At time , terminal A in the circuit shown in the figure is connected to B by a keyand an
alternating current , with Aand ω = 500 rad s⁻¹ starts flowing in it with
𝑡𝑡 = 0
the initial direction
𝐼𝐼 (
s
𝑡𝑡
h
)
o
=
wn
𝐼𝐼
0i
c
n
o s
th
(
e
𝜔𝜔
f
𝑡𝑡
i
)
gure. A
𝐼𝐼
t0 t
=
=
1
, the key is switched from B to D. Now
7𝜋𝜋
onwards only A and D are connected. A total charge Q flows from the battery to charge the
6𝜔𝜔
capacitor fully. If C=20µF, R= 10 Ω and the battery is ideal with emf of 50V, identify the correct
statement (s).
B D
A
50 V
C=20 µF
R=10 Ω
A light source, which emits two wavelengths and , is used in a
Young’s double slit experiment. If recorded fringe widths for and are and © number of
𝜆𝜆1 = 400 𝑛𝑛𝑛𝑛 𝜆𝜆2 = 600 𝑛𝑛𝑛𝑛
fringes for them within a distance yon one side of the central maximum are m and m ,
𝜆𝜆1 𝜆𝜆2 𝛽𝛽1 𝛽𝛽2 1 2
respectively, then
One end of a taut string of length 3m along the x axis is fixed at x=0.The speed of the waves in
the string is . The other end of the string is vibrating in the y direction so that
stationary waves are −s1et up in the string. The possible waveform(s) of these stationary waves
100 𝑛𝑛𝑠𝑠
is(are)
A parallel plate capacitor has a dielectric slab of dielectric constant K between its plates that
covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor
is C while that of the portion with dielectric in between is C . When the capacitor is charged, the
1
plate area covered by the dielectric gets charge Q and the rest of the area gets charge Q . The
1 2
electric field in the dielectric is E and that in the other portion is E . Choose the correct
1 2
option/options, ignoring edge effects.
Q 1 E 1
Q 2 E 2
Let , and be the respective electric fields at a distance r from a point charge
Q, an infinitely long wire with constant linear charge density λ, and an infinite plane with
𝐸𝐸1(𝑟𝑟) 𝐸𝐸2(𝑟𝑟) 𝐸𝐸3(𝑟𝑟)
uniform surface charge density σ. If at a given distance r , then
0
𝐸𝐸1(𝑟𝑟0) = 𝐸𝐸2(𝑟𝑟0) = 𝐸𝐸3(𝑟𝑟0)
A student is performing an experiment using a resonance column and a tuning fork of
frequency . He is told that the air in the tube has been replaced by another gas (assume
that the column −re1mains filled with the gas). If the minimum height at which resonance occurs is
244𝑠𝑠
, the gas in the tube is
( 0.350±0.005) 𝑛𝑛
(Useful information: mole ; mole .The molar
1⁄2 −1⁄2 1⁄2 −1⁄2
masses Min grams are g√iv1e6n7 i𝑅𝑅n 𝑅𝑅th=e o6p4ti0o n𝐽𝐽s. Take the v√al1u4e0s 𝑅𝑅o𝑅𝑅f = 5f9o0r 𝐽𝐽each gas as given there.)
10
�𝑀𝑀
( )
Heater of an electric kettle is made of a wire of length L and diameter d. It takes 4 minutes to
raise the temperature of 0.5kgwater by 40K. This heater is replaced by a new heater having two
wires of the same material, each of length L and diameter 2d.The way these wires are connected
is given in the options. How much time in minutes will it take to raise the temperature of the
same amount of water by 40 K?
