16-06-2024

1001CJA106220240005 JA

PART-1 : PHYSICS

SECTION-I (i)

1) In the Arctic region hemispherical houses called Igloos are made of ice. It is possible to maintain
a temperature inside an Igloo as high as 20°C because :-

(A) ice has high thermal conductivity.

(B) ice has low thermal conductivity.
(C) ice has high specific heat capacity.
(D) ice has higher density than water.

2) Let the temperature of a room be 6°C in winter and 27°C in summer. Find the weight of the air in
the room (of volume 10 m3) in two seasons. (Assume the air pressure in the room is equal to 100 kPa
during the two seasons.) Air has a molar mass of 30 gm/mol.

(A) 12.9 kg and 12 kg

(B) 13.1 kg and 11.8 kg
(C) 12.6 kg and 12 kg
(D) 11.8 kg and 11.2 kg

3) Two similar flywheels are made of the same metal, and the linear dimensions of the second one
are twice as large as the linear dimensions of the first one. The angular velocity of both the wheels is
the same.

(A) The kinetic energy of the second one is four times that of the first one.
(B) The kinetic energy of the second one is eight times that of the first one.
(C) The kinetic energy of the second one is thirty two times that of the first one.
(D) The kinetic energy of the second one is sixteen times that of the first one.

4) Three thin plates made of same material, each of thickness t, are welded together as shown.
Knowing the total mass of the assembly in m, compute its mass moment of inertia about the z-axis.

Assume t<<b.

(A)
(B)

(C)

(D)

5) Two identical bodies are made of a material for which the heat capacity increases with
temperature. One of these is held at a temperature of 100°C while the other one is kept at 0°C. If the
two are brought into contact, then, assuming no heat loss to the environment, the final temperature
that they will reach is :-

(A) 50°C
(B) less than 50°C
(C) more than 50°C
(D) 0°C

6) A cylindrical steel rod of length 0.10 m and thermal conductivity 50 W.m–1.K–1 is welded end to end
to copper rod of thermal conductivity 400 W.m–1.K–1 and of the same area of cross section but 0.20 m
long. The free end of the steel rod is maintained at 100°C and that of the copper rod at 0°C.
Assuming that the rods are perfectly insulated from the surrounding, the temperature at the junction
of the two rods is :-

(A) 20°C
(B) 30°C
(C) 40°C
(D) 50°C

SECTION-I (ii)

1) An ideal gas undergoes a quasi-static cycle process as shown in the figure, where AB, CD, EF are
isothermal processes with temperatures T1, T2 and T3 respectively. BC, DE and FA are adiabatic
processes. It is known that T1 = 800K, T2 = 400K, T3 = 200K; the system absorbs heat Q1 = 1600J

during the AB process, and Q2 = 800J during the CD process.

(A) The heat Q3 released by the system during the isothermal process of EF is 800 J.
(B) The work done by the gas during the whole cycle is 1200 J.
(C) The efficiency of the cycle is 66.66% approx.
(D) There is no heat released during process BC.
2) A sphere of radius 20 cm is kept at a constant temperature of 400 K. A highly conducting sheet of
radius 40 cm is kept at a distance of 30 cm from the centre of the sphere. Assume both the bodies

radiate like black bodies. ( W/m2K4)

(A)
The steady state temperature of sheet is .

(B)
The steady state temperature of sheet is .
(C) The heat radiated by sheet is equal on both sides.
(D) 20% of energy radiated by sphere falls on the sheet.

3) Two compressors adiabatically compress a diatomic gas. First, one compressor works,
compressing the gas from volume V0 to intermediate volume V1. Then the compressed gas is cooled
to the initial temperature isochorically after which the second compressor enters operation,
compressing the gas to volume V2. At what volume V1 is the total work of both compressors minimum
and what is it equal to? The volumes V0 and V2 are assumed to be fixed, the initial gas pressure is P0.

