MikaStars39/nano-eval huggingface dataset:
One mole of a monoatomic ideal gas initially at a pressure of $2.00$ bar and at $273$  K is taken to final pressure of $4.00$ bar by a reversible path defined by $\dfrac{P}{V} $= constant. taking $C_{v}$ equal to 12.5 J $mol^{-1}$$K^{-1}$, A correct conclusion for this process is(are)?

Options:
A. $$\Delta E = 9.0 kJ$$
B. $$W_{sys}=-3.4 kJ$$
C. $$\Delta E = 9.7 kJ$$
D. $$\Delta E = 9.5 kJ$$
E. $$Q_{sys}=6.8 kJ$$
F. $$\Delta E = 10.3 kJ$$
G. $ \Delta E$=$10.2 kJ$
H. $$\Delta E = 10.0 kJ$$
I. $$\Delta E = 10.5 kJ$$
Answer: G
An excess of liquid Hg was added to $10^{-3}$M acidified solution of $Fe^{3+}$ ions. It was found that only 4.6% of the ions remained as $Fe^{3+}$ at equilibrium at $25^o$C. Calculate $E^o$ for $2Hg/Hg_2^{2+} $ at $25^o$C for,$2Hg + 2Fe^{3+}  \rightleftharpoons  Hg_2^{2+} + 2Fe^{2+}$ and $E^o_{Fe^{2+} / Fe^{3+}} = - 0.7712V$.

Options:
A. -0.8912 V
B. -0.9922 V
C. -0.8812 V
D. -0.7721 V
E. None of these
F. -0.8712 V
G. -0.8012 V
H. -0.7812 V
I. -0.7821 V
J. -0.7912 V
Answer: J
$P$ and $Q$ are two points on the axis and the perpendicular bisector respectively of an electric dipole. Both the points are far away from the dipole and at equal distances from it. If $\overrightarrow{E_P} = \overrightarrow{E_Q}$ are fields at $P$ and $Q$, then

Options:
A. $$\overrightarrow{E_P} = \overrightarrow{E_Q}$$
B. $$\overrightarrow{E_P} = \dfrac{3}{2}\overrightarrow{E_Q}$$
C. $$\overrightarrow{E_P} = \dfrac{1}{3}\overrightarrow{E_Q}$$
D. $$\overrightarrow{E_P} = 3\overrightarrow{E_Q}$$
E. $$\overrightarrow{E_P} = \dfrac{1}{2}\overrightarrow{E_Q}$$
F. $|E_Q| = \dfrac{1}{2} |E_P|$, and $\overrightarrow{E_Q}$ is perpendicular to $\overrightarrow{E_P}$.
G. $$\overrightarrow{E_P} = -2\overrightarrow{E}_P$$
H. $$\overrightarrow{E_P} = \dfrac{4}{3}\overrightarrow{E_Q}$$
I. $$\overrightarrow{E_P} = 2\overrightarrow{E_Q}$$
Answer: A
The series combination of resistance R and inductance L is connected to an alternating source of e. m. f. e = 311 sin (100 $\pi t$). If the peak value of wattless current is 0.5 A and the impedance of the circuit is 311 $\Omega$, the power factor will be -

Options:
A. $$\frac{1}{\sqrt 5}$$
B. $$\frac{1}{\sqrt 3}$$
C. $$\frac{\sqrt 3}{2}$$
D. \frac{\sqrt 2}{2}
E. \frac{1}{\sqrt 6}
F. \frac{2}{3}
G. \frac{\sqrt 2}{3}
H. \frac{1}{\sqrt 2}
I. $$\frac{1}{2}$$
J. \frac{\sqrt 3}{3}
Answer: I
When two moles of Hydrogen atoms join together to form a mole of hydrogen molecules in closed rigid vessel with diathermic walls:$H(g)+H(g)\longrightarrow { H }_{ 2 }(g)$

Options:
A. $$w \neq 0$$
B. $${ q }_{ system }=positive\quad $$
C. w \neq -10
D. $$ { q }_{ surrounding }=negative\quad $$
E. $$w< 0$$
F. $$w = 0$$
G. $$w = -10$$
H. w \neq 10
I. $$\Delta U=negative$$
J. $$w > 0$$
Answer: E
From equation 1 and 2,
$${ CO }_{ 2 }\rightleftharpoons CO+1/2{ O }_{ 2 }$$
[${ K }_{ d }={ 9.1\times 10 }^{ -13 }$ at ${ 1000 }^{ 0 }$C] (eq. 1)
$${ H }_{ 2 }O\rightleftharpoons { H }_{ 2 }+1/2{ O }_{ 2 }$$
[${ K }_{ d }={ 7.1\times 10 }^{ -12 }$ at ${ 1000 }^{ 0 }$C] (eq. 2)
The equilibrium constant for the reaction$CO_{ 2 }+{ H }_{ 2 }\rightleftharpoons CO+{ H }_{ 2 }O$ at the same temperature, is:

Options:
A. 0.78
B. 1.28
C. 2.87
D. 1.62
E. 0.72
F. 1.78
G. 16.2
H. 2.81
I. 0.81
J. 2.0
Answer: A
Consider three identical massless spring.The left end of the first,Second and the spring are affixed to a wall, to a block of mass m placed on a horizontal floor (not frictionless) and to a block of mass m placed on a horizontal frictionless floor respectively as shown in the figure.THe right end of the each spring is pulled by a gradually increasing force.When magnitude of the force becomes $ F_1 $ extensions in these spring becomes $ x_1, x_2 and x_3 $ respectively.Which of the following statements is correct?

Options:
A. $$ x_1 > x_2 > x_3 $$
B. $$x_1 = x_2 > x_3$$
C. $$x_1 < x_2 = x_3$$
D. $$x_1 < x_2 < x_3$$
E. $$x_1 = x_2 < x_3$$
F. $$ x_1 = x_2 = x_3 $$
G. $$ x_1 > x_2 = x_3 $$
H. $$x_1 > x_2 < x_3$$
Answer: F
When metallic copper is shaken with a solution of a copper salt, the reaction $ Cu + Cu^{2+} \rightleftharpoons 2Cu^{+} $ proceeds. When equilibrium is established at $298\ K$, $[Cu^{2+}]/[Cu^{+}]^{2} = 2.647 \times 10^{6}\ M^{-1}$.If the standard potential of the $ Cu^{2+} |Cu$ half-cell is $+0.2376\ V$,  what is the standard potential of $ Cu^{+} |Cu $ half-cell? (Given : $2.303\ RT/F = 0.06\ ,\ log\ 2 = 0.3\ ,\ log\ 3 = 0.48$)

Options:
A. $$0.4502\ V$$
B. $$0.3802\ V$$
C. $$0.6752\ V$$
D. $$0.5302\ V$$
E. $$0.4302\ V$$
F. $$0.3302\ V$$
G. $$0.151\ V$$
H. $$-0.3732\ V$$
I. $$0.4802\ V$$
J. $$0.5102\ V$$
Answer: E
What is the pH of 0.10 M KCN solution at $ 25^oC $ for HCN , $ K_a = 6.2 \times 10^{-10} $ ?

Options:
A. 11
B. 4
C. 3
D. 7
E. 13
F. 12
G. 6
H. 10
I. 5
J. 9
Answer: I
Three non conducting infinite planar sheets are parallel to the y-z plane. Each sheet has a uniform surface charge density. The first sheet, with a negative surface charge density $-\sigma$, passes through the x-axis at x=1 m.  The second sheet has an unknown surface charge density and passes through x-axis at x = 2m. The third sheet has a negative surface charge density $-3\sigma$ and passes through x-axis at x = 4m. The net electric field due to the sheets is zero at x
= 1.5 m.

