A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original
position after hitting the surface. The force on the ball during the collision is proportional to the
length of compression of the ball. Which one of the following sketches describes the variation of
its kinetic energy with time most appropriately? The figures are only illustrative and not to
the scale.
𝐾𝐾 𝑡𝑡
A wire, whichpasses through the hole in a small bead, is bent in the form of quarter of a circle.
The wire is fixed vertically on ground as shown in the figure. The bead is released from near the
top of the wire and it slides along the wire without friction. As the bead moves from A to B, the
force it applies on the wire is
A
90°
B
During an experiment with ametre bridge, the galvanometer shows a null point when the
jockey is pressed at using a standard resistance of ,as shown in the figure. The least
count of the scale used in the metre bridge is . The unknown resistance is
40.0 𝑐𝑐𝑚𝑚 90 𝛺𝛺
1 𝑚𝑚𝑚𝑚
R 90 Ω
40.0 cm
Charges Q, 2Q and 4Q are uniformly distributed in three dielectric solid spheres 1, 2 and 3 of
radii R/2, R and 2R, respectively, as shown in figure. If magnitudes of the electric fields at point
P at a distance R from the center of spheres 1, 2 and 3 are E , E and E respectively, then
1 2 3
P P P
R R R
2Q 4Q
Q
R/2
2R
Sphere 1 Sphere 2 Sphere 3
A point source S is placed at the bottom of a transparent block of height 10 mm and refractive
index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found
that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54
mm on the top of the block. The refractive index of the liquid is
Liquid
Block
S
Parallel rays of light of intensity are incident on a spherical black body kept in
surroundings of temperature 300 −K2. Take Stefan-Boltzmann constant
𝐼𝐼 = 912 𝑊𝑊𝑚𝑚
and assume that the energy exchange with the surroundingsis only
through radiati−o8n. The− 2fin−a4l steady state temperature of the black body is close to
𝜎𝜎 = 5.7 ×10 𝑊𝑊𝑚𝑚 𝐾𝐾
A metal surface is illuminated by light of two different wavelengths 248nm and 310 nm. The
maximum speeds of the photoelectrons corresponding to these wavelengths areu and
1
u ,respectively. If the ratio and hc = 1240 eV nm, the work function of the metal is
2
nearly
𝑢𝑢1:𝑢𝑢2 = 2:1
If is the wavelength of X-ray line of copper (atomic number 29) and is the
wavelength of the X-ray line of molybdenum (atomic number 42), then the ratio is
𝜆𝜆𝐶𝐶𝑢𝑢 𝐾𝐾𝛼𝛼 𝜆𝜆𝑀𝑀𝑀𝑀
close to
𝐾𝐾𝛼𝛼 𝜆𝜆𝐶𝐶𝑢𝑢/𝜆𝜆𝑀𝑀𝑀𝑀
A planet of radius (radius of Earth) has the same mass density as Earth. Scientists
1
dig a well of depth 𝑅𝑅on= it1 0a×nd lower a wire of the same length and of linear mass density
𝑅𝑅
into it. If the wire is not touching anywhere, the force applied at the top of the wire
5
by −a3 perso−n1 holding it in place is (take the radius of Earth = and the acceleration due
10 𝑘𝑘𝑘𝑘𝑚𝑚
to gravity on Earth is 10 ms ̶ 2) 6
6×10 𝑚𝑚
A glass capillary tube is of the shape of a truncated cone with an apex angle so that its two
ends have cross sections of different radii. When dipped in water vertically, water rises in it to a
𝛼𝛼
height h, where the radius of its cross section is . If the surface tension of water is S, its density