(A)

(B)

(C)

(D)

4) 2 moles of an ideal monoatomic gas, with an initial temperature of T0 = 200K expands in the
process where gas pressure directly proportional to its volume. Take the universal gas constant
equal to 25/3 J/(mol‧K). Its volume has increased to 1.5 times in this process.

(A) Molar heat capacity is 50/3 J/mol K.

(B) Final temperature of gas is 450 K.
(C) Work done by gas is 6000 J.
(D) Final pressure is 1.5 times the initial pressure.

5) Two spherical black bodies A and B having radii rA and rB where rB = 2rA emit radiation with peak
intensities at wavelength 400 nm and 800 nm respectively. If their temperature are TA and TB
respectively in Kelvin scale, their emissive powers are EA and EB and energies emitted per second PA
and PB then :- (both have same emissivity)

(A)
(B)

(C)

(D)

6) A hollow hemisphere of mass m and radius R lies as shown in the figure. The axis AC is passing
through the centre C of the hollow hemisphere and line AB is tangent on the hollow hemisphere.

Then choose the correct option(s).

Moment of inertia about an axis passing through the centre C and parallel to the line AB is
(A)
.
Moment of inertia about an axis passing through the centre of mass of hollow hemisphere and
(B)
parallel to the line AB is .

(C)
Moment of inertia about the line AB is .

(D)
Moment of inertia about the line AB is .

SECTION-II

1) A uniform horizontal rod is at rest on a rough inclined plane as shown. The total contact force (in
N) exerted by the inclined surface on the rod is _____ N. The string on the left is massless,

inextensible and vertical.

2) In a long smooth heat-insulated pipe there are heat-insulated pistons of mass m1 and m2, between
which a monatomic gas is located in volume V0 at pressure P0. Pistons are released. Determine
the maximum speeds of piston of mass m1 (in m/s), if the gas mass is much less than the mass of
each piston. The surrounding is perfect vacuum. (Given : P0 = 105 Pa, V0 = 1/3 litre, m1 = 1 kg, m2 =

1/3 kg)
3) In the ocean there is a boat with a piece of ice weighing 1 kg at 0°C on board. Determine the work
(in J) that is done by atmosphere on melting ice till it melts completely. (Given : ρice = 900 kg/m3,
ρwater = 1000 kg/m3, Patm = 100 kPa)

4) Three metallic blocks A, B and C have masses m, m and 2m respectively. Specific heat of A. B and
C are C, 2C and C respectively. Initial temperature of A. B and C are 10°C, 5°C and 5°C respectively.
Now the blocks are connected by 3-identical rods as shown. Find the final temperature of block A on

Celsius scale. (Neglect any heat loss from system.)

5) The combined mass center of the motorcycle and the cyclist is located at G. Find the smallest
acceleration for which the cyclist can perform a “wheelie” that is, raise the front wheel off the

ground. (g = 10 m/s2)

6) A small sphere (outside diameter = 50 mm) with a surface temperature of 277°C is located at
the geometric centre of a large sphere (inside diameter = 250 mm) with an inner surface
temperature of 7°C. Calculate how much of emission (in %) from the inner surface of the large
sphere is incident upon the outer surface of the small sphere; assume that both sides approach black
body behaviour.

PART-2 : CHEMISTRY

SECTION-I (i)

1) Ion exchange Resins are used to soften water. They contain sodium ions which get exchanged
with 'hard' ions like Ca2+ and Mg2+. Resins are not 100% efficient i.e. all sodium ions present in the
resin do not get exchanged at once and may need repeated passage of a solution through the column
to attain full efficiency.
Molecular formula of a commercial ion exchange resin is C7H6SO3Na. A 100 cm3 solution containing
0.3 mol L–1 of Mg2+ is passed through a column of ion-exchange resin weighing 20 g only once. What
are the molarities of Mg2+ and Na+, respectively, in the solution obtained after passing through the
column, if the exchange efficiency is only 25%.