Options:
A. The electric field at x = 6m is \(\frac{\sigma}{\epsilon_0}\hat i\)
B. The electric field at x = 3m is $\frac{-\sigma}{\epsilon_0}\hat i$
C. The electric field at x = 6m is \(\frac{-2\sigma}{\epsilon_0}\hat i\)
D. The electric field at x = 6m is $\frac{-\sigma}{\epsilon_0}\hat i$
E. The electric field at x = 6m is $\frac{2\sigma}{\epsilon_0}\hat i$
F. The electric field at x = 6m is \(\frac{-5\sigma}{\epsilon_0}\hat i\)
G. The electric field at x = 6m is $\frac{-4\sigma}{\epsilon_0}\hat i$
H. The surface charge density on the second sheet is $+2\sigma$
I. The electric field at x = -2m is $\frac{\sigma}{\epsilon_0} \hat i$
J. The electric field at x = 6m is $\frac{-3\sigma}{\epsilon_0}\hat i$
Answer: D
What would be the period of the free oscillations of the system shown here if mass $\displaystyle M_{1}$ is pulled down a little? Force constant of the spring is k, the mass of fixed pulley is negligible and movable pulley is smooth

Options:
A. $$\displaystyle T=2\pi\sqrt{\frac{3M_{2}+M_{1}}{k}}$$
B. \displaystyle T=2\pi\sqrt{\frac{M_{1}+2M_{2}}{k}}
C. $$\displaystyle T=2\pi\sqrt{\frac{M_{1}+4M_{2}}{k}}$$
D. $$\displaystyle T=2\pi\sqrt{\frac{2M_{1}+3M_{2}}{k}}$$
E. $$\displaystyle T=2\pi\sqrt{\frac{M_{2}+4M_{1}}{k}}$$
F. $$\displaystyle T=2\pi\sqrt{\frac{4M_{1}+3M_{2}}{k}}$$
G. $$\displaystyle T=2\pi\sqrt{\frac{M_{1}+3M_{2}}{k}}$$
H. \displaystyle T=2\pi\sqrt{\frac{4M_{2}+M_{1}}{k}}
I. $$\displaystyle T=2\pi\sqrt{\frac{M_{2}+3M_{1}}{k}}$$
J. $$\displaystyle T=2\pi\sqrt{\frac{M_{1}+M_{2}}{k}}$$
Answer: E
Three particles each of mass m are located at vertices of an equilateral triangle ABC. They starts moving with equal speed v each along the medians of the triangle and collide at its centroid G. If, after collision, A comes to rest and B retraces its path along GB, then C

Options:
A. moves with a speed v/3 along CG.
B. moves with a speed v along AG.
C. moves with a speed 2v along CG.
D. moves with a speed 3v/4 along CG.
E. moves with a speed v along CG.
F. moves with a speed v/2 along CG.
G. moves with a speed 3v/2 along CG.
H. moves with a speed v along BG.
I. moves with a speed v/2 along BG.
J. also comes to rest.
Answer: H
The instantaneous values of current and voltage inan A.c. circuit ate I = 4 sin $\omega t$ and E = 100 cos
$(\omega t + \dfrac{\pi}{3})$ respectively. Thee phase difference between voltage and current is

Options:
A. \dfrac{5\pi}{6}
B. \dfrac{\pi}{6}
C. \dfrac{\pi}{4}
D. $$\dfrac{\pi}{3}$$
E. \dfrac{2\pi}{3}
F. $$\dfrac{7\pi}{6}$$
G. \dfrac{4\pi}{3}
H. \dfrac{\pi}{2}
Answer: D
A certain compound on burning in air forms three oxides. One of the oxides turned lime water milky, the other turned anhydrous $CuSO_4$ blue and third formed a solution of $pH=9$. Compound is formed of :

Options:
A. $S,N$ and $H$
B. $S,C$ and $Na$
C. $S,H$ and $K$
D. $S,H$ and $C$
E. $S,C$ and $H$
F. $S,N$ and $C$
G. S,H and O
H. $S,H$ and $Mg$
I. $S,H$ and $Na$
J. $S,H$ and $Cl$
Answer: I
Predict which of the following reactions would proceed spontaneously at 298 K ? Take $E^0_{Fe^{2+}/Fe} = -0.44V$

Options:
A. $$Cd(s)\, +\, Co^{2+}(aq)\, \rightarrow\, Cd^{2+}(aq)\, +\, Co(s)$$ $$E^{\ominus}_{Fe^{2+}/Fe}\, =\, -\, 0.48\, V$$
B. $$Cd(s)\, +\, Co^{2+}(aq)\, \rightarrow\, Cd^{2+}(aq)\, +\, Co(s)$$ $$E^{\ominus}_{Fe^{2+}/Fe}\, =\, -\, 0.54\, V$$
C. $$Zn^{2+}(aq)\, +\, H_{2}\, \rightarrow\, Zn(s)\, +\, 2H^{\oplus}$$
D. $$Co(s)\, +\, Fe^{2+}(aq)\, \rightarrow\, Co^{2+}(aq)\, +\, Fe(s)$$ $$E^{\ominus}_{Fe^{2+}/Fe}\, =\, -\, 0.54\, V$$
E. $$Cd(s)\, +\, Co^{2+}(aq)\, \rightarrow\, Cd^{2+}(aq)\, +\, Co(s)$$ $$E^{\ominus}_{Fe^{2+}/Fe}\, =\, -\, 0.34\, V$$
F. $$Cd(s)\, +\, Co^{2+}(aq)\, \rightarrow\, Cd^{2+}(aq)\, +\, Co(s)$$ $$E^{\ominus}_{Fe^{2+}/Fe}\, =\, -\, 0.58\, V$$
G. $Co(s)\, +\, Fe^{2+}(aq)\, \rightarrow\, (Co^{2+}(aq)\, +\, Fe(s)$$E^{\ominus}_{Co^{+2}/\, Co}\, =\, -\, 0.28\, V$
H. $$Cd(s)\, +\, Co^{2+}(aq)\, \rightarrow\, Cd^{2+}(aq)\, +\, Co(s)$$
$$E^{\ominus}_{Fe^{2+}/Fe}\, =\, -\, 0.44\, V$$
Answer: H
The density of air at Normal temperature and pressure is $1.293 \ g/litre$ and $\gamma = 1.36.$ What is the frequency of a tuning fork which emits sound of wavelength $66.5 \ cm$ at $17 ^\circ C$?

Options:
A. 526 Hz
B. 306 Hz
C. 406 Hz
D. 516 Hz
E. 546 Hz
F. 566 Hz
G. 536 Hz
H. 586 Hz
I. 556 Hz
J. 506 Hz
Answer: E
A mass of 1 kg is acted upon by a single force F = $(4\hat{i} - 4 \hat{j})$N. Under this force it is displaced from (0, 0) to (1m, 1m). If intially the speed of the particle was 2 m/s, What is its final speed approximately?

Options:
A. 7.2 m/s
B. 6.2 m/s
C. 6.3 m/s
D. 4.5 m/s
E. 5.8 m/s
F. 6.6 m/s
G. 6.5 m/s
H. 6.0 m/s
I. 8.6 m/s
J. 6.4 m/s
Answer: J
Molarity of $H_2SO_4$ is $0.8M$ and its density is $1.06gcm^{-3}$. What will be the concentration of the solution in terms of molality $(m)$and mole fraction $(x)$?

Options:
A. m=0.145, x=0.805
B. $$m=0.5, x=0.45$$
C. m=0.0145, x=0.815
D. $$m=0.815, x=0.0145$$
E. m=0.0145, x=0.810
F. $$m=0.4, x=0.4$$
G. m=0.0145, x=0.85
H. $$m=0.145, x=0.815$$
I. $$m=0.145, x=0.45$$
Answer: H
The degree of dissociation of HI at a particular temperature is 0.8. Find the volume of 1.5 M sodium thiosulphate solution required to react completely with the iodine present at equilibrium in acidic conditions, when 0.135 mol each of $H_2$ and $I_2$ are heated at 440 K in a closed vessel of capacity 2.0 L :

Options:
A. V = 162 mL
B. V = 126 mL
C. $$V = 288mL$$
D. V = 216 mL
E. V = 180 mL
F. None of these
G. $$V = 72 mL$$
H. $$V = 144 mL$$
I. V = 108 mL
J. V = 120 mL
Answer: H
Two gases P and Q both decolourise aqueous bromine but only one of them gives white ppt with Tollen's reagent. P and Q are likely to be

