02-02-2025

9610ZJA801443240024 JA

PART 1 : PHYSICS

SACTION-I

1) A body of mass m was slowly hauled up the hill by a force F which at each point was directed
along a tangent to the path. The work done by this force, if the height of the hill is h, the length of its

base is ℓ and the coefficient of friction is µ, is :

(A) mgh – μmgℓ

(B) mgh + μmgℓ
(C) mgh + μmg
(D) Can't determined

2) A block of mass kg is released from the top of an inclined smooth surface as shown in figure. If
spring constant of spring is 100 N/m and block comes to rest after compressing the spring by 1m,
then the distance travelled by block before it comes to rest is :-

(A) 1 m
(B) 1.25 m
(C) 2.5 m
(D) 5 m

3) In figure equal :-

(A)
(B)
(C)
(D)
-

4) If E, M, L and G denotes energy, mass, angular momentum and universal gravitational constant,
respectively, then EL2/M5G2 represents the unit of :-

(A) Length
(B) Mass
(C) Time
(D) Angle

5) For the given situation, If bob of pendulum is released at position 'A'. Then find time taken by it

from position 'A' to position 'B' :

(A) T/6
(B) T/3
(C) T/10
(D) None of these

6) A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing
a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its
equilibrium position, it oscillates simple harmonically. The period of oscillation will be :-

(h = initial length of cylinder)

(A)

(B)

(C)

(D)

7) A hollow sphere of mass M and radius r slips on a rough horizontal plane. At some instant it has
translational velocity v0 and angular velocity about the centre v0/2r as shown in the figure. The
translational velocity after the sphere starts pure rolling
(A) 4v0/5 in forward direction
(B) 6v0/7 in backward direction
(C) 6v0/7 in forward direction
(D) 5v0/4 in backward direction

8) A disc has mass 9m. A hole of radius is cut from it an shown in the figure. The moment of
inertia of remaining part about an axis passing through the centre 'O' of the disc and pependicular ot

the plane of the disc is :-

(A) 8 mR2
(B) 4 mR2

(C)
mR2

(D)
mR2

9) A string is wrapped around a cylinder of mass M. One end of the string is held and the cylinder is

released from rest then acceleration of the cylinder is :-

(A)

(B) g
(C) g/3
(D) g/2

10)

In the figure shown string is massless and inextensible. Pulley and spring are massless. When the
string is cut, ratio of acceleration of block 2 to acceleration of block 1 is :
(A) 1
(B) 2
(C) ∞
(D) 8

11) A body takes time t to reach the bottom of an inclined plane of angle θ with the horizontal if
plane is smooth. If the plane is made rough, time taken now is 2t. The coefficient of friction of the
rough surface is :-

(A)
tan θ

(B)
tan θ

(C)
tan θ

(D)
tan θ

12) A particle of unit mass undergoes one-dimensional motion such that its velocity varies according
to v(x) = βx–2n Where β and n are constant and x is the position of the particle. The acceleration of
the particle as a function of x is given by:

(A) –2n β2x–2n–1

(B) –2n β2x–4n–1
(C) –2n β2x–2n+1
(D) –2n β2e–4n+1

13) The engine of a motorcycle can produce a maximum acceleration 5 m/s2. Its brakes can produce
a maximum retardation 10 m/s2. If motorcyclist start from point A and reach at point B. What is the
minimum time in which it can cover if distance between A and B is 1.5 km. (Given : that motorcycle
comes to rest at B)

(A) 30 sec
(B) 15 sec
(C) 10 sec
(D) 5 sec

14) Two spherical vessel of equal volume, are connected by a narrow tube. The apparatus contains
an ideal gas at one atmosphere and 300 K. Now if one vessel is immersed in a bath of constant
temperature 600 K and the other in a bath of constant temperature 300 K. Then the common

pressure will be :-

(A) 1 atm

(B)
atm

(C)
atm

(D)
atm

15) A light steel wire of length L and area of cross-section A is hanging vertically downward from a
ceiling. It cools to the room temperature 20°C from the initial temperature 100ºC. Calculate the
weight which should be attached at its lower end such that its length remains same (α is the linear
thermal expansion coefficient)