(A) 0.13 and 0.26 M

(B) 0.26 M and 0.17 M
(C) 0.17 M and 0.26 M
(D) 0.21 M and 0.14 M

2) When a certain weight of solid potassium permanganate was treated in mild basic medium with
an excess of hydrogen peroxide at 1 atm, 0°C, the volume of oxygen formed was 168 L. What is the
weight in kg of potassium permanganate used ?
[Molar mass of KMnO4 = 158 gm/mole]

(A) 3.16 kg
(B) 0.158 kg
(C) 0.790 kg
(D) 7.90 kg

3) An element "A" of periodic table contains exactly 20 electrons having ℓ = 2. If "A" forms
predominantly acidic oxides only and both A and A2+ are paramagnetic, then which of the following
is correct about "A".

(A) Maximum value of degeneracy in valence shell of A is 7

(B) A can form oxide of the type AO4
(C) Valence electrons of A are always greater than 6
(D) EA1 of A > EA1 of oxygen atom

4) Which of the following oxy-salts will involve inter-molecular hydrogen bonding?

(A) NaH2PO2
(B) Na4HP3O10
(C) Na2H2P2O5
(D) Na2HPO3

5) Which of the following represent correct order of heat of combustion?

(A)
>

(B)
>

(C)
>

(D)
> CH2 = CH2

6) Which of the following can work like electron donating group in conjugation.

(A)
(B)

(C) –F

(D)

SECTION-I (ii)

1) Which of the following statement(s) is/are correct :

(A)
Larger the value of of a gas, large would be excluded volume.

(B)
for a non-ideal gas.
(C) Gas following P(V – b) = RT can be liquified at lower temperatures.
Liquification of NH3(g) is difficult than H2O(g) because size of NH3 molecule is greater than H2O
(D)
molecule.

2) Select incorrect statement(s) among the following :

(A) Kinetic energy of each molecule of O2 at 400 K is greater than each molecule of H2 at 300 K.
(B) Vrms of He gas at 300 K is twice to that of CH4 gas at 600 K.
(C) Non-ideal gas behaves ideally at high temperature and high pressure.
(D) Gases with higher values of van der Waal's constants 'a' and 'b' are easier to liquify.

3) Consider the following reaction.

Given ;
(i) "A" is isostructural with P4O10 and "B" is isostructural with BF3
(ii) "B" acts as a Lewis acid & "C" acts as Lewis base which contains only N and H atoms and is
isoelectronic with H2O2.
Then which of the following is/are correct?

(A) NN(PCl3)2 & PSCl3 both contains 2pπ-3dπ bonds.

(B) (σ bonds / π bonds) ratio is more in "A" as compared to NN(PCl3)2
(C) All bond lengths in cationic part of [BCl4][PCl4] are same.
(D) Both "C" and H2O2 are non-planar

4) Which of the following set(s) contain only acidic oxides and of p-block only.
(A) NO2 , Br2O5 , In2O3 , CO2
(B) CrO3 , I2O7 , P4O6 , Mn2O7
(C) N2O3 , SeO2 , XeO3 , SO3
(D) Cl2O , P4O10 , SO2 , N2O5

5) Which of the following are more basic than

(A)

(B)

(C)

(D)

6) Which of the following represent correct order :

(A)
Stability of alkene

(B)
(C=O) bond length

(C)

Resonance energy
(D)
(C = C) bond strength

SECTION-II

1) 10 litre, 2M H2O2 solution is left open till oxygen gas liberated can burn 20 litre CO gas at STP
completely. Another 30 litre, 1M H2O2 solution is mixed in the above resulting solution. Calculate
the volume strength of H2O2 solution formed.

