MikaStars39/nano-eval huggingface dataset:
The speed of a particle moving along a circle of '$R$' depends on the distance covered as $v=as$ where '$a$' is constant, then acceleration of the particle or

Options:
A. $$ \dfrac {a^2s^2}{R} $$
B. $$a^2s\left( 1+\dfrac { s^{ 2 } }{ 3R^2} \right)^{1/2}$$
C. $$a^2s\left( 1-\dfrac { s^{ 2 } }{ 2R^2} \right)^{1/2}$$
D. $$a^2s\left( 1-\dfrac { s^{ 2 } }{ 3R^2} \right)^{1/2}$$
E. $$ a^2s $$
F. $$a^2s\left( 1-\dfrac { s^{ 2 } }{ 4R^2} \right)^{1/2}$$
G. $$ a^{ 2 }s\left( 1+\dfrac { s^{ 2 } }{ R^2} \right)^{1/2} $$
H. $$a^2s\left( 1-\dfrac { s^{ 2 } }{ R^2} \right)^{1/2}$$
I. $$a^2s\left( 1+\dfrac { s^{ 2 } }{ 2R^2} \right)^{1/2}$$
J. $$ \dfrac {a^2s^2}{S} $$
Answer: E
A light beam travelling in the x-direction is described by the electric field ${ E }_{ y }=(300V{ m }^{ -1 })sin\quad \omega (t-x/c).$ An electron is constrained to move along the y-direction with a speed of $2.0\times  {10}^7 { m }^{ -1 }$ Find the maximum electric force and the maximum magnetic force on the electron

Options:
A. $$4.2 \times 10^{-18} N, 1.8 \times 10^{-8}N$$
B. Zero, Zero
C. $$4.8\times {10}^{-17} N, 3.2\times 10^{-18}N$$
D. $$4.8 \times 10^{-17} N, 1.6 \times 10^{-18}N$$
E. $$4.8 \times 10^{-17} N, 8.0 \times 10^{-18}N$$
F. $$4.8\times {10}^{-17} N, zero$$
G. $$4.8 \times 10^{-17} N, 4.0 \times 10^{-18}N$$
H. $$4.8 \times 10^{-17} N, 1.2 \times 10^{-18}N$$
I. $$4.8 \times 10^{-17} N, 2.4 \times 10^{-18}N$$
J. $$4.2\times {10}^{-18} N, 1.8\times 10^{-8}N$$
Answer: C
Calculate the coefficient of real expansion from the following data. weight of the specific gravity bottle = 40.25 gm, weight of the bottle + liquid at $ 25^0 C = 70.50 gm, $ weight of the bottle + liquid at $ 100^0 C = 69.20 gm $ coefficient of linear expansion of glass $ = 0.00009/^0 C $

Options:
A. $$ 6.26 \times 10^{-4} \ ^0 C $$
B. $$ 8.26 \times 10^{-4} \ ^0 C $$
C. $$5.26 \times 10^{-4} \ ^0 C$$
D. $$ 4.26 \times 10^{-4} \ ^0 C $$
E. $$9.26 \times 10^{-4} \ ^0 C$$
F. $$1.76 \times 10^{-4} \ ^0 C$$
G. $$1.26 \times 10^{-4} \ ^0 C$$
H. $$3.26 \times 10^{-4} \ ^0 C$$
I. $$7.26 \times 10^{-4} \ ^0 C$$
J. $$ 2.26 \times 10^{-4} \ ^0 C $$
Answer: J
What is the activation energy for a reaction if its rate doubles when the temperature is aised from ${ 20 }^{ \circ  }C$ to ${ 35 }^{ \circ  }C$ ? ( R = 8.314 J ${ mol }^{ -1 }{ K }^{ -1 }$)

Options:
A. $269 kJ$ ${ mol }^{ -1 }$
B. 279 kJ ${ mol }^{ -1 }$
C. 273 kJ ${ mol }^{ -1 }$
D. $34.7 kJ$ ${ mol }^{ -1 }$
E. 171 kJ ${ mol }^{ -1 }$
F. 271 kJ ${ mol }^{ -1 }$
G. 278 kJ ${ mol }^{ -1 }$
H. $15.1 kJ$ ${ mol }^{ -1 }$
I. $342 kJ$ ${ mol }^{ -1 }$
J. 267 kJ ${ mol }^{ -1 }$
Answer: A
A copper sphere of $10cm$ diameter is lowered into a water filled hemispherical copper vessel of $20cm$ diameter so that the sphere and the vessel becomes concentric. Electrical conductivity of water is $\sigma={10}^{-3}S/m$. The electrical resistance between the sphere and the vessel is

Options:
A. $$1593.6\Omega$$
B. $$1682.4\Omega$$
C. $$1580.6\Omega$$
D. $$1505.4\Omega$$
E. $$1603.4\Omega$$
F. $$1591.6\Omega$$
G. $$1650.4\Omega$$
H. $$1450\Omega$$
I. $$1489.6\Omega$$
J. $$1601.6\Omega$$
Answer: F
The waves are represented by the following equations 
$y_{1} =  5 \sin 2 \pi  (10t-0.1x)$  
$y_2 = 10 \sin 2\pi  (20t-0.2x)$ 
Ratio of intensities $\dfrac {I_{2} }{I_{1}}$ will be :

Options:
A. $$64$$
B. $$8$$
C. 32
D. $$4$$
E. $$36$$
F. $$1$$
G. $$16$$
H. $$2$$
Answer: G
An amount of $1$ mole of a gas is changed from its initial state($20$L, $2$ atm) to final state($4$L, $10$ atm), respectively. If the change can be represented by a straight line in P-V curve, the maximum temperature achieved by the gas in the process is?($R=0.08$L-atm/K-mol)

Options:
A. $$1100^oC$$
B. $$800^oC$$
C. $$900^oC$$
D. $627$K
E. $$1200^oC$$
F. $900$K
G. $$1173^oC$$
H. $$1000^oC$$
I. $$827^oC$$
Answer: C
The displacement $y$ of a particle on a straight line is given by $y=f(x,l)$ as a function of time. Which of the following functions does not represent wave motion?

Options:
A. $$y=A\sin{(2kx-\omega t)}$$
B. $$y=A\sin{(kx-\omega t)}$$
C. $$y=A\sin{(kx+\omega t+\cfrac{\pi}{10})}$$
D. $$y=A\sin{(kx-\omega^2 t)}$$
E. $$y=A\sin{(kx-\omega t+\cfrac{\pi}{2})}$$
F. $$y=A\sin{({k}^{2}{\omega}^{2}-{\omega}^{2}{t}^{2})}$$
G. $$y=A\sin{(kx+\omega t)}$$
H. $$y=\sin ^{ 2 }{ (kx-\omega t) } $$
I. $$y=A\sin{(kx+\omega^2 t)}$$
J. $$y=A\sin{(2kx+\omega t)}$$
Answer: B
The van der Waals equation of state represents an approximate behaviour of gases at high pressure and is given by $\displaystyle \left ( P + \frac{an^2}{V^2} \right ) (V - nb) = n RT$ where a and b are constants having different values for different gases. In a special case if $a = 0, b = 0$ van der Waals equation will take the shape of ideal gas equation. For ethane $(C_{2}H_{6})$ gas $a = 0.554 \ Jm^3 mol^{-2} $ and $b = 6.38 \times 10^{-5}\ m^3mol^{-1}$. 1.8 mole of ethane allowed to expand from $2 \times 10^{-3} m^3 $ to $4 \times 10^{-3} m^3$ isothermally at $300 K$.Find the work done using van der Waal's equation in an isothermal condition.