(A) 20 AYα
(B) 80 AYα
(C) 100 AYα
(D) AYα

16) A cylinder of radius R made of a material of thermal conductivity K1 is surrounded by a

cylindrical shell of inner radius R and outer radius 2R made of a material of thermal conductivity K2.
The two ends of the combined system are maintained at two different temperatures. There is no loss
of heat across the cylindrical surface and the system is in the steady state. The effective thermal
conductivity of the system will be :-

(A)

(B)

(C)

(D)

17) The above p-v diagram represents the thermodynamic cycle of an engine, operating with an ideal
monoatomic gas. The amount of heat, extracted from the source in a single cycle is :
(A) p0v0

(B)

(C)

(D) 4p0v0

18) A wooden wheel of radius R is made of two semicircular parts (see figure). The two parts are
held together by a ring made of a metal strip of cross sectional area S and Length L. L is slightly less
than 2πR. To fit the ring on the wheel, it is heated so that its temperature rises by ΔT and it just
steps over the wheel. As it cools down to surrounding temperature, it presses the semicircular parts
together. If the coefficient of linear expansion of the metal is a, and its Young's modulus is Y, the

force that one part of the wheel applies on the other part is :

(A) 2SYαΔT
(B) 2 π SYαΔT
(C) SYαΔT
(D) π SYαΔT

19) The acceleration a of the vertical U-tube is :-

(A) g
(B) g/2
(C) 2g
(D) Zero

20) A tank is filled upto a height h with a liquid and is placed on a platform of height h from the
ground. To get maximum range xm a small hole is punched at a distance of y from the free surface of
the liquid. Then

(A) for
(B) for
(C) for
(D) for

SECTION-II

1) Two wires A and B are of the same materials. Their lengths in the ratio 1 : 2 and diameter are in
the ratio 2 : 1 when stretched by force FA and FB respectively, they get equal increase in their
lengths. Then FA = NFB. Find N

2)

The square of the angular velocity ω of a certain wheel increases linearly with the angular
displacement during 100 rev of the wheel's motion as shown. The time t required for the increase is

, then x is ?

3) A wooden block of mass 0.9 kg is suspended from the ceiling of a room by thin wires. A bullet of
mass 0.1 kg moving horizontally with a speed of 10 ms–1 strikes the block and sticks to it. What is the
height (cm) to which the block rises ? Take g = 10ms–2:

4) A uniformly thick plate in the shape of an arrow head has dimensions as shown. Find the distance

of centre of mass from point O (in cm)

5) A normal human eye can see an object making an angle of 1.8° at the eye approximate height of
object which can be seen by an eye placed at a distance of 1 m from the object is . Find the

value of x.

PART 2 : CHEMISTRY

SECTION-I

1) Correct order of ionic radii is:

(A)
(B)
(C)
(D)

2) Which of the following structures is correctly drawn according to VSEPR theory-

(A)

(B)

(C)

(D)

3) Thiols (C4H10S)can be deodorized by reaction with household bleach (NaOCl) according to

following unbalanced reaction.
C4H10S + NaOCl → C8H18S2 + NaCl + H2O
How many g of thiol can be deodrized by74.5 g NaOCl ? (S = 32)

(A) 90
(B) 45
(C) 22.5
(D) 180

4) An ideal gas is taken around the cycle ABCA as shown in P-V diagram. The net work done by the

gas during the cycle is equal to:

(A) 12P1V1
(B) 6P1V1
(C) 5P1V1
(D) P1V1

5) In which of the following case carbon-carbon dissociation energy is maximum

(A) Ethane
(B) Ethene
(C) Butene
(D) Ethyne

6) A heat engine absorbs 760 kJ heat from a source at 380K. It rejects 650 kJ of heat to sink at 280
K, it represents which cycle?