2) The vaccine needs to be stored in dry ice. The dry ice, which is solid Carbon dioxide occupies
2840 L when it completely sublimes at 27°C and 1 atm.
If this dry ice is to be obtained from limestone having 80% CaCO3 content, what mass (in kg) of
limestone will be required ?
[Use : R = 0.08 atm lit/K-mole]

3) Number of species which have more bond order than CO2 between any two atoms among the
following.
HN3 , OCN– , , CO , SNF3 , P4S7 , D3PO4 , CH3CN ,

4) Consider the following orders

(i) (Dipole moment)

(ii) BF3 > BMe3 (Boiling point)
(iii) H2O2 > H2O (Strength of H-bond)
(iv) Cl2 > F2 (Ease of liquefaction)
(v) C(CN)4 < C6H6 (Ratio of π bonds to σ bonds)
(vi) (Bond order)
Number of correct order(s) will be :-

5) How many of the following represent aromatic compound

(i) (ii) (iii)

(iv) (v) (vi)

(vii) (viii) (ix)

6) How many of the following can evolve CO2 gas when react with NaHCO3 :

(i) (CH3)3C–COOH (ii) (iii) Me–SO3H

(iv) (v) (vi)

(vii) HC ≡ CH (viii)

PART-3 : MATHEMATICS

SECTION-I (i)

1) Let α, β and γ be the length of sides of a triangle ΔPQR. The possible value of

is equal to (where [.] represent greatest integer function)

(A) 2
(B) 5
(C) 4
(D) 1

2) Let x, y, z are unequal even numbers then minimum value of

is

(A) 64
(B) 128
(C) 256
(D) 512

3) If P = (tan3n+1θ – tanθ) and then

(A) P = 3Q
(B) P = 2Q
(C) 2P = Q
(D) 3P = Q

4) The complete interval of values of x in satisfying the inequations

cos x.cos 2x > and 3cos x > 1, is

(A)

(B)

(C)

(D)

5) Equation has

(A) 2018 real roots

(B) 1008 real and 1010 imaginary roots
(C) 2018 imaginary roots
(D) 1010 real and 1008 imaginary roots

6) If α, β are roots of equation P(x2 – x) + x + 5 = 0 and P1 and P2 are two values of P obtained from

, then values of is-

(A) 64514
(B) 65719
(C) 67543
(D) 61217

SECTION-I (ii)

1) Let x, y, z are real numbers such that x, y, z ∈ and x > y > z. If x + y + z = , tanx +
tany + tanz = 5 and tanx tany tanz = 1, then
(A)

(B)

(C)

(D)

2) Let a, b, c be the roots of px3 + qx2 + rx + s = 0 where p, q, r, s are non zero real numbers and a,

b, c are positive real numbers. If determinant , then

(A) value of 'D' is r3p–3

(B) minimum value of 'D' is 27s2p–2

(C) r is always greater than

(D) If 3p + q = 0 and a2 + b2 + c2 = 5, then value of 'D' is 8.

3) Consider a sequence {an} with a1 = 2 and (for all n ≥ 3, terms of the sequence being
distinct). Given that a2 and a5 are positive integers and a5 ≤ 162 then the possible value(s) of a5 can
be

(A) 2
(B) 32
(C) 64
(D) 162

4) Suppose 'a' and 'b' are integers and b ≠ –1. If the quadratic equation x2 + ax + b + 1 = 0 has a
positive integral root, then which of the following is/are true -

(A) the other root is also a positive integer

(B) the other root is an integer
(C) a2 + b2 is a prime number
(D) a2 + b2 has a factor other than 1 and itself.

5) If then possible simplification of is/are

(A) 2cosx
(B) –2cosx
(C) 2sinx
(D) –2sinx
6) If , |x + 1|, |x – 1| are the first three terms of an arithmetic progression then

(A) Sum of first 50 terms of the AP can be 970

(B) 50th term of the AP can be 40
(C) Sum of first 50 terms of an AP can be 2400
(D) 50th terms of the AP can be 97

SECTION-II

1) If the quadratic equation ax2 + bx + c = 0 has equal roots and 4a – 2b + c = 0, then value of

2) Let a, b, c be three real numbers such that and c ∈ R. If

& , then value of is

equal to

3) Let the sum of all the solutions of the equation is S, then the value

of , is

4) In a triangle ABC(with usual notations), a = 5, b = 4 and then the value

of is

5) If a, b, c are positive real numbers such that a + b + c = 1, then the minimum value of

is

6) Sum of all the integral roots of , is λ then is

 ANSWER KEYS

PART-1 : PHYSICS

SECTION-I (i)