Options:
A. ${ H }_{ 2 }C={ CH }_{ 2 }$ & ${ CH }_{ 3 }-{ CH }_{ 2 }-C\equiv C-C{ H }_{ 3 }
B. ${ H }_{ 2 }C={ CH }_{ 2 }$ & ${ CH }_{ 3 }-C\equiv C-{ CH }_{ 3 }$
C. $HC\equiv CH$ & ${ CH }_{ 3 }-C\equiv CH$
D. ${ CH }_{ 3 }-{ CH }_{ 2 }-C\equiv CH$ & ${ CH }_{ 3 }-C\equiv C-C{ H }_{ 3 }$
E. ${ H }_{ 2 }C={ CH }_{ 2 }$ & ${ CH }_{ 3 }-{ CH }_{ 2 }-C\equiv C-{ CH }_{ 3 } $
F. ${ H }_{ 2 }C={ CH }_{ 2 }$ & ${ CH }_{ 3 }-C\equiv C-C{ H }_{ 3 }$
G. $HC\equiv CH$ & ${ CH }_{ 3 }-{ CH }_{ 2 }-C\equiv CH$
Answer: D
The force of attraction between two charges separated by certain distance in air is F1. If the space between the charges is completely filled with dielectric of constant 4 the force becomes F2. If half of the distance between the charges is filled with same dielectric the force between the charges is F3. Find F1:F2:F3 is:

Options:
A. 16 : 9 : 4
B. 36 : 18 : 16
C. 36 : 9 :16
D. 9 : 4 : 2
E. 16 : 4 : 8
F. 36 : 18 : 8
G. 9 : 36 : 16
H. 18 : 8 : 4
I. 16 : 9 : 2
J. 4 : 1 : 2
Answer: C
A light ray is incident on a transparent sphere of index $= \sqrt {2}$, at an angle of incidence $= 45^{\circ}$. What is the deviation of a tiny fraction of the ray, which enters the sphere, undergoes two internal reflections and then refracts out into air?

Options:
A. $$120^{\circ}$$
B. $$180^{\circ}$$
C. $$330^{\circ}$$
D. $$285^{\circ}$$
E. $$315^{\circ}$$
F. $$240^{\circ}$$
G. $$345^{\circ}$$
H. $$270^{\circ}$$
I. $$300^{\circ}$$
J. $$255^{\circ}$$
Answer: H
A light body of mass m and  a heavy body of mass M are tied at the ends of  a string which passes through a frictionless tube as shown in figure. The light body moves round a circular path in a horizontal plane. The length of the string from the mass m to the tope of the tube is $l $and makes an angle $\theta $ with the verticle. The frequency of rotation of the light body for which the heavy body remains stationry will be:

Options:
A. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{mg}{(M+2m)l}$$
B. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{Mg}{ml}$$
C. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{M}{m}$$
D. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{mg}{(M+4m)l}$$
E. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{mg}{ML}$$
F. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{mg}{(M+3m)l}$$
G. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{mg}{(M+m)l}$$
H. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{mg}{(M-m)l}$$
I. $$\displaystyle n= \frac{1}{2\pi}\sqrt \frac{mg}{(M+5m)l}$$
Answer: E
A block of mass $2kg$ and sp gravity $\cfrac{5}{2}$ is attached with a spring constant $K=100N/m$ and is half dipped in the water. If extension in the spring is $1cm$. The force exerted by the bottom of the tank on block is $(g=10m/{s}^{2}$)

Options:
A. $$20N$$
B. $$15N$$
C. $$16N$$
D. 18N
E. 17N
F. 23N
G. 25N
H. $$19N$$
I. 14N
J. 22N
Answer: A
At $298$K, the equilibrium constant for the reaction$Zn^{2+}+4NH_3\rightleftharpoons [Zn(NH_3)_4]^{2+}$ is $10^9$.If $E^o_{[Zn(NH_3)_4]^{2+}/[Zn^{2+}+4NH_3]}=-1.03$V then the value of $E^o_{Zn/Zn^{2+}}$ will be:

Options:
A. -0.9545 V
B. $+1.1$V
C. $-1.1$V
D. -0.7745 V
E. none of these
F. -0.7445 V
G. -0.8245 V
H. $-0.7645$V
I. -0.7845 V
J. -0.9245 V
Answer: H
A particle suspended from a vertical spring oscillates $10$ times per second. At the highest point of oscillation, the spring becomes unstretched. Find the speed when the spring is stretched by $0.20$cm. (Take, $g=\pi^2m/s^2$).

Options:
A. $$12.14cm/s$$
B. $$14.3cm/s$$
C. $$15.2cm/s$$
D. $$15.6cm/s$$
E. $$15.9cm/s$$
F. $$110.25cm/s$$
G. $$16.7cm/s$$
H. $$15.1cm/s$$
I. $$15.4cm/s$$
J. $$15.8cm/s$$
Answer: I
A small sphere is given vertical velocity of magnitude $v_o =5\ ms^{-1}$ and it swings in a vertical plane about the end of a massless string. The angle $\theta$ with the vertical at which string will break, knowing that it can withstand a maximum tension equal to twice the weight of the sphere, is

Options:
A. $$\cos^{-1}\dfrac{2}{3}$$
B. $$\cos^{-1}\left(\dfrac{1}{3}\right)$$
C. $$\cos^{-1}\left(\dfrac{1}{6}\right)$$
D. $$\cos^{-1}\left(\dfrac{1}{8}\right)$$
E. $$\cos^{-1}\left(\dfrac{3}{5}\right)$$
F. $$60^o$$
G. $$\cos^{-1}\left(\dfrac{1}{4}\right)$$
H. $$30^o$$
I. $$\cos^{-1}\left(\dfrac{1}{5}\right)$$
J. $$\cos^{-1}\left(\dfrac{1}{2}\right)$$
Answer: G
A small bead with mass $M$ is attached to a very light string hung from a ceiling. The string can be torn by a force exceeding the value of $10\ Mg$ where $g$ is acceleration due to gravity. The bead on the string may oscillate harmonically with a period $T_{0}$. A student decides to perform another experiment. He takes the attached bead a side from its equilibrium and pushes it in such a way that the bead performs a full revolution in a horizontal plane. The time of one revolution of the bead is $T$. What is the expression for T?

Options:
A. $$T=\dfrac{T_{0}}{\sqrt{14}}$$
B. $$T=\dfrac{T_{0}}{\sqrt{7}}$$
C. $$T=\dfrac{T_{0}}{\sqrt{10}}$$
D. $$T=\dfrac{T_{0}}{\sqrt{17}}$$
E. $$T=\dfrac{T_{0}}{\sqrt{9}}$$
F. $$T=\dfrac{T_{0}}{\sqrt{6}}$$
G. $$T=\dfrac{T_{0}}{\sqrt{11}}$$
H. $$T=\dfrac{T_{0}}{\sqrt{8}}$$
Answer: A
Ethyl chloride $(C_{2}H_{5}Cl)$ is prepared by:-$C_{2}H_{4(g)}+HCl_{(g)}\rightarrow C_{2}H_{5}Cl_{(g)};\ \ \Delta H=-72.3 kJ$ if 98 g of $C_{2}H_{4}$ and 109.5 g $HCl$ are used at 300 K, then find $\Delta E$.

Options:
A. -209.41 kJ
B. -65.8 kJ
C. -66.3 kJ
D. -70.3 kJ
E. -65.3 kJ
F. -69.3 kJ
G. -72.3 kJ
H. -209.10 kJ
I. 208.4 kJ
J. -190 kJ
Answer: A
A $10kg$ block is placed on a horizontal surface. The coefficient of friction between them is $0.2$ . A horizontal force P=15 N first acts on it in the eastward direction. Later, in addition to P a second horizontal force Q=20 N acts on it in the northward direction. Choose the correct option of the following.