(A) Reversible cycle

(B) Non-reversible cycle
(C) Impossible cycle
(D) None of these

7) For the reaction: A + B C + D, the initial concentraction of A and B is equal, but the
equilibrium concentration of C is twice that of equilibrium concentration of A. The equilibrium
constant is -

(A) 4
(B) 9
(C) 1/4
(D) 1/9

8) When 0.02 moles of NaOH are added to a litre of buffer solution, its pH changes from 5.75 to
5.80. What is its buffer capacity >

(A) 0.4
(B) 0.05
(C) – 0.05
(D) 2.5
9) The molarity of nitrous acid at which its pH becomes 2. (Ka = 4.5 × 10–4) :

(A) 0.3333
(B) 0.4444
(C) 0.6666
(D) 0.2222

10) Choose the option where parent chain selection is correct?

(A)

(B)

(C)

(D)

11) Which of the following reactions results in the formation of a pair of diastereomers ?

(A)

(B)

(C)

(D)

12) Which is most basic in aqueous solution ?

(A) CH3NH2
(B) (CH3)2NH
(C) (CH3)3N
(D) Ph–NH2

13) Which of the following is correct order of percentage enolic content?

(A)

(B)

(C)

(D)

14) Number of isomeric forms of C7H9N having benzene ring will be-

(A) 7
(B) 6
(C) 5
(D) 4

15) Find total number of electron present in 1.6 gm of CH4.

(A)
(B)
(C)
(D)

16) A wavelength of 400 nm of an electromagnetic radiation is not correspond to :

(A) Frequency = 7.5 × 1014 Hz

(B) Wave number = 2.5 × 106m–1
(C) Velocity = 3 × 108 m/s
(D) λ = 40Å

17) The electronic configuration of an element is 1s2 2s2 2p6 3s2 3p4. The atomic number and the
group number of the element ‘X’ which is just below the above element in the periodic table are
respectively.

(A) 24 & 6
(B) 24 & 15
(C) 34 & 16
(D) 34 & 8

18) Which of the following species is polar as well as planar?

(A) SF2
(B) S8
(C) XeO3
(D) SF4

19) For the reaction: 7A + 13B + 15C 17P

If 15 moles of A, 26 moles of B & 30.5 moles of C are taken initially then liniting reactant is

(A) A
(B) B
(C) C
(D) None of these

20) One litre of saturated solution contains AgSCN and AgBr. Calculate the ratio of in
the solution. [Ksp of AgSCN = 1.0 × 10–22; Ksp of AgBr = 5 × 10–13]

(A) 1.5
(B) 0.5
(C) 2
(D) 3.5

SECTION-II

1)

When 0.04 moles of base is added to 250 ml of a buffer solution, pH changes from 4 to 4.16. The
buffer capacity of the system is

2) The total number of hyperconjugative structures for the following carbocation is ?

3) In how many of the following cases, the negative charge is delocalised?

(i) (ii) (iii) (iv)

(v) (vi) (vii) (viii)

(ix) (x)

4) Number of carbon containing functional group in the molecule given below are

5) The compound Na2CO3.x H2O has 50% H2O by mass. The value of x is

PART 3 : MATHEMATICS

SECTION-I

1) The equation has :-

(A) only one solution

(B) two solutions
(C) no solution
(D) more than two solutions

2)

(A)

(B)

(C)

(D)

2 2
3) If ax + by = 10 is the chord of minimum length of the circle (x – 10) + (y – 20) = 729 and the
chord passes through (5,15), then the value of (4a + 2b) is

(A) 2
(B) 3
(C) 4
(D) 5

4) If the circle x2 + y2 + 6x + 8y + h = 0 bisects the circumference of x2 + y2 + 2x – 6y – k = 0, then h

+ k is equal to

(A) 38
(B) –38
(C) 42
(D) 48

5) If the straight-line 2x + y – 3 = 0 intersects the circle x2 + y2 = 3 at points P and Q, the coordinates

of the point of intersection of the tangents drawn at P and Q to the circle x2 + y2 = 3 is (α, β) then (β –
α) is –

(A) 0
(B) –1
(C) 1
(D) 2

6) A straight line with slope 2 and y-intercept 5 touches the circle, at a

point Q. Then the coordinates of Q are

(A)
(B)
(C)
(D)

7) The ellipse has a double contact with a circle at the extremity of latus rectum and
point of contacts lie in first and fourth quadrant then the product of length of intercepts made by
the circle on coordinate axes is

(A)

(B)

(C)
(D)

8) Let S and S' be two foci of the ellipse . If a circle describe on SS' as diameter
intersects the ellipse in real and distinct points, then the eccentricity e of the ellipse satisfies.