Options:
A. None of these
B. $$n RT \log_e \displaystyle \dfrac{V_2 - nb}{V_1 - nb} - an^2 \left [ \dfrac{1}{V_1} - \dfrac{1}{V_2} \right ]$$
C. $$n RT \log_e \displaystyle \dfrac{V_1 - nb}{V_2 - nb} + an^2 \left [ \dfrac{1}{V_1} - \dfrac{1}{V_2} \right ]$$
D. $$n RT \log_e \displaystyle \dfrac{V_2 - nb}{V_1 - nb} + an^2 \left [ \dfrac{1}{V_1} - \dfrac{1}{V_2} \right ]$$
E. $$n RT \log_e \displaystyle \dfrac{V_2 + nb}{V_1 + nb} + an^2 \left [ \dfrac{1}{V_2} - \dfrac{1}{V_1} \right ]$$
F. $$n RT \log_e \displaystyle \dfrac{V_1 + nb}{V_2 + nb} - an^2 \left [ \dfrac{1}{V_2} - \dfrac{1}{V_1} \right ]$$
G. $$n RT \log_e \displaystyle \dfrac{V_1 - nb}{V_2 - nb} - an^2 \left [ \dfrac{1}{V_2} - \dfrac{1}{V_1} \right ]$$
H. $$n RT \log_e \displaystyle \dfrac{V_2 - nb}{V_1 - nb} + an^2 \left [ \dfrac{1}{V_2} - \dfrac{1}{V_1} \right ]$$
I. $$n RT \log_e \displaystyle \dfrac{V_1 - nb}{V_2 - nb} + an^2 \left [ \dfrac{1}{V_2} - \dfrac{1}{V_1} \right ]$$
J. $$n RT \log_e \displaystyle \dfrac{V_1 + nb}{V_2 + nb} + an^2 \left [ \dfrac{1}{V_2} - \dfrac{1}{V_1} \right ]$$
Answer: H
The conductivity of 0.01 M HA is $3.8\times { 10 }^{ -5 }{ \Omega  }^{ -1 }{ cm }^{ -1 }$ and the conductivity of solution formed by mixing 100 ${ cm }^{ 3 }$ of 0.01 M HA and 1 ${ cm }^{ 3 }$ 1M NaOH is $80.0\times { 10 }^{ -5 }{ \Omega  }^{ -1 }{ cm }^{ -1 }$ calculate ${ \lambda  }_{ mA^{ - } }^{ \infty  }$ if ${ \lambda  }_{ mNa^{ + } }^{ \infty  }=50{ \Omega  }^{ -1 }{ cm }^{ 2 }{ mol }^{ 1 }$

Options:
A. $$73.2 \times { 10 }^{ -5 }{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ -1 }$$
B. $$78.2 \times { 10 }^{ -5 }{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ -1 }$$
C. $$74.2 \times { 10 }^{ -5 }{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ -1 }$$
D. $$75.0 \times { 10 }^{ -5 }{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ -1 }$$
E. $$30.0\times { 10 }^{ -5 }{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ 1 }$$
F. $$30.0{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ 1 }$$
G. $$75.2 \times { 10 }^{ -5 }{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ -1 }$$
H. $$76.2\times { 10 }^{ -5 }{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ 1 }$$
I. $$77.2 \times { 10 }^{ -5 }{ \Omega }^{ -1 }{ cm }^{ 2 }{ mol }^{ -1 }$$
Answer: H
A $1.025\space g$ sample containing a weak acid $HX$ (mol. wt. = 82) is dissoved in $60\space mL$ water and titrated with $0.25\space M\space NaOH$. When half of the acid was neutralised the $pH$ was found to be $5.0$ and at the equivalent point the $pH$ is $9.0$. Calculate weight percentage of $HX$ in sample.

Options:
A. $$75\mbox{%}$$
B. $$51\mbox{%}$$
C. $$80\mbox{%}$$
D. $$56\mbox{%}$$
E. $$48\mbox{%}$$
F. $$58\mbox{%}$$
G. $$60\mbox{%}$$
H. $$54\mbox{%}$$
I. $$50\mbox{%}$$
J. $$52\mbox{%}$$
Answer: C
At $27^{o}C$ following equation is established ${ H }_{ 2 }+S\rightleftharpoons { H }_{ 2 }S$ ${K}_{c}=0.08$. If $0.3$ mole of hydrogden and $2$ mole of sulphur are heated to $2L$ container. What will be the concentration of ${H}_{2}S$ at equilibrium.

Options:
A. $$0.022$$
B. $$0.044$$
C. $$0.099$$
D. $$0.066$$
E. $$0.055$$
F. $$0.133$$
G. $$0.11$$
H. $$0.033$$
I. $$0.122$$
J. $$0.08$$
Answer: G
Two exactly similar wires of steel and copper are stretched by equal forces. If the total elongation is 2cm, then how much is the elongation in steel and copper wire respectively? Given Y$_{steel}$ =20 x 10$^{11}$ dyne/cm$^2$ , $Y_{copper}$ = 12 x 10$^{11}$ dyne/cm$^2$.

Options:
A. 0.80 cm; 1.20 cm
B. 1.05 cm; 0.95 cm
C. 0.90 cm; 1.10 cm
D. 1.10 cm; 0.90 cm
E. 0.95 cm; 1.05 cm
F. 0.85 cm; 1.15 cm
G. 1.25 cm; 0.75 cm
H. 0.75 cm ; 1.25 cm
I. 1.15 cm; 0.85 cm
Answer: G
A sphere of radius $0.1m$ and mass $8\pi kg$ is attached to the lower end of a steel wire of length $5.0m$ and diameter ${ 10 }^{ -3 }m$. The wire is suspended from $5.22m$ high ceiling of a room. When the sphere is made to swing as a simple pendulum, it just grazes the floor at its lower point. Calculate the velocity of the sphere at the lowest point. (Y for steel=$1.994\times { 10 }^{ 11 }N/{ m }^{ 2 }$)

Options:
A. $$7.9m{ s }^{ -1 }$$
B. $$8.6m{ s }^{ -1 }$$
C. $$8.2m{ s }^{ -1 }$$
D. $$8.0m{ s }^{ -1 }$$
E. $$6.5m{ s }^{ -1 }$$
F. $$8.8m{ s }^{ -1 }$$
G. $$7.5m{ s }^{ -1 }$$
H. $$9.1m{ s }^{ -1 }$$
I. $$8.3m{ s }^{ -1 }$$
J. $$8.5m{ s }^{ -1 }$$
Answer: F
Using bond energy data, calculate heat of formation of isoprene. Given bond energy of $C-H=98.8\ kcal,H-H=104\ kcal,C-C=83\ kcal,C=C=147\ kcal$ and $\Delta H$ of $C(s)\rightarrow C(g)=171\ kcal$.

Options:
A. $$28.6\ kcal\ mol^{-1}$$
B. $$20.6\ kcal\ mol^{-1}$$
C. $$28.1\ kcal\ mol^{-1}$$
D. $$28.4\ kcal\ mol^{-1}$$
E. $$206\ kcal\ mol^{-1}$$
F. $$28.3\ kcal\ mol^{-1}$$
G. $$283\ kcal\ mol^{-1}$$
H. $$28.5\ kcal\ mol^{-1}$$
I. $$28.2\ kcal\ mol^{-1}$$
Answer: F
$0.01 M$ NH$_4$OH is $50\%$ neutralised by using $HCl$ solution. The $pOH$ value of final solution will be :$(p{K_{NH_4^ + }} = 9.26)$

Options:
A. 8.74
B. 6.75
C. 7.76
D. 6.76
E. 8.26
F. 7.26
G. 6.74
H. 7.74
I. 4.74
J. 5.74
Answer: G
A sonometer wire of length 0.1 m under a tension of 2 kg weight, when sounded together with a tuning fork of frequency 502 Hz, produces 2 beats per second. When the fork is loaded with a little wax and sounded together, no beats are heard. Then, the linear density in kg$ ms{-1} $ of the wire is $ (g = 10 ms ^{-2} ) $

Options:
A. $$0.05 \times 10^{-3}$$
B. $$ 0.02 \times 10^{-3} $$
C. $$0.04 \times 10^{-3}$$
D. $$0.06 \times 10^{-3}$$
E. $$ 2 \times 10^{-3} $$
F. $$0.01 \times 10^{-3}$$
G. $$0.03 \times 10^{-3}$$
H. $$ 0.2 \times 10^{-3} $$
I. $$ 20 \times 10^{-3} $$
J. $$0.025 \times 10^{-3}$$
Answer: B
For the change 
$C_{diamond} \rightarrow C_{graphite} : \Delta H = -1.89 kJ$.
If 6 g of diamond and 6 g graphite are separately burnt to yield $CO_2$ the heat librated in first case is

Options:
A. less than in the second case by $14.34 kJ$
B. less than in the second case by $1.98 kJ$
C. less than in the second case by $1.94 kJ$
D. less than in the second case by $11.34 kJ$
E. more than in the second case by $0.945 kJ$
F. less than in the second case by $2.835 kJ$
G. less than in the second case by $3.78 kJ$
H. less than in the second case by $3.87 kJ$
I. less than in the second case by $1.89 kJ$
Answer: E
One mole of ammonium carbamate dissociate as shown below at $500K$
${NH}_{3}COO{NH}_{4(s)\rightleftharpoons  2{NH}_{3}(g}+{CO}_{2}(g)$. If the pressure exerted by the released gases is $3.0$ atm, the value of ${K}_{p}$ is (standard state $=1atm$)