is ρ, and its contact angle with glass is , the value of h will be (g is the acceleration due to
𝑏𝑏
gravity)
𝜃𝜃
h
Consider the partition to be rigidly fixed so that it does not move. When equilibrium is
achieved, the final temperature of the gases will be
Now consider the partition to be free to move without friction so that the pressure of gases in
both compartments is the same. Then total work done by the gases till the time they achieve
equilibrium will be
If the piston is pushed at a speed of 5 mms ̶ 1, the air comes out of the nozzle with a speed of
If the density of air is and that of the liquid , then for a given piston speed the rate
(volume per unit time) at which the liquid is sprayed will be proportional to
𝑏𝑏𝑎𝑎 𝑏𝑏ℓ
When d≈a but wires are not touching the loop, it is found that the net magnetic field on the
axis of the loop is zero at a height h above the loop. In that case
Consider , and the loop is rotated about its diameter parallel to the wires by 30° from
the position shown in the figure. If the currents in the wires are in the opposite directions, the
𝑑𝑑 ≫ 𝑎𝑎
torque on the loop at its new position will be (assume that the net field due to the wires is
constant over the loop)
Four charges Q 1 , Q 2 , Q 3 and Q 4 of same magnitude are fixed along the x axis at
and , respectively. A positive charge q is placed on the positive y axis at a distance
. Four options of the signs of these charges are given in List I. The direction of t𝑥𝑥he= fo−r2ce𝑎𝑎s,
−on𝑎𝑎 t,h+e 𝑎𝑎ch arge+ q2 𝑎𝑎is given in List II. Match List I with List II and select the correct answer using
𝑏𝑏the> c0ode given below the lists.
q (0, b)
Q Q Q Q
1 2 3 4
( ̶2a, 0) ( ̶a, 0) (+a,0) (+2a,0)
List I List II
P. all positive 1.
Q.𝑄𝑄1,𝑄𝑄2, 𝑄𝑄po 3 s,i𝑄𝑄ti 4 ve; negative 2.+𝑥𝑥
R.𝑄𝑄1,𝑄𝑄2 positive; 𝑄𝑄3,𝑄𝑄4 negative 3.−𝑥𝑥
S.𝑄𝑄1,𝑄𝑄4 positive; 𝑄𝑄2,𝑄𝑄3 negative 4.+𝑦𝑦
𝑄𝑄1,𝑄𝑄3 𝑄𝑄2,𝑄𝑄4 −𝑦𝑦
Four combinations of two thin lenses are given in List I. The radius of curvature of all
curved surfaces is r and the refractive index of all the lenses is 1.5. Match lens combinations in
List I with their focal length in List II and select the correct answer using the code given below
the lists.
List I List II
P.
1.
2𝑟𝑟
Q.
2.
𝑟𝑟/2
R.
3.
−𝑟𝑟
S.
4.
𝑟𝑟
Choices:
A block of mass m 1 = 1 kg another mass m 2 = 2 kg, are placed together (see figure) on an
inclined plane with angle of inclination θ. Various values of θ are given in List I. The coefficient
of friction between the block m and the plane is always zero. The coefficient of static and
1
dynamic friction between the block m and the plane are equal to µ = 0.3. In List II expressions
2
for the friction on block m are given. Match the correct expression of the friction in List II with
2
the angles given in List I, and choose the correct option. The acceleration due to gravity is
denoted by g.
[Useful information :
o o o
tan(5.5 ) ≈ 0.1; tan(11.5 ) ≈ 0.2; tan(16.5 ) ≈ 0.3]
m
1
m
2
θ
List I
List II
P. θ = 5o 1.
Q.θ = 10o 2.𝑚𝑚2𝑘𝑘 sin𝜃𝜃
R.θ = 15o 3.(𝑚𝑚1+𝑚𝑚2)𝑘𝑘 sin𝜃𝜃
S.θ = 20o
4.𝜇𝜇𝑚𝑚2𝑘𝑘 cos𝜃𝜃
𝜇𝜇(𝑚𝑚1+𝑚𝑚2)𝑘𝑘 cos𝜃𝜃
Choices:
A person in a lift is holding a water jar, which has a small hole at the lower end of its side.
