14-09-2025

7101CJA101021250019 JA

PHYSICS

SECTION - I (i)

1) Two conducting spheres of external radii r and 3rhave wall thickness 0.05r each. Point charges q
and 2q are placed at the centres of these sphere respectively. The spheres are kept at great
separation so that they do not affect charge distribution on the surface of each other and their
interaction energy is negligible. Deduce expression for the minimum amount of work done by an
external agency to exchange the position of the charges. The holes made to enable exchange of

charges are so small that their size can be neglected.

(A)

(B)

(C)

(D)

2) A conducting balloon of radius a is charged to a potential V0 and held at a large height above the
earth surface. The large height of the balloon from the earth ensures that charge distribution on the
surface of the balloon remains unaffected by the presence of the earth. It is earthed through a
resistance R and a valve is opened. The gas inside the balloon escapes through the valve and the size
of the balloon decreases. The rate of decreases in radius of the balloon is controlled in such a
manner than potential of the balloon remains constant . Assume electric permittivity of the
surrounding air equal to that of free space and charge does not leak to the surrounding air.

How much heat is dissipated in the resistance R until radius of the balloon becomes
half?

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

3) Electric field given by the vector is present in the x-y plane. A small ring of
mass M carrying charge +Q, which can slide freely on a smooth non conducting rod, is projected
along the rod from the point (0, ℓ) such that it can reach the other end of the rod. Assuming there is
no gravity in the region. What minimum velocity should be given to the ring (in m/s)? If in S.I. unit

(A) 2
(B) 3
(C) 2.5
(D) 3.5

4) A conducting wire of length and cross-section area A taut between two fixed grounded metallic
supports is vibrating in its fundamental mode with a frequency and amplitude a in presence of a
uniform magnetic field of induction B pointing normal to the plane of vibration of the string. Find an
expression for electromotive force induced in the wire as a function of time t assuming the wire

straight at t = 0.

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

SECTION - I (ii)

1) A block string system is shown in the figure. Mass of block A, B and C is 5 Kg, 2 Kg, and 2 Kg
respectively. Mass of rod is given as 1 kg and its length is 1 m. A wave transverse is transmitted in
the rod in between the block B and C once in forward and once backward direction. Choose the

correct statements.
 Time for string wave to reach one end to other end in forward direction is
(A)

Time for waves to reach one end to other end is same for both in forward and backward
(B)
direction
(C) Time for wave to reach one end to other end is different for forward and backward direction
Time for string wave to reach one to other end in backward direction is
(D)

2) In the circuit shown, ratio of capacitors is and terminal voltage of the battery connected
at the left end is V0 . Given that the potentials of nodes 1, 2, 3, 4, 5 and 6 are in geometric
progression, when the terminal voltage of the battery connected at the right most end is V. (in

steady state condition) Then:-

(A)
V may be
(B) V may be

(C)
V may be
(D) V cannot be concluded with this data

3) A loop is made by joining the ends of a flexible, inextensible and light conduction wire of length
2 with a thin insulating coating. The loop passes through two frictionless hooks, one of the hooks is
attached to the ceiling and from the other hook a small rubber block of mass is suspended as shown
in the figure. Below the ceiling, a horizontal magnetic field of induction B is established. Now a
gradually increasing current is made to circulate in the loop. For all values of current in the loop,
forces of magnetic interaction between different parts of the wire are negligible as compared to

other forces. Choose the correct options.

(A) Shape of the wire loop will be circular.

 Maximum height upto which the block can be lifted is
(B)

(C)
Current in the loop to lift the block to a height is
Gravitational potential energy of Block and earth is increasing due to work done by magnetic
(D)
force.

SECTION - I (iii)

1) Column-I shows different sets of standing waves in a string of length L whose ends are fixed or
free according to respective figure & Column-II shows possible equations for them where symbols
have usual meaning.