Options:
A. 11
B. 3
C. 7
D. 6
E. 10
F. 12
G. 5
H. 8
I. 4
J. 9
Answer: I
For a reaction $2A+B \rightarrow $ product ,doubling the initial concentration of both the reactants increases the rate by a factor of 8, and doubling the concentration of B alone doubles the rate.The rate for the reaction is

Options:
A. $$\gamma =k[A]^{ 3 }[B]$$
B. $$\gamma =k[A]^{ 2 }[B]$$
C. $$\gamma =k[A]^{ 1.5 }[B]^{ 2 }$$
D. $$\gamma =k[A]^{ 3 }[B]^{ 2 }$$
E. $$\gamma =k[A]^{ 2 }[B]^{ 3 }$$
F. $$\gamma =k[A][B]$$
G. $$\gamma =k[A][B]^{ 2 }$$
H. $$\gamma =k[A]^{ 2 }[B]^{ 1.5 }$$
I. $$\gamma =k[A][B]^{ 1.5 }$$
J. $$\gamma =k[A]^{ 2 }[B]^{ 2 }$$
Answer: G
One mole of an ideal monoatomic gas expends isothermally against constant external pressure of $1$ atm from initial volume of $1L$ to a state where its final pressure become equal to external pressure. If initial temperature of gas is $300\ K$ then total entropy change of system in the above process is:$ \left[ R=0.082\ L\ atmmol^{ -1 }K^{ -1 }=8.3Jmol^{ -1 }K^{ -1 } \right]$

Options:
A. $$\dfrac{1}{2} R\ell n\left( 246 \right)$$
B. $$\dfrac{1}{2} R\ell n\left( 24.6 \right)$$
C. $$ R\ell n\left( 24.6 \right) $$
D. $$R\ell n\left( 2.46 \right)$$
E. $$R\ell n\left( 246 \right)$$
F. $$0$$
G. $$\dfrac{3}{2} R\ell n\left( 2.46 \right)$$
H. $$\dfrac{1}{2} R\ell n\left( 2490 \right)$$
I. $$ \dfrac { 3 }{ 2 } R\ell n\left( 24.6 \right) $$
J. $$ R\ell n\left( 2490 \right) $$
Answer: C
One mole of a monatomic ideal gas initially at a pressure of 2.00 bar and a temperature of 273 K is taken to a final pressure of 4.00 bar by a reversible path defined by p/V = constant. Taking $C_V$ to be equal to $12.5 J mol^{-1} K^{-1}$, the value of $\triangle U/ w$ for this process is calculated to be

Options:
A. $$+ 2.0$$
B. $$- 1.0$$
C. $$- 2.0$$
D. $$- 4.0$$
E. $$+ 3.0$$
F. $$+ 1.5$$
G. $$- 3.0$$
H. $$- 2.5$$
I. $$- 1.5$$
Answer: G
Three identical balls of mass $m$ and radius $R$ are placed on frictionless horizontal $x - y$ plane. Ball $A$ at $(0, 0)$, Ball $B$ at $(\sqrt {2} R, -\sqrt {2}R)$. Ball $A$ is suddenly given an impulse $ec {P} = \sqrt {2} mV\hat {i}$. If collision between balls $A$ and $B$ is perfectly elastic while between $B$ and $C$ is head on and perfectly inelastic, then

Options:
A. The $y$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {5v}{4\sqrt {2}}$
B. The $y$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {7v}{4\sqrt {2}}$
C. The $x$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {3v}{2\sqrt {2}}$.
D. The $y$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {3v}{2\sqrt {2}}$.
E. The $y$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {v}{2\sqrt {2}}$.
F. The $x$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {v}{2\sqrt {2}}$.
G. The $y$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {v}{\sqrt {2}}$
H. The $y$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {5v}{2\sqrt {2}}$.
I. The $y$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {3v}{4\sqrt {2}}$
J. The $y$ component of relative velocity of ball $A$ with respect to ball $C$ after a long time is $\dfrac {v}{4\sqrt {2}}$
Answer: D
A chemical reaction proceeds into the following stepsstepI,  $2A X$ faststep II,  $X + B Y$  slowstep III, $Y + B$ Product fastThe rate law for the overall reaction is ?

Options:
A. Rate = $k[A][B]^4$
B. Rate = $k[A]^2[B]^2$
C. Rate = $k[A]^2$
D. Rate = $k[A][B]^2$
E. Rate = $k[B]^2$
F. Rate = $k[A][B]$
G. Rate = $k[A]^2[B]$
H. Rate = $k[A][B]^3$
Answer: E
A particle which can move along x-axis is released from rest at the position $x = x_0.$ The potential energy (U) of the block is described below :
$$U=\begin{Bmatrix} -ax &;x<0 \    bx^2 &;x\geq0  \end{Bmatrix}$$
Which of the following statements is/are correct :

Options:
A. The magnitude of acceleration is a continuous function of time
B. The subsequent motion is simple harmonic and periodic
C. The subsequent motion is simple harmonic
D. The subsequent motion is periodic and simple harmonic
E. The subsequent motion is simple harmonic but not periodic
F. The speed is a continuous function of time
G. The subsequent motion is simple harmonic and not periodic
H. The magnitude of acceleration is a continuous function of position
Answer: F
In a hypothetical reaction, $A(aq) \rightleftharpoons 2B(q) + C(aq) (1^{st}$ order decomposition)$'A'$ is optically active (dextro-rototory) while $'B'$ and $'C'$ are optically inactive but $'B'$ takes part in a titration reaction (fast reaction) with $H_{2}O_{2}$. Hence, the progress of reaction can be monitored by measuring rotation of plane polarised light or by measuring volume of $H_{2}O_{2}$ consumed in titration.In an experiment the optical rotation was found to be $\theta = 40^{\circ}$ at $t = 20\ min$ and $\theta = 10^{\circ}$ at $t = 50\ min$. from start of the reaction. If the progress would have been monitored by titration method, volume of $H_{2}O_{2}$ consumed at $t = 15\ min$. (from start) is $40\ ml$ then volume of $H_{2}O_{2}$ consumed at $t = 60$ min will be:

Options:
A. $$90\ ml$$
B. $$60\ ml$$
C. $$80\ ml$$
D. $$70\ ml$$
E. $$52.5\ ml$$
F. $$57.5\ ml$$
G. $$65\ ml$$
H. $$75\ ml$$
I. $$55\ ml$$
J. $$62.5\ ml$$
Answer: B
A force $F=-k(x \hat i+y \hat j)$ here k is constant acts on a particle moving is the x-y plane.Starting from the origin,the particle is taken to (a,a) and then to $(a/ \sqrt{2},0)$.The total work done by the force F on the particle is

Options:
A. $$5ka^2/4$$
B. $$-5ka^2/4$$
C. $$2ka^2$$
D. $$-(ka^2/4)$$
E. $$-2ka^2$$
F. $$-3ka^2/8$$
G. $$-3ka^2/4$$
H. $$3ka^2/4$$
I. $$5ka^2/8$$
Answer: H
For a very dilute solution of a non-volatile solution in  a volatile liquid, the variation of a V.P of Solution (P) with $\frac { { M }_{ 1 } }{ { M }_{ 2 } } $ can be shown as below where $M_1$ & $M_2$ are m.wts, of solvent and solute respectively. Which of the following is true of given wights of solvent ($W_1$) and solute ($W_2$)

Options:
A. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-P^{ 0 }\left(\frac { { w }_{ 2 } }{ { w }_{ 1 } + w_2 } \right)$$
B. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-P^{ 0 }\left(\frac { { w }_{ 1 } }{ { w }_{ 2 } + w_1 } \right) $$
C. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-P^{ 0 }\left(\frac { { w }_{ 1 } }{ { w }_{ 2 } - w_1 } \right)$$
D. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-P^{ 0 }\left(\frac { { w }_{ 2 } }{ { w }_{ 1 } - w_2 } \right) $$
E. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-{ P }^{ 0 }(\frac { { w }_{ 2 } }{ { w }_{ 1 } } )$$
F. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-\frac { { w }_{ 2 } }{ { w }_{ 1 }{ P }^{ 0 } } $$
G. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-\frac { { w }_{ 1 } }{ { w }_{ 2 }{ P }^{ 0 } } $$
H. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-{ P }^{ 0 }(\frac { { w }_{ 1 } }{ { w }_{ 2 } } )$$
I. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-{ P }^{ 0 }\left(\frac { { w }_{ 1 } }{ { w }_{ 2 } + w_1 } \right)$$
J. $$\frac { dP }{ d({ { M }_{ 1 } }/{ { M }_{ 2 }) } } =-{ P }^{ 0 }\left(\frac { { w }_{ 1 } + w_2 }{ { w }_{ 2 } } \right)$$
Answer: H
An electrical device draws 2kW power from ac mains voltage 223 V(rms). The current differs lags in phase by $\phi = tan^{-1} \lgroup -\frac{3}{4} \rgroup$ as compared to voltage. The resistance R in the circuit is

Options:
A. $$24 \Omega$$
B. $$30 \Omega$$
C. $$21 \Omega$$
D. $$15 \Omega$$
E. $$23 \Omega$$
F. $$20 \Omega$$
G. $$19 \Omega$$
H. $$22 \Omega$$
I. $$18 \Omega$$
J. $$25 \Omega$$
Answer: F
The heat capacity of one more an ideal is found to be $C_V=3R(1+aRT)/2$ where 'a' is constant. The equation obeyed by this gas during a reversible adiabatic expansion is?