When the lift is at rest, the water jet coming out of the hole hits the floor of the lift at a distance
of 1.2 m from the person. In the following, state of the lift’s motion is given in List I and the
distance where the water jet hits the floor of the lift is given in List II. Match the statements from
List I with those in List II and select the correct answer using the code given below the list.
List I List II
P. Lift is accelerating vertically up. 1. m
Q. Lift is accelerating vertically down with an 2. 𝑑𝑑 = 1.2 m
acceleration less than the gravitational acceleration.
𝑑𝑑 > 1.2
3. m
R. Lift is moving vertically up with constant speed.
S. Lift is falling freely. 4. N𝑑𝑑o< w1a.t2e r leaks out of the jar
The acidic hydrolysis of ether (X) shown below is fastest when
[Figure]
Isomers of hexane, based on their branching, can be divided into three distinct
classes as shown in the figure.
[Figure]
The correct order of their boiling point is
The major product in the following reaction is
[Figure]
Hydrogen peroxide in its reaction with KIO 4 and NH 2 OH respectively, is acting as a
The product formed in the reaction of SOCl 2 with white phosphorous is
Under ambient conditions, the total number of gases released as products in the final
step of the reaction scheme shown below is
For the identification of β-naphthol using dye test, it is necessary to use
For the elementary reaction M → N, the rate of disappearance of M increases by a
factor of 8 upon doubling the concentration of M. The order of the reaction with respect
to M is
For the process
H O (l) → H O (g)
2 2
at T = 100°C and 1 atmosphere pressure, the correct choice is
Assuming 2s-2p mixing is NOT operative, the paramagnetic species among the
following is
The product X is
The correct statement with respect to product Y is
M1, Q and R, respectively are
Reagent S is
The value of d in cm (shown in the figure), as estimated from Graham’s law, is
The experimental value of d is found to be smaller than the estimate obtained using
Graham’s law. This is due to
Different possible thermal decomposition pathways for peroxyesters are shown
below. Match each pathway from List I with an appropriate structure from List II and
select the correct answer using the code given below the lists.
List-I List-II
P. Pathway P
Q. Pathway Q
R. Pathway R
S. Pathway S
Codes:
P Q R S
Match the four starting materials (P, Q, R, S) given in List I with the corresponding
reaction schemes (I, II, III, IV) provided in List II and select the correct answer using the
code given below the lists.
List-I List-II
Codes:
P Q R S
Match each coordination compound in List-I with an appropriate pair of
characteristics from List-II and select the correct answer using the code given below the
lists.
{en = H NCH CH NH ; atomic numbers: Ti = 22; Cr = 24; Co = 27; Pt = 78}
2 2 2 2
List-I List-II
P. [Cr(NH 3 ) 4 Cl 2 ]Cl 1. Paramagnetic and exhibits ionisation isomerism
Q. [Ti(H 2 O) 5 Cl](NO 3 ) 2 2. Diamagnetic and exhibits cis-trans isomerism
R. [Pt(en)(NH 3 )Cl]NO 3 3. Paramagnetic and exhibits cis-trans isomerism
S. [Co(NH 3 ) 4 (NO 3 ) 2 ]NO 3 4. Diamagnetic and exhibits ionisation isomerism
Codes:
P Q R S
Match the orbital overlap figures shown in List-I with the description given in List-II
and select the correct answer using the code given below the lists.
List-I List-II
1. p─d π antibonding
2. d─d σ bonding
3. p─d π bonding
4. d─d σ antibonding
Codes:
P Q R S
The function is the solution of the differential equation
𝑦𝑦=𝑓𝑓(𝑥𝑥)
4
𝑑𝑑𝑦𝑦 𝑥𝑥𝑦𝑦 𝑥𝑥 + 2𝑥𝑥
+ 2 = 2
in satisfying Then 𝑑𝑑𝑥𝑥 𝑥𝑥 − 1 √1−𝑥𝑥
(−1,1) 𝑓𝑓(0)= 0.