List-I List-II

(P) (1)
y = A sin cos ωt

(Q) (2)
y = A sin cos ωt

(R) (3)
y = A cos cos ωt

(S) (4)
y = A cos cos ωt

Codes:
(A) P → 4;Q → 2;R → 1;S → 3
(B) P → 2;Q → 1;R → 3;S → 4
(C) P → 2;Q → 3;R → 4;S → 1
(D) P → 4;Q → 2;R → 3;S → 1

2) A long current carrying wire applies F force on current carrying finite wire as shown in figure is

length of finite wire Now, different situation are given in column-I

and in column-II net force on finite wire of total length ℓ is given :
 Column-I Column-II

(A) (P) 2F

(B) (Q) 5F/4

(C) (R) F

(D) (S) 0

(A) A → Q;B → P;C → P;D → P

(B) A → Q;B → S;C → Q;D → Q
(C) A → R;B → P;C → R;D → Q
(D) A → R;B → P;C → Q;D → Q

3) A uniform magnetic field B0 exists perpendicular to the plane of the figure. A positively charged
particle having charge q and mass m is projected with velocity u into the field. The particle moves in
the plane of the figure. During its course of motion the particle is subjected to a friction force which
varies with velocity as , where k is a positive constant and is instantaneous velocity.
 List–I List–II

(A) Distance travelled before it stops (P)

(B) (Q)
Radius of curvature of t =

(C) (R)
Radius of curvature at t =

(D) (S)
Speed at t =

(T)

(U)

(A) A → Q;B → P;C → R;D → T

(B) A → Q;B → R;C → P;D → T
(C) A → S;B → P;C → R;D → U
(D) A → S;B → R;C → P;D → U

SECTION - II

1) In a region of space, a uniform static magnetic field of induction B established above the x-z plane
and another uniform static magnetic field of induction B2 (> B1) is established below the x-z-pane.
Both the fields are in the positive z-direction. A particle of mass m and charge q is projected from
the origin with velocity v making an angle θ with x-axis as shown in the figure. Find average velocity
of the particle in a large time interval.
B1 = 1T
B2 = 2T

rad
2) Two identical plates with area A and mass m are connected by 3 insulating springs as shown. The
springs are symmetrically located w.r.t. center. In equilibrium the separation between plates is d0.
The lower plates if fixed and upper can move. When the battery is connected as shown. The
equilibrium separation decreases to d. d & d0 << . If the upper disc is slightly displaced from it's
equilibrium and released calculate the angular frequency of its oscillation (in rad/s). There is no
resistance in the circuit.
Take m = 200 gm, spring constant = 100 N/m, d0 = 1.2 mm, d = 1mm, A = 100 cm2.

3) A soap bubble of mass m (excluding air inside) blown with the help of a capillary tube is given an
unknown amount of charge. Surface tension of the soap solution is σ. When the bubble acquires a
stable radius, an additional small amount of air is blown into the bubble from the tube and then the
tube is left open. Because of this operation, the soap bubble starts oscillations. If the soap bubble

retains its spherical shape during the oscillations, then the time period is . Find x.

4) A wire segment is bent into the shape of an Archimedes spiral (see figure). The equation that

describes the curve in the range 0 ≤ θ ≤ π is r(θ) = for 0 ≤ θ ≤ π

where θ is the angle from x-axis in radians. Point P is at the origin. I is the current. The magnetic

field strength at point P is , the value of b is:

5) The velocity of blood flow in an artery can be measured using Doppler shifted ultrasound.
Suppose sound with frequency 1.5 × 106 Hz is reflected straight back by red blood cell moving at 1
m/s in blood. Assume that the velocity of sound in blood is 1500 m/s and the sound is incident at a
very small angle as shown in the figure. The difference (in KHz) between the original frequency of
sound and the frequency of sound received after reflection (to the nearest integer) is

6) A metallic rod of mass m and length L(thick line in the figure below) can slide without friction on
two perpendicular wires (thin lines in the figure). Entire arrangement is located in the horizontal
plane. A constant magnetic field of magnitude B exists perpendicular to this plane in the downward
direction. The wire have negligible resistance compared to the rod whose resistance is R, Initially,
the rod is along one of the wire so that one end of it is at the junction of the two wires (see Fig.(a)).