In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium
making an angle θ with the horizontal floor. The coefficient of friction between the wall and the
ladder is and that between the floor and the ladder is . The normal reaction of the wall on
the ladder is N and that of the floor is N . If the ladder is about to slip, then
𝜇𝜇1 1 2 𝜇𝜇2
µ
1
θ
µ
2
A transparent thin film of uniform thickness and refractive index n 1 = 1.4 is coated on the
convex spherical surface of radius Rat one end of a long solid glass cylinder of refractive index
n = 1.5, as shown in the figure. Rays of light parallel to the axis of the cylinder traversing
2
through the film from air to glass get focused at distance f from the film, while rays of light
1
traversing from glass to air get focused at distance f from the film. Then
2
n
1
Air n
2
Two ideal batteries of emfV 1 and V 2 and three resistancesR 1 , R 2 andR 3 are connected as
shown in the figure. The current in resistanceR would be zero if
2
V
1 R
1
R
2
V
2
R
3
Airplanes A and B are flying with constant velocity in the same vertical plane at angles 30°
and 60° with respect to the horizontal respectively as shown in figure. The speed of A is
. At time , an observer in A finds B at a distance of 500 m. This observer
sees B mov−i1ng with a constant velocity perpendicular to the line of motion of A. If at , A
100√3 𝑛𝑛𝑠𝑠 𝑡𝑡 = 0 𝑠𝑠
just escapes being hit by B, in seconds is
𝑡𝑡 = 𝑡𝑡0
𝑡𝑡0
A
B
30o 60o
During Searle’s experiment, zero of the Vernier scale lies between and
of the main scale. The 20th division of the Vernier scale exactly coinc−id2es with
3.20×10 𝑛𝑛
one of the −m²ain scale divisions. When an additional load of 2 kg is applied to the wire, the zero
3.25×10 𝑛𝑛
of the Vernier scale still lies between and of the main scale but
now the 45th division of Vernier scale coincides w−2ith one of the main −s2cale divisions. The length
3.20×10 𝑛𝑛 3.25×10 𝑛𝑛
of the thin metallic wire is 2 m and its cross-sectional area is . The least count of the
Vernier scale is . The maximum percentage error in t−h7e Y2oung’s modulus of the
8×10 𝑛𝑛
wire is −5
1.0×10 𝑛𝑛
A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal
frictionless surface. Three forces of equal magnitude F = 0.5 N are applied simultaneously along
the three sides of an equilateral triangle XYZ with its vertices on the perimeter of the disc (see
figure). One second after applying the forces, the angular speed of the disc in is
−1
𝑟𝑟𝑟𝑟𝑟𝑟𝑠𝑠
F
X
o
Y
F
Z
F
Two parallel wires in the plane of the paper are distance X 0 apart. A point charge is moving
with speed u between the wires in the same plane at a distance X from one of the wires. When
1
the wires carry current of magnitude I in the same direction, the radius of curvature of the path of
the point charge is R . In contrast, if the currents I in the two wires have directions opposite to
1
each other, the radius of curvature of the path is R . If = 3, the value of is
2
𝑋𝑋0 𝑅𝑅1
𝑋𝑋1 𝑅𝑅2
To find the distance d over which a signal can be seen clearly in foggy conditions, a railways
engineer uses dimensional analysis and assumes that the distance depends on the mass density ρ
of the fog, intensity (power/area) of the light from the signal and its frequency f. The engineer
finds that . The value of is
𝑆𝑆
1⁄𝑛𝑛
𝑟𝑟isproportionalto𝑆𝑆 𝑛𝑛
A galvanometer gives full scale deflection with 0.006 A current. By connecting it to a 4990
Ω resistance, it can be converted into a voltmeter of range 0 – 30 V. If connected to a Ω
2𝑛𝑛
resistance, it becomes an ammeter of range 0 – 1.5 A. The value of is
249
Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A
block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always
parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the
block when it reaches Q is Joules. The value of n is (take acceleration due to gravity =
)
(𝑛𝑛×10)
−2
1 0 𝑛𝑛𝑠𝑠 Q
4 m
90o
O 3 m P
A rocket is moving in a gravity free space with a constant acceleration of 2 along +x
direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is throw−2n from the
𝑛𝑛𝑠𝑠
left end of the chamber in +x direction with a speed of 0.3 relative to the rocket. At the
same time, another ball is thrown in –x direction with a speed −o1f 0.2 from its right end
𝑛𝑛𝑠𝑠
relative to the rocket. The time in seconds when the two balls hit each other −is1
𝑛𝑛𝑠𝑠
a = 2 ms ̶ 2
0.3 ms ̶ 1 0.2 ms ̶ 1 x
4 m
A horizontal circular platform of radius 0.5 m and mass 0.45 kg is free to rotate about its
axis. Two massless spring toy-guns, each carrying a steel ball of mass 0.05 kg are attached to the
platform at a distance 0.25 m from the centre on its either sides along its diameter (see figure).
Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in
opposite directions. After leaving the platform, the balls have horizontal speed of 9 with
respect to the ground. The rotational speed of the platform in after the balls le−a1ve the
𝑛𝑛𝑠𝑠
platform is −1
𝑟𝑟𝑟𝑟𝑟𝑟𝑠𝑠
A thermodynamic system is taken from an initial state i with internal energy to
the final state f along two different paths iaf and ibf, as schematically shown in the figure. The
𝑈𝑈𝑖𝑖 = 100 𝐽𝐽
work done by the system along the paths af, ib and bf are and
respectively. The heat supplied to the system along the path iaf, ib and bf are and
𝑊𝑊𝑟𝑟𝑓𝑓 = 200 𝐽𝐽,𝑊𝑊𝑖𝑖𝑖𝑖 = 50 𝐽𝐽 𝑊𝑊𝑖𝑖𝑓𝑓 =
respectively. If the internal energy of the system in the state b is and
100 𝐽𝐽 𝑄𝑄𝑖𝑖𝑟𝑟𝑓𝑓,𝑄𝑄𝑖𝑖𝑖𝑖
, the ratio is
𝑄𝑄𝑖𝑖𝑓𝑓 𝑈𝑈𝑖𝑖 = 200 𝐽𝐽 𝑄𝑄𝑖𝑖𝑟𝑟𝑓𝑓 =
5 00 𝐽𝐽 𝑄𝑄𝑖𝑖𝑓𝑓/𝑄𝑄𝑖𝑖𝑖𝑖
a f
P
i b
V
Answer : 2
PART – 2 CHEMISTRY
SECTION – 1 (One or More Than One Options Correct Type)
This section contains 10 multiple choice type questions. Each question has four
choices
The correct combination of names for isomeric alcohols with molecular formula
C H O is/are
4 10
In the reaction shown below, the major product(s) formed is/are
An ideal gas in a thermally insulated vessel at internal pressure = P 1 , volume = V 1
and absolute temperature = T expands irreversibly against zero external pressure, as
1
shown in the diagram. The final internal pressure, volume and absolute temperature of
the gas are P , V and T , respectively. For this expansion,
2 2 2
Hydrogen bonding plays a central role in the following phenomena:
In a galvanic cell, the salt bridge
Upon heating with Cu 2 S, the reagent(s) that give copper metal is/are
The correct statement(s) for orthoboric acid is/are
For the reaction:
I – + ClO – + H SO → Cl – + HSO – + I
3 2 4 4 2
The correct statement(s) in the balanced equation is/are:
The pair(s) of reagents that yield paramagnetic species is/are
Consider all possible isomeric ketones, including stereoisomers, of MW = 100. All
these isomers are independently reacted with NaBH (NOTE: stereoisomers are also
4
reacted separately). The total number of ketones that give a racemic product(s) is/are
A list of species having the formula XZ 4 is given below.
XeF , SF , SiF , BF –, BrF –, [Cu(NH ) ]2+, [FeCl ]2–, [CoCl ]2– and [PtCl ]2–.
4 4 4 4 4 3 4 4 4 4
Defining shape on the basis of the location of X and Z atoms, the total number of
species having a square planar shape is
Among PbS, CuS, HgS, MnS, Ag 2 S, NiS, CoS, Bi 2 S 3 and SnS 2 , the total number of
BLACK coloured sulfides is
The total number(s) of stable conformers with non-zero dipole moment for the
following compound is (are)
Consider the following list of reagents:
Acidified K Cr O , alkaline KMnO , CuSO , H O , Cl , O , FeCl , HNO and Na S O .
