrac { { \rho }_{ 0 }r }{ { 4\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 3 } -\dfrac { r }{ R } \right) $$
J. $$\dfrac { 4\pi { \rho }_{ 0 }r }{ { 3\varepsilon }_{ 0 } } \left( \dfrac { 5 }{ 3 } -\dfrac { r }{ R } \right) $$
Answer: I
The rise in temperature of an ideal gas ($\displaystyle Y ={ 5 }/{ 3 }$) when at $\displaystyle { 27 }^{ o }C$ it is adiabatically compressed to $\dfrac{8}{27}$ of its original volume is:

Options:
A. $$\displaystyle { 398 }K$$
B. $$\displaystyle { 225 }K$$
C. $$\displaystyle { 450 }K$$
D. $$\displaystyle { 300 }K$$
E. $$\displaystyle { 402}K$$
F. $$\displaystyle { 375 }K$$
G. $$\displaystyle { 325 }K$$
H. $$\displaystyle { 377 }K$$
I. $$\displaystyle { 350 }K$$
Answer: F
A satellite is revolving around earth with orbital speed $v_{0}$. If it stops suddenly, find the speed with which it will strike the surface of earth. $(v_{e} =$ escape velocity).

Options:
A. $$\sqrt{v_{e}^{2} - 4v_{0}^{2}}$$
B. $$\dfrac {v_e^{2}}{v_0}$$
C. $$\sqrt {v_{e}^{2} - 2v_{0}^{2}}$$
D. $$\sqrt {v_{e}^{2}} - 2v_{0}^{2}$$
E. $$\sqrt{v_{e}^{2} - 5v_{0}^{2}}$$
F. $$v_{0}$$
G. $$\sqrt{v_{e}^{2} - 3v_{0}^{2}}$$
H. $$\sqrt {v_{e}^{2} - v_{0}^{2}}$$
I. $$\sqrt{2v_{e}^{2} - v_{0}^{2}}$$
J. $$\sqrt{v_{e}^{2} + v_{0}^{2}}$$
Answer: C
The electric field strength in a region is given as $\bar { E } =\quad \cfrac { x\bar { i } +y\bar { j }  }{ { x }^{ 2 }+{ y }^{ 2 } } $. Find the net charge inside a sphere of radius $a$ with its centre at origin.

Options:
A. $$2\pi { \varepsilon }_{ 0 }a$$
B. $$6\pi { \varepsilon }_{ 0 }a$$
C. $$6\pi{\varepsilon}_{0}Ra$$
D. $$4\pi{\varepsilon}_{0}Ra$$
E. $$4\pi{\varepsilon}_{0}r$$
F. $$2\pi{\vatorepsilon}_{0}Ra$$
G. $$4\pi { \varepsilon }_{ 0 }a$$
H. $$8\pi { \varepsilon }_{ 0 }a$$
Answer: D
A volume of $100\ ml$ of $M-NaCl$ solution , $100\ ml$ of $2\ M-MgCl_{2}$ solution and $300\ ml$ of $4\ M-Mg (NO_{3})_{2}$ solution is mixed together and the mixture is diluted to $21$. Which of the following is the correct final concentration of ions?

Options:
A. $$NO_{3}^{-}=1.1\ M$$
B. $$Cl^{-}=0.2\ M$$
C. $$NO_{3}^{-}=1.6\ M$$
D. $$NO_{3}^{-}=1.5\ M$$
E. $$NO_{3}^{-}=1.3\ M$$
F. $$NO_{3}^{-}=1.4\ M$$
G. $$NO_{3}^{-}=1.7\ M$$
H. $$NO_{3}^{-}=1.2\ M$$
Answer: H
Two solid compounds $X$ and $Y$ dissociates at a certain temperature as follows$X(s) \rightleftharpoons A(g) + 2B(g); K_{P_{1}} = 9\times 10^{-3} atm^{3}$$Y(s) \rightleftharpoons 2B(g) + C(g); K_{P_{2}} = 4.5\times 10^{-3} atm^{3}$The total pressure of gases over a mixture of $X$ and $Y$ is?