Options:
A. $$TV^{3/2}e^{aRT/2}=constant$$
B. $$TV^{3/2}e^{3aRT/2}=constant$$
C. $$TV^{3/2}e^{2aRT}=constant$$
D. $$TV^{3/2}e^{2aRT/3}=constant$$
E. $$TV^{3/2}=constant$$
F. $$TV^{3/2}e^{4aRT/3}=constant$$
G. $$TV^{3/2}e^{aRT}=constant$$
H. $$TV^{3/2}e^{aRT/4}=constant$$
I. $$TV^{3/2}e^{aRT/3}=constant$$
J. $$TV^{3/2}e^{5aRT/3}=constant$$
Answer: G
Calculate the heat of formation of KOH from the following data in K.Cal.
$K_{(g)}+H_2O+aq\rightarrow KOH_{(aq)}+\dfrac{1}{2}H_2$;
$\Delta H=-48.4$K.Cal
$H_{2(g)}+\dfrac{1}{2}O_{2(g)}\rightarrow H_2O_{(l)}; \Delta H=-68.44$ K.Cal
$KOH_{(s)}+aq\rightarrow KOH_{(aq)}; \Delta H=-14.0$K.Cal

Options:
A. $$+108.83$$
B. $$+130.85$$
C. $$-108.83$$
D. $$+102.43$$
E. $$-130.45$$
F. $$-102.84$$
G. $$-102.43$$
H. $$+102.83$$
I. $$-130.85$$
J. $$-102.83$$
Answer: H
A spool of inner radius $R$ and outer radius $3R$ has a moment of inertia $ = MR^2 $ about an axis passing through its geometric centre, where $M$ is the mass of the spool. A thread wound on the inner surface of the spool is pulled horizontally with a constant force $= Mg$. Find the acceleration of the point on the thread which is being pulled assuming that the spool rolls purely on the floor.

Options:
A. $$18 m/s^2$$
B. $$ 16  m/s^2 $$
C. $$12  m/s^2$$
D. $$14 m/s^2$$
E. $$24 m/s^2$$
F. $$22  m/s^2$$
G. $$ 8.4  m/s^2 $$
H. $$ 40  m/s^2 $$
I. $$ 3.33  m/s^2 $$
J. $$20  m/s^2$$
Answer: B
Two bubbles collapse to form a single bubble. If $V$ is the subsequent change in the volume contained air and $S$ the change in total surface area. $T$ is the surface tension and $P$ atmosphere pressure. Which of the following relations is correct?(assume temperature is constant)

Options:
A. $$3PV+2ST=0$$
B. $$2PV+3ST=0$$
C. $$4PV+3ST=0$$
D. $$5PV + 3ST = 0$$
E. $$5PV + 2ST = 0$$
F. $$2PV + 4ST = 0$$
G. $$3PV+4ST=0$$
H. $$3PV+5ST=0$$
I. $$4PV+2ST=0$$
Answer: G
A solid sphere, a cone and a cylinder are floating in water. All have same mass, density and radius. Let $ f_{1},f_{2}$ and $f_{3} $ are the fraction of their volumes inside the water and $h_{1}, h_{2} $ and $h _{3}$ depths inside water Then

Options:
A. $$h_{2} < h_{1}$$
B. $$h_{3}< h_{2}$$
C. $$h_{3} > h_{1}$$
D. $$h_{3} < h_{1}$$
E. $$h_{3} = h_{2}$$
F. $$f_{3} > f_{2} >f_{1}$$
G. $$f_{1}=f_{2}=f_{3}$$
H. $$h_{3} > h_{2}$$
I. $$h_{2} < h_{3}$$
Answer: B
The half life decomposition of ${N_2}{O_5}$ in a first order reaction represented by 
${N_2}{O_5} \to {N_2}{O_4} + \frac{1}{2}{O_2}$ 
After 20 min the volume of ${O_2}$ produced is 10 ml and at the end of the reaction 40 ml. The value of the rate constant is $\left[ {Given\,\,2.303\,\,\log \,\left( {\frac{4}{3}} \right) = 0.28} \right]$

Options:
A. $$0.0145{\min ^{ - 1}}$$
B. $$0.0014{\min ^{ - 1}}$$
C. $$0.14{\min ^{ - 1}}$$
D. $$14{\min ^{ - 1}}$$
E. $$0.145{\min ^{ - 1}}$$
F. $$0.00014{\min ^{ - 1}}$$
G. $$1.4{\min ^{ - 1}}$$
H. $$0.014{\min ^{ - 1}}$$
Answer: D
The values of x and y in the following redox reaction 
$x Cl_2 + 6OH^- \rightarrow ClO_3^- + yCl^- + 3H_2O $ are

Options:
A. x = 4, y = 2
B. x = 4, y = 3
C. x = 5, y = 2
D. x = 5, y = 6
E. x = 5, y = 1
F. x = 2, y = 4
G. x =3 , y = 5
H. x = 5, y = 3
I. x = 5, y = 4
J. x = 4, y = 5
Answer: H
A small ball of mass $m$ is released from rest at a height $h_1$, above ground at time $t= 0$. At time $t = t _ { 0 }$ the ball again comes to rest at a height $h_2$. Consider the ground to be perfectly rigid  and neglect friction, In the time interval from $t = 0 \text { to } t = t _ { 0 }$. Pick up the Incorrect statement

Options:
A. Work done by ground on ball for duration of contact is $m g \left( h _ { 2 } - h _ { 1 } \right)$
B. Average acceleration of the ball is zero
C. Net work done on the ball by air forces except gravity is $\operatorname { mg } \left( h _ { 2 } - h _ { 1 } \right)$
D. Work done by gravity on ball is $m g \left( h _ { 1 } - h _ { 2 } \right)$
Answer: B
A block is suspended by n ideal spring constant K. If the block is pulled down by constant force F and if maximum displacement of block from it's equilibrium position is z. then

Options:
A. $$ z = \dfrac{ F }{ 2K } $$
B. work done by force F is equal to 2Fz
C. $$z = \dfrac{ F }{ 3K }$$
D. $$z=\dfrac{ 5F }{ 3K }$$
E. $$z = \dfrac{ 4F }{ 3K }$$
F. $$z=\dfrac { F }{ K } $$
G. $$z=\dfrac{ 3F }{ K }$$
H. $$z=\dfrac { 2F }{ K } $$
I. increase in potential energy of the spring is $\dfrac { 1 }{ 2 } K{ z }^{ 2 }$
J. $$z = \dfrac{ 3F }{ 2K }$$
Answer: F
A particle initially at rest starts moving from point A on the surface of a fixed smooth hemisphere of radius r as shown. The particle looses its contact with hemisphere at point B. C is centre of the hemisphere. The equation relating $\alpha$ and $\beta$ is:

Options:
A. $$3sin\alpha = 2sin\beta$$
B. $$2sin\beta = 3cos\alpha$$
C. $$2sin\gamma = 3cos\alpha$$
D. $$3sin\beta = 2cos\alpha$$
E. $$3sin\alpha = 2cos\beta$$
F. $$2sin\alpha = 3cos\beta$$
G. $$3sin\gamma = 2cos\delta$$
Answer: D
A uniform circular wire loop is connected to the terminals of a battery. The magnetic field induction at the centre due to ABC portion of the wire will be (length of ABC = ${ l }_{ 1 }$, length of ADC = ${ l }_{ 2 }$)