Q. 1 2 3 4 5 6
A. B A C D C A

SECTION-I (ii)

Q. 7 8 9 10 11 12
A. A,C,D B,C,D A,C A,B,D A,B A,C

SECTION-II

Q. 13 14 15 16 17 18
A. 5.00 5.00 11.11 6.00 9.37 to 9.38 4.00

PART-2 : CHEMISTRY

SECTION-I (i)

Q. 19 20 21 22 23 24
A. C C D B D C

SECTION-I (ii)

Q. 25 26 27 28 29 30
A. A A,B,C,D C,D C,D B,C A,B,C

SECTION-II

Q. 31 32 33 34 35 36
A. 13.93 or 13.94 14.79 6.00 2.00 6.00 3.00

PART-3 : MATHEMATICS

SECTION-I (i)

Q. 37 38 39 40 41 42
A. A C B C A A

SECTION-I (ii)

Q. 43 44 45 46 47 48
A. A,B,D A,B,D B,D B,D B,C A,C,D
 SECTION-II

Q. 49 50 51 52 53 54
A. 4.50 3.00 4.50 1.20 27.00 1.50
 SOLUTIONS

PART-1 : PHYSICS

2)

3)

4)

5)
S2 > S1
⇒ T – 0 > 100 – T
T > 50

6)

5T = 100
T = 20°C
7)

⇒ Q3 = 4 × 200 = 800 J
w = 1600 + 800 – 800 = 1600 J

8)

= πR2σ(T4) × 2
⇒ Ωr2 × (4004) = 2πR2T4
2π(1 – cos 53°) × 0.04(4004) = 2π × 0.16 T4

9)
 10)

TVm – 1 = constant

T = 2.25 T0 = 450

11)

λATA = λBTB
400 TA = 800 TB

12)
 13) T + F = mg = 10

14) m1v1 = m2v2

15) W = PΔV

16)

mC(10 – T) = m × 2C × (T – 5) + 2m × C × (T – 5)
⇒ 10 – T = 2T – 10 + 2T – 10 = 4T – 20
5T = 30
T = 6°C

17)

ma × 0.8 = mg × 0.75

18)

Fraction

PART-2 : CHEMISTRY

19)

Molar mass of the resin = 193

2 molecules of the resin take up one Mg2+ ion.
386 g of resin takes up 1 mol of Mg2+ ions.
1g of the resin can take 2.59 × 10–3 mol of Mg2+
20g of the resin can take 51.8 × 10–3 mol of Mg2+ or 51.8 m mol.
Exchange efficiency is only 25% → 0.25 × 51.8 m mol of Mg2+ is exchanged = 12.95 m mol of Mg2+ is
exchanged.
Initial amount of Mg2+ is 100 × 0.3 = 30 m mol.
Amount exchanged = 12.95 m mol.
Amount left = 30 – 12.95 = 17.05 m mol.

Molarity = = 0.17 M.

Na+ molarity is twice the molarity of Mg2+ exchanged = = 0.259 M.

 20)

2KMnO4 + 3H2O2 → 2MnO2 + 2KOH + 2H2O + 3O2

Volume at STP = 168 L hence mol of O2 = = 7.5 mol

2 mol KMnO4 = 3 mol O2.

Hence mol KMnO4 required = = 5 mol

= 5 × 158 = 790 g = 0.790 kg.

25)

Theoretical.

26)

Theoretical.

31)

CO(g) + O2(g) → CO2(g)

20 litre CO gas required 10 litre of O2 gas at STP for combustion.

∴ Volume strength of H2O2 solution = = 21.7 V

After mixing, volume strength of resulting solution

= = 13.9375 V.

32)

Moles of CO2 needed = = 118.33 mole

Moles of CaCO3 needed = 118.33 mole
∴ Weight of CaCO3 needed = 118.33 × 100 = 11833 gm

∴ Mass of limestone required = × 100

= 14791.67 gm = 14.79 kg.