Options:
A. $$0.6\ atm$$
B. $$0.5\ atm$$
C. $$0.55\ atm$$
D. $$4.5\ atm$$
E. $$0.75\ atm$$
F. $$0.65\ atm$$
G. $$0.40\ atm$$
H. $$0.9\ atm$$
I. $$0.45\ atm$$
J. None of these
Answer: I
A chain of mass M= '9kg' and length L='10m' initially rest on a horizontal frictionless surface. If it is slightly pushed down the horizontal surface due to which the chain start sliding down, then calculate closest value of the rate at which work is done on the chain by the gravitational force at the instant when one third of the chain is hanging vertical. (Neglect all dissipative forces)

Options:
A. 325 W
B. 173 W
C. 275 W
D. 290 W
E. 150 W
F. 200 W
G. 273 W
H. 300 W
I. 100 W
J. 250 W
Answer: H
Three discs $A, B$ and $C$ having radii $2m, 4m$ and $6m$ respectively are coated with carbon black on their outer surface. The wavelengths corresponding to the maximum intensity are $300nm, 400nm$ and $500nm$ respectively. The power radiated by them are ${Q}_{A}, {Q}_{B}$ and ${Q}_{C}$ respectively. Then

Options:
A. ${Q}_{C} > {Q}_{B} > {Q}_{A}$
B. ${Q}_{B} = {Q}_{C}$
C. $${Q}_{A}={Q}_{B}={Q}_{C}$$
D. ${Q}_{B}$ is maximum
E. ${Q}_{A} = {Q}_{B}$
F. ${Q}_{C}$ is maximum
G. ${Q}_{A} > {Q}_{B} > {Q}_{C}$
H. ${Q}_{A} = {Q}_{C}$
I. ${Q}_{A}$ is maximum
Answer: D
Mark out the correct statement(s)

Options:
A. in fission process, BE per nucleon of reactant nuclide is equal to the binding energy per nucleon of product nuclide
B. in fusion process, BE per nucleon of reactant nuclide is equal to the binding energy per nucleon of product nuclide
C. in fusion process, BE per nucleon of reactant nuclide is equal to or less than the binding energy per nucleon of product nuclide
D. in fusion process, BE per nucleon of reactant nuclide is greater than or equal to the binding energy per nucleon of product nuclide
E. in fission process, BE per nucleon of reactant nuclide is greater than the binding energy per nucleon of product nuclide
F. in both fission and fusion processes, the mass of reactant nuclide is greater than the mass of product nuclide
G. in fusion process, BE per nucleon of reactant nuclide is less than the binding energy per nucleon of product nuclide
H. in fusion process, BE per nucleon of reactant nuclide is greater than the binding energy per nucloen of product nuclide
Answer: G
At ordinary temperature,the molecule of an ideal gas have only translation and rotational kinetic energies,At high temperature,they may have vibrational energy.As a result of this, at higher temperature molar specific heat capacity at constant volume $ C_v $ for a diatomic gas is

Options:
A. $$\frac{6R}{2}$$
B. $$ = \frac {5R}{2}  $$
C. $$= \frac{9R}{2}$$
D. $$ = \frac {3R}{2} $$
E. $$\frac{7R}{2}$$
F. $$\frac{3R}{2}$$
G. $$= \frac{7R}{2}$$
H. $$\frac{4R}{2}$$
I. $$ > \frac {3R}{2} $$
J. $$ > \frac {5R}{2}  $$
Answer: B
In a reaction $\displaystyle \Delta H$ and $\displaystyle \Delta S$ both are positive. In which of the following cases, the reaction would be spontaneous?

Options:
A. All of these.
B. $$\displaystyle T\Delta S>\Delta H$$
C. $$\displaystyle \Delta H=T\Delta S$$
D. $$\Delta H < \frac{T\Delta S}{2}$$
E. $$\displaystyle \Delta H>T\Delta S$$
F. $$\Delta H = \frac{T\Delta S}{2}$$
Answer: E
A voltmeter has an internal resistance $r$ and can measure up to $v$ volts. A high resistance $R$ is connected in series with it and now it is able to measure up to $V$ volts. Then which of the following is the correct relation between the $r,R,v$ and $V$?