The rod is given an initial angular speed ω

such that it slides with its two ends always in contact with the two wires (see Fig. (b)), and just
comes to rest in an aligned position with the border wire (see fig. (c)). Determine the value of ω.
Neglect the self-inductance of the system.

If your answer is then find x + y

CHEMISTRY

SECTION - I (i)

1)

In which of the following compounds back bonding is not possible :-

(A) BF3
(B) AℓF3
(C) BCℓ3
(D) N(SiH3)3

2) Compare π bond strength between B and N in given two compounds :-

(I)
(II)

(A) There is no π-bond character between B and N

(B) Same in I and II
(C) I > II
(D) II > I

3) Statement-1: In FCC unit cell, packing efficiency is more when all tetrahedral voids are filled
with spheres of minimum possible size as compared with packing efficiency when all octahedral
voids are filled in similar way.
Statement-2: Tetrahedral voids are more in the number than octahedral voids in FCC.

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for
(B)
statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true.

4) An aqueous solution containing Na+, Sn2+, Cl– & ions, all at unit concentration, is
electrolysed between a silver anode and a platinum cathode. What changes occur at the electrodes
when current is passed through the cell?

Given :

(A) Sn2+ is reduced and Cl– is oxidized

(B) Ag is oxidized and Sn2+ is reduced
(C) Sn2+ is reduced and Sn2+ is oxidized
(D) H+ is reduced and Sn2+ is oxidised

SECTION - I (ii)

1) Select the correct statements:

The molecularity of an elementary reaction indicates how many reactant molecules take part in
(A)
the step
The rate law of an elementary reaction can be predicted by simply seeing the stoichiometry of
(B)
reaction
The slowest elementary step in sequence of the reactions governs the overall rate of formation
(C)
of product
 A rate law is often derived from a proposed mechanism by imposing the steady state
(D)
approximation or assuming that there is a pre-equilibrium

2) Which of the following statement(s) is/are incorrect regarding chemisorption ?

Its extent increases with increase in pressure and it may change to multilayer adsorption at
(A)
high pressure.
(B) Enthalpy of adsorption is high in this case.
(C) Its extent increases with increase in surface area of adsorbent.
(D) If is reversible in natures.

3) Consider the following system.

Three different aqueous solution each having volume 100 ml are taken and kept in contact as shown.

After sufficient time (Consider temp constant &

100% dissociation of strong electrolyte)

(A)
Volume of urea solution will be

(B)
Volume of AlCl3 solution will be
(C) There will be no change in volume of KCl solution.
(D) Volume of both KCl and AlCl3 solutions will increase.

SECTION - I (iii)

1) Match the following

Column - II (Rate
constants)
Column - I (First order reactions)

(A) (P)
V is the volume of gas

Hydrolysis of sucrose [acid - catalysed]

(B) r is angle of rotation of (glucose + (Q)
fructose)

Hydrolysis of ester
(C) (acid - catalysed) V is the volume of (R)
NaOH
 (D) (S)
V is the volume of KMnO4 used

(T)

(A) A → P;B → Q;C → R;D → T

(B) A → Q;B → T;C → S;D → R
(C) A → P;B → R;C → P;D → S
(D) A → P,Q;B → R;C → S,T;D → Q

2) Match the ion perfection in solids with the characteristic features :

List-I List-II

(P) Schottky defect (1) Cations occupy interstitial sites

Conductivity increases due to

(Q) Frenkel defect (2)
presence of free electrons

Metal excess defect Compound represented as A1+δX,

(R) (3)
due to extra cation. where A is metal atom

Impurity substitutional
(S) (4) Non-stoichiometric defect
defect

(5) Density of solid may decrease

(A) P → 2,3;Q → 1,3;R → 3,4,5;S → 5
(B) P → 1,4;Q → 2;R → 2,4;S → 2,3
(C) P → 5;Q → 1;R → 1,2,3,4;S → 5
(D) P → 5;Q → 1;R → 1;S → 1,4

3) In a conductometric titration, small volume of titrant of higher concentration is added stepwise to

a larger volume of titrate of much lower concentration, and the conductance is measured after each
addition.
The limiting ionic conductivity (Λ0) values (in mS m2 mol–1) for different ions in aqueous solutions are
given below :

