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.

A spring of force constant $800 \mathrm{~N/m}$ has an extension of 5 cm. The work done in extending it from 5 cm to 15 cm is
(1) 16 J
(2) 8 J
(3) 32 J
(4) 24 J

A ball whose kinetic energy is $E$, is projected at an angle of $45^\circ$ to the horizontal. The kinetic energy of the ball at the highest point of its flight will be
(1) $E$
(2) $E / \sqrt{2}$
(3) $E / 2$
(4) zero

Two identical particles move towards each other with velocity $2v$ and $v$ respectively. The velocity of centre of mass is
(1) $v$
(2) $v/3$
(3) $v/2$
(4) zero

Initial angular velocity of a circular disc of mass $M$ is $\omega_1$. Then two small spheres of mass $m$ are attached gently to diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?
(1) $\left( \frac{M+m}{M} \right) \omega_1$
(2) $\left( \frac{M+m}{m} \right) \omega_1$
(3) $\left( \frac{M}{M+4m} \right) \omega_1$
(4) $\left( \frac{M}{M+2m} \right) \omega_1$

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) $T$

Length of a string tied to two rigid supports is 40 cm. Maximum length (wave length in cm) of a stationary wave produced on it is
(1) 20
(2) 80
(3) 40
(4) 120

Tube A has both ends open while tube B has one end closed, otherwise they are identical. The ratio of fundamental frequency of tube $A$ and $B$ is
(1) $1 : 2$
(2) $1 : 4$
(3) $2 : 1$
(4) $4 : 1$

A tuning fork arrangement (pair) produces 4 beats/sec with one fork of frequency 288 cps. A little wax is placed on the unknown fork and it then produces 2 beats/sec. The frequency of the unknown fork is
(1) 286 cps
(2) 292 cps
(3) 294 cps
(4) 288 cps

A wave $y = a\sin(\omega t - kx)$ on a string meets with another wave producing a node at $x = 0$. Then the equation of the unknown wave is
(1) $y = a\sin(\omega t + kx)$
(2) $y = -a\sin(\omega t + kx)$
(3) $y = a\sin(\omega t - kx)$
(4) $y = -a\sin(\omega t - kx)$

On moving a charge of 20 coulombs by $2\mathrm{~cm}$, $2\mathrm{~J}$ of work is done, then the potential difference between the points is
(1) 0.1 V
(2) 8 V
(3) 2 V
(4) 0.5 V

If a charge $q$ is placed at the centre of the line joining two equal charges $Q$ such that the system is in equilibrium then the value of $q$ is
(1) $Q/2$
(2) $-Q/2$
(3) $Q/4$
(4) $-Q/4$

If there are $n$ capacitors in parallel connected to $V$ volt source, then the energy stored is equal to
(1) $CV$
(2) $\frac{1}{2}\mathrm{n}CV^2$
(3) $CV^2$
(4) $\frac{1}{2n}CV^2$

Capacitance (in $F$) of a spherical conductor with radius 1 m is
(1) $1.1 \times 10^{-10}$
(2) $10^{-6}$
(3) $9 \times 10^{-9}$
(4) $10^{-3}$

A wire when connected to 220 V mains supply has power dissipation $\mathrm{P}_1$. Now the wire is cut into two equal pieces which are connected in parallel to the same supply. Power dissipation in this case is $P_2$. Then $P_2 : P_1$ is
(1) 1
(2) 4
(3) 2
(4) 3