Options:
A. $$\frac{v}{R} = \frac{V + v}{R}$$
B. $$\frac{v}{r}=\frac{V}{R}$$
C. $$\frac{v}{R} = \frac{V + r}{v}$$
D. $$\frac{v}{R} = \frac{V + v}{r}$$
E. $$\frac{v}{R}=\frac{V-v}{r}$$
F. $$\frac{v}{r}=\frac{V-v}{R}$$
G. $$\frac{v}{r} = \frac{V + v}{R}$$
H. $$\frac{v}{R} = \frac{V + R}{r}$$
I. $$\frac{v}{R}=\frac{V}{r}$$
J. $$\frac{v}{r} = \frac{V + R}{v}$$
Answer: E
Two identical sleeves A and B each of mass m can slide without friction
on a fixed horizontal rod. A load C of mass 2m is suspended from the
mid-point of a light inextensible cord oflength $ I =3 \sqrt {3} $ m connecting the
sleeves A and B. Initially the sleeves are at a separation I and the load
is held motionless. What can you conclude for subsequent motion after
the load is released? Acceleration of free fall is $ 10m/s^2 $

Options:
A. Before the sleeves collide, the net force on the system consisting of sleeves A and B and load C is 20m.
B. Before the sleeves collide, the net force on the system consisting of sleeves A and B and load C is 15m.
C. Before the sleeves collide, the net force on the system consisting of sleeves A and B and load C is 5m.
D. Maximum speed of the sleeves is $ \sqrt{10} m/s $
E. Before the sleeves collide, the net force on the system consisting of sleeves A and B and load C is 12m.
F. Before the sleeves collide, the net force on the system consisting of sleeves A and B and load C is 10m.
Answer: D
$V_1$ ml of $NaOH$ of molarity $X$ and $V_2$ ml of $Ba(OH)_2$ of molarity $\cfrac{y}{2}$ are mixed together. Mixture is completely neutralized by 100 ml $\cfrac{0.1}{2}MH_2SO_4$. If $\cfrac{V_1}{V_2}=\cfrac{1}{4}$ and $\cfrac{x}{y}=4$, what fraction of acid is neutralized by $Ba(OH)_2$,

Options:
A. 0.4
B. 0.5
C. 0.75
D. 0.25
E. 0.67
F. 0.2
G. 0.6
H. 0.7
I. 0.33
J. 0.8
Answer: B
Two blocks \( A \) and \( B \), each of mass \( m \), are connected by a light thread passing over a smooth, massless pulley located at the top of a wedge of mass \( m \). Block \( A \) rests on the left inclined plane of the wedge, which makes an angle of \( 37^\circ \) with the horizontal, and block \( B \) rests on the right inclined plane of the wedge, which makes an angle of \( 53^\circ \) with the horizontal. The wedge lies on a smooth horizontal surface and is pulled to the right with a constant acceleration \( a \). Friction is absent, and both blocks remain in contact with the wedge. Given \( g \) as the acceleration due to gravity, find the normal reaction acting on block \( B \).

Options:
A. $$\frac{m}{5}\left( {3g + 3a} \right)$$
B. $$\frac{m}{5}\left( {3g + 2a} \right)$$
C. $$\frac{m}{5}\left( {4g + 2a} \right)$$
D. $$\frac{m}{5}\left( {4g - 3a} \right)$$
E. $$\frac{m}{5}\left( {3g + 5a} \right)$$
F. $$\frac{m}{5}\left( {3g - 5a} \right)$$
G. $$\frac{m}{5}\left( {3g - 4a} \right)$$
H. $$\frac{m}{5}\left( {4g + 3a} \right)$$
I. $$\frac{m}{5}\left( {3g + 4a} \right)$$
Answer: I
Heptene decomposses according to following reaction $C_{7}H_{14}\overset{\Delta }{\rightarrow}2C_{2}H_{4}+C_{3}H_{6}$Rate constant was found to be $1\times 10^{-4}sec^{-1}$. In what time molar ration of heptene to ethene in the reaction mixture will attain the value 1?[log 2 = 0.3010, log 3=0.4771]

Options:
A. 8100 min
B. 4065 min
C. 8110 min
D. 4050 min
E. 228 min
F. 4675 min
G. 1736 min
H. 4060 min
I. 8 min
J. 4055 min6
Answer: I
A horizontal cylinder open from one end closed from the other, is rotated with a constant angular velocity '$\omega$' about a vertical axis passing through its open end. If outside air pressure is $P_0$, the temperature is $T$, and the molar mass of air is $m$, then find the air pressure as a function of the distance $'r'$ from the rotational axis, : (Assume molar mass is independent of $\gamma$) :