Ions Ag+ K+ Na+ H+ Cl– OH– CH3COO–

Λ0 6.2 7.4 5.0 35.0 7.2 7.6 16.0 19.9 4.1

For different combinations of titrates and titrants given in List-I, the graphs of 'conductance' versus
'volume of titrant' are given in List-II.
Match each entry in List-I with the appropriate entry in List-II and choose the correct option.
List-I List-II

Titrate : KCl
(P) (1)
Titrant : AgNO3
 Titrate : AgNO3
(Q) (2)
Titrant : KCl

Titrate : NaOH
(R) (3)
Titrant : HCl

Titrate : NaOH
(S) (4)
Titrant : CH3COOH

(5)

(A) P → 4;Q → 3;R → 2;S → 5

(B) P → 2;Q → 4;R → 3;S → 1
(C) P → 3;Q → 4;R → 2;S → 5
(D) P → 4;Q → 3;R → 2;S → 1

SECTION - II

1) For any acid catalysed reaction

half life period is independent of of concentration of A at given pH. At given concentration of A, half
life period is 10 min at pH = 2 and half life time is 1000 min at pH = 3. If the rate law expression of
reaction is
= k[A]x[H+]y,
then calculate the value of x+y.

2) A student carried out the experiment to measure the pressure of carbon dioxide in the space
above the carbonated soft drink in a bottle. Initially weight of soft drink in bottle was found to be
452.2 g then its weight again measured after CO2 escaped out completely was found to be 450 gm.
Volume of the soft drink is 450 mL. If Henry’s law constant for CO2 in water is 3.4 ×103 atm, find the
approximate pressure (in atm) of CO2 over the soft drink in bottle before it was opened. [Given :
density of soft drink = 1 gm mL–1]

3) The vapour pressure of two miscible liquid A and B are 300 mm and 500 mm Hg respectively. In a
flask 10 mole of A is mixed with 20 moles of B. When B is added in A, A starts polymerizing into a
completely insoluble solid by following first order kinetics. After 200 min, 1 mole of a nonvolatile
solute is dissolved which arrest process of polymerization completely. Final vapour pressure of
solution is 450 mm of Hg, then what will be value of rate constant in unit hr–1 in polymerization
process.

Given ℓn =1

4) For a cell,
4B(s) + 3O2(g) → 2B2O3(g) ; Eºcell = 1.50 volts
What is molar entropy (in JK–1mol–1) of oxygen gas at 300K ? (Take 1F = 96500C)
Given :

5) The molar conductivity of NH4Cl, OH– and CI– at infinite dilution is 150, 200 and 75 Ω–1 cm2 mol–1,
respectively. If the molar conductivity of a 0.01 M–NH4OH solution is 22 Ω–1 cm2 mol–1,then its degree
of dissociation is (Give answer upto 2 decimal places)

6) Marbles of diameter 10 mm each are to be arranged on a flat surface so that their centres lie
within the area enclosed by four lines of length each 40 mm. What is the maximum number of
marbles according to the above condition.

MATHS

SECTION - I (i)

1) Point 'O' is the centre of the ellipse with major axis AB and minor axis CD. Point F is one focus of
the ellipse. If OF =6 and the diameter of the inscribed circle of triangle OCF is 2, then the product
(AB) (CD) is equal to

(A) 52
(B) 65
(C) 78
(D) None of these

2) If f(x) and g(x) are derivative and antiderivative of a function h(x) where = 3 and

cos x dx = 5. Let p(x) be a polynomial of degree 4 such that p(1) = 7 and attains its
local minimum value 3 at both x = 2 and x = 3. If the local maximum value of p(x) is equal to λ then

the value of is :
(A)

(B)

(C)

(D) None of these

3) If where C is the
integration constant, then AB is equal to

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

4) Let f(x) be a differentiable function satisfying Then the

value of f(2) is

(A) e2

(B)

(C) e + e2
(D) 2e2

SECTION - I (ii)

1) The complete set of values of 'a' for which the inequality is true are

(A) [0,1]
(B) (–∞,–1]
(C) [0,∞)
(D) (–∞,–1] ∪ [0,∞)

2) Let

Let g : [0, 1] → be the function defined by

Then, which of the following statements is/are TRUE ?