Options:
A. When both P and Q act, the direction of the force of friction acting on the block will be \( tan^{-1} \left ( \frac{4}{5} \right ) \) west of south.
B. When the block moves, its direction of motion will be $tan^{-1} \displaystyle \left ( \frac{4}{3} \right )$ east of north.
C. If the block moves, its acceleration will be 0.5 m/s$^2$.
D. When both P and Q act, the direction of the force of friction acting on the block will be $tan^{-1} \displaystyle \left ( \frac{3}{4} \right )$ west of south.
E. When both P and Q act, the direction of the force of friction acting on the block will be $tan^{-1} \left ( \frac{4}{3} \right )$ east of north.
F. When both P and Q act, the direction of the force of friction acting on the block will be $tan^{-1} \left ( \frac{3}{5} \right )$ west of north.
G. When both P and Q act, the direction of the force of friction acting on the block will be \( tan^{-1} \left ( \frac{3}{5} \right ) \) east of north.
H. The block will not move when only P acts, but will move when both P and Q act.
I. When both P and Q act, the direction of the force of friction acting on the block will be $tan^{-1} \left ( \frac{3}{5} \right )$ west of south.
J. When both P and Q act, the direction of the force of friction acting on the block will be \( tan^{-1} \left ( \frac{5}{3} \right ) \) east of south.
Answer: D
A block of mass $M$, moving with a velocity $v_{0}$ on a horizontal table strikes a spring of spring constant $k$ at $t=0$ and beings to experience a frictional force. The co-efficient of friction is variable and is given by $\mu=bx$, where $x$ is the compression in the spring. The loss in mechanical energy when the block first comes momentarily to rest is

Options:
A. $$\dfrac{gbMv_{0}^{2}}{2k}$$
B. $$\dfrac{Mgbv_{0}^{3}}{2(k+gb)}$$
C. $$\dfrac{Mgbv_{0}^{2}}{2(k+gb)}$$
D. $$\dfrac{Mgbv_{0}^{2}}{(k+4gb)}$$
E. $$\dfrac{Mgbv_{0}^{2}}{(2k+Mgb)}$$
F. $$\dfrac{Mgbv_{0}^{2}}{(k+gb)}$$
G. $$\dfrac{Mgbv_{0}^{2}}{(k+2gb)}$$
H. $$Zero$$
I. $$\dfrac{Mgbv_{0}^{2}}{(k+3gb)}$$
J. $$\dfrac{Mgbv_{0}^{2}}{2(2k+gb)}$$
Answer: J
In which of the following option (s) all species contains X-O-X bond(s) in structure ($X =$ central atom)?

Options:
A. $$H_2S_2O_5, S_3O_9, S_2O_6^{2-}$$
B. $$S_3O_9, P_4O_6, Si_2O_8^{6-}$$
C. $$N_2O_5, N_2O, N_2O_4$$
D. $$S_3O_9, P_4O_6, Si_2O_7^{6-}$$
E. $$S_3O_9, H_5P_3O_{10}, Si_2O_7^{6-}$$
F. S_3O_9, H_5P_3O_{10}, Si_2O_8^{6-}
Answer: D
You are in a paddleboat that is floating alongside a wooden block in a river. You are asked to perform following three experiments. I. Row your boat $20 m$ downstream ahead of the wooden block. II. Row your boat upstream to go $20 m$ away from the wooden block. III. Row your boat to move along the stream together with the block and simultaneously shift $20 m$ towards a bank. In all these experiments, you row your boat at a constant speed relative to water expending energies $ E_1 , E_2$ and $E_3 $ respectively. Which of the following statements is correct?

Options:
A. $$ E_1 < E_2 < E_3 $$
B. $$E_2 < E_3 < E_1$$
C. $$ E_1 = E_2 < E_3 $$
D. $$ E_2 < E_1 < E_3 $$
E. $$ E_1 = E_2 =E_3 $$
F. $$ E_1 < E_2 = E_3 $$
Answer: E
For reaction $A \rightarrow B,$ the rate constant $k_1 = A_1 (e^{-E_{a_1} / RT})$ and for the reaction $X \rightarrow Y,$ the rate constant $k_2 = A_2 (e^{-E_{a_2}} / RT).$ If $A_1 = 10^9. A_2 = 10^{10}$ and $E_{a_1} = 1200 \,cal / mol,$ then the temperature at which $k_1 = k_2$ is: $(Given; R = 2\,cal / K-mol)$

Options:
A. $$\dfrac{300 \times 1.303}{1.303}K$$
B. $$\dfrac{300 \times 2.303}{1.303}K$$
C. $$300\,K$$
D. $$\dfrac{300 \times 2.303}{2}K$$
E. $$None of these$$
F. $$\dfrac{300 \times 1.303}{2}K$$
G. $$\dfrac{300}{2.303}K$$
H. $$\dfrac{300 \times 1.303}{2.303}K$$
I. $$\dfrac{300 \times 2.303}{2.303}K$$
J. $$300 \times 2.303 \,K$$
Answer: G
The lower end of a 4m long uniform rod AB is pulled with constant speed v = 4m/s. The speed of centre of mass of the rod at $ \theta =6{  0}^{ 0 } $ will be $ \frac { n }{ \sqrt {3  }  } m/s  $ . Find the value of n

Options:
A. $$3$$
B. $$1$$
C. $$\frac{4}{\sqrt{3}}$$
D. $$\frac{5}{\sqrt{3}}$$
E. 4
F. $$\frac{7}{\sqrt{3}}$$
G. $$\frac{3}{\sqrt{3}}$$
H. $$2$$
I. $$\frac{6}{\sqrt{3}}$$
J. $$\frac{2}{\sqrt{3}}$$
Answer: B
The sphere at P is given a downward velocity $v_0$ and swings in a vertical plane at the end of a rope of $l = 1 m$ attached to a support at O. The rope breaks at angle $30 ^ { \circ }$ from horizontal, knowing that it can withstand a maximum tension equal to three times the weight the sphere. Then the value of $v_0$ will be : ($g = \pi ^ { 2 } m / s ^ { 2 } )$)

Options:
A. \frac { 3 \pi } { 5 } \mathrm { m } / \mathrm { s }
B. \frac { 2 \pi } { 5 } \mathrm { m } / \mathrm { s }
C. \frac { 3 \pi } { 8 } \mathrm { m } / \mathrm { s }
D. $$\frac { 2 \pi } { 3 } \mathrm { m } / \mathrm { s }$$
E. \frac { 3 \pi } { 6 } \mathrm { m } / \mathrm { s }
F. $$\sqrt { \frac { 3 } { 4 } } \pi \mathrm { m } / \mathrm { s }$$
G. \frac { 3 \pi } { 7 } \mathrm { m } / \mathrm { s }
H. $$\sqrt { \frac { 3 } { 2 } } \pi \mathrm { m } / \mathrm { s }$$
I. none of these
J. \frac { 3 \pi } { 4 } \mathrm { m } / \mathrm { s }
Answer: H
Orthophosphoric acid $({ H }_{ 3 }{ PO }_{ 4 })$ is a weak tribasic mineral acid,${ H }_{ 3 }{ PO }_{ 4 }\rightleftharpoons { H }^{ + }+{ H }_{ 2 }{ PO }_{ 4 }^{ - };\quad { K }_{ { a }_{ 1 } }={ 10 }^{ -3 }\quad $${ H }_{ 2 }{ PO }_{ 4 }^{ - }\rightleftharpoons { H }^{ + }+{ H }{ PO }_{ 4 }^{ -2 };\quad { K }_{ { a }_{ 2 } }={ 10 }^{ -7 }\quad $$HPO_{ 4 }^{ -2 }\rightleftharpoons { H }^{ + }+{ PO }_{ 4 }^{ -3 };\quad { K }_{ { a }_{ 3 } }={ 10 }^{ -13 }$In a solution of $0.1M$ ${ H }_{ 3 }{ PO }_{ 4 }$, concentration of ${ PO }_{ 4 }^{ 3- }$ is:

Options:
A. $${ 10 }^{ -12 }$$
B. $${ 10 }^{ -25 }$$
C. $${ 10 }^{ -26 }$$
D. $${ 10 }^{ -24 }$$
E. $${ 10 }^{ -15 }$$
F. $${ 10 }^{ -21 }$$
G. $${ 10 }^{ -27 }$$
H. $${ 10 }^{ -22 }$$
I. $${ 10 }^{ -20 }$$
J. $${ 10 }^{ -23 }$$
Answer: J
pH of 0.1 M solution of $NaA$ (sodium salt of a weak acid $HA$) is 8.92. Calculate ${ pK }_{ a }$ of $HA$. If a drop of HPh $\left( { pK }_{ in }=9.52 \right) $ be added to the above solution, predict whether the pink colour will  visible or not under the medical fact that our eyes can see the pink colour if the mole % of ionized form of indicator is 25% or more.