(A)
(B)

(C)

(D) None of these

9) The tangent to the hyperbola xy = c2 at the point P intersects the x–axis at T and the y–axis at T'-
the normal to the hyperbola at P intersects the x–axis at N and the y–axis at N'. The areas of the

triangles PNT and are Δ and Δ' respectively. Then is

(A) equal to 1
(B) depends on t
(C) depends on c
(D) equal to 2

10) A cat moves two steps forward or one step upward in a grid having milk at one corner as shown
in adjacent diagram. If cat starts from diagonally opposite corner then number of ways in which it

can reach to the milk is -

11
(A) C6
8
(B) C6
8
(C) C3
8
(D) C3.2!.2!2!

11) The common tangent to parabola and is

(A)
(B)
(C)
(D)

12) If α and β be the roots of 3x2 + 6x – 2 = 0 with α > β. If an = 3αn – 4βn for n ≥ 1 then find the

value of is

(A) 1
(B) –2
(C) –3
(D) 4

13)
If atleast one root of the equation x2 – 2ax + a2 – 1 = 0 is positive, then

(A)
(B)
(C)
(D)

14) Two consecutive sides of a parallelogram are 3x + 5y = 0 and 5x + 3y = 0. If the equation to one
diagonal is 15y + 105x = 32 then the equation of the other diagonal is

(A) 12y + 13x = 0

(B) 11y + 13x = 0
(C) 22y + 15x = 0
(D) None of the above

15) A ray of light coming from the point is incident on the line x = l at the point A. The ray
gets reflected on the line x = 1 and meets x-axis at the point B. If the angle of incidence is 60°, then
the line AB passes through the point :

(A)

(B)

(C)

(D)

16) The sum upto 11-terms is :-

(A)

(B)

(C)

(D)

17) If the equation sin2x + |sinx| + a = 0 has two distinct roots in [0, π] then number of integers in
the range of 'a' is equal to :-

(A) 1
(B) 2
(C) 3
(D) 4
18) The general value of satisfies the equation

is

(A)

(B)

(C)

(D)

19) The expression (wherever defined) equal to

(A)

(B) sinθ

(C)
(tan27θ – tanθ)

(D)
(tan9θ – tan3θ)

20) Find the sum of the infinite series

(A) 21
(B) 23
(C) 25
(D) 27

SECTION-II

1)

Let
and x is prime number greater than 2}
and then number of elements in is

2) Consider the following frequency distribution :

Class 0-6 6-12 12-18 18-24 24-30

Frequency a b 12 9 5

If mean = and median = 14, then the value (a – b)2 is equal to _____.

3) There are 4 letters and 4 directed envelopes. The number of ways in which all the letters can be
put in wrong envelope is

4) In the given figure of squares, 6 A's should be written in such a manner that every row contains at
least one 'A'.
In How many number of ways is it possible?

5) The coefficient of x50 in the expansion of

is then is
 ANSWER KEYS

PART 1 : PHYSICS

SACTION-I

Q. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A. B D D D B A A B A B A B A C B B B A B A

SECTION-II

Q. 21 22 23 24 25
A. 8 7 5 1 1

PART 2 : CHEMISTRY

SECTION-I

Q. 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
A. D C D C D B A A D D B B C C B D C A B B

SECTION-II

Q. 46 47 48 49 50
A. 1 5 5 3 6

PART 3 : MATHEMATICS

SECTION-I

Q. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
A. C B B B B D A B C C A C C B B D B A C B

SECTION-II

Q. 71 72 73 74 75
A. 3 4 9 26 1002
 SOLUTIONS

PART 1 : PHYSICS

1) By using W = ΔKE ⇒ Wf + Wg + Wext = 0 ⇒ –

⇒ Wext =

2) wnet = ΔK.E.

3) ⇒
⇒

4)
∴ Angle

5) θ = ωt

t= .