2 2 7 4 4 2 2 2 3 3 3 2 2 3
The total number of reagents that can oxidise aqueous iodide to iodine is
The total number of distinct naturally occurring amino acids obtained by
complete acidic hydrolysis of the peptide shown below is
In an atom, the total number of electrons having quantum numbers ,
and is
𝑛𝑛 = 4 |𝑛𝑛𝑙𝑙| =
1 𝑛𝑛𝑠𝑠 = − 1 �
2
If the value of Avogadro number is 6.023 × 1023 mol–1 and the value of Boltzmann
constant is 1.380 × 10–23J K–1, then the number of significant digits in the calculated
value of the universal gas constant is
A compound H 2 X with molar weight of 80 g is dissolved in a solvent having density of
0.4 g ml –1. Assuming no change in volume upon dissolution, the molality of a 3.2
molar solution is
MX 2 dissociates into M 2+ and X – ions in an aqueous solution, with a degree of
dissociation (α) of 0.5. The ratio of the observed depression of freezing point of the
aqueous solution to the value of the depression of freezing point in the absence of ionic
dissociation is
Answer: 2
PART – 3 MATHEMATICS
SECTION – 1 (One or More Than One Options Correct Type)
This section contains 10 multiple choice type questions. Each question has four
choices
Let and be two matrices such that . Further, if and , then
2 2 4
𝑀𝑀 𝑁𝑁 3×3 𝑀𝑀𝑁𝑁 =𝑁𝑁𝑀𝑀 𝑀𝑀 ≠𝑁𝑁 𝑀𝑀 = 𝑁𝑁
For every pair of continuous functions such that
𝑓𝑓,𝑚𝑚:[0,1 ]=→ ℝ ,
the correct statement(sm) iasx(a {r𝑓𝑓e()𝜋𝜋 :) :𝜋𝜋 ∈[0,1]} max {𝑚𝑚(𝜋𝜋):𝜋𝜋 ∈[0,1]}
Let be given by
𝑓𝑓:(0,∞)→ℝ
𝜋𝜋 1
−�𝑡𝑡+ 𝑡𝑡� 𝑟𝑟𝑡𝑡
𝑓𝑓(𝜋𝜋)= �1 𝑒𝑒 .
𝜋𝜋 𝑡𝑡
Then
Let and let be given by
𝑟𝑟 ∈ ℝ 𝑓𝑓: ℝ→ℝ
.
5
Then 𝑓𝑓(𝜋𝜋)= 𝜋𝜋 −5𝜋𝜋+𝑟𝑟
Let be a continuous function and let be defined as
𝑓𝑓:[𝑟𝑟,𝑖𝑖]→[1, ∞) 𝑚𝑚: ℝ→ℝ
0 if 𝜋𝜋 <𝑟𝑟,
⎧
𝜋𝜋
⎪
⎪
� 𝑓𝑓(𝑡𝑡)𝑟𝑟𝑡𝑡 if 𝑟𝑟 ≤𝜋𝜋 ≤𝑖𝑖,
𝑚𝑚(𝜋𝜋)= 𝑟𝑟
⎨
𝑖𝑖
⎪
⎪
� 𝑓𝑓(𝑡𝑡)𝑟𝑟𝑡𝑡 if 𝜋𝜋 >𝑖𝑖.
Then ⎩ 𝑟𝑟
Let be given by
𝜋𝜋 𝜋𝜋
𝑓𝑓:(−2, 2 )→ ℝ
.
3
Then 𝑓𝑓(𝜋𝜋)= (log(sec𝜋𝜋+ tan𝜋𝜋))
From a point perpendiculars and are drawn respectively on the lines
and . If is such that is a right angle, then the possible value(s) of is(are)
𝑃𝑃(𝜆𝜆,𝜆𝜆,𝜆𝜆), 𝑃𝑃𝑄𝑄 𝑃𝑃𝑅𝑅 𝑦𝑦 =𝜋𝜋, 𝑧𝑧=1
𝑦𝑦=−𝜋𝜋, 𝑧𝑧=−1 𝑃𝑃 ∠𝑄𝑄𝑃𝑃𝑅𝑅 𝜆𝜆
Let , and be three vectors each of magnitude and the angle between each pair of them is .