Options:
A. Pink colour will be visible in a weaker base
B. Pink colour will be visible in a neutral solution
C. Pink colour will be visible in a weaker acid
D. Pink colour will not be visible
E. Pink colour will not be visible in a weaker acid
F. Pink colour will be visible in a stronger acid
G. Pink colour will be visible in a stronger base
H. Pink colour will be visible
Answer: H
A certain mass of gas undergoes a process given by $dU=\dfrac{dW}{2}$. If the molar heat capacity of the gas for this process is $\dfrac{15}{2}R$, then the gas is :

Options:
A. data insufficient
B. polyatomic
C. diatomic
D. monoatomic
Answer: C
Decreasing order of mass of pure NaOH in each of the aqueous solution.
(I) $50$g of $40\%$(W/W) NaOH
(II) $50$ml of $50\%$(W/V) NaOH $(d_{sol}=1.2g/ml)$
(III) $50$g of $15$M NaOH $(d_{sol}=1$ g/ml).

Options:
A. I, II, III
B. II, III, I
C. III $=$ II $=$ I
D. III, I, II
E. II, I, III
F. I, III, II
G. III, II, I
Answer: A
In what direction, can the reaction::$2NaCl + Fe_{2}(SO_{4})_{3}\rightleftharpoons 2FeSO_{4} + Cl_{2} + Na_{2}SO_{4}$ proceed spontaneously?(Given: $E_{Fe^{3+}/ Fe^{2+}}^{\circ} =0.77\ V; E_{Cl^{-}/Cl_{2}}^{\circ} = 1.36\ V$).

Options:
A. first backward and then forward
B. backward
C. backward and then forward
D. can't say
E. first forward and then backward
F. forward
Answer: F
Sulphide ion reacts with solid sulphur
$$ \mathrm { s } _ { ( \mathrm { aq } ) } ^ { 2 - } + \mathrm { S } _ { ( \mathrm { s } ) } \rightleftharpoons  \mathrm { S } _ { 2 ( \mathrm { aq } ) } ^ { 2 - } ; \mathrm { K } _ { 1 } = 10 $$
$$ \mathrm { s } _ { ( \mathrm { aq } ) } ^ { 2 - } + \mathrm { 2S } _ { ( \mathrm { s } ) } \rightleftharpoons  \mathrm { S } _ { 3 ( \mathrm { aq } ) } ^ { 2 - } ; \mathrm { K } _ { 2 } = 130 $$
The equilibrium constant for the formation of $ S _ { 3 } ^ { 2 - } ( a q ) $ and $S _ {2 } ^ { 2 - } ( a q )$ and sulphur is

Options:
A. 11
B. 130
C. 1000
D. 1300
E. 1.3
F. 1.0
G. 13
H. 100
Answer: A
We now define an algorithm: The definition of a(n) is the sequence derived from the Molien series for the 6-dimensional complex representation of the double cover of the second Janko group (J2), which is an important object in the study of finite simple groups and modular forms. Given the input x_list (a series of values): [56, 57, 58, 59, 60, 61, 62, 63, 64, 65], determine the corresponding output sequence y_list.

Options:
A. [314, 335, 372, 381, 445, 462, 500, 538, 598, 622]
B. [322, 340, 377, 390, 446, 453, 507, 539, 596, 615]
C. [317, 331, 370, 383, 440, 459, 510, 535, 590, 614]
D. [316, 333, 373, 384, 441, 458, 506, 532, 590, 613]
E. [315, 334, 374, 386, 442, 455, 503, 530, 592, 614]
F. [313, 332, 371, 385, 448, 461, 502, 536, 593, 616]
G. [318, 336, 375, 382, 443, 460, 508, 533, 595, 618]
H. [321, 338, 379, 387, 444, 457, 504, 529, 589, 612]
I. [320, 339, 380, 391, 450, 467, 515, 541, 601, 625]
J. [319, 337, 378, 388, 447, 456, 501, 534, 597, 620]
Answer: D
What is the key difference between a simple VAR and an Electrolux refrigerator?

Options:
A. Generator
B. Power switch
C. Drip tray
D. Cooling agent
E. Condenser coil
F. Liquid pump
G. Absorber
H. Insulation type
I. Door seal
Answer: F
The acoustic impedance of the load at the end of a rigid tube is known to be three times the acoustic impedance of the air within the tube. Find the minimum length of the tube such that the input acoustic resistance equals the input acoustic reactance when a plane wave of 500 Hz is propagating through it.

Options:
A. 0.400m.
B. 0.175m.
C. 0.243m.
D. 0.315m.
E. 0.280m.
F. 0.455m.
G. 0.295m.
H. 0.350m.
I. 0.162m.
J. 0.218m.
Answer: J
Write the electrode reactions and the cell reaction for the battery $\mathrm{Zn}$ | Zn$Cl_{\mathrm{2}}$(0.1mol/dm$^{3}$), AgCl(solid)|Ag, and the electromotive force of this battery at 25°C is().

Options:
A. $$
= 1. 1 5 0 \mathrm{~ V}
$$
B. $$
= 1. 0 9 0 \mathrm{~ V}
$$
C. $$
= 1. 1 0 4 \mathrm{~ V}
$$
D. $$
= 1. 1 2 1 \mathrm{~ V}
$$
E. $$
= 1. 0 6 8 \mathrm{~ V}
$$
F. $$
= 1. 0 2 3 \mathrm{~ V}
$$
G. $$
= 0. 9 8 5 \mathrm{~ V}
$$
H. $$
= 1. 0 7 5 \mathrm{~ V}
$$
I. $$
= 1. 0 5 9 \mathrm{~ V}
$$
J. $$
= 1. 0 8 2 \mathrm{~ V}
$$
Answer: J
Given a uniform cone with a radius of $r$ and a height of $h$, find the condition _____ under which its inertia ellipsoid with respect to the centroid degenerates into a sphere.

Options:
A. $$
h=5r
$$
B. $$
h=4r
$$
C. $$
h=3r
$$
D. $$
h=\pi r
$$
E. $$
h=r/2
$$
F. $$
h=r^2
$$
G. $$
h=2r
$$
H. $$
h=\sqrt{2}r
$$
I. $$
h=r^3
$$
J. $$
h=\frac{1}{2}r
$$
Answer: G
The increase in height and weight is _________.

Options:
A. Rapidly rises after adolescence
B. Remains constant with age
C. Decreases significantly with maturity
D. Slows down with age
E. Increases steadily with age
F. Peaks during teenage years
G. Fluctuates randomly over time
H. Accelerates in young adulthood
I. Completely reverses with aging
J. Stops completely after childhood
Answer: D
As the plano-convex lens in the Newton's ring device is slowly translated upward, the thickness of the air film gradually increases and a ____ change in the ring interference fringes can be seen.

Options:
A. Expand towards the edges
B. Contract towards the center
C. Stand still
D. Shift towards the left
E. Darken in color
F. Brighten progressively
G. Flatten and disappear
H. Slide to the right
I. expanding abroad
J. Rotate counterclockwise
Answer: B
What is the most sensitive stimulus for increasing the secretion of antidiuretic hormone?

Options:
A. Increased renal blood flow
B. Increased plasma colloid osmotic pressure
C. Increased circulating blood volume
D. Increased body temperature
E. Decreased heart rate
F. Increased arterial blood pressure
G. Increased urine production
Answer: B
The Indian Institute of Management, Bangalore is the work of which of the following architects?

Options:
A. D. Lee
B. S. Patel
C. R. Rival
D. T. Carter
E. L. Nguyen
F. B. Dorsey
G. K. Johnson
H. A. Monroe
I. C. Kolya
J. M.Islam
Answer: F
Which of the following reflections have positive feedback regulation ( ).

Options:
A. Tendon reflex
B. Decompression reflex
C. Intestinal absorption reflex
D. Flexor reflex
E. Vasomotor tone reflex
F. Extensor muscle reflex
G. Gastrointestinal peristalsis reflex
H. Cardiac output reflex
I. Pulmonary stretch reflex
J. Urinary reflex
Answer: J
What is the mechanism of phenazopyridine in the treatment of Parkinson's disease?