6) at equillibrium position
pA – patm A = 0
when disturbed by x
pAh = p' A(h – x)

p' =
net force

(Fnet) = (p' – patm) A = A

= ≈ x<<h k=

=ω=

T = 2π
7)
Angular momentum can be conserved about lowest point

0
mv r + = mv'r +

v' =

8) Ibig =

Ismall =
Iremaining = Ibig – Ismall

9)
Mg - T = Ma

TR = Iα =

10)

Before cutting spring force (Fs)

Fs = 2mg – mg = mg
After cutting,

a2 = g

11) μ = tanθ = tan θ = tanθ

12)

we are given velocity of the particle

v(x) = βx–2n We know acceleration a =

a = βx–2n (βx–2n)
= β2 x–2n(–2n)x–2n–1
= – 2nβ2 x–2n–1–2n
a = –2nβ2x–4n–1

13)

........(i)

from equation (i) and (ii)

t1 + t2 = 30 sec

14) Conservation of moles

ni = nf

P= atm

15) strain = αΔT = 80α

Y= ⇒ F = 80 AYα

16)
17)
Heat is released in B → C (isochoric) & C → D (Isobaric)
Qreleased =

18)

Y= ⇒stress = Y × strain

⇒ F = YSαΔT
Therefore force by one part on other part = 2F = 2SYαΔT

19) tan θ =

20)

Velocity of liquid through orifice,

and time taken by liquid to reach the ground
21)

22)

ω2 = ω02 + 2αθ
1600 = 900 + 2α(100 × 2π)

23)

E text in x = 0
pxi = pxf
m1v1 + m2v2 = m3v3
0.1 × 10 + 0 = 1 × v
v = 1 m/s
By WET

mv2 = mgh

(1)2 = 10h

h= m = 5 cm

24)

Note that the centre of mass of a uniform thick triangular sheet is at the centroid which
divides a median in a ratio 2:1. The required centre of mass must be on the line of symmetry
passing through O and the vertex to the right. Consider the shape of arrowhead to be obtained
by cutting the triangular part to the left (with base 6 cm and height 3 cm) from the uncut
triangular sheet with base 6 cm and height 9 cm. The ratio of masses of these two is 3:1, the
 thickness being uniform.

25)

PART 2 : CHEMISTRY

26)

27)

(A) incorrect, θ is less than 109°28'

(B) incorrect position of lone pair
(C) correct square planar structure with θ = 90°
(D) θ incorrect because it is square planar
with two lone pair at Xe and θ = 90°

28) 2C4H10S + NaOCl → C6H18S2 + NaCl + H2O

mass of C4H10S = 2 × × 90 = 180

29)

w=

30) Ethyne contain bond

31)

Efficiency of engine =
For given cycle

η1 < η
Hence cycle is non-reversible cycle.
32) A + B C + D
t=0 a a – –
t = teq a–x a–x x x
x = 2(a – x)

=4

33) Buffer capacity = = 0.4

34) pH = 2
[H⊕] = 10–2
10–2 =
KaC = 10–4
4.5 × 10–4 × C = 10–4

C= = 0.2222

35)

36)

37)

Secondary amine is most basic in aqueous solution among aliphatic amines.

If R = (–CH3)
Order is : 2° amine > 1° amine > 3° amine > NH3
38)

39)

40) Ne– =

41)

(A) ν =

(B)
(C) Fact
(D) Fact

42)

1s2 2s2 2p6 3s2 3p4 = sulphur (16)

Period no = 3
Group no = 16
Element X which is just below sulphur = Se(34)
Period no = 4
Group no = 16

43)

Class notes

44) 7A + 13B + 15C 17P

for L.R.

So, B is min. that is L.R.

45) AgSCN Ag+ + SCN–

 a+b a
AgBr Ag+ + Br–
a+b b

=2

46) Buffer capacity

47)

No. of α-hydrogen → No. of hyperconjugative structures.

α-H = 5
No. of hyperconjugative structures = 5.

48) 5 (i, ii, iv, vi, ix)

49) Theory based.

50) Molar mass of Na2CO3.x H2O

= (2×23 + 12 + 3×16) + 1 8x
= 106 + 18 x

% of H2O present
or 36x = 106 + 18x or 18x = 106 or x = 6.