Options:
A. $$P=P_0e^{(m\omega^2 r^2 /4RT)}$$
B. $$P=P_0e^{(m\omega^2 r^2 /RT)}$$
C. $$P=P_0e^{(m\omega^2 r /2RT)}$$
D. $$P=P_0e^{(m\omega^2 r^2 /2RT)}$$
E. $$P=P_0e^{(m\omega^2 r^2 /3RT)}$$
F. $$P=P_0e^{-m\omega^2 r^2 /RT}$$
G. $$P=P_0e^{-m\omega^2 r^2 /4RT}$$
H. $$P=P_0e^{(m\omega^2 r^2 /5RT)}$$
I. $$P=P_0e^{(m\omega^2 r /RT)}$$
J. $$P=P_0e^{-m\omega^2 r^2 /2RT)}$$
Answer: D
Brass scale of a Barometer gives correct reading at $0^{\circ} C$. Coefficient of linear expansion of brass is $18\times 10^{-6}/ ^{\circ}C$. If the barometer reads 76 cm at $20^{\circ}C$ ,the correct reading is ($ \gamma_{Hg} =18 \times 10^{-5}/^{\circ} C$):

Options:
A. 76.264 cm
B. 75.9cm
C. 75.7cm
D. 76.542 cm
E. 75.8cm
F. 76.124 cm
G. 76.042 cm
H. 76.426 cm
I. 76.642 cm
J. 76.142 cm
Answer: C
For the reactions:
$2NOCl(g)\rightleftharpoons 2NO(g)+{ Cl }_{ 2 }(g)$, $\Delta { H }^{ o }=18kcal$ and $\Delta { S }^{ o }=30 cal/K$ at 300 K. The equilibrium constant ${ K }_{ P }^{ o }$ of the reaction at 300 K is

Options:
A. $${ e }^{ 10 }$$
B. $${ e }^{ -13 }$$
C. $${ e }^{ 12 }$$
D. $${ e }^{ -10 }$$
E. $${ e }^{ 15 }$$
F. $${ e }^{ 18 }$$
G. $${ e }^{ -12 }$$
H. $${ e }^{ -15 }$$
I. $${ e }^{ -18 }$$
Answer: E
The freezing point of a 0.4 m aqueous solution of a weak acid ( $\alpha = 0.25$ )  would be  
( Given $K_{f}$  ( water ) = $1.85^{\circ}C$)

Options:
A. $$-0.9375^{\circ}C$$
B. $$-1.3875^{\circ}C$$
C. $$-0.5^{\circ}C$$
D. $$-0.0925^{\circ}C$$
E. $$-1.86^{\circ}C$$
F. $$-1.325^{\circ}C$$
G. $$-1.36^{\circ}C$$
H. $$-1.35^{\circ}C$$
I. $$-1.85^{\circ}C$$
J. $$-0.925^{\circ}C$$
Answer: I
Two points P and Q perform SHM along the same straight line, greatest distance between P and Q noticed by an observer studying the motion is found to be A (A=amplitude). Hence, we can deduce that in this case P leads Q ( or Q leads P) by a phase angle of

Options:
A. $$\pi$$
B. $$\dfrac { 3\pi }{ 8 }$$
C. $$\dfrac { \pi }{ 3 }$$
D. $$\dfrac { 2\pi }{ 3 }$$
E. $$\dfrac{\pi}{6}$$
F. $$\dfrac{5\pi}{12}$$
G. $$\dfrac{3\pi}{4}$$
H. $$\dfrac{\pi}{4}$$
I. $$\dfrac{5\pi}{6}$$
J. $$\dfrac { \pi }{ 2 }$$
Answer: J
The density of vapours of a substance of molar mass 18 g at 1 atm pressure and 500 K is 0.36 Kg $M^{-3}$. the value of compressibility factor Z for the vapour will be $(Take R = 0.082 L atm mole^{-1}K^{-1})$

Options:
A. 0.69
B. 0.89
C. 0.79
D. 0.92
E. 0.74
F. 0.82
G. 1.22
H. 0.76
I. 1.42
J. 1.45
Answer: A
A particle performs a horizontal circle of radius r in a funnel type vessel of frictionless surface with a half cone angle $  $. The mass of the particle is m, then the speed of the particle must be