Options:
A. $$\dfrac { { \mu }_{ 0 } { il }_{ 1 } {l}_{2}}{ 8\pi R^{ 2 }(l_{ 1 }+{ l }_{ 2 }{ ) } }$$
B. $$\dfrac { { \mu }_{ 0 } { il }_{ 1 } {l}_{2}}{ 5\pi R^{ 2 }(l_{ 1 }+{ l }_{ 2 }{ ) } }$$
C. $$\dfrac { { \mu }_{ 0 }{ il }_{ 1 }{ l }_{ 2 } }{ 2R(l_{ 1 }+{ l }_{ 2 }{ ) }^{ 2 } } $$
D. $$\dfrac { { \mu }_{ 0 } { il }_{ 1 } {l}_{2}}{ 4\pi R^{ 2 }(l_{ 1 }+{ l }_{ 2 }{ ) } } $$
E. $$\dfrac { { \mu }_{ 0 } { il }_{ 1 } {l}_{2}}{ 3\pi R^{ 2 }(l_{ 1 }+{ l }_{ 2 }{ ) } }$$
F. $$\dfrac { { \mu }_{ 0 } { il }_{ 1 } {l}_{2}}{ 7\pi R^{ 2 }(l_{ 1 }+{ l }_{ 2 }{ ) } }$$
G. $$Zero$$
H. $$\dfrac { { \mu }_{ 0 } { il }_{ 1 } {l}_{2}}{ 6\pi R^{ 2 }(l_{ 1 }+{ l }_{ 2 }{ ) } }$$
I. $$\dfrac { { \mu }_{ 0 } i(l_{ 1 }+{ l }_{ 2 }{ ) }\quad }{ 2\pi R\quad l_{ 1 }{ l }_{ 2 } } $$
J. $$\dfrac { { \mu }_{ 0 } { il }_{ 1 } {l}_{2}}{ 2\pi R^{ 2 }(l_{ 1 }+{ l }_{ 2 }{ ) } }$$
Answer: D
An electron flying in vacuum into a uniform magnetic field of $  B = 128\pi  \times 10^-4 T $ at a right angle to the magnetic field moves in a circle of radius $  R = 2cm $. The potential difference which accelerated the electron before it enters  the magnetic field is V, Find V. Take $ m = 9.1  \times  10^31 kg $ & $ e = 1.6  \times 10^{-19 }C $  :-

Options:
A. $$ 78 kV $$
B. $$ 14 kV $$
C. $$14.3 kV$$
D. $$14.1 kV$$
E. $$ 57.6 kV $$
F. $$14.6 kV$$
G. $$ 25kV $$
H. $$14.5 kV$$
I. $$14.2 kV$$
J. $$14.4 kV$$
Answer: I
Which will be the least stable resonating structure:

Options:
A. $$CH_2=CH - \overset{\delta Y}{C}H - \overset{\circleddash}{C}H - O - CH_3$$
B. $$CH_2=CH - \overset{\circleddash}{C}H - \overset{\ominus}{C}H - O - CH_3$$
C. $$CH_2=CH - \overset{\circleddash}{C}H - \overset{\oplus}{C}H - O - CH_3$$
D. $$CH_2=CH - \overset{\ominus}{C}H - CH = \overset{\circleddash}{O} - CH_3$$
E. $$CH_2=CH - \overset{\ominus}{C}H - \overset{\circleddash}{C}H - O - CH_3$$
F. $$\overset{\circleddash}{C}H_2- \overset{\oplus}{C}H - CH =CH - OCH_3$$
G. $$CH_2=CH - \overset{\circleddash}{C}H - CH = \overset{\oplus}{O} - CH_3$$
H. $$CH_2=CH - \overset{\oplus}{C}H - \overset{\circleddash}{C}H - O - CH_3$$
I. $$\overset{\circleddash}{C}H_2-CH = CH - CH = \overset{\oplus}{O}-CH_3$$
Answer: H
Two opposite forces $F_1 = 120N \:and \:F_2 = 80N$ act on an heavy elastic plank of modulus of elasticity $y=2\times 10^{11}N/m^2$ and length $L = 1m$ placed over a smooth horizontal surface. The cross-sectional area of plank is $A = 0.5m^2$. If the change in the length of plank is (in nm )

Options:
A. 5
B. 0.1
C. 1.2
D. 0.8
E. 0.5
F. 2
G. 0.3
H. 0.4
I. 0.2
J. 1
Answer: J
In the complete combustion of butanol ($C_{4}H_{9}OH$), if $\Delta H$ is the enthalpy of combustion and $\Delta E$ is the heat of combustion at constant volume, then:

Options:
A. \Delta H = \Delta E
B. \Delta H = \Delta E + 3R T
C. $\Delta H,\Delta E$ relation can't be predicted
D. $$\Delta H>\Delta E$$
E. \Delta H = \Delta E + 2R T
F. \Delta H = \Delta E - R T
G. $$\Delta H<\Delta E$$
H. \Delta H = \Delta E + R T
Answer: D
A buffer solution of $pH=9.00$ is made by dissolving ammonium chloride and ammonia in water. How many moles of ammonia chloride must be added to $1.0L$ of $0.25 M$  ammonia to prepare this buffer ?
$N{H_3}(aq) + {H_2}O\leftrightharpoons N{H^ - }_4(aq) + O{H^ - }(aq);$ ${K_b} = 1.8*{10^{ - 5}}$

Options:
A. 0.55 mol
B. $2.2$ mol
C. $0.25$ mol
D. 0.30 mol
E. 0.35 mol
F. 0.50 mol
G. $0.45$ mol
H. $0.65$ mol
I. 0.15 mol
J. 0.40 mol
Answer: C
A 60 mm radius wheel is connected to a fixed support D by two links AB and BD. At the instant shown, the velocity of the centre A of the wheel is 30 m/s to the left. Rod AB and BD have equal length $\ell $=5 m.
The angular velocity of rod AB will be

Options:
A. $6\sqrt { 2 } $ rad/sec
B. $7\sqrt { 2 } $rad/sec
C. $5\sqrt { 2 } $rad/sec
D. 4$\sqrt { 2 } $ rad/sec
E. $2\sqrt { 2 } $ rad/sec
F. $8\sqrt{2}$ rad/sec
G. None of these
H. $3\sqrt { 2 } $ rad/sec
Answer: H
If a $g$ is the mass of $NaHC_{2}O_{4}$ required to neutralize $100\ mL$ of $0.2\ M\ NaOH$ and $b\ g$ that required to reduce $100\ mL$ of $0.02\ M\ KMnO_{4}$ in acidic medium, then

Options:
A. $$2a = b$$
B. $$a = 6b$$
C. $$a = 4b$$
D. $$a = 7b$$
E. $$a = 3b$$
F. None of these
G. $$a = 5b$$
H. $$a = 2b$$
I. $$a = b$$
Answer: H
How many m moles of sucrose should be dissolved in 500 grams of water so as to get a solution which has a difference of $ 103.57^0 C $ between boiling point and freezing point ? 
$$ (K_f = 1.86 K kg mol^{-2}, K_b = 0.52 K Kg mol^{-1} ) $$

Options:
A. 900 m moles
B. 850 m moles
C. 800 m moles
D. 650 m moles
E. 750 m moles
F. 825 m moles
G. 700 m moles
H. 500 m moles
I. 775 m moles
Answer: E
Mixture $X=0.02$ mol of $[Co(NH_3)_5SO_4]Br$ and 0.02 mol of $[Co(NH_3)_5Br]SO_4$ was prepared in 2L of solution.1L of mixture X + excess $AgNO_3\rightarrow Y$1L of mixture X + excess $BaCl_2\rightarrow Z$Number of moles of Y and Z are :

Options:
A. $$0.02, 0.04$$
B. $$0.01, 0.04$$
C. $$0.03, 0.02$$
D. $$0.02, 0.03$$
E. $$0.02, 0.01$$
F. $$0.01, 0.01$$
G. $$0.01, 0.02$$
H. $$0.01, 0.03$$
I. $$0.02, 0.02$$
Answer: F
A wheel of radius $1\ m$ is spinning with angular velocity which varies with line ad $5t^{2}\ rad/s.$  The acceleration of the point on the wheel's rim at $t=1\ s$, has the magnitude equal to

Options:
A. $$30\ m/s^{2}$$
B. $$29\sqrt{5}\ m/s^{2}$$
C. $$25\ m/s^{2}$$
D. $$5 \sqrt{29}\ m/s^{2}$$
E. $$20\ m/s^{2}$$
F. $$15\ m/s^{2}$$
G. $$15 \sqrt{5}\ m/s^{2}$$
H. $$10\ m/s^{2}$$
I. $$10 \sqrt{5}\ m/s^{2}$$
J. $$5 \sqrt{5}\ m/s^{2}$$
Answer: C
Two block of masses of $1\ kg$ and $2\ kg$ are connected by a metal wire going over a smooth pulley. The breaking stress of metal is $\dfrac {40}{3\pi}\times 10^{6}N/m^{2}$. What should be the minimum radius of wire used if it should not break?
$$(g = 10\ m/s^{2})$$

Options:
A. $$1.6\ mm$$
B. $$1.65 mm$$
C. $$1.75 mm$$
D. $$0.5\ mm$$
E. $$1.5\ mm$$
F. $$2\ mm$$
G. $$1.2\ mm$$
H. $$1.4\ mm$$
I. $$1\ mm$$
J. $$1.3\ mm$$
Answer: I
A simplified flute shown in the figure. It is open at end $D$.
There is also a large opening at $A$ (near the mouth piece) and there are two holes at $B$ and $C$ [$AB=BD$ and $BC=CD$]. The distance $AD=34\ cm$. The speed of sound is $340\ m/s$. What frequency do you expect to hear when you blow keeping hole $C$ closed?

