16. An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$. It has a spherical cavity of radius $R / 2$ with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point $P$, which is at a distance $2 R$ from the axis of the cylinder, is given by the expression $\frac { 23 \rho R } { 16 k \varepsilon _ { 0 } }$. The value of $k$ is [Figure]

ANSWER : 6

1. A cylindrical cavity of diameter $a$ exists inside a cylinder of diameter $2 a$ as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density $J$ flows along the length. If the magnitude of the magnetic field at the point $P$ is given by $\frac { N } { 12 } \mu _ { 0 } a J$, then the value of $N$ is

[Figure]
ANSWER : 5

1. A lamina is made by removing a small disc of diameter $2 R$ from a bigger disc of uniform mass density and radius $2 R$, as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I _ { O }$ and $I _ { P }$, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac { I _ { P } } { I _ { O } }$ to the nearest integer is [Figure]

ANSWER : 3

1. A circular wire loop of radius $R$ is placed in the $x - y$ plane centered at the origin $O$. A square loop of side $a ( a \ll R )$ having two turns is placed with its center at $z = \sqrt { 3 } R$ along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of $45 ^ { \circ }$ with respect to the $z$-axis. If the mutual inductance between the loops is given by $\frac { \mu _ { 0 } a ^ { 2 } } { 2 ^ { p / 2 } R }$, then the value of $p$ is

[Figure]
ANSWER : 7

1. A proton is fired from very far away towards a nucleus with charge $Q = 120 e$, where $e$ is the electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm ) of the proton at its start is: (take the proton mass, $m _ { p } = ( 5 / 3 ) \times 10 ^ { - 27 } \mathrm {~kg}$; $\mathrm { h } / e = 4.2 \times 10 ^ { - 15 } \mathrm {~J} . \mathrm { s } / \mathrm { C } ; \frac { 1 } { 4 \pi \varepsilon _ { 0 } } = 9 \times 10 ^ { 9 } \mathrm {~m} / \mathrm { F } ; 1 \mathrm { fm } = 10 ^ { - 15 } \mathrm {~m}$ )

PART II : CHEMISTRY

SECTION 1 : Single Correct Answer Type

This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

21. In allene $\left( \mathrm { C } _ { 3 } \mathrm { H } _ { 4 } \right)$, the type(s) of hybridisation of the carbon atoms is (are)
(A) $s p$ and $s p ^ { 3 }$
(B) $s p$ and $s p ^ { 2 }$
(C) only $s p ^ { 2 }$
(D) $s p ^ { 2 }$ and $s p ^ { 3 }$

ANSWER : B

1. For one mole of a van der Waals gas when $b = 0$ and $T = 300 \mathrm {~K}$, the $P V$ vs. $1 / \mathrm { V }$ plot is shown below. The value of the van der Waals constant $a \left( \mathrm {~atm} . \mathrm { liter } ^ { 2 } \mathrm {~mol} ^ { - 2 } \right)$ is [Figure]
   (A) 1.0
   (B) 4.5
   (C) 1.5
   (D) 3.0

ANSWER: C

1. The number of optically active products obtained from the complete ozonolysis of the given compound is [Figure]
   (A) 0
   (B) 1
   (C) 2
   (D) 4

ANSWER : A

1. A compound $\mathrm { M } _ { \mathrm { p } } \mathrm { X } _ { \mathrm { q } }$ has cubic close packing (ccp) arrangement of X . Its unit cell structure is shown below. The empirical formula of the compound is [Figure]
   (A) MX
   (B) $\mathrm { MX } _ { 2 }$
   (C) $M _ { 2 } X$
   (D) $M _ { 5 } X _ { 14 }$

ANSWER : B

1. The number of aldol reaction(s) that occurs in the given transformation is [Figure]
   (A) 1
   (B) 2
   (C) 3
   (D) 4
2. The colour of light absorbed by an aqueous solution of $\mathrm { CuSO } _ { 4 }$ is
   (A) orange-red
   (B) blue-green
   (C) yellow
   (D) violet