Options:
A. $$ v={rgtan \theta } $$
B. $$ v=\sqrt {rg/tan \theta } $$
C. $$v=\sqrt { \frac{rg}{2 \cdot tan \theta} }$$
D. $$v=\sqrt {r^2/gtan \theta }$$
E. $$v=\sqrt {rg \cdot tan \theta }$$
F. $$v=\sqrt { \frac{rg}{tan \theta} }$$
G. $$v=\sqrt { \frac{r}{g \cdot tan \theta} }$$
H. $$v=\sqrt { \frac{r}{gtan \theta} }$$
I. $$ v=\sqrt { \frac { 1 }{ {rgtan \theta } } } $$
J. $$ v=\sqrt {rgtan \theta } $$
Answer: B
A liquid rises to a height of $50cm$ in a capillary tube of diameter $0.04mm$. If density of liquid is $0.08\times {10}^{3}kg$ ${m}^{-3}$ angle of contact is ${20}^{o}$, the surface tension of the liquid is ($\cos { { 20 }^{ o } } =0.94)$

Options:
A. $$4.15 \times { 10 }^{ -3 }\ \cfrac { N }{ m }$$
B. $$4.27 \times 10^{-2}\ \cfrac{N}{m}$$
C. $$4.15 \times 10^{-2}\ \cfrac{N}{m}$$
D. $$0.417\ \cfrac { N }{ m } $$
E. $$4.17 \ \cfrac { N }{ m } $$
F. $$4.17\times { 10 }^{ -2 }\ \cfrac { N }{ m } $$
G. $$4.27 \times 10^{-3}\ \cfrac { N }{ m }$$
H. $$4.25\times { 10 }^{ -3 }\ \cfrac { N }{ m } $$
I. $$4.15 \times { 10 }^{ -4 }\ \cfrac { N }{ m }$$
J. $$4.15 \times 10^{-3}\ \cfrac { N }{ m }$$
Answer: F
A transverse travelling wave represented by a equation: $ y = (1 cm) sin (2 \pi x-20 \pi t): $ Where  x is in meters and t is seconds, is travelling on a string of mass per unit length 0.01 kg/m. What is the kinetic energy of a portion of the string of length equal one wavelength ? $ (take \  \pi = \sqrt {10} ) $

Options:
A. $$ 2 \times 10^{-3} J $$
B. $$6 \times 10^{-3} J$$
C. $$7 \times 10^{-3} J$$
D. $$1.5 \times 10^{-3} J$$
E. $$ 10^{-3} J $$
F. $$8 \times 10^{-3} J$$
G. $$ 4 \times 10^{-3} J $$
H. $$ \frac {1}{2} \times 10^{-3} J $$
I. $$5 \times 10^{-3} J$$
J. $$3 \times 10^{-3} J$$
Answer: E
A wire of length 2.5 km in part of circuit of resistance 35 $\Omega$ oriented along E-W direction falls under gravity to the ground from a height of 10 m, then the current in the wire is ($g=9.8 \ m/s^2$ and $B_H = 5 \times 10^{-5} T$)

Options:
A. 0.015 A
B. 0.03 A
C. 0.02 A
D. 2 A
E. 0.005 A
F. 0.025 A
G. 0.2 A
H. 0.01 A
I. 0.04 A
J. 0.012 A
Answer: C
The electric field in an electromagnetic wave is given by 
$$E = 100\,\cos (15 \times {10^8}t - 5x)V{m^{ - 1}}$$
The average energy density of E.M. wave is (in S.I. unit)

Options:
A. $$\dfrac{{{{10}^{ - 5}}}}{{48\pi }}$$
B. $\dfrac{{{{10}^{ - 5}}}}{{18\pi }} $
C. $$\frac{{{{10}^5}}}{{72\pi }}$$
D. $$\frac{{{{10}^{ - 5}}}}{{36\pi }}$$
E. $\dfrac{{{{10}^{ - 5}}}}{{50\pi }}$
F. $$\frac{{{{10}^5}}}{{36\pi }}$$
G. \(\dfrac{{{{10}^{ - 5}}}}{{54\pi }}\)
H. $$\dfrac{{{{10}^{ - 5}}}}{{42\pi }}$$
I. \(\dfrac{{{{10}^{ - 5}}}}{{64\pi }}\)
J. $$\frac{{{{10}^{ - 5}}}}{{72\pi }}$$
Answer: D
Two projectiles are throw simultaneously in the same plane from the same point, their velocities are $v_{1}$ and $v_{2}$ at angles $\theta_{1}$ and $\theta_{2}$ respectively from the horizontal.If $v_{1} \cos \theta_{1}=v_{2} \cos\theta_{2}$, then choose the incorrect statement :