PART-3 : MATHEMATICS

37)

α + β > γ, β + γ > α, γ + α + β

Also, we hence, (α + β + γ)
 ...(1)
using A.M. ≥ H.M.

...(2)
from (1) and (2)

38)

= (x – y)2(y – z)2(z – x)2 = 256 (Min. value)

x, y, z are unequal even numbers
|x – y| = 2, |y – z| = 2, |z – x| = 4

39)

Now as

∴ P = 2Q

40)

4cosx.cos2x + 1 > 0
4cosx (2cos2x – 1) + 1 > 0
8cos3x – 4cosx + 1 > 0

Let cosx = t

∴ ∴ 1 ≥ cosx > ∴

41)

let equation has roots a + ib and a – ib,

+....

...+ ...(1)

also

...+ ...(2)
(2) – (1)

⇒ b = 0 so roots cannot be imaginary hence 2018 real roots

42)

Px2 + (1 – P)x + 5 = 0

P2 – 16p + 1 = 0
P1 + P2 = 16 P1P2 = 1

= ((P1 + P2)2 – 2P1P2)2 – 2

= (256 – 2)2 – 2
= 64514
 43)

Given t1 + t2 + t3 = 5
t1t2t3 = 1

and x+y+z=

⇒ 5 – 1 = Σt1t2 – 1
⇒ Σt1t2 = 5

44)

D = (ab + bc + ca)3
∵ a,b,c are roots of px3 + qx2 + rx + s = 0

a+b+c=

ab + bc + ca =

abc
∵ AM ≥ GM

(ab + bc + ca)3 ≥ 27a2b2c2

If 3p + q = 0 ⇒
and a2 + b2 +c2 = 5 then (a + b + c)2
= Σa2 + 2Σab
9 = 5 + 2 Σab
⇒ Σab = 2, D = 8

45)

a1, a2, a3, a4, ..... are in G.P.

Let a2 = x ⇒
⇒ with common ratio

common ratio

Given ⇒x≤6

and x and are only in x, must be even and then only will be an integer
⇒

46)

x2 + ax + b + 1 = 0
α + β = –a αβ = b + 1
If one root is integer then other root is also integer.
a2 + b2 = (α + β)2 + ((αβ) – 1)2
= α2 + β2 + α2β2 + 1
a2 + b2 = (1 + α2)(1 + β2)
Thus a2 + b2 has a factor other than 1 and itself.

47)

⇒ ⇒

⇒ ⇒ 2x ∈ (π, 2π)

= 2|sinx| = 2|cosx|
= 2sinx = –2cosx

48)

C-1 x≥1

2x + 2 = –1
x = –6 (rejected)
C-2 x ∈ [–1, 1]

2x + 2 = +1–x

⇒ ⇒
 ,

S50 = 970

C-3 x ∈ (–∞, –1]

–2x – 2 = +1–x

–2x – 2 = +1

⇒ ⇒ x = –2
–1, 1, 3 ..... A.P.
T50 = –1 + 49(2) = 97

49)

ƒ(–2) = 0 and both roots are equal so root is –2.

So SOR = –4, POR = 4

i.e. and

∴ ⇒ ⇒ 4.50

50)

Each term independently zero

⇒ ⇒

51)

⇒
⇒ x = 9, 27
or |x – 3| = 1 ⇒ x = 2, 4
or |x – 3| = 0 ⇒ x = 3
Sum of solutions = 3 + 2 + 4 + 9 + 27 = 45

52)
⇒ ⇒ ⇒

⇒ c2 = 36

53)

Also ⇒

⇒ hence

54)

Let log5x = t
t2 + t3 + 1 – t = 1 + t
t3 + t2 – 2t = 0
t = 0, t2 + t – 2 = 0
t = 1, t = –2
log5x = 0, log5x = 1, log5x = –2

x=1 x=5
sum = 6