Options:
A. $$1500\ Hz$$
B. $$500\ Hz$$
C. $$1250\ Hz$$
D. $$750\ Hz$$
E. $$1000\ Hz$$
F. $$875\ Hz$$
G. $$950\ Hz$$
H. $$1125\ Hz$$
I. $$250\ Hz$$
J. $$625\ Hz$$
Answer: E
The time period of a particle in simple harmonic motion is $T$. Assume potential energy at mean position to be zero. After a time of $T/6$ it passes its mean position, its

Options:
A. velocity will be half its maximum velocity
B. displacement will be nearly $71\%$ of its amplitude
C. acceleration will be nearly $86\%$ of its maximum acceleration
D. displacement will be nearly $58\%$ of its amplitude
E. displacement will be nearly 86% of its amplitude
F. acceleration will be half its maximum acceleration
G. displacement will be half its amplitude
H. $$KE=PE$$
I. acceleration will be nearly 71% of its maximum acceleration
J. displacement will be nearly 63\% of its amplitude
Answer: A
A block of mass 1 kg rests inside a cubical vessel, as shown in the figure which is moving with a velocity of v = $\sqrt { 69 } t$ in horizontal direction , where t is in second. All the surface are smooth. The block is at rest w.r.t the cube, the magnitude of net force exerted by cube on the block is (g = 10 m/${ s }^{ 2 }$)

Options:
A. $\sqrt { 59 } $ N
B. $\sqrt { 78 } N$
C. $\sqrt { 79 } $ N
D. $\sqrt{69} N$
E. $\sqrt { 89 } $ N
F. 13 N
G. None of these
H. $\sqrt { 99 } N$
Answer: F
An ideal gas (1 mol, monatomic) is in the initial state $ P $ (see Fig. 2.111 ) on an isothermal $ A $ at temperature $ T_{0} . $ It is brought under a constant volume $ \left(2 V_{0}\right) $ process to $ Q $ which lies on an adiabatic $ B $ intersecting the isothermal $ A $ at $ \left(P_{0}, V_{0}, T_{0}\right) . $ The change in the internal energy of the gas during the process is (in terms of $ T_{0} $ ) $ \left(2^{2 / 3}=\right.1.587)$

Options:
A. $$-1.1935 T_{0}$$
B. $$-2.3 T_{0}$$
C. $$2.3 T_{0} $$
D. $$-4.6 T_{0}$$
E. $$-1.587 T_{0}$$
F. $$4.6 T_{0}$$
G. $$-3.174 T_{0}$$
H. $$-3.561 T_{0}$$
Answer: D
Two plates $A$ and $B$ of equal surface area are placed one on top of the other to form a composite plate of the same surface area. The thickness of $A$ and $B$ are $4.0$cm and $6.0$cm respectively. The temperature of the exposed surface of plate $A$ is $-10$$^{o}$C and that of the exposed surface of plate B is $10$$^{0}$C. Neglect heat loss from the edges of the composite plate, the temperature of the contact surface is $T_{1}$ if the plates A and B are made of the same material and $T_{2}$ if their thermal conductivities are in the ratio $2:3$ then

Options:
A. $$T_{2}= -3^{0}C$$
B. $$T_{2}= -2^{0}C$$
C. $$T_{2}= -6^{0}C$$
D. $$T_{2}= -1^{0}C$$
E. $$T_{2}= -5^{0}C$$
F. $$T_{2}= 1^{0}C$$
G. $$T_{1}= -2^{0}C$$
H. $$T_{1}= -4^{0}C$$
I. $$T_{2}= -4^{0}C$$
J. $$T_{2}= 0^{0}C$$
Answer: J
For a gaseous homogeneous reaction,
2A (g) + B (g) $ \rightleftharpoons  $ 2C(g) +2D (g)  
$ \triangle {  G}^{  0} $ =0.693 RT at TK. $ { K }_{p  }^{  0} $ for the reaction :
(Given : $ \ell n  $ 2= 0.693)

Options:
A. 2.5
B. 50
C. 20
D. 5
E. 25
F. 500
G. 0.25
H. 0.05
I. 2.0
J. 0.5
Answer: B
If rotation of the earth stopped suddenly then find the rise in temperature of the earth. Assume total rational energy is converted into thermal energy (specific heat of earth $= 0.15$, radius of earth $6400\ km$)

Options:
A. $$78.76^\circ C$$
B. $$76.78^\circ C$$
C. $$6.876^\circ C$$
D. $$77.76^\circ C$$
E. $$79.76^\circ C$$
F. $$0.6876^\circ C$$
G. $$77.86^\circ C$$
H. $$16.876^\circ C$$
I. $$86.76^\circ C$$
J. $$68.76^\circ C$$
Answer: A
The torque of a force $F$ was defined. If $F = 3 \hat {i} + 5 \hat {j} -  2 \hat {k} N,$ acting at $7 \hat {i} - 2 \hat {j} +  5 \hat {k}$ what are the torques about :

Options:
A. none of these
B. the point ($0, 1, -1$)
C. the origin
D. both a and b is correct
E. the point ($1, -1, 0$)
F. the point ($1, -1, 1$)
G. the point ($1, 0, 0$)
H. the point ($0, 1, 0$)
I. the point ($1, 0, -1$)
J. the point ($0, -1, 0$)
Answer: J
Two sources of sound $A$ and $B$ produces the wave of $350 HZ$, they vibrate in the same phase. The particle P is vibrating under the influence of these two waves, if the amplitudes at the point $P$ produced by the two waves is $0.3 mm $ and $0.4 mm$, then the resultant amplitude of the point $P$ will be when $AP - BP = 25 mm$ and the velocity of sound is $350 m/sec$

Options:
A. $$0.8 mm$$
B. $$0.6 mm$$
C. $$0.4 mm$$
D. $$0.3 mm$$
E. $$0.7 mm$$
F. $$0.9 mm$$
G. $$0.5 mm$$
H. $$0.2 mm$$
I. $$0.1 mm$$
Answer: G
A particle with charge $-40.0 \,nC$ is on the x axis at the point with coordinate $x = 0.A$ second particle, with charge $-20.0 \,nC$, is on the x axis at $x = 0.500 \,m$
Is the point at a finite distance where the electric field is zero

Options:
A. Between $x=0.500 \,m$ and $x=0.750 \,m$
B. Between $x=0$ and $x=0.250 \,m$
C. to the right of $x=0.500 \,m?$
D. to the left of $x = 0$
E. Between $x=0.250 \,m$ and $x=0.750 \,m$
F. to the right of $x=0.500 \,m$
G. to the left of $x = 0$,
H. Between $x=0.250 \,m$ and $x=0.500 \,m$
I. at $x=0.5m$
J. Between $x=0$ and $x=0.500 \,m$
Answer: J
Two vertical walls separated by a distance of 2 meters wall A is smooth while wall B is rough with a coefficient of friction $\mu$ = 0.5. A uniform rod is propped between them the length of the longest rd that can be probed between the walls is equal to:

Options:
A. $$\frac{\sqrt{16}}{2} m$$
B. $$\frac{\sqrt{13}}{2} m$$
C. $$\frac{\sqrt{18}}{2} m$$
D. $$\frac{\sqrt{19}}{2} m$$
E. 2 m
F. $$2 \sqrt{2} m$$
G. $$\frac{\sqrt{15}}{2} m$$
H. $$\sqrt{5} m$$
I. $$\frac{\sqrt{21}}{2} m$$
J. $$\frac{\sqrt{17}}{2} m$$
Answer: J
Given below are the half cell reactions:$\eqalign{  & {\text{M}}{{\text{n}}^{2 + }} + 2\,{e^ - } \to {\text{Mn,}}\,\,\,\,\,\,\,\,{{\text{E}}^0} =  - 1.18\,{\text{V}}  \cr   & {\text{2(M}}{{\text{n}}^{3 + }} + {e^ - } \to {\text{M}}{{\text{n}}^{2 + }}),{{\text{E}}^0} =  + 1.51\,{\text{V}} \cr} $
$$The\,{{\text{E}}^0}\,for\,3\,{\text{M}}{{\text{n}}^{2 + }} \to {\text{Mn + 2M}}{{\text{n}}^{3 + }}will\,be$$

Options:
A. -0.35 V, the reaction will not occur
B. -0.33 V, the reaction will not occur
C. -0.32 V, the reaction will not occur
D. -2.69 V, the reaction will no occur
E. -0.33 V, the reaction will not occur.
F. -0.33 V, the reaction will occur
G. -2.69 V, the reaction will Occur
H. -0.34 V, the reaction will occur
I. -0.34 V, the reaction will not occur
J. -0.32 V, the reaction will occur
Answer: F
The charge supplied by source varies with time t as $Q=at-bt^2$. The total heat produced in resistor $2R$ is?