ANSWER : A

1. The carboxyl functional group $( - \mathrm { COOH } )$ is present in
   (A) picric acid
   (B) barbituric acid
   (C) ascorbic acid
   (D) aspirin

ANSWER : D

1. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [ $a _ { 0 }$ is Bohr radius]
   (A) $\frac { \mathrm { h } ^ { 2 } } { 4 \pi ^ { 2 } m a _ { 0 } ^ { 2 } }$
   (B) $\frac { \mathrm { h } ^ { 2 } } { 16 \pi ^ { 2 } m a _ { 0 } ^ { 2 } }$
   (C) $\frac { \mathrm { h } ^ { 2 } } { 32 \pi ^ { 2 } m a _ { 0 } ^ { 2 } }$
   (D) $\frac { \mathrm { h } ^ { 2 } } { 64 \pi ^ { 2 } m a _ { 0 } ^ { 2 } }$

ANSWER: C

1. Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?
   (A) $\mathrm { HNO } _ { 3 } , \mathrm { NO } , \mathrm { NH } _ { 4 } \mathrm { Cl } , \mathrm { N } _ { 2 }$
   (B) $\mathrm { HNO } _ { 3 } , \mathrm { NO } , \mathrm { N } _ { 2 } , \mathrm { NH } _ { 4 } \mathrm { Cl }$
   (C) $\mathrm { HNO } _ { 3 } , \mathrm { NH } _ { 4 } \mathrm { Cl } , \mathrm { NO } , \mathrm { N } _ { 2 }$
   (D) $\mathrm { NO } , \mathrm { HNO } _ { 3 } , \mathrm { NH } _ { 4 } \mathrm { Cl } , \mathrm { N } _ { 2 }$

ANSWER : B

1. As per IUPAC nomenclature, the name of the complex $\left[ \mathrm { Co } \left( \mathrm { H } _ { 2 } \mathrm { O } \right) _ { 4 } \left( \mathrm { NH } _ { 3 } \right) _ { 2 } \right] \mathrm { Cl } _ { 3 }$ is
   (A) Tetraaquadiaminecobalt (III) chloride
   (B) Tetraaquadiamminecobalt (III) chloride
   (C) Diaminetetraaquacobalt (III) chloride
   (D) Diamminetetraaquacobalt (III) chloride

ANSWER: D

SECTION II : Multiple Correct Answer(s) Type

This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

21. $\mathrm { NiCl } _ { 2 } \left\{ \mathrm { P } \left( \mathrm { C } _ { 2 } \mathrm { H } _ { 5 } \right) _ { 2 } \left( \mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \right) \right\} _ { 2 }$ exhibits temperature dependent magnetic behaviour (paramagnetic/ diamagnetic). The coordination geometries of $\mathrm { Ni } ^ { 2 + }$ in the paramagnetic and diamagnetic states are respectively
(A) tetrahedral and tetrahedral
(B) square planar and square planar
(C) tetrahedral and square planar
(D) square planar and tetrahedral

ANSWER : C

1. In the cyanide extraction process of silver from argentite ore, the oxidizing and reducing agents used are
   (A) $\mathrm { O } _ { 2 }$ and CO respectively.
   (B) $\mathrm { O } _ { 2 }$ and Zn dust respectively.
   (C) $\mathrm { HNO } _ { 3 }$ and Zn dust respectively.
   (D) $\mathrm { HNO } _ { 3 }$ and CO respectively.

ANSWER : B

1. The reaction of white phosphorus with aqueous NaOH gives phosphine along with another phosphorus containing compound. The reaction type; the oxidation states of phosphorus in phosphine and the other product are respectively
   (A) redox reaction; - 3 and - 5
   (B) redox reaction; + 3 and + 5
   (C) disproportionation reaction; - 3 and + 5
   (D) disproportionation reaction; - 3 and + 3

Zero Marks to all

24. The shape of $\mathrm { XeO } _ { 2 } \mathrm {~F} _ { 2 }$ molecule is
(A) trigonal bipyramidal
(B) square planar
(C) tetrahedral
(D) see-saw