(A) The minimum value of is
The maximum value of is 1 +
(B)

(C) The function attains its maximum at more than one point
(D) The function attains its minimum at more than one point

3) Let A (–1,0) and B (2,0) be two points on the x-axis. A point M is moving in x y-plane (other than x-
axis) in such a way that ∠MBA = 2 ∠MAB, then the point M moves along a conic whose

(A) coordinate of vertices are (± 3, 0).

(B) length of latus-rectum equals 6.
(C) eccentricity equals 2.
equation of directrices are
(D)

SECTION - I (iii)

1) The function is differentiable and continuous and for all x in the interval [4, 8]

If
Identify correct statements

List - I List - II

(P) (1) 0
No. of solution of is

(Q) (2) 1
is

(R) (3) 4
is

(S) No. of solution of f(x) = e–x (4) 3

(5) 8
(A) P → 2;Q → 4;R → 5;S → 2
(B) P → 2;Q → 3;R → 4;S → 5
(C) P → 4;Q → 3;R → 2;S → 1
(D) P → 2;Q → 4;R → 5;S → 1

2)

Let ~ {1} & f(1) = 1, then

Match Column-I with Column-II.

List-I List-II

(P) f(x) is increasing in (1) x ∈ (1,∞)

(Q) f(x) is decreasing in (2)

x ∈ (0, ) ∪ (5, ∞)

(R) f(x) has a local minima in (3) x ∈ (0,1)

(S) f(x) = x has only one solution in (4) x ∈ (0,2)

(5) x ∈ (0, 1) ∪ (4, 5)

(A) P → 3;Q → 4;R → 1;S → 5
(B) P → 3;Q → 1;R → 4;S → 4
(C) P → 4;Q → 1;R → 4;S → 5
(D) P → 4;Q → 1;R → 2;S → 4

3) Match the following :

List - I
List - II

If the distance between two parallel tangents having slope m drawn

(P) (1) 5
to the hyperbola is 2, then is equal to

If a variable line has its intercepts on the coordinate axes e and e’,

(Q) where and are the eccentricities of a hyperbola and its conjugate (2) 4
hyperbola, then the lines always touches the circle x2 + y2 = r2, where
r is equal to

If the equation of lines touching both parabolas y2 - 4x = 0 and x2 +

(R) (3) 2
32y = 0 is ax - 2y + b = 0 (a, b ∈ R), then (a + b) is equal to

(S) If the mid-point of a chord of the ellipse is M (0, 3) and (4) 1

the length of chord is then p is equal to

(5) 0
(A) P → 3;Q → 2;R → 1;S → 4
(B) P → 2;Q → 3;R → 4;S → 1
(C) P → 2;Q → 3;R → 4;S → 5
(D) P → 4;Q → 3;R → 1;S → 2

SECTION - II

1) A variable circle S of radius r cuts a rectangular hyperbola H in four distinct point A, B, C and D.
If O is the centre of H then the value of is

2) From a variable point P, two tangents are drawn to the parabola y2 = 4ax and these tangents meet
the co-ordinate axes in concyclic points. If locus of P is x = λ1a, then λ1 is equal to

3) Let P(x) be a real polynomial of degree 3 which vanishes at x = -3. Let P(x) have local minima at x

= 1, local maxima at x = -1 and , then the sum of all the coefficients of the
polynomial P(x) is equal to __

4) A tangent to the curve y = 1 - x2 is drawn so that the abscissa x0 of the point of tangency belongs
to the interval (0, 1]. The tangent at x0 meets the x-axis and y-axis at A & B respectively. If the

minimum area of the triangle OAB, where O is the origin is then find the value of λ – μ

5) The greatest integer less than or equal to is ________.