Options:
A. both will have the same maximum height
B. both will have the same horizontal component of velocity
C. both will have the same horizontal component of acceleration
D. none of these.
E. both will have the same vertical component of acceleration
Answer: D
The equation of the transverse wave travelling in a rope is given by $\mathrm{y}=5 \sin (4\mathrm{t}-0.02\mathrm{x})$ where $y$ and $x$ are in meters and $t$ is in seconds. Calculate the intensity of the wave if the density of rope material is $1250\  \mathrm{k}\mathrm{g}/\mathrm{m}^{3}$

Options:
A. $$10\ kJm^{-2}s^{-1}$$
B. $$1500\ kJm^{-2}s^{-1}$$
C. $$2000\ kJm^{-2}s^{-1}$$
D. $$3\ kJm^{-2}s^{-1}$$
E. $$1250\ kJm^{-2}s^{-1}$$
F. $$40\ kJm^{-2}s^{-1}$$
G. $$2500\ kJm^{-2}s^{-1}$$
H. $$500\ kJm^{-2}s^{-1}$$
I. $$1000\ kJm^{-2}s^{-1}$$
J. $$100\ kJm^{-2}s^{-1}$$
Answer: C
when a foil of atomic number $Z=40$ was used in an $\alpha  - $ scattering experiment, the number of $\alpha  - $ particles scattered at $30^0$ was found to be $200$ per minute. If the foil is replaced by another $(Z=20)$ foil of same thickness, the number of $\alpha  - $particles scattered per minute at $30^0$ is nearly equal to

Options:
A. $$15$$
B. $$40$$
C. $$10$$
D. $$60$$
E. $$30$$
F. $$35$$
G. $$25$$
H. $$45$$
I. $$20$$
J. $$50$$
Answer: I
A 2 kg block situated on a smooth fixed incline is connected to a spring of negligible mass, with spring constant $K= 500 Nm^{-1}$, via frictionless pulley. The block is released. The distance travelled by the block before coming to rest is (Assume initially spring is in relaxed state)

Options:
A. $$9 cm$$
B. $$4 cm$$
C. $$6 cm$$
D. $$3 cm$$
E. $$2 cm$$
F. $$5 cm$$
G. $$7 cm$$
H. $$10 cm$$
I. $$8 cm$$
J. $$1 cm$$
Answer: J
a ball dropped from a certain height, falls in the influence of uniform gravity, strikes the grounds and repeatedly rebounds elastically . During a time interval t = 8 sec from it was dropped,  it covers a distance s =20 m and reaches the initial position . How many collisions 'n' during this time, did the ball make with ground ?$(g = 10m/{s^2})$

Options:
A. $$4$$
B. $$10$$
C. $$8$$
D. $$14$$
E. 6
F. $$12$$
Answer: E
A whistle emitting a sound of frequency 440 Hz is tied to a string of length 1.5 m and rotated with an angular velocity of 20 $rad/s$ in the horizontal plane. Then, the range of frequencies heard by an observer stationed at a large distance from the whistle will be :$\displaystyle (V_s = 330 ms^{-1})$

Options:
A. 400.0 Hz to 483.0 Hz
B. 400.0 Hz to 484.0 Hz.
C. 403.3 Hz to 480.0 Hz.
D. 400.0 Hz to 484.0 Hz
E. 403.3 Hz to 480.0 Hz
F. 403 Hz to 483.0 Hz
G. 403.3 Hz to 483.0 Hz
H. 403.3 Hz to 484.0 Hz
I. 400.0 Hz to 480.0 Hz
J. 403 Hz to 484.0 Hz
Answer: J
Which of the following statement(s) is (are) correct?

Options:
A. The oxidation state of nitrogen in $HN_{3}$ is $-3$.
B. In silver atom, $22$ electrons have a spin of one type and $26$ of the opposite type.
C. In silver atom, $24$ electrons have a spin of one type and $24$ of the opposite type.
D. In silver atom, $22$ electrons have a spin of one type and $25$ of the opposite type.
E. In silver atom, $23$ electrons have a spin of one type and $24$ of the opposite type.
F. In silver atom, $23$ electrons have a spin of one type and $25$ of the opposite type.
G. In silver atom, $25$ electrons have a spin of one type and $22$ of the opposite type.
H. The electronic configuration of $Cr$ is $[Ar]3d^{5} 4s^{1}$ (Atomic Number of $Cr=24$).
I. In silver atom, $24$ electrons have a spin of one type and $23$ of the opposite type.
J. The magnetic quantum number may have a negative value.
Answer: E
If the displacement of a particle as a function of velocity is given by $x=2-v$ (k is a positive constant) then velocity of the particle as a function of time t will be (given its position is zero at t=0)