Options:
A. $$\dfrac{a^3R}{54b}$$
B. $$\dfrac{a^3R}{12b}$$
C. $$\dfrac{a^3R}{24b}$$
D. $$\dfrac{a^2R}{6b}$$
E. $$\dfrac{a^3R}{9b}$$
F. $$\dfrac{a^3R}{27b}$$
G. None of these
H. $$\dfrac{a^3R}{15b}$$
I. $$\dfrac{a^3R}{18b}$$
J. $$\dfrac{a^3R}{3b}$$
Answer: F
A rope of length L and mass M is being pulled on a rough horizontal floor by a constant horizontal force $F=Mg$. The force is acting at one end of the rope in the same direction as the length of the rope. The coefficient of kinetic friction between rope and floor is $1/2$. Then, the tension at the midpoint of the rope is

Options:
A. $$\dfrac{Mg}{3}$$
B. $$\dfrac{Mg}{8}$$
C. $$\dfrac{3Mg}{4}$$
D. $$\dfrac{Mg}{5}$$
E. $$\dfrac{Mg}{2}$$
F. $$\dfrac{2Mg}{5}$$
G. $$\dfrac{Mg}{6}$$
H. $$\dfrac{Mg}{4}$$
I. $$\dfrac{3Mg}{8}$$
Answer: E
The indicator phenol red is half in the ionic form when $pH$ is $7.2$. If the ration of the undissociated from to the ionic form is $1:5$, find the $pH$ of the solution. With the same $pH$ for solution, if indicator is altered such that the ratio of undissociated form to dissociated form becomes $1:4$, find the $pH$ when $50$% of the new indicator is in ionic form.

Options:
A. $$pH=7.3;7.9$$
B. $$pH=7.1;7.7$$
C. $$pH=8.3;6.9$$
D. $$pH=7.9;6.9$$
E. $$pH=7.3;8.3$$
F. $$pH=7.9;8.3$$
G. $$pH=7.3;6.9$$
H. $$pH=7.9;7.3$$
I. $$pH=6.9;8.3$$
Answer: H
Two rods of different materials and identical cross sectional area, are joined face to face at one end and their free ends are fixed to the rigid walls. If the temperature of the surroundings is increased by ${ 30 }^{ o }C$, the magnitude of the displacement of the joint of the rod is (length of rods ${l}_{1}={l}_{2}=1$ unit, ratio of their young's moduli ${ Y }_{ 1 }/{ Y }_{ 2 }=2$, coefficient of linear expansion are ${ \alpha  }_{ 1 }$ and ${ \alpha  }_{ 2 }$)

Options:
A. $$10\left( { \alpha }_{ 2 }-{ 2\alpha }_{ 1 } \right) $$
B. $$5\left( \alpha_{2} - 2\alpha_{1} \right)$$
C. $$5\left( 2\alpha_{2} - \alpha_{1} \right)$$
D. $$10\left( { \alpha }_{ 1 }-{ \alpha }_{ 21 } \right) $$
E. $$10\left( 2\alpha_{2} - \alpha_{1} \right)$$
F. $$5\left( { \alpha }_{ 2 }-{ \alpha }_{ 1 } \right) $$
G. $$10\left( \alpha_{1} - \alpha_{2} \right)$$
H. $$5\left( { 2\alpha }_{ 1 }-{ \alpha }_{ 2 } \right) $$
I. $$5\left( \alpha_{1} - 2\alpha_{2} \right)$$
J. $$10\left( 2\alpha_{1} - \alpha_{2} \right)$$
Answer: A
The value of ${ log }_{ 10 }$K for a reaction $A\leftrightharpoons B$ is ::
(Given-${ \Delta  }_{ r }{ H }^{ \circ  }_{ 298\quad K }=-54.07KJ\quad mol^{ -1 }$, ${ \Delta  }_{ r }{ S }^{ \circ  }_{ 298\quad K }=10KJ^{ -1 }\quad mol^{ -1 }$ and $R=8.314JK^{ -1 }\quad mol^{ -1 }$; 2303*8314*298=5705)

Options:
A. 5
B. 7
C. 12
D. 105
E. 10
F. 15
G. 150
H. 95
I. 9
J. 100
Answer: A
Three forces $\vec{F}_{1} = (2 \hat{i} + 4 \hat{j}) \, \text{N}$, $\vec{F}_{2} = (2 \hat{j} - \hat{k}) \, \text{N}$, and $\vec{F}_{3} = (\hat{k} - 4\hat{i} - 2 \hat{j}) \, \text{N}$ are applied on an object of mass $1 \, \text{kg}$ at rest at the origin. The position of the object at $t = 2 \, \text{s}$ will be

Options:
A. $$8 m$$
B. $$1 m$$
C. $$3 m$$
D. $$4 m$$
E. $$6 m$$
F. $$7 m$$
G. $$9 m$$
H. $$5 m$$
Answer: D
$1\ litre$ of an ideal gas $(\gamma = 1.5)$ at $300\ K$ is suddenly compressed to half its original volume.
The gas is now expanded isothermally to achieve its original volume of $1\ litre$. Calculate the work done by the gas.

Options:
A. $$104\ J$$
B. $$96\ J$$
C. $$88\ J$$
D. $$100\ J$$
E. $$92\ J$$
F. $$106\ J$$
G. $$98\ J$$
H. $$115\ J$$
I. $$102\ J$$
Answer: G
The correct statement about the magnetic properties of $[Fe(CN)_6]^{3-}$ and $[FeF_6]^{3-}$ is: $(Z=26)$.

Options:
A. Both are paramagnetic
B. Both are ferromagnetic
C. $[Fe(CN)_6]^{3-}$ is diamagnetic, $[FeF_6]^{3-}$ is paramagnetic
D. $[Fe(CN)_6]^{3-}$ is ferromagnetic, $[FeF_6]^{3-}$ is paramagnetic
E. Both complexes are paramagnetic
F. Both complexes are diamagnetic
G. $[Fe(CN)_6]^{3-}$ is ferromagnetic, $[FeF_6]^{3-}$ is diamagnetic
H. Both complexes are ferromagnetic
I. Both are diamagnetic
J. $[Fe(CN)_6]^{3-}$ is paramagnetic, $[FeF_6]^{3-}$ is diamagnetic
Answer: A
In the situation shown, the block can execute free oscllation (no damping) with angular frequency $ \omega _{1} $. In presence of week damping, it executes damped $ SHM $ with angular frequency $ \omega _{2} $. When it is subjected to a sinusoidal force, it executes forced oscillation with maximum amplitude at angular frequency $ \omega _{3} $ (assume damping is present)

Options:
A. $$\omega _{1} < \omega _{2} < \omega _{3}$$
B. $$ \omega _{1} > \omega _{2} = \omega _{3} $$
C. $$\omega _{1} = \omega _{3} > \omega _{2}$$
D. $$\omega _{1} > \omega _{3} > \omega _{2}$$
E. $$\omega _{1} < \omega _{2} > \omega _{3}$$
F. $$ \omega _{1} > \omega _{2} > \omega _{3} $$
G. $$\omega _{1} = \omega _{2} > \omega _{3}$$
H. $$ \omega _{1} = \omega _{2} = \omega _{3} $$
I. $$\omega _{1} < \omega _{3} = \omega _{2}$$
Answer: F
A steel ball is dropped from a height of 1 m on to a horizontal non-conducting surface. Every time it bounces, it reaches 80% of its previous height. Nearly by how much will the temperature of the ball rise after 4 bounces? Specific heat capacity of the $ball \:(steel) =0.1 \:cal/g-^o\! \!C$. Neglect loss in heat to the surroundings and the floor.