𝜋𝜋
If is a𝜋𝜋⃗ no𝑦𝑦⃗nzero𝑧𝑧 ⃗vector perpendicular to and a
√
n2d is a nonzero vector perpendicular to and
3
, then
𝑟𝑟⃗ 𝜋𝜋⃗ 𝑦𝑦⃗×𝑧𝑧⃗ 𝑖𝑖�⃗ 𝑦𝑦⃗
𝑧𝑧⃗×𝜋𝜋⃗
A circle passes through the point and is orthogonal to the circles and
2 2
.𝑆𝑆 Then (0,1) (𝜋𝜋−1) +𝑦𝑦 =16
2 2
𝜋𝜋 +𝑦𝑦 =1
Let be a symmetric matrix with integer entries. Then is invertible if
𝑀𝑀(A ) the 2fir×st2 column of is the transpose of the second r𝑀𝑀ow of
Let be positive integers such that is an integer. If are in geometric progression and the
𝑖𝑖
arithmetic mean of is , then the value of
𝑟𝑟,𝑖𝑖,𝑐𝑐 𝑟𝑟 𝑟𝑟,𝑖𝑖,𝑐𝑐
𝑟𝑟,𝑖𝑖,𝑐𝑐 𝑖𝑖+2
2
𝑟𝑟 +𝑟𝑟−14
is 𝑟𝑟+1
Let be an integer. Take distinct points on a circle and join each pair of points by a line
segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the
𝑛𝑛≥2 𝑛𝑛
number of red and blue line segments are equal, then the value of is
𝑛𝑛
Let be positive integers such that . Then
the number of such distinct arrangements ( is
𝑛𝑛1 < 𝑛𝑛2 < 𝑛𝑛3 < 𝑛𝑛4 < 𝑛𝑛5 𝑛𝑛1 + 𝑛𝑛2 + 𝑛𝑛3 + 𝑛𝑛4+ 𝑛𝑛5 =20
𝑛𝑛1, 𝑛𝑛2, 𝑛𝑛3, 𝑛𝑛4, 𝑛𝑛5)
Let and be respectively given by and . Define
by 2
𝑓𝑓:ℝ→ ℝ 𝑚𝑚:ℝ→ ℝ 𝑓𝑓(𝜋𝜋)=|𝜋𝜋|+ 1 𝑚𝑚(𝜋𝜋)= 𝜋𝜋 +1
ℎ:ℝ→ ℝ
max {𝑓𝑓(𝜋𝜋),𝑚𝑚(𝜋𝜋)} if 𝜋𝜋 ≤0,
ℎ(𝜋𝜋)=�
min {𝑓𝑓(𝜋𝜋),𝑚𝑚(𝜋𝜋)} if 𝜋𝜋 >0.
The number of points at which is not differentiable is
ℎ(𝜋𝜋)
The value of
1
2
3 𝑟𝑟 2 5
�4𝜋𝜋 � 2(1−𝜋𝜋 ) � 𝑟𝑟𝜋𝜋
0 𝑟𝑟𝜋𝜋
is
The slope of the tangent to the curve at the point is
5 2 2 2
(𝑦𝑦− 𝜋𝜋 ) =𝜋𝜋(1+𝜋𝜋 ) (1,3)
The largest value of the nonnegative integer for which
𝑟𝑟
1−𝜋𝜋
1−√𝜋𝜋
−𝑟𝑟𝜋𝜋+sin(𝜋𝜋−1)+ 𝑟𝑟 1
𝜋𝜋li→m1� � =
is 𝜋𝜋+ sin(𝜋𝜋−1)−1 4
Let be defined by . The number of points satisfying
the equation −1
𝑓𝑓:[0,4𝜋𝜋]→ [0,𝜋𝜋] 𝑓𝑓(𝜋𝜋)= cos (cos𝜋𝜋) 𝜋𝜋 ∈ [0,4𝜋𝜋]
10−𝜋𝜋
𝑓𝑓(𝜋𝜋)=
Is 10
For a point in the plane, let and be the distances of the point from the lines
and 𝑃𝑃 respective𝑟𝑟l 1 y(. 𝑃𝑃T)he ar𝑟𝑟e 2 a( 𝑃𝑃o)f the region consisting of all𝑃𝑃 points lying in the first
quadrant of the plane and satisfying is
𝜋𝜋−𝑦𝑦=0 𝜋𝜋+𝑦𝑦=0 𝑅𝑅 𝑃𝑃
2≤𝑟𝑟1(𝑃𝑃)+ 𝑟𝑟2 (𝑃𝑃)≤4 ,
Let and be three non-coplanar unit vectors such that the angle between every pair of them is
𝜋𝜋
. If 𝑟𝑟���⃗, 𝑖𝑖���⃗, �𝑐𝑐�⃗ where and are scalars, then the value of is 3
2 2 2
𝑝𝑝 + 2𝑞𝑞 + 𝑟𝑟
𝑟𝑟���⃗ × 𝑖𝑖���⃗ + 𝑖𝑖�⃗ × �𝑐𝑐�⃗ =𝑝𝑝𝑟𝑟���⃗ +𝑞𝑞𝑖𝑖�⃗+ 𝑟𝑟𝑐𝑐��⃗ , 𝑝𝑝,𝑞𝑞 𝑟𝑟 𝑞𝑞 2