Options:
A. Suppressing central adrenaline activity
B. Blocking central cholinergic receptors
C. Inhibiting spinal cholinergic receptors
D. Activating spinal serotonergic pathways
E. Stimulating peripheral dopamine receptors
F. Suppressing peripheral acetylcholine activity
G. Inhibiting cortical adrenaline release
H. Enhancing spinal dopamine release
I. Enhancing peripheral glutamate activity
J. Enhancing central glutamate transmission
Answer: B
According to Chinese standards,in a single-pipe unequal-spaced hot water heating system, the relative difference in pressure loss between each parallel branch should not exceed what value (excluding the common section)?

Options:
A. 5%
B. 25%
C. 20%
D. 22%
E. 12%
F. 18%
G. 28%
H. 30%
I. 10%
J. 15%
Answer: J
Which of the following statements about gender and sexual characteristics is incorrect?

Options:
A. Structural abnormalities in sex chromosomes do not produce intersex conditions.
B. The number and structure of sex chromosomes do not affect sexual development.
C. Sex hormone levels have no influence on sexual characteristic formation.
D. The development of sexual characteristics is not affected by androgens or estrogens.
E. The normal development of sexual characteristics is unrelated to sex hormones.
F. Abnormalities in the number and structure of sex chromosomes affect sexual development but do not affect sexual characteristics.
G. Androgens and estrogens play no role in the development of sexual features.
H. Differences in sex chromosomes have no impact on sexual characteristic development.
I. Sexual characteristics develop independently of sex hormone effects.
J. The presence of androgens and estrogens does not affect sexual development.
Answer: F
Under what circumstances is a dog most likely to develop Gastric Dilatation-Volvulus (GDV)?

Options:
A. Change in diet
B. Obstructed airways
C. Eating too rapidly
D. Overeating
E. Drinking cold water
F. Excessive stress
G. Allergic reaction
H. Exercise on an empty stomach
I. Drinking hot water
J. Parasite infection
Answer: D
Regarding soil and water conservation measures, what is considered the most effective technical measure?

Options:
A. applying mulch layers to cut down on surface evaporation
B. growing cover crops to enhance soil nitrogen content
C. applying fertilizers to replenish essential soil nutrients
D. planting deep-rooted trees to stabilize riverbanks
E. using contour plowing to slow down water runoff on hills
F. building raised beds to prevent soil compaction in gardens
G. constructing check dams to impede river flow erosion
H. building terraces to reduce the impact of water flow on slopes
I. installing pipes to channel surface water runoff efficiently
J. using windbreaks to minimize the impact of strong winds
Answer: H
What US president is mentioned by name in the opening theme song of TV's 'All in the Family'?

Options:
A. Ronald Reagan
B. Herbert Hoover
C. Richard Nixon
D. Lyndon B. Johnson
E. Franklin Roosevelt
F. Harry Truman
G. Calvin Coolidge
H. John F. Kennedy
I. Gerald Ford
J. Dwight D. Eisenhower
Answer: B
What is the primary objective of management in managerial economics?

Options:
A. maximization of social welfare
B. minimization of costs
C. minimization of risk exposure
D. maximization of market share
E. maximization of employee satisfaction
F. maximization of the firm's value
G. maximization of customer satisfaction
H. minimization of environmental impact
I. maximization of production output
J. maximization of profits
Answer: F
A beam of monochromatic light is incident at $i={50}^{o}$ on one face of an equilateral prism, the angle of emergence is ${40}^{o}$, then the angle of minimum deviation is:

Options:
A. $$\le {45}^{o}$$
B. $$\le {40}^{o}$$
C. $$\ge{30}^{o}$$
D. $${30}^{o}$$
E. $$\ge{40}^{o}$$
F. $$\ge{32}^{o}$$
G. $$< {30}^{o}$$
H. $$\le {42}^{o}$$
I. $$\ge{35}^{o}$$
J. $$\le {30}^{o}$$
Answer: J
The cold junction of a thermocouple is at $0^{o}C$ . The thermo e.m.f produced in the thermocouple is given by an equation $E=16T - 0.04T^{2} V$,  where T is the temperature of the hot junction.  The temperature of inversion and neutral temperature of the thermocouple are

Options:
A. $$200^{0}C,300^{0}C$$
B. $$200^{0}C,400^{0}C$$
C. 300^{0}C,450^{0}C
D. 450^{0}C,300^{0}C
E. 350^{0}C,200^{0}C
F. 350^{0}C,400^{0}C
G. $$200^{0}C,450^{0}C$$
H. $$400^{0}C,200^{0}C$$
I. $$300^{0}C,200^{0}C$$
J. 300^{0}C,400^{0}C
Answer: H
A mirror $\left( {\mu  = \frac{3}{2}} \right)$ is $10cm$ thick. An object is placed $15cm$ in front of it. The position of image from the front surface is

Options:
A. 18.36cm
B. 21.66cm
C. 18.33cm
D. $$28.34cm$$
E. $$35cm$$
F. 18.35cm
G. 21.65cm
H. $$21.67cm$$
I. 18.34cm
J. $$10.34cm$$
Answer: J
The focal length \( f \) of a mirror is given by
\[
\frac{1}{f} = \frac{1}{u} + \frac{1}{v},
\]
where \( u \) and \( v \) represent the object and image distances, respectively.
If small changes \( \Delta u \), \( \Delta v \), and \( \Delta f \) occur in \( u \), \( v \), and \( f \), respectively, what is the expression for the relative error \( \frac{\Delta f}{f} \) in terms of \( \Delta u \), \( \Delta v \), \( u \), and \( v \)?

Options:
A. $$\cfrac{\Delta f}{f}=\cfrac{\Delta u}{v}+\cfrac{\Delta v}{u}$$
B. $$\cfrac{\Delta f}{f}=\cfrac{\Delta u}{u}+\cfrac{\Delta v}{v}+\cfrac{\Delta U}{u+v}+\cfrac{\Delta v}{u+v}$$
C. $$
\frac{\Delta f}{f} = \frac{\Delta u}{u} + \frac{\Delta v}{v} + \frac{\Delta U}{u + v} - \frac{\Delta v}{u}
$$
D. $$\cfrac{\Delta f}{f}=\cfrac{\Delta u}{u}+\cfrac{\Delta v}{v}+\cfrac{\Delta U}{u}+\cfrac{\Delta v}{u}$$
E. $$\cfrac{\Delta f}{f}=\cfrac{\Delta u}{u}+\cfrac{\Delta v}{v}+\cfrac{\Delta U}{u+v}-\cfrac{\Delta v}{u+v}$$
F. $$\cfrac{\Delta f}{f}=\cfrac{\Delta u}{u}+\cfrac{\Delta v}{v}-\cfrac{\Delta U}{u+v}+\cfrac{\Delta v}{u+v}$$
G. $$\cfrac{\Delta f}{f}=\cfrac{\Delta u}{u}+\cfrac{\Delta v}{v}$$
H. $$\cfrac{\Delta f}{f}=\cfrac{\Delta u}{u}+\cfrac{\Delta v}{v}+\cfrac{\Delta U}{u}-\cfrac{\Delta v}{v}$$
I. $$\cfrac{\Delta f}{f}=\cfrac{\Delta u}{u}+\cfrac{\Delta v}{v}+\cfrac{\Delta u}{u+v}-\cfrac{\Delta v}{u+v}$$
Answer: B
A biconvex lens of focal length 15 cm is in front of a plane mirror. The distance between the lens and the mirror is 10 cm. A small object is kept at a distance of 30 cm from the lens. The final image is :

Options:
A. Real and at a distance of 20 cm from the mirror
B. Real and at a distance of 14 cm from the mirror
C. Virtual and at a distance of 20 cm form the mirror
D. Real and at a distance of 18 cm from the mirror
E. Virtual and at a distance of 14 cm from the mirror
F. Real and at a distance of 17 cm from the mirror
G. Real and at distance of 16 cm from the mirror
H. Real and at distance of 19 cm from the mirror
I. Virtual and at a distance of 16 cm from mirror
J. Virtual and at a distance of 18 cm from the mirror
Answer: G
Two identical buggies move one other due to inertia (without friction) with the same velocity $v_{0}$. A man of mass $m$ rides the rear buggy. At a certain moment the man jumps into the front buggy with a velocity $u$ relative to his buggy. If mass of each buggy is equal to $M$ and velocity of buggies after jumping of man are $v_{rear}$ and $v_{front}$. Then

