STATEMENT-1:
The numbers b₁, b₂, b₃, b₄ are neither in A.P. nor in G.P.
and
STATEMENT-2:
The numbers b₁, b₂, b₃, b₄ are in H.P.

STATEMENT-1:
(p² - q)(b² - ac) ≥ 0
and
STATEMENT-2:
b ≠ pa or c ≠ qa

STATEMENT-1:
If line L₁ is a chord of circle C, then line L₂ is not always a diameter of circle C.
and
STATEMENT-2:
If line L₁ is a diameter of circle C, then line L₂ is not a chord of circle C.

13) Let a solution y = y(x) of the differential equation
xdy - ydx = 0

satisfy y(2) =	2

STATEMENT-1:

y(x) = sec(sec-1x -	π	)
	6

and
STATEMENT-2:
y(x) is given by

Answer:
A → (s) , B → (p,q) , C → (r), D → (p,q,q)

Answer:
A → (r) , B → (q,s), C → (r), D → (p,r)

Answer:
A → (p) , B → (s), C → (q), D → (q)

36) The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density ρ(r) [charge per unit volume] is dependent only on the radial distance r from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.
Figure :

The electric field at r = R is

37) The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density ρ(r) [charge per unit volume] is dependent only on the radial distance r from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.
Figure :

For a = 0, the value of d (maximum value of ρ as shown in the figure) is

38) The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density ρ(r) [charge per unit volume] is dependent only on the radial distance r from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.
Figure :

The electric field within the nucleus is generally observed to be linarly dependent on r. This implies

39) A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant k which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity V₀ = V₀i The coefficient of friction is μ
Figure :

The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is

40) A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant k which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity V₀ = V₀i The coefficient of friction is μ
Figure :

The centre of mass of the disk undergoes simple harmonic motion with angular frequency ω equal to

41) A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant k which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity V₀ = V₀i The coefficient of friction is μ
Figure :

The maximum value of V₀ for which the disk will roll without slipping is

42) Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS.

Column I	Column II
(A) Potential energy of a simple pendulum (y axis) as a function of displacement (x axis)	(p)
(B) Displacement (y axis) as a function of time (x-axis) for a one dimensional motion at zero or constant acceleration when the body is moving along the positive x-direction	(q)
(C) Range of a projectile (y axis) as a function of its velocity (x axis) when projected at a fixed angle	(r)
(D) The square of the time period (y axis) of a simple pendulum as a function of its length (x axis)	(s)

Answer:
A → (p,s) , B → (q,r,s) , C → (s), D → (q)

43) An optical component and an object S placed along its optic axis are given in Column I. The distance between the object and the component can be varied. The properties of images are given in Column II. Match all the properties of images from Column II with the appropriate components given in Column I. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS.

Column I	Column II
(A)	(p) Real Image
(B)	(q) Virtual image
(C)	(r) Magnified image
(D)	(s) Image at infinity

Answer:
A → (p,q,r,s) , B → (q) , C → (p,q,r,s), D → (p,q,r,s)

44) Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS.

Column I	Column II
(A) An insulated container has two chambers separated by a valve. Chamber I contains an ideal gas and the Chamber II has vacuum. The valve is opened.	(p) The temperature of the gas decreases
(B) An ideal monoatomic gas expands to twice its original volume such or remains constant that its pressure P ∝  1   , is the volume of the gas V2	(q) The temperature of the gas increases or remains constant
(C) An ideal monoatomic gas expands to twice its original volume such that its pressure P ∝  1   , where V is its volume V4/3	(r) The gas loses heat
(D) An ideal monoatomic gas expands such that its pressure P and volume V follows the behaviour shown in the graph	(s) The gas gains heat

Answer:
A → (q) , B → (p,r) , C → (p,s), D → (q,s)

58) A tertiary alcohol H upon acid catalysed dehydration gives a product I. Ozonolysis of I leads to compounds J and K. Compound J upon reaction with KOH gives benzyl alcohol and a compound L, whereas K on reaction with KOH gives only M.

Compound H is formed by the reaction of

59) A tertiary alcohol H upon acid catalysed dehydration gives a product I. Ozonolysis of I leads to compounds J and K. Compound J upon reaction with KOH gives benzyl alcohol and a compound L, whereas K on reaction with KOH gives only M.

The structure of compound I is

60) A tertiary alcohol H upon acid catalysed dehydration gives a product I. Ozonolysis of I leads to compounds J and K. Compound J upon reaction with KOH gives benzyl alcohol and a compound L, whereas K on reaction with KOH gives only M.

The structures of compounds J, K and L respectively, are

The number of atoms in this HCP unit cell is

The volume of this HCP unit cell is

The empty space in this HCP unit cell is

64) Match the compounds in Column I with their characteristic test(s)/reaction(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS

Column I	Column II
(A)	(p) Sodium fusion extract of the compound gives Prussian blue colour with FeSO4
(B)	(q) Gives positive FeCl3 test
(C)	(r) Gives white precipitate with AgNO3
(D)	(s) Reacts with aldehydes to form the corresponding hydrazone derivative

Answer:
A → (r,s) , B → (p,q) , C → (p,q,r), D → (p)

Answer:
A → (p) , B → (q) , C → (p,r), D → (p,s)

Answer:
A → (q) , B → (s) , C → (p,q,r), D → (p,q,r)