√3
2
� 𝑓𝑓(𝑥𝑥) 𝑑𝑑𝑥𝑥
√3
− 2
is
The following integral
𝜋𝜋
2
17
�(2 cosec 𝑥𝑥) 𝑑𝑑𝑥𝑥
𝜋𝜋
4
is equal to
Coefficient of in the expansion of is
11 2 4 3 7 4 12
Let be a function which is continuous on and is differentiable on with
Let
𝑓𝑓:[0,2]→ℝ [0,2] (0,2) 𝑓𝑓(0)=
1.
2
𝑥𝑥
𝐹𝐹(𝑥𝑥)= � 𝑓𝑓(√𝑡𝑡 ) 𝑑𝑑𝑡𝑡
0
for . If for all , then equals
′ ′
The common tangents to the circle and the parabola touch the circle at the points
and the parabola at the points . T2hen 2the area of the quadrilate2ral is
𝑥𝑥 +𝑦𝑦 =2 𝑦𝑦 =8𝑥𝑥
𝑃𝑃(,A𝑄𝑄) 𝑅𝑅,𝑆𝑆 𝑃𝑃𝑄𝑄𝑅𝑅𝑆𝑆
For , the equation has
In a triangle the sum of two sides is and the product of the same two sides is If ,
where is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the2 triang2le is
𝑥𝑥 𝑦𝑦. 𝑥𝑥 − 𝑐𝑐 =𝑦𝑦
𝑐𝑐
Six cards and six envelopes are numbered and cards are to be placed in envelopes so
that each envelope contains exactly one card and no card is placed in the envelope bearing the same
1,2,3,4,5,6
number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number
of ways it can be done is
Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every
girl is at least one more than the number of girls ahead of her, is
The quadratic equation with real coefficients has purely imaginary roots. Then the equation
𝑝𝑝(𝑥𝑥)= 0
𝑝𝑝�𝑝𝑝(𝑥𝑥)�= 0
has
The v(a2l𝑎𝑎u,e0 )of is
Choices: 𝑟𝑟
If , then the tangent at and the normal at to the parabola meet at a point whose ordinate is
𝑠𝑠𝑡𝑡 =1 𝑃𝑃 𝑆𝑆
The value of is
1
𝑘𝑘�2�
The value of is
′ 1
𝑘𝑘 �2�
The probability that is odd, is
𝑥𝑥1+𝑥𝑥2+ 𝑥𝑥3
The probability that are in an arithmetic progression, is
𝑥𝑥1,𝑥𝑥2,𝑥𝑥3
Let ;
2kπ 2𝑘𝑘𝜋𝜋
𝑧𝑧𝑘𝑘 = cos�10�+ 𝑖𝑖 sin�10� 𝑘𝑘 =1,2,…,9.
List I List II
P. For each there exists a such that 1. True
𝑧𝑧𝑘𝑘 𝑧𝑧𝑗𝑗 𝑧𝑧𝑘𝑘 ∙ 𝑧𝑧𝑗𝑗
Q. There exists a such that 2. False
has no solution𝑘𝑘 ∈in {t1h,e2 ,s…et, 9o}f complex n𝑧𝑧u 1 m∙𝑧𝑧be=rs.