Options:
A. $$v_{front}=v_{o}-\dfrac{mM}{(m+M)^{2}}u$$
B. $$v_{front}=v_{o}-\dfrac{mM}{(m+M+1)}u$$
C. $$v_{front}=v_{o}+\dfrac{mM}{(m+M)^{2}}u$$
D. $$v_{rear}=v_{o}-\dfrac{m}{m+M}u$$
E. $$v_{front}=v_{o}-\dfrac{mM}{(m+M)}u$$
F. $$v_{rear}=v_{o}+\dfrac{m}{m+M}u$$
G. v_{front}=v_{o}+\dfrac{mM}{(m+2M)^{2}}u
H. $$v_{front}=v_{o}-\dfrac{mM}{(m+2M)^{2}}u$$
I. $$v_{front}=v_{o}-\dfrac{mM}{(m+2M)}u$$
Answer: C
An object \( O \) is kept in front of a converging lens of focal length \( 30 \, \text{cm} \), behind which there is a plane mirror at \( 15 \, \text{cm} \) from the lens. Based on this setup, which of the following statements about the final image is correct?

Options:
A. The final image is formed at 58 cm from the lens towards right of it
B. The final image is formed at 70 cm from the lens towards right of it
C. The final image is formed at 55 cm from the lens towards right of it
D. The final image is formed at 45 cm from the lens towards right of it
E. The final image is formed at 60 cm from the lens towards right of it
F. The final image is virtual
G. The final image is formed at 50 cm from the lens towards right of it
H. The final image is formed at 57 cm from the lens towards right of it
I. The final image is real
J. The final image is at 60 cm from lens towards left of it
Answer: E
A cylindrical pipe of radius $r$ is rolling towards a frog sitting on the horizontal ground. Centre of the pipe is moving with a constant velocity $v$. To save itself, the frog jumps off and passes over the pipe touching it only at the top. Denoting air-time of the frog by $T$, horizontal range of the jump by $R$ and acceleration due to gravity by $g$, which of the following conclusions can you make?

Options:
A. $$T\ge4\sqrt{\dfrac{r}{g}}$$
B. $$R=4(\sqrt{gr}-v)\sqrt{\dfrac{r}{g}}$$
C. $$R=4(\sqrt{gr}-3v)\sqrt{\dfrac{r}{g}}$$
D. $$R=4(\sqrt{gr}-2v)\sqrt{\dfrac{r}{g}}$$
E. $$T=4\sqrt{\dfrac{r}{g}}$$
F. $$R\ge4(\sqrt{gr}-v)\sqrt{\dfrac{r}{g}}$$
G. $$R=4(\sqrt{gr}-v)\sqrt{\dfrac{r}{g}} + v\sqrt{\dfrac{r}{g}}$$
H. $$R\ge4(\sqrt{gr}+v)\sqrt{\dfrac{r}{g}}$$
I. $$R=4(\sqrt{gr}+v)\sqrt{\dfrac{r}{g}}$$
Answer: F
An inclined plane of length $5.60m$ making an angle of ${ 45 }^{ o }$ with the horizontal is placed in an uniform electric field $E=100V{ m }^{ -1 }$. A particle of mass $1kg$ and charge ${10}^{-2}C$ is allowed to slide down from rest position from maximum height of slope. If the coefficient of friction is $0.1$, the time taken by the particle to reach the bottom is ______

Options:
A. $$None of these$$
B. $$1s$$
C. $$1.30s$$
D. $$1.40s$$
E. $$2s$$
F. $$1.20s$$
G. $$1.60s$$
H. $$1.71s$$
I. $$1.41s$$
J. $$1.50s$$
Answer: B
Two particles are projected from the same point on the ground with same speed simultaneously at angle $ \theta_1$ and $\theta_2 $ with the horizontal .they strike the ground at same point after $ t_1$ and $t_2 $ respectively , then

Options:
A. $$\frac{t_1}{t_2} = \frac{cos \theta_1}{sin \theta_2}$$
B. $$ \frac {t_1}{t_2} = tan \theta_1 $$
C. \frac{t_1}{t_2} = \frac{\sin \theta_1}{\tan \theta_2}
D. $$\frac{t_1}{t_2} = \frac{cos \theta_1}{cos \theta_2}$$
E. $$\frac{t_1}{t_2} = \tan \theta_2$$
F. \frac{t_1}{t_2} = \frac{\sin \theta_1}{\cos \theta_2}
G. $$\frac{t_1}{t_2} = \frac{\tan \theta_1}{\sin \theta_2}$$
H. All of these
I. $$ \frac {t_1}{t_2} = \frac {sin \theta_1}{sin \theta_2} $$
J. $$ \frac {t_1}{t_2} =1 $$
Answer: B
Water is filled in a container upto a height of $5cm$.There is a concave mirror of radius of curvature $40$cm .A person just above the surface observes images of $'A'$.The distance of image from water surface is $(\mu_{water}=1.33)$:

Options:
A. $$6.8cm$$
B. $$7.8cm$$
C. $$5.4cm$$
D. $$7.4cm$$
E. $$8.4cm$$
F. $$8.8cm$$
G. $$7.6cm$$
H. $$7.2cm$$
I. $$8.2cm$$
J. $$9.8cm$$
Answer: F
A point moves with constant acceleration and $v_1, v_2$ and $v_3$ denote the average velocities in the three successive intervals $t_1, t_2$ and $t_3$ of time. Which of the following relations is correct?

Options:
A. $$\dfrac{v_1-v_2}{v_2-v_3}=\dfrac{t_1-t_2}{t_1+t_3}$$
B. $$\dfrac{v_1-v_2}{v_2-v_3}=\dfrac{t_1-t_2}{t_1-t_3}$$
C. $$\dfrac{v_1-v_2}{v_2-v_3}=\dfrac{t_1-t_2}{t_2-t_3}$$
D. $$\dfrac{v_1-v_2}{v_2-v_3}<\dfrac{t_1-t_2}{t_2+t_3}$$
E. $$\dfrac{v_1-v_2}{v_2-v_3}=\dfrac{t_1+t_2}{t_2+t_3}$$
F. $$\dfrac{v_1-v_2}{v_2-v_3}=\dfrac{t_1+t_2}{t_1+t_3}$$
G. $$\dfrac{v_1-v_2}{v_2-v_3}>\dfrac{t_1+t_2}{t_1+t_3}$$
H. $$\dfrac{v_1-v_2}{v_2-v_3}=\dfrac{t_1+t_2}{t_1-t_3}$$
Answer: E
A set of $ n  $ identical cubical blocks lies at rest parallel to each other along a line on a smooth horizontal surface. The separation between the near surfaces of any two adjacent blocks is $ L $. The block at one end is given a speed $ v $ towards the next one at time
t = 0
. All collisions are completely inelastic, then

Options:
A. The centre of mass of the system will have a final speed $\frac{v}{2n+1}$
B. The centre of mass of the system will have a final speed $ \frac{v}{n+1} $
C. The centre of mass of the system will have a final speed $ \frac{v}{n+2} $
D. The centre of mass of the system will have a final speed $ \frac{v}{2n} $
E. The last block starts moving at $ t= \frac{(n-1)L}{v} $
F. The centre of mass of the system will have a final speed $ \frac{v}{n-1} $
G. The last block starts moving at $ t=\frac{n(n-1)L}{2v} $
H. The centre of mass of the system will have a final speed $ \frac{v}{n}$
I. The centre of mass of the system will have a final speed $ v $
J. The centre of mass of the system will have a final speed $ \frac{v}{2n-1} $
Answer: H
A particle starts from origin at time t=0 with velocity $8\hat { j } { ms }^{ -1 }$ and moves in x-y plane with constant acceleration of $\left( -2\hat { i } +4\hat { j }  \right) ms^{ -2 }$ the position coordinates at which velocity of particle becomes parallel to y-axis is