ANSWER : D

1. For a dilute solution containing 2.5 g of a non-volatile non-electrolyte solute in 100 g of water, the elevation in boiling point at 1 atm pressure is $2 ^ { \circ } \mathrm { C }$. Assuming concentration of solute is much lower than the concentration of solvent, the vapour pressure ( mm of Hg ) of the solution is (take $K _ { \mathrm { b } } = 0.76 \mathrm {~K} \mathrm {~kg} \mathrm {~mol} ^ { - 1 }$ )
   (A) 724
   (B) 740
   (C) 736
   (D) 718

ANSWER:A

1. The compound that undergoes decarboxylation most readily under mild condition is

(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
ANSWER : B

27. Using the data provided, calculate the multiple bond energy ( $\mathrm { kJ } \mathrm { mol } ^ { - 1 }$ ) of a $\mathrm { C } \equiv \mathrm { C }$ bond in $\mathrm { C } _ { 2 } \mathrm { H } _ { 2 }$. That energy is (take the bond energy of a $\mathrm { C } - \mathrm { H }$ bond as $350 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 }$.) $2 \mathrm { C } ( \mathrm { s } ) + \mathrm { H } _ { 2 } ( \mathrm {~g} ) \longrightarrow \mathrm { C } _ { 2 } \mathrm { H } _ { 2 } ( \mathrm {~g} )$
$$\begin{aligned} & \Delta H = 225 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 } \\ & \Delta H = 1410 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 } \\ & \Delta H = 330 \mathrm {~kJ} \mathrm {~mol} ^ { - 1 } \end{aligned}$$
(A) 1165
(B) 837
(C) 865
(D) 815

ANSWER : D

1. The major product $\mathbf { H }$ of the given reaction sequence is [Figure]

(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
ANSWER : A

SECTION II : Paragraph Type

This section contains $\mathbf { 6 }$ multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Questions 29 and 30

Bleaching powder and bleach solution are produced on a large scale and used in several household products. The effectiveness of bleach solution is often measured by iodometry.

29. Bleaching powder contains a salt of an oxoacid as one of its components. The anhydride of that oxoacid is
(A) $\mathrm { Cl } _ { 2 } \mathrm { O }$
(B) $\mathrm { Cl } _ { 2 } \mathrm { O } _ { 7 }$
(C) $\mathrm { ClO } _ { 2 }$
(D) $\mathrm { Cl } _ { 2 } \mathrm { O } _ { 6 }$

ANSWER : A

1. 25 mL of household bleach solution was mixed with 30 mL of 0.50 M KI and 10 mL of 4 N acetic acid. In the titration of the liberated iodine, 48 mL of $0.25 \mathrm {~N} \mathrm { Na } _ { 2 } \mathrm {~S} _ { 2 } \mathrm { O } _ { 3 }$ was used to reach the end point. The molarity of the household bleach solution is
   (A) 0.48 M
   (B) 0.96 M
   (C) 0.24 M
   (D) 0.024 M

ANSWER : C

Paragraph for Questions 31 and 32

The electrochemical cell shown below is a concentration cell. $\mathrm { M } \mid \mathrm { M } ^ { 2 + }$ (saturated solution of a sparingly soluble salt, $\left. \mathrm { MX } _ { 2 } \right) \left| \left| \mathrm { M } ^ { 2 + } \left( 0.001 \mathrm {~mol} \mathrm { dm } ^ { - 3 } \right) \right| \mathrm { M } \right.$ The emf of the cell depends on the difference in concentrations of $\mathrm { M } ^ { 2 + }$ ions at the two electrodes. The emf of the cell at 298 K is 0.059 V .

31. Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction, as shown in the given scheme. [Figure]
(A) $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { OH }$ and $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { COOH }$
(B) $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { COOH }$ and $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { OH }$
(C) $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { OH }$ and $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { OH }$
(D) $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { OH }$ and $\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { COOH }$

ANSWER : BD

1. Choose the correct reason(s) for the stability of the lyophobic colloidal particles.
   (A) Preferential adsorption of ions on their surface from the solution
   (B) Preferential adsorption of solvent on their surface from the solution
   (C) Attraction between different particles having opposite charges on their surface
   (D) Potential difference between the fixed layer and the diffused layer of opposite charges around the colloidal particles

ANSWER : AD

1. Which of the following molecules, in pure form, is (are) unstable at room temperature ?

(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]
ANSWER : BC

31. The solubility product $\left( K _ { s p } ; \mathrm { mol } ^ { 3 } \mathrm { dm } ^ { - 9 } \right)$ of $\mathrm { MX } _ { 2 }$ at 298 K based on the information available for the given concentration cell is (take $2.303 \times \mathrm { R } \times 298 / \mathrm { F } = 0.059 \mathrm {~V}$ )
(A) $1 \times 10 ^ { - 15 }$
(B) $4 \times 10 ^ { - 15 }$
(C) $1 \times 10 ^ { - 12 }$
(D) $4 \times 10 ^ { - 12 }$

ANSWER : B

1. The value of $\Delta G \left( \mathrm {~kJ} \mathrm {~mol} ^ { - 1 } \right)$ for the given cell is (take $\left. 1 \mathrm {~F} = 96500 \mathrm { C } \mathrm { mol } ^ { - 1 } \right)$
   (A) - 5.7
   (B) 5.7
   (C) 11.4
   (D) - 11.4

ANSWER : D

Paragraph for Questions 33 and 34

In the following reaction sequence, the compound $\mathbf { J }$ is an intermediate. [Figure]
J $\left( \mathrm { C } _ { 9 } \mathrm { H } _ { 8 } \mathrm { O } _ { 2 } \right)$ gives effervescence on treatment with $\mathrm { NaHCO } _ { 3 }$ and a positive Baeyer's test.

33. The compound I is
(A) [Figure]
(B) [Figure]
(C) [Figure]
(D) [Figure]

ANSWER : A

1. The compound $\mathbf { K }$ is
   (A) [Figure]
   (B) [Figure]
   (C) [Figure]
   (D) [Figure]

ANSWER : C

SECTION III : Multiple Correct Answer(s) Type

This section contains $\mathbf { 6 }$ multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

34. Which of the following hydrogen halides react(s) with $\mathrm { AgNO } _ { 3 } ( \mathrm { aq } )$ to give a precipitate that dissolves in $\mathrm { Na } _ { 2 } \mathrm {~S} _ { 2 } \mathrm { O } _ { 3 } ( \mathrm { aq } )$ ?
(A) HCl
(B) HF
(C) HBr
(D) HI

ANSWER : ACD

1. For an ideal gas, consider only $P$ - $V$ work in going from an initial state $X$ to the final state $Z$. The final state $\boldsymbol { Z }$ can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct ? [take $\Delta S$ as change in entropy and $w$ as work done] [Figure]
   (A) $\Delta S _ { \mathrm { x } \rightarrow \mathrm { z } } = \Delta S _ { \mathrm { x } \rightarrow \mathrm { y } } + \Delta S _ { \mathrm { y } \rightarrow \mathrm { z } }$
   (B) $w _ { \mathrm { x } \rightarrow \mathrm { z } } = w _ { \mathrm { x } \rightarrow \mathrm { y } } + w _ { \mathrm { y } \rightarrow \mathrm { z } }$
   (C) $w _ { x \rightarrow y \rightarrow z } = w _ { x \rightarrow y }$
   (D) $\Delta S _ { \mathrm { x } \rightarrow \mathrm { y } \rightarrow \mathrm { z } } = \Delta S _ { \mathrm { x } \rightarrow \mathrm { y } }$

ANSWER : AC

SECTION III : Integer Answer Type

This section contains $\mathbf { 5 }$ questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).

