A thin uniform rod, pivoted at O, is rotating in the horizontal
plane with constant angular speed o, as shown in the figure.
Z
At time t = 0, a small insect starts from O and moves with
V -
constant speed v with respect to the rod towards the other
0
end. It reaches the end of the rod at t = T and stops. The
angular speed of the system remains w throughout. The
magnitude of the torque (l) on the system about O, as a
function of time is best represented by which plot ?.
Consider a thin spherical shell of radius R with its centre at the origin, carrying uniform positive
surface charge density. The variation of the magnitude of the electric field |E(r)| and the
electric potential V(r) with the distance r from the centre, is best represented by which graph ?
In the determination of Young's modulus
by using Searle's method, a wire of
tld2
length L = 2 m and diameter d == 0.5 mmis used. For a load M = 2.5 kg, an extensionl = 0.25 mm in
the length of the wire is observed. Quantities d and l are measured using a screw gauge and.
a micrometer, respectively. They have the same pitch of0.5 mm. The number of divisions on
their circular scale is 100. The contributions to the maximum probable error of the Y.
measurement
A small block is connected to one end of a massless spring of un-stretched length 4.9 m.
The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless
surface. The block is stretched by0.2 m and released from rest at t = 0. It then executes simple
harmonic motion with angular frequency @ =
T
- rad/s. Simultaneously att = 0, a small pebble
3
is projected with speedv from point P at an angle of 45 as shown in the figure. Point P is at a
horizontal distance of 10 m from O. If the pebble hits the block at t = 1 s, the value of v is
(take g = 10 m/s2)
45
10 m.
Young's double slit experiment is carried out by using green, red and blue light, one color at
a time. The fringe widths recorded are g, , and ,, respectively. Then,
A small mass m is attached to a massless string whose other end is fixed at P as shown in
the figure. The mass is undergoing circular motion in the x-y plane with centre at O and
constant angular speed . If the angular momentum of the system, calculated about O and P
are denoted by Lo and L,respectively, then
m
A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas
(atomic mass = 40 amu) is kept at 300 K in a container. The ratio of the rms speeds
Vrms(helium)
is
Vrms (argon)
Two large vertical and parallel metal plates having a separation of 1 cm are connected to a
DC voltage source of potential difference X. A proton is released at rest midway between the
two plates. It is found to move at 45 to the vertical JUST after release. Then X is nearly.
A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive
index n of the first lens is 1.5 and that of the second lens is 1.2. Both the curved surfaces are
of the same radius of curvature R = 14 cm. For this bi-convex lens, for an object distance of
40 cm, the image distance will be
n =1.5.
n =1.2
R=14 cm
A cubical region of side a has its centre at the origin. It encloses three fixed point charges,
--q at (0, --a/4,0), +3q at (0,0,0) and --q at (0,+a/4,0). Choose the correct option(s)..
b
3q
For the resistance network shown in the figure, choose the correct option(s).
I2
2 Q
2 Q
Ww
S
2 Q
10
4 Q
WW
4 Q
Q
4 Q
T
12 V
A small block of mass of 0.1 kg lies on a fixed inclined plane PQwhich makes an angle 0 with
the horizontal. A horizontal force of 1 N acts on the block through its center of mass as shown
in the figure. The block remains stationary if (take g = 10 m/s2)
1 N
0
0
Consider the motion of a positive point charge in a region where there are simultaneous.
uniform electric and magnetic fields E = E, j and B = B, j. At time t = 0, this charge has.
velocity in the x-y plane, making an angle 0 with the x-axis. Which of the following option(s)
is(are) correct for time t > 0 ?
A person blows into open-end of a long pipe. As a result, a high-pressure pulse of air travels
down the pipe. When this pulse reaches the other end of the pipe,.
An infinitely long solid cylinder of radius R has a uniform volume charge density p. 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 2R from the axis of the.
23pR . The value of k is
cylinder, is given by the expression.
16keo
4Z
R
R/
2R->
y
ANSWER : 6
A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure
If the magnitude of the magnetic field at the point P is given by -.
N
5 Ho aJ , then the value ot N is
20
ANSWER : 5
PHYSICS
A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass
density and radius 2R, as shown in the figure. The moment of inertia of this lamina about.
axes passing through O and P is I, and Ip, respectively. Both these axes are perpendicular
2R
2R P
0
ANSWER : 3
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 << R) having two turns is placed with its center at z =3 R along the axis of
the circular wire loop, as shown in figure. The plane of the square loop makes an angle of45
Ho a2
the value of p is
Z
3 R
ANSWER : 7
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, = (5/3) 102' kg;.
h/e = 4.2 x 101 J.s/C;
1
= 9 x 10 m/F; 1 fm = 10 m)
4TE0
ANSWER : 7
PART II : CHEMISTRY
SECTION I : Single Correct Answer Type
This section contains 10 multiple choice questions. Each question has four choices
In allene (C,H4), the type(s) of hybridisation of the carbon atoms is (are)
For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. 1/V plot is shown
below. The value of the van der Waals constant a (atm.liter2 mol-2) is
[Graph not to scale]
PV (liter-atm mol'l)
24.6
23.
20.1
0
2.0
3.0
1
(mol liter')
V
The number of optically active products obtained from the complete ozonolysis of the given
compound is
CH₃
H
CH₃ - CH = CH - C
CH = CH C CH = CHCH3
H
CH₃
A compound M,Xg has cubic close packing (ccp) arrangement of X. Its unit cell structure is
shown below. The empirical formula of the compound is
M =
X =
The number of aldol reaction(s) that occurs in the given transformation is
HO
-OH
CH3CHO + 4HCHO --
conc. aq. NaOH
HO--
OH
The colour of light absorbed by an aqueous solution of CuSO, is.