Options:
A. $$v=1-e^{-t}$$
B. $$v=2(e^{-t})$$
C. $$v=2(1+e^{-2t})$$
D. $$v=2e^{-2t}$$
E. $$v=2(1+e^{-t})$$
F. $$v=e^{-t}$$
G. $$v=2(1-e^{-2t})$$
H. $$v=2(1-e^{-t})$$
I. $$v=2(e^{-2t})$$
Answer: B
Two immiscible liquids P and Q of different densities are contained in a wide U-tube. The heights of the two liquids above the horizontal line xx' which cuts the boundary between the liquids are $H_P$  and $H_Q$ respectively. The U-tube is transported to a planet where the acceleration of free fall is 2/3 that on the earth, where the liquids do not evaporate and where the heights of liquid (measured relative to xx') are $h_P$ and $h_Q$ respectively. Which of the given statements is correct?

Options:
A. Both iquid levels rise up so that $h_P - H_Q = H_P - H_Q$
B. Both liquid levels rise up so that $h_P/H_P = h_Q/H_Q$
C. Both liquid levels rise up so that $h_P + H_Q = H_P + h_Q$
D. Both liquid levels fall such that $h_P + H_Q = H_P + h_Q$
E. Both liquid levels fall such that \( h_P - H_Q = H_P - h_Q \)
F. Both liquid levels fall such that \( h_P - H_P = H_Q - h_P \)
G. Both liquid levels fall such that \( h_P - H_P = H_Q - h_Q \)
H. Both liquid levels rise up so that $h_P + H_Q = H_P + H_Q$
I. The liquid P falls and liquid Q rises such that $\dfrac{h_P}{h_Q} = \dfrac{2}{3} \frac{H_P}{H_Q}$
J. The liquid levels are unchanged, i.e. , $h_p = H_P$ and $h_Q = H_Q$
Answer: B
A shell is projected from a level ground with a velocity of $20\ m/s$ at $45^{o}$ to the horizontal. When the shell is at he height point is breaks into two equals fragments. One of the fragment whose initial velocity after the explosion is zero falls vertically downward. At what distance from the point of projection does the other fragment fall? $(g=10\ ms^{-2})$

Options:
A. $$30\ m$$
B. $$35\ m$$
C. $$80\ m$$
D. $$25\ m$$
E. $$70\ m$$
F. $$20\ m$$
G. $$60\ m$$
H. $$50\ m$$
I. $$90\ m$$
J. $$40\ m$$
Answer: A
Which sequence of reactions would be best for the conversion of toluene into para-bromopheny lacetic acid? (NBS = N-bromosuccinimide)

Options:
A. i) Br_2 & FeBr_3; \, \, ii) NBS \,in \,CCl_4;\, \, iii) \,Mg \,in \,ether \,followed \,by \,CO_2;\, \, iv) \,H_3O^{(+)} \,heat
B. $i)\, Br_2$ & $FeBr_3; \, \, ii)\, NBS \,in \,CCl_4;\, \, iii)\, NaCN \, in \,ethanol;\, \, iv)\, H_3O^{(+)}$ 7 $heat$
C. i) Br_2 & FeBr_3; \, \, ii) \,KMnO_4 & heat; \, \, iii) \,NaCN \,in \,ethanol; \, \, iv) \,H_3O^{(+)} 7 \,cold
D. $i)\, Br_2$ & $FeBr_3; \, \, ii)\, KMnO_4$ & $heat;\, \, iii)\, Mg\, in\, ether\, followed\, by\, CO_2;\, \, iv)\, H_3O^{(+)}$
E. i) Br_2 & FeBr_3; \, \, ii) \,NBS \,in \,CCl_4;\, \, iii) \,Mg \,in \,ether \,followed \,by \,CO_2;\, \, iv) \,H_3O^{(+)} \,cold
Answer: B
Emf of the cell
$$sn\left( s \right) { sn }^{ 2+ }\left( 0.050M \right) ||{ H }^{ + }\left( 0.020M \right) |{ H }_{ 2 }\left( g \right)$$
$$\left( 1bar \right) |Pt\left( s \right) $$
at 298 K is [ given ${ E }^{ \circ  }{ sn }^{ 2+ }/sn=-0.16V$ ]