PART 3 : MATHEMATICS

51)

x–3>0&x–3≠1
⇒x>3&x≠4 ...(1)

x2 + x + 1 = x2 + 3x

⇒ (Rejected)
Hence no solution

52) + + + ...... +

53) It is chord with a given middle point i.e.

x + y = 20

54) Radical axes of two circles is

4x + 14y + h + k = 0
it passes through (–1, 3)
⇒ h + k = –38

55) Let R(α, β) be the point of intersection of the tangents drawn at P and Q to the given circle.
Then PQ is the chord of the contact of tangents drawn from R to x2 + y2 = 3
So, its equation is
αx + βy – 3 = 0 ...(i)
It is given that the equation of PQ is
2x + y – 3 = 0 ...(ii)
Since (i) and (ii) represent the same line, we have

Or α = 2, β = 1
Hence, β – α is equal to –1.

56) Equation of tangent y = 2x + 5

and
 and

57)
By symmetry centre of circle lies on X-axis

∴ Normal at P is

∴ radius =
∴ equation of circle is

y - intercept

∴ product

58) Radius of the circle having SS' as diameter is r = ae

If it cuts an ellipse, then r > b
 59)

put y = 0; x = 2ct(T)

similarly normal is

put

60)

Number of horizontal moves = 3 (each of two steps)

Number of vertical moves = 5 (each of one step)
⇒ required number of ways = number of arrangements

of HHHVVVVV
61) Equation of tangent to is
where m is the slope of tangent.
If this is tangent to then

will have two equal roots

So,

62) 3an + 6an – 1 – 2an – 2 = 0

Put n = 99

63)

a>0
so mouth of parabola is upward
D>0
4a2 – 4(a2 – 1) > 0
4>0
a ϵ R,
Case I : both roots positive

f(0) > 0
a ϵ (–∞, –1) (1, ∞) ...(i)

a > 0 ... (ii)

from (i) and (ii)
a ϵ (1, ∞) ... (iii)
Case II : one root positive
f(0) < 0
a ϵ (–1, 1) ...(iv)
(iii) (iv) a ϵ (–1, ∞)

64)
On solving 3x + 5y = 0 and 15y + 105x = 32
We will get vertex A
(3x + 5y = 0) × 3
15y + 105x = 32
from equations

96x = 32 ⇒

Similarly

⇒
⇒ Equation of OD

65)

line AP
x=1

line AB

point

point satisfying

66)

= = = =

67) Let t = |sin x| ∀ x \

⇒ t ∈ [0, 1]
a = –t2 – t
⇒ a ∈ {0, –1, –2}
for a = 0, x has two values in [0, π]
for a = –1, x has two values in [0, π]
for a = –2, x has one value in [0, π]

68)

69)
70) Sum, …(1)

⇒ …(2)
Subtracting (2) from (1) ;

Let, …(3)

⇒ …(4)
Subtracting (4) from (3) ;

⇒ ( ∵ sum of infinite G.P.)

Hence,

⇒
⇒ S = 23

71)

A = {1, 2, 3, 4, 5}
B = {3, 5, 7} U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C = {5, 10}

= 10 – 7 = 3

72)

Class Frequency
 0-6 a 3 3a

6-12 b 9 9b

12-18 12 15 180

18-24 9 21 189

24-30 5 27 135

N=(26+a+b) (504+3a+9b)

Mean =
⇒ 66a + 198b + 11088 = 309a + 309b + 8034
⇒ 243a + 111b = 3054
⇒ → (1)

Now, Median = 12 +

⇒
⇒ → (2)
From equation (1) & (2)
a = 8, b = 10
∴ (a – b)2 = (8 – 10)2 = 4

73) Such problems are called problems of derangement. Hence, using the formula of
derangement. The required number of ways of placing all letters in wrong envelope

74) There are 8 squares and 6 'A' in given figure. First we can put 6 'A' in these 8 squares by
number of ways.

According to question, atleast one 'A' should be included in each row. So after subtracting

these two cases, number of ways are

75)

Subtract above equations,

[Sum of G.P]

Coefficient of x50 in S = coefficient of x50 in