35. The reversible expansion of an ideal gas under adiabatic and isothermal conditions is shown in the figure. Which of the following statement(s) is (are) correct? [Figure]
(A) $T _ { 1 } = T _ { 2 }$
(B) $T _ { 3 } > T _ { 1 }$
(C) $w _ { \text {isothermal } } > w _ { \text {adiabatic } }$
(D) $\Delta U _ { \text {isothermal } } > \Delta U _ { \text {adiabatic } }$
ANSWER : ACD or AD

36. The substituents $\mathbf { R } _ { \mathbf { 1 } }$ and $\mathbf { R } _ { \mathbf { 2 } }$ for nine peptides are listed in the table given below. How many of these peptides are positively charged at $\mathrm { pH } = 7.0$ ? [Figure]

Peptide	$\mathbf { R } _ { \mathbf { 1 } }$	$\mathbf { R } _ { \mathbf { 2 } }$
1	H	H
II	H	$\mathrm { CH } _ { 3 }$
III	$\mathrm { CH } _ { 2 } \mathrm { COOH }$	H
IV	$\mathrm { CH } _ { 2 } \mathrm { CONH } _ { 2 }$	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$
V	$\mathrm { CH } _ { 2 } \mathrm { CONH } _ { 2 }$	$\mathrm { CH } _ { 2 } \mathrm { CONH } _ { 2 }$
VI	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$
VII	$\mathrm { CH } _ { 2 } \mathrm { COOH }$	$\mathrm { CH } _ { 2 } \mathrm { CONH } _ { 2 }$
VIII	$\mathrm { CH } _ { 2 } \mathrm { OH }$	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$
IX	$\left( \mathrm { CH } _ { 2 } \right) _ { 4 } \mathrm { NH } _ { 2 }$	$\mathrm { CH } _ { 3 }$

ANSWER : 4

36. The given graphs / data I, II, III and IV represent general trends observed for different physisorption and chemisorption processes under mild conditions of temperature and pressure. Which of the following choice(s) about I, II, III and IV is (are) correct? [Figure]
(A) I is physisorption and II is chemisorption
(B) I is physisorption and III is chemisorption
(C) IV is chemisorption and II is chemisorption
(D) IV is chemisorption and III is chemisorption

ANSWER : AC

1. For the given aqueous reactions, which of the statement(s) is (are) true ? [Figure]
   (A) The first reaction is a recux reaction.
   (B) White precipitate is $\mathrm { Zn } _ { 3 } \left[ \mathrm { Fe } ( \mathrm { CN } ) _ { 6 } \right] _ { 2 }$.
   (C) Addition of filtrate to starch solution gives blue colour.
   (D) White precipitate is soluble in NaOH solution.

ANSWER : ACD

1. With respect to graphite and diamond, which of the statement(s) given below is (are) correct ?
   (A) Graphite is harder than diamond.
   (B) Graphite has higher electrical conductivity than diamond.
   (C) Graphite has higher thermal conductivity than diamond.
   (D) Graphite has higher C-C bond order than diamond.

ANSWER : BD

1. With reference to the scheme given, which of the given statment(s) about $\mathbf { T } , \mathbf { U } , \mathbf { V }$ and $\mathbf { W }$ is (are) correct ? [Figure]
   (A) T is soluble in hot aqueous NaOH
   (B) $\mathbf { U }$ is optically active
   (C) Molecular formula of $\mathbf { W }$ is $\mathrm { C } _ { 10 } \mathrm { H } _ { 18 } \mathrm { O } _ { 4 }$
   (D) $\mathbf { V }$ gives effervescence on treatment with aqueous $\mathrm { NaHCO } _ { 3 }$

ANSWER : ACD

37. The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes a nuclear reaction yielding element $\mathbf { X }$ as shown below. To which group, element X belongs in the periodic table?
$${ } _ { 29 } ^ { 63 } \mathrm { Cu } + { } _ { 1 } ^ { 1 } \mathrm { H } \rightarrow 6 { } _ { 0 } ^ { 1 } \mathrm { n } + \alpha + 2 { } _ { 1 } ^ { 1 } \mathrm { H } + \mathrm { X }$$
ANSWER : 8