The carboxyl functional group (COOH) is present in
The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is
[a, is Bohr radius]
h2
h2
h2
h2
Which ordering of compounds is according to the decreasing order of the oxidation state of
nitrogen?
As per IUPAC nomenclature, the name of the complex [Co(H,O)(NH₃),]CIis
Identify the binary mixture(s) that can be separated into individual compounds, by differential
extraction, as shown in the given scheme.
NaOH (aq)
Compound 1
Compound 2
Binary mixture containing
Compound 1 and Compound 2
NaHCO3 (aq)
Compound 1
Compound 2
Choose the correct reason(s) for the stability of the lyophobic colloidal particles..
Which of the following molecules, in pure form, is (are) unstable at room temperature ?.
Which of the following hydrogen halides react(s) with AgNO,(aq) to give a precipitate that
dissolves in Na,S,O(aq) ?
For an ideal gas, consider only P-Vwork in going from an initial state X to the tinal state Z.
following choice(s) is (are) correct ? [take S as change in entropy and w as work done]
x
P (atmosphere)
Z
V (liter)
The substituents R, and R, for nine peptides are listed in the table given below. How many of
these peptides are positively charged at pH = 7.0 ?
H, N-CH-CO-NH-CH-CO-NH-CH-CO-NH-CH-COO
H
R,
R,
H
Peptide
R
R2
1
H
H
CH,
H
III
CH,COOH
H
IV
CH,CONH
(CH2) NH2
v
CH,CONH,
CH,CONH,
vI
hN'(H5)
(CH,),NH2
VII
CH,COOH
CH,CONH
VIII
CH,OH
HN"(HS)
IX
(CH) NH2
CHg
ANSWER : 4
The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons,
undergoes a nuclear reaction yielding element X as shown below. To which group, element
X belongs in the periodic table ?.
63 Cu+|H->6'n+a+2|H+X
ANSWER : 8
CHEMISTRY
When the following aldohexose exists in its D-configuration, the total number of stereoisomers
in its pyranose form is
CHO
CH2
CHOH
CHOH
CHOH
CH2OH
ANSWER : 8
39. 29.2% (w/w) HCI stock solution has a density of 1.25 g mL1 . The molecular weight of HCI
is 36.5 g mol⁻¹. The volume (mL) of stock solution required to prepare a 200 mL solution of.
0.4 M HCI is
ANSWER : 8
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/s and ty/o respectively.
[t18]
What is the value of
-x10?(take l0g12 = 0.3)
[tno]
ANSWER : 9
PART III : MATHEMATICS
SECTION I : Single Correct Answer Type
This section contains 10 multiple choice questions. Each question has four choices
The total number of ways in which 5 balls of different colours can be distributed among.
3 persons so that each person gets at least one ball is.
Let f(x)=
x
, xeR,
0, x=0
then f is
The functionf: [0, 3] -> [1, 29], detined by f(x) = 2x3 -- 15x2 + 36x + 1, is
If lim
- ax-b =4, then
x->8
x+1
Let z be a complex number such that the imaginary part of z is nonzero and a = z2 + z + 1 is
real. Then a cannot take the value
1
1
3|4
The ellipse E,
-1 is inscribed in a rectangle R whose sides are parallel to the
9
4
coordinate axes. Another ellipse E, passing through the point (0, 4) circumscribes the rectangle
R. The eccentricity of the ellipse E, is
2
3
1
3|4
Let P = [a;] be a 3x3 matrix and let Q=[by], where by = 2i+jay for 1 i, j 3. If the
determinant of P is 2, then the determinant of the matrix Q is
The integral
The point P is the intersection of the straight line joining the points Q(2,3,5) and
R(1, -- 1, 4) with the plane 5x -- 4y -- z = 1. If S is the foot of the perpendicular drawn from the
point T(2, 1, 4) to QR, then the length of the line segment PS is
The locus of the mid-point of the chord of contact of tangents drawn from points lying on the
straight line 4x -- 5y = 20 to the circle x2 + y2 = 9 is
Let 0, e [0, 2] be such that
0
0
2 cos0(1- sing)= sin?0
tan
+ cot
coso -1,
2
3
tan(2-0 )>0 and-1< sin 0 <
2
Then o cannot satisfy
r|2
4
4t
3t
3
>d>0 (v)
2
Let S be the area of the region enclosed by y = e*2, y = 0, x = 0, and x = 1. Then
A ship is fitted with three engines E,, E, and E,. The engines function independently of each
other with respective probabilities
and -- . For the ship to be operational at least two of
2' 4
its engines must function. LetX denote the event that the ship is operational and let X, X, and.
X, denote respectively the events that the engines E,, E, and E, are functioning. Which of the.
following is (are) true ?
7
Tangents are drawn to the hyperbola
- 1, parallel to the straight line 2x -- y = 1. The
9
4
points of contact of the tangents on the hyperbola are.
(222)
91
9
1
If y (x) satisfies the differential equation y - y tan x = 2x sec x and y(0) = 0, then.
t2
82
18
t2
4n
2t2
Let f: IR -> IRbe defined asf(x) =|x|+x2 -1. The total number of points at whichfattains.
either a local maximum or a local minimum is
ANSWER : 5
The value of 6 + Iog
is
32
3/2 V
32V
32
ANSWER : 4
Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local.
minimum at x = 3. If p(1) = 6 and p(3) = 2, then p'(0) is
ANSWER : 9
It a, b and are unit vectors satistying |a-5| +|5-| +|-a] = 9, then
2a + 5b+5c|is
ANSWER : 3
Let S be the focus of the parabola y2 = 8x and let PQ be the common chord of the circle
ANSWER : 4