Options:
A. $$0.028^{\circ}\! C$$
B. $$0.059^{\circ}\! C$$
C. $$0.018^{\circ}\! C$$
D. $$0.59^{\circ}\! C$$
E. $$1.4^{\circ}\! C$$
F. $$0.016^{\circ}\! C$$
G. $$0.042^{\circ}\! C$$
H. $$0.017^{\circ}\! C$$
I. $$0.014^{\circ}\! C$$
J. $$0.056^{\circ}\! C$$
Answer: I
The time taken by block-bullet system to move from $y=\dfrac{mg}{k}$ (initial equilibrium position) to $y=0$ (natural length of spring) is : [A represents the amplitudes of motion]

Options:
A. $$\sqrt{\dfrac{4m}{3k}}\left[\cos^{-1}\left(\dfrac{2mg}{3kA}\right)-\cos^{-1}\left(\dfrac{5mg}{3kA}\right)\right]$$
B. $$\sqrt{\dfrac{4m}{3k}}\left[\cos^{-1}\left(\dfrac{mg}{3kA}\right)-\cos^{-1}\left(\dfrac{3mg}{2kA}\right)\right]$$
C. $$\sqrt{\dfrac{4m}{3k}}\left[\cos^{-1}\left(\dfrac{mg}{4kA}\right)-\cos^{-1}\left(\dfrac{3mg}{3kA}\right)\right]$$
D. $$\sqrt{\dfrac{4m}{3k}}\left[\cos^{-1}\left(\dfrac{mg}{3kA}\right)-\cos^{-1}\left(\dfrac{4mg}{3kA}\right)\right]$$
E. $$\sqrt{\dfrac{4m}{3k}}\left[\cos^{-1}\left(\dfrac{mg}{3kA}\right)-\cos^{-1}\left(\dfrac{5mg}{3kA}\right)\right]$$
F. $$\sqrt{\dfrac{3k}{4m}}\left[\cos^{-1}\left(\dfrac{mg}{3kA}\right)-\cos^{-1}\left(\dfrac{4mg}{3kA}\right)\right]$$
G. $$\sqrt{\dfrac{4m}{6k}}\left[\sin^{-1}\left(\dfrac{4mg}{3kA}\right)-\sin^{-1}\left(\dfrac{mg}{3kA}\right)\right]$$
H. $$none\ of\ the\ above$$
I. $$\sqrt{\dfrac{4m}{3k}}\left[\cos^{-1}\left(\dfrac{mg}{2kA}\right)-\cos^{-1}\left(\dfrac{4mg}{3kA}\right)\right]$$
J. $$\sqrt{\dfrac{4m}{3k}}\left[\cos^{-1}\left(\dfrac{2mg}{3kA}\right)-\cos^{-1}\left(\dfrac{3mg}{2kA}\right)\right]$$
Answer: D
An open pipe is in resonance in its $2^nd$ harmonic with tuning fork of frequency $f_1$. Now it is closed at one end. If the frequency of the tuning fork is increased slowly from $f_1$ then again a resonance is obtained with a frequency $f_2$. If in this case, the pipe vibrates harmonics then

Options:
A. $$n = 5, f_2 = \dfrac{4}{3}f_1$$
B. $$n = 5, f_2 = \dfrac{4}{5}f_1$$
C. $$n = 5, f_2 = \dfrac{5}{3}f_1$$
D. $$n = 5, f_2 = \dfrac{5}{6}f_1$$
E. $$n = 5, f_2 = \dfrac{5}{4}f_1$$
F. $$n = 5, f_2 = \dfrac{6}{5}f_1$$
G. $$n = 3, f_2 = \dfrac{3}{4}f_1$$
H. $$n = 3, f_2 = \dfrac{4}{5}f_1$$
I. $$n = 3, f_2 = \dfrac{5}{4}f_1$$
J. $$n = 5, f_2 = \dfrac{3}{4}f_1$$
Answer: E
The equilibrium constant for some reactions are given below against each of the reaction :(i)$ 2N_2 +5O_2 \leftrightharpoons 2N_2O_5; K = 5 \times 10^{-27} $(ii) $N_2 +O_2 \leftrightharpoons 2NO; K= 2 \times 10^{-15} $(iii) $ N_2 +2O_2\leftrightharpoons 2NO_2 ; K = 1.5 \times 10^{-29} $Which of the following statements is correct?

Options:
A. The least stable oxide is $ NO_2 $
B. The least stable oxide is $ N_2O_5 $
C. The least stable oxide is NO
D. The most stable oxide is NO
E. The stability order is $ NO_2 > NO > N_2O_5 $
F. The most stable oxide is $ NO_2 $
G. The most stable oxide is $N_2O_5$
H. The stability order is $ N_2O_5 > NO > NO_2 $
I. The stability order is $ N_2O_5 > NO_2 > NO_2 $
J. The stability order is $ NO > N_2O_5 > NO_2 $
Answer: D
Statement -1:
Two longitudinal waves given by equation $y_{1}$(x,t) = 2a sin $(\omega - kx)$ and $y_{2}$(x,t) = a sin $(2\omega - 2kx)$ will have equal intensity.
Statement -2:
Intensity of waves of given frequency in the same medium is proportional to the square of amplitude only.

Options:
A. Statement -1 is false, statement -2 is true; statement -2 is not the correct explanation of statement -1
B. Statement -1 is true, statement -2 is true; statement -2 is not correct explanation of statement-1.
C. Statement -1 is true, statement -2 is true; statement -2 is not correct explanation of statement-1
D. Statement -1 is false, statement -2 is true
E. Statement -1 is true, statement -2 is true; statement -2 is the correct explanation of statement -1
F. Statement -1 is true, statement -2 is false; statement -2 is the correct explanation of statement -1
G. Statement -1 is false, statement -2 is false
H. Statement -1 is false, statement -2 is true.
I. Statement -1 is true, statement -2 is false
J. Statement -1 is true, statement -2 true; statement -2 is the correct explanation of statement -1
Answer: I
Two particles A and B start moving due to their mutual interaction only. If at any time 't', $\vec{a_{A}}$ and $\vec{a_{B}}$ are their respective accelerations, $\vec{v}_{A}$ and $\vec{v}_{B}$ are their respective velocities, and upto that time $W_{A}$ and $W_{B}$ are the work done on A and B respectively by the mutual force. $m_{A}$ and $m_{B}$ are their masses respectively, then which of the following is always correct.

Options:
A. $$\vec{a}_{A} + \vec{a}_{B} = 0$$
B. $$m_{A}\vec{v}_{A} + m_{B}\vec{v}_{B}$$
C. $$m_{A} \vec{v}_{A} + m_{B} \vec{v}_{B} = \vec{0}$$
D. $$\vec{v}_{A} + \vec{v}_{B} = 0$$
E. $$m_{A} \vec{v}_{A} + m_{B} \vec{v}_{B} = \vec{2}$$
F. $$m_{A} \vec{v}_{A} + m_{B} \vec{v}_{B} = \vec{1}$$
G. $$W_{A} + W_{B} =0$$
H. $$m_{A} \vec{v}_{A} + m_{B} \vec{v}_{B} = 0$$
I. $$m_{A} \vec{v}_{A} - m_{B} \vec{v}_{B}$$
Answer: C
A ray of light along the line $x = 3 $ is reflected at the ellipse $\dfrac{{{x^2}}}{{25}} + \dfrac{{{y^2}}}{{16}} = 1$, then the slope of the reflected ray is

Options:
A. $$\dfrac{4}{{15}}$$
B. $$\dfrac{3}{15}$$
C. $$\dfrac{3}{{10}}$$
D. $$\dfrac{3}{{20}}$$
E. $$\dfrac{{15}}{4}$$
F. $$\dfrac{5}{{12}}$$
G. $$\dfrac{12}{25}$$
H. $$\dfrac{15}{4}$$
I. $$\dfrac{{15}}{8}$$
J. $$\dfrac{8}{{15}}$$
Answer: A
Three large identical conducting parallel plates carrying chrage +Q, -Q, and +2Q, respectively, are placed as shown in figure. If $E_A, E_B$ and $ E_C$ refers to the magnitudes of the electric fields at points A, B, C, respectively, then :

Options:
A. $E_A=0$ and $E_C > E_B$
B. $$E_A = 0$$ and $$E_B > E_C$$
C. $$E_A>E_B>E_C$$
D. $E_A=E_B$ and $E_C=0$
E. $$E_A=E_B>E_C$$
F. $E_A=0$ and $E_B=E_C$
G. $E_A=0$ and $E_B < E_C$
Answer: F