Options:
A. 0.059 V
B. 0.058 V
C. 0.098 V
D. 0.06 V
E. 0.18 V
F. 0.10 V
G. 0.065 V
H. 0.061 V
I. 0.12 V
J. 0.07 V
Answer: D
A mass of diatomic gas $(\gamma = 1.4)$ at a pressure of $2\ atmospheres$ is compressed adiabatically so that its temperature rises from $27^{o}C$ to $927^{o}C.$ The pressure of the gas in the final state is

Options:
A. $$69.7\ atm$$
B. $$256\ atm$$
C. $$7.29\ atm$$
D. $$8\ atm$$
E. $$72.9\ atm$$
F. $$28\ atm$$
G. $$71.3\ atm$$
H. $$56.7\ atm$$
I. $$70.3\ atm$$
Answer: B
3.2 g of oxygen (At wt.=16) and 0.2 g of hydrogen (At wt,=1) are placed in a 1.12 L flask at ${ 0 }^{ \circ  }C$. The total pressure of the gas mixture will be:

Options:
A. 3 atm
B. 9 atm
C. 4 atm
D. 2 atm
E. 6 atm
F. 0.25 atm
G. 0.5 atm
H. 8 atm
I. 0.75 atm
J. 7 atm
Answer: H
The value of equilibrium constant for the following reaction at $300\ K$ and constant pressure, for the following reaction, is:$ A(g) + B(g) \rightleftharpoons  C(g) + D(g) + E(g)$Given : $ \Delta E^{\circ} = -30\  kcal$ and $\Delta S^{\circ} = 100\ cal/K $

Options:
A. $$ \dfrac { 1 }{ { e }^{ 2 } } $$
B. $$ \dfrac { 1 }{ e } $$
C. $$ e $$
D. $$\dfrac { 1 }{ { e }^{ 6 } }$$
E. $$\dfrac { 1 }{ { e }^{ 9 } }$$
F. $$\dfrac { 1 }{ { e }^{ 4 } }$$
G. $$\dfrac { 1 }{ { e }^{ 5 } }$$
H. $$\dfrac { 1 }{ { e }^{ 3 } }$$
I. $$\dfrac { 1 }{ { e }^{ 7 } }$$
J. $$ { e }^{ 2 } $$
Answer: B
A trolley is moving with a velocity $V$. There is a smooth circular tube fixed in the trolley having a small sphere resting at the bottom. Now the trolley is uniformly decelerated until it comes to stop. During the deceleration the sphere moves by an angle $\theta $ and acquires sufficient speed so that it just completes the vertical circle in the tube. The magnitude of the deceleration of the trolley is (Consider the vertical circle is just completed when velocity at the top most point is zero, take acceleration due to gravity $=g$ ):

Options:
A. $$\displaystyle \frac{g}{sin^{2} \theta} $$
B. $$\displaystyle \frac{2g}{sin\theta} $$
C. $$\frac{2g}{tan\theta}$$
D. $$\frac{2g}{cos\theta}$$
E. $$\displaystyle \frac{2g}{cos\theta} $$
F. $$\frac{3g}{sin\theta}$$
G. $$\frac{2g}{sin(2\theta)}$$
H. $$\displaystyle \frac{g}{sin \theta} $$
I. $$\frac{3g}{sin^{2} \theta}$$
J. $$\frac{2g}{tan(2\theta)}$$
Answer: B
One mole of monoatomic ideal gas expands adiabatically at initial temperature T against a constant external pressure of $1$ atm from $1$L to $2$L. Find out the final temperature ($R=0.0821$L $K^{-1}$ $mol^{-1}$).

Options:
A. $$T-\dfrac{4}{3 \times 0.082}$$
B. $$T-\dfrac{5}{3 \times 0.082}$$
C. $$T-\dfrac{2}{2 \times 0.082}$$
D. $$T-\dfrac{1}{2 \times 0.082}$$
E. $$T-\dfrac{2}{3\times 0.0821}$$
F. $$T-\dfrac{1}{3 \times 0.082}$$
G. $$T+\dfrac{2}{3\times 0.082}$$
H. $$T-\dfrac{3}{3 \times 0.082}$$
I. $$\dfrac{T}{(2)^{\dfrac{5}{3}-1}}$$
J. $$T-\dfrac{2}{3\times 0.08}$$
Answer: E