𝑧𝑧𝑘𝑘 𝑧𝑧 3. 1
R. equals
|1−𝑧𝑧1||1−𝑧𝑧2| ⋯|1−𝑧𝑧9|
10
S. equals 4. 2
9 2𝑘𝑘𝜋𝜋
1− ∑𝑘𝑘=1cos�10�
P Q R S
List I List II
P. The number of polynomials with non-negative integer coefficients 1. 8
of degree , satisfying and , is
𝑓𝑓(𝑥𝑥)
Q. The number of points in the interval 1 at which 2. 2
≤2 𝑓𝑓(0)=0 ∫0 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑥𝑥 =1
attains its maximum value, is
R. 2 2 equals
[−√13,�13] 𝑓𝑓(𝑥𝑥)=
3. 4
sin(𝑥𝑥 )+c2os(𝑥𝑥 )
2 3𝑥𝑥
∫−2 (1+𝑒𝑒 𝑥𝑥 ) 𝑑𝑑𝑥𝑥 4. 0
1
S. 2 1+𝑥𝑥 equals
�∫
−
1 cos2𝑥𝑥 log�1−𝑥𝑥� 𝑑𝑑𝑥𝑥�
2
1
2 1+𝑥𝑥
�∫0 cos2𝑥𝑥 log�1−𝑥𝑥� 𝑑𝑑𝑥𝑥�
P Q R S
List I List II
1. 1
P. Let . Then
−1 √3 1 2
𝑦𝑦(𝑥𝑥)=cos(3 e cqousals 𝑥𝑥),𝑥𝑥 ∈[−1,1],𝑥𝑥 ≠± 2 𝑦𝑦(𝑥𝑥)�(𝑥𝑥 −
2
𝑑𝑑 𝑦𝑦(𝑥𝑥) 𝑑𝑑𝑦𝑦(𝑥𝑥)
Q. Let ( be the vertices of a regular polygon of sides 2. 2
1) 𝑑𝑑𝑥𝑥 2 +𝑥𝑥 𝑑𝑑𝑥𝑥 �
with its centre at the origin. Let be the position vector of the point
𝐴𝐴1,𝐴𝐴2,…,.𝐴𝐴 I 𝑛𝑛 f 𝑛𝑛>2) , then th𝑛𝑛e
minimum value of is𝑛𝑛 −1
𝑎𝑎����𝑘𝑘⃗
𝑛𝑛−1
𝐴𝐴𝑘𝑘, 𝑘𝑘 =1,2,…,𝑛𝑛 |∑𝑘𝑘=1( 𝑎𝑎����𝑘𝑘⃗×𝑎𝑎����𝑘𝑘��+���1⃗ )|= |∑𝑘𝑘=1( 𝑎𝑎����𝑘𝑘⃗∙𝑎𝑎����𝑘𝑘��+���1⃗ )| 3. 8
R. If the normal from the point on the ellipse is
𝑛𝑛 2 2
𝑥𝑥 𝑦𝑦
perpendicular to the line , then the value of is
𝑃𝑃(ℎ,1) 6 + 3 =1
S. Number of positive solutions satisfying the equation 4. 9
𝑥𝑥+𝑦𝑦=8 ℎ
is
−1 1 −1 1 −1 2
tan �2𝑥𝑥+1�+ tan �4𝑥𝑥+1�= tan �𝑥𝑥 2�
P Q R S
Let , , and be defined by
𝑓𝑓1:ℝ→ℝ 𝑓𝑓2:[0,∞)→ℝ 𝑓𝑓3:ℝ→ℝ 𝑓𝑓4:ℝ→[0,∞)
𝑓𝑓1(𝑥𝑥)=�
|𝑥𝑥|
𝑥𝑥
if 𝑥𝑥 <0,
𝑒𝑒; if 𝑥𝑥 ≥0;
2
𝑓𝑓2(𝑥𝑥)=𝑥𝑥
𝑓𝑓3(𝑥𝑥)=�
sin𝑥𝑥 if 𝑥𝑥 <0,
and 𝑥𝑥 if 𝑥𝑥 ≥0
𝑓𝑓2�𝑓𝑓1(𝑥𝑥)� if 𝑥𝑥 <0,
𝑓𝑓4(𝑥𝑥)=�
𝑓𝑓2�𝑓𝑓1(𝑥𝑥)�−1 if 𝑥𝑥 ≥0.
List II List I
P. is 1. onto but not one-one
Q.𝑓𝑓4
is 2. neither continuous nor one-one
R.
𝑓𝑓3
is 3. differentiable but not one-one
S.
𝑓𝑓2 𝑀𝑀is𝑓𝑓1
4. continuous and one-one
𝑓𝑓 2 P Q R S