38. When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its pyranose form is [Figure]

ANSWER : 8

1. $29.2 \% ( \mathrm { w } / \mathrm { w } ) \mathrm { HCl }$ stock solution has a density of $1.25 \mathrm {~g} \mathrm {~mL} ^ { - 1 }$. The molecular weight of HCl is $36.5 \mathrm {~g} \mathrm {~mol} ^ { - 1 }$. The volume $( \mathrm { mL } )$ of stock solution required to prepare a 200 mL solution of 0.4 M HCl is

ANSWER : 8

1. An organic compound undergoes first-order decomposition. The time taken for its decomposition to $1 / 8$ and $1 / 10$ of its initial concentration are $t _ { 1 / 8 }$ and $t _ { 1 / 10 }$ respectively. What is the value of $\frac { \left[ \mathrm { t } _ { 1 / 8 } \right] } { \left[ \mathrm { t } _ { 1 / 10 } \right] } \times 10 ? \quad \left( \right.$ take $\left. \log _ { 10 } 2 = 0.3 \right)$

ANSWER : 9

PART III : MATHEMATICS

SECTION I : Single Correct Answer Type

This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

40. Which of the given statement(s) about $\mathbf { N } , \mathbf { O } , \mathbf { P }$ and $\mathbf { Q }$ with respect to $\mathbf { M }$ is (are) correct ? [Figure]
M [Figure]
N [Figure]
0 [Figure]
P [Figure]
Q
(A) $\mathbf { M }$ and $\mathbf { N }$ are non-mirror image stereoisomers
(B) M and O are identical
(C) $\mathbf { M }$ and $\mathbf { P }$ are enantiomers
(D) $\mathbf { M }$ and $\mathbf { Q }$ are identical

ANSWER : ABC

PART III : MATHEMATICS

SECTION 1 : Single Correct Answer Type

This section contains $\mathbf { 8 }$ multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Moment of inertia of a circular wire of mass $M$ and radius $R$ about its diameter is
(1) $\mathrm{MR}^2 / 2$
(2) $MR^2$
(3) $2MR^2$
(4) $\mathrm{MR}^2 / 4$

The kinetic energy needed to project a body of mass $m$ from the earth surface (radius $R$) to infinity is
(1) $\mathrm{mgR}/2$
(2) $2\mathrm{mgR}$
(3) $\mathrm{mgR}$
(4) $\mathrm{mgR}/4$

Energy required to move a body of mass $m$ from an orbit of radius $2R$ to $3R$ is
(1) $\mathrm{GMm}/12\mathrm{R}^2$
(2) $\mathrm{GMm}/3\mathrm{R}^2$
(3) $\mathrm{GMm}/8\mathrm{R}$
(4) $\mathrm{GMm}/6\mathrm{R}$

The escape velocity of a body depends upon mass as
(1) $m^0$
(2) $m^1$
(3) $m^2$
(4) $m^3$

A cylinder of height 20 m is completely filled with water. The velocity of efflux of water (in $\mathrm{ms}^{-1}$) through a small hole on the side wall of the cylinder near its bottom is
(1) 10
(2) 20
(3) 25.5
(4) 5

Two spheres of the same material have radii 1 m and 4 m and temperatures 4000 K and 2000 K respectively. The ratio of the energy radiated per second by the first sphere to that by the second is
(1) $1 : 1$
(2) $16 : 1$
(3) $4 : 1$
(4) $5 : 3$

At what temperature is the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at $47^\circ\mathrm{C}$?
(1) 80 K
(2) 73 K
(3) 3 K
(4) 20 K

1 mole of a gas with $\gamma = 7/5$ is mixed with 1 mole of a gas with $\gamma = 5/3$, then the value of $\gamma$ for the resulting mixture is
(1) $7/5$
(2) $2/5$
(3) $24/16$
(4) $12/7$

If a spring has time period $T$, and is cut into $n$ equal parts, then the time period of each part will be
(1) $T\sqrt{n}$
(2) $T/\sqrt{n}$
(3) $nT$
(4) $T$