Options:
A. (8 m, 32 m)
B. (32 m, 16 m)
C. (16 m, 24 m)
D. (16 m, 8 m)
E. (32 m,32 m)
F. (8 m, 16 m)
G. (16 m,32 m)
H. (16 m,16 m)
Answer: H
A concave mirror with its optic axis vertical and mirror facing upward is placed at the bottom of the water tank. The radius of curvature of mirror is $40\ cm$ and refractive index for water $\mu =4/3$. The tank is $20\ cm$ deep and if a bird is flying over the tank at a height $60\ cm$ above the surface of water, the position of image of a bird is :

Options:
A. $$4.2\ cm$$
B. $$4.75\ cm$$
C. $$5.75\ cm$$
D. $$5.2\ cm$$
E. $$4.25\ cm$$
F. $$3.2\ cm$$
G. $$3.75\ cm$$
H. $$4.5\ cm$$
I. $$4.23\ cm$$
Answer: G
A  particle  thrown  up vertically reaches its highest point in time $ t_{1} $ and return  to the ground is further time $ t_{2} $. The air  resistance exerts a constant force on the particle opposite to its direction of motion, then

Options:
A. $$t_{1} >7{t_{2}}$$
B. $$ t_{1} < t_{2} $$
C. $$ 2t_{1} < t_{2} $$
D. $$ t_{1} > t_{2} $$
E. $$t_{1} \approx 2t_{2}$$
F. $$ t_{1} \approx \frac{t_{2}}{2} $$
G. $$ t_{1} = 3t_{2} $$
Answer: B
The motion of a particle along a straight line is described by equation $x=8+12t -t^3$, where x is in metre and f in second. The retardation of the particle when its velocity becomes zero is

Options:
A. $$6 ms ^{-2}$$
B. $$36 ms^{-2}$$
C. $$24 ms ^{-2}$$
D. $$Zero$$
E. $$8 ms^{-2}$$
F. $$12 ms ^{-2}$$
G. $$14 ms ^{-2}$$
H. $$18 ms^{-2}$$
I. $$16 ms^{-2}$$
J. $$10 ms^{-2}$$
Answer: A
A particle is projected from an inclined plane $\displaystyle OP_{1}$ from A with velocity $\displaystyle v_{1}= 8 \ ms^{-1}$ at an angle $\displaystyle 60^{\circ}$ with horizontal. An another particle is projected at the same instant from B with velocity $\displaystyle v_{2}= 16\ ms^{-1}$ and perpendicular to the plane $\displaystyle OP_{2}$. After time $\displaystyle 10\sqrt{3}\ sec $ there separation was minimum and found to be $70\ m$. Then find distance AB.

Options:
A. $$200\ m$$
B. $$300\ m$$
C. $$275\ m$$
D. $$250\ m$$
E. $$325\ m$$
F. $$350\ m$$
G. $$225\ m$$
H. $$400\ m$$
I. $$375\ m$$
Answer: D
For the one dimensional motion described by $x=t-\cos t(t>0)$ where $x$ is position and $t$ is time. Which of the following options is incorrect?

Options:
A. $0\le v\le 2$ for all $t >0$
B. $-1\le a\le 0$ for all $t >0$
C. $0\le a\le 1$ for all $t >0$
D. $-2\le a\le 2$ for all $t >0$
E. $$-1 \le a \le 2$$
F. $-1\le v\le 1$ for all $t >0$
G. $-1\le a\le 1$ for all $t >0$
H. At $t-0, x=-1$ and $v=+1$
I. $$-2 \le a \le 0$$
Answer: G
There is a dust particle on a glass slab of thickness t and refractive index $\mu  = 1.5$. When seen from the top side of the slab, the dust particle appears at a distance of 6 cm. From another side, it appears at a distance of 6 cm. On another side it appears at 4 cm, find the thickness t of the glass slab

Options:
A. $$15 cm$$
B. $$18 cm$$
C. $$12 cm$$
D. $$20 cm$$
E. $$14 cm$$
F. $$10 cm$$
G. $$13 cm$$
H. $$21 cm$$
I. $$11 cm$$
J. $$17 cm$$
Answer: A
A plano-convex lens is made of a material of refractive index u = 1.5. The radius of curvature of curved surface of the lens is 20 cm. If its plane surface is silvered, the focal length of the slivered lens will be

Options:
A. 50 cm
B. 45 cm
C. 35 cm
D. 80 cm
E. 20 cm
F. 30 cm
G. 70 cm
H. 60 cm
I. 10 cm
J. 40 cm
Answer: J
The focal length of a biconvex lens is 20 cm and its refractive index is 1.5. If the radii of curvatures of two surfaces of lens are in the ratio 1 : 2, then the larger radius of curvature is :

Options:
A. 15 cm
B. 30 cm
C. 40 cm
D. 10 cm
E. 45 cm
F. 28 cm
G. 25 cm
H. 35 cm
I. 22 cm
J. 20 cm
Answer: B
Two masses $m_{1}=1\ kg$ and $m_{2}= 2\ kg$ are connected by a light in extensible string and suspended by means of a weightless pulley. Assuming that both the masses starts from rest, the distance travelled by the centre of mass in two second is $(Take\ g=1\ ms^{-2} )$

Options:
A. $$\dfrac{2}{3}m$$
B. \dfrac{16}{9}m
C. $$\dfrac{10}{9}m$$
D. $$\dfrac{25}{9}m$$
E. $$\dfrac{30}{9}m$$
F. $$\dfrac{20}{9}m$$
G. $$\dfrac{1}{3}m$$
H. $$\dfrac{40}{9}m$$
Answer: F
A current of 2.68 A is passed for one hour through an aqueous solution of $CuSO_4$ using copper electrodes. Calculate the change in mass of cathode and that of the anode. (At. mass of copper = 63.5).

Options:
A. Increase in the mass of cathode = 6.348g
B. Decrease in the mass of anode = 3.176g
C. Increase in the mass of cathode = 6.350g
D. The ratio between the change in the mass of cathode to anode is 1:2
E. Decrease in the mass of anode = 3.172g
F. Increase in the mass of cathode = 3.174g
G. No change in the mass of electrodes
H. Decrease in the mass of anode = 3.178g
I. Decrease in the mass of anode = 3.174g
J. Decrease in the mass of anode = 3.180g
Answer: I
A diverging lens of focal length $f_{1}$ is placed in front of an coaxially with a concave mirror of focal length $f_{2}$. Their separation is $d$. A parallel beam of light incident on the lens returns as a parallel beam from the arrangement. Then, returns as a parallel beam from the arrangement. Then which of the following are correct

Options:
A. $$d = 3|f_{2}| - |f_{1}|$$
B. $$d = 2|f_{2}| - 2|f_{1}|$$
C. $$d = |f_{2}| + |f_{1}|$$
D. The beam diameter of the incident and reflected beams must be the same.
E. $$d = 2|f_{2}| + |f_{1}|$$
F. $$d = 2|f_{2}| - |f_{1}|$$
G. $$d = |f_{2}| - 2|f_{1}|$$
H. If the entire arrangement is immersed in water, the conditions will remain unaltered.
I. $$d = |f_{2}| + 2|f_{1}|$$
J. $$d = |f_{2}| - |f_{1}|$$
Answer: F
A ball thrown upward returns to the ground with the initial speed of u. The ball is at a height of 80 m at two times, interval being 6 s. Find u by assuming $g=\quad 10m{ s }^{ 2 }$

Options:
A. 75 m/s
B. 80 m/s
C. 95 m/s
D. 60 m/s
E. 90 m/s
F. 59 m/s
G. 50 m/s
H. 97 m/s
I. 65 m/s
J. 55 m/s
Answer: G
Focal length of glass $\left(\mu =\dfrac{3}{2}\right)$ lens is $20\ cm$. Find its focal length in liquid of refractive index $2$ is

Options:
A. $$-15\ cm$$
B. $$80\ cm$$
C. $$-10\ cm$$
D. $$-20\ cm$$
E. $$40\ cm$$
F. $$10\ cm$$
G. $$60\ cm$$
H. $$-40\ cm$$
I. -30\ cm
Answer: C