6) If , (k > 0) then the value

of must be
 ANSWER KEYS

PHYSICS

SECTION - I (i)

Q. 1 2 3 4
A. A C A B

SECTION - I (ii)

Q. 5 6 7
A. A,B A,B A,B,C

SECTION - I (iii)

Q. 8 9 10
A. B A A

SECTION - II

Q. 11 12 13 14 15 16
A. 2.00 30.00 12.00 8.00 2.00 19.00

CHEMISTRY

SECTION - I (i)

Q. 17 18 19 20
A. B D D C

SECTION - I (ii)

Q. 21 22 23
A. A,B,C,D A,D B,C

SECTION - I (iii)

Q. 24 25 26
A. B C C

SECTION - II

Q. 27 28 29 30 31 32
A. 3.00 6.78 to 6.80 0.30 110.00 0.08 18.00

MATHS
 SECTION - I (i)

Q. 33 34 35 36
A. B B D B

SECTION - I (ii)

Q. 37 38 39
A. D A,B,C B,C,D

SECTION - I (iii)

Q. 40 41 42
A. D B D

SECTION - II

Q. 43 44 45 46 47 48
A. 4.00 1.00 8.00 1.00 5.00 1.60
 SOLUTIONS

PHYSICS

1)

dr = 0.05r

= = wext

2) ∴

Also,

Heat =

Heat =

3) For the electric field,

from energy conservation

v = 2 m/s

4)
=

=
= uBdx

Now, y = a sin

= a sin cos

= sin cos

Now, sin cos dx

5) Acceleration of system a = 5 m/s2

In case of backward direction wave transmission :
Tension at point from r distance from C in the string BC T = 15 – 5r

⇒
In cace of forward direction wave transmission :
Tension at point from r distance from B in the string BC T = 10 + 5r

⇒
⇒

6)
Apply nodal at P

7) (B)

(C)
2T cos θ = mg

⇒ put value of "θ".

sinθ

⇒
⇒

8) (1) λ = L as x = 0 & x = L are nodes

∴ terms should be either sin or sin as k = = ⇒ λ=L

∴ (2) is suitable for (P)
(2) λ = 2L as x = 0 & x = L are nodes

∴ terms should be either sin or sin

as k = = ⇒ λ = 2L
∴ (1) is suitable for (Q)
(3) λ = 4L as x = 0 is node

∴ terms should be either cos or cos

as k = = ⇒ l = 4L
∴ (3) is suitable for (3)
(4) λ = 2L as x = 0 is node

∴ terms should be either cos or cos

as k = = = λ = 2L
∴ (4) is suitable for (4)

9)
r2, r1 are distance of current of finite wire from long wire.

10)

m(v – u) = – ks

v=u– =0

11)
r1 = and r2 =
In one cycle, displacement will be in the x-direction

Δx = 2r1 sinθ – 2r2 sinθ

= 2(r1 – r2)sinθ

Time spent in B1, t1 =

Time spent in B2, t2 =

∴ Avg. velocity

vavg. =

12) mg = 3kx1

mg +

0 0
3k(d – d) (1 + ) – 3k(d – d + x) = man

3k
13) Equilibrium situation:

After air is blown:-

r<<R

→
Similar to F = kx
K = 16π 6
whole bubble of mass 'm' can be considered as a spring of mass 'm'

14)

15)

Let C be the speed of sound.

f'=

f"=

f"=
Frequency difference is given as

Δf = f – f " = f

Δf =

Δf =

Δf =
Δf = 2KHz

16) A metallic rod of mass m and length L (thick line

about IAoR
CHEMISTRY

17) → ionic compound

therefore it dissociate is

18)

In compound (I) N form II-bond with Si & B both, while is compound II N will form π-bond with
only B

19) Packing effeciency when THV are filled in FCC

Stat 1

P.ETV. = ,R=
PE when OHV are filled in FCC, a = R.

P.EOV =
P.E.TV > P.E.O.V.
Stat - 2 THV = 8, OHV = 4.

20) At anode either Ag can oxidised to Ag+ or Sn2+ to Sn4+ or Cℓ– to Cl2 or
Their respective oxidation potential values are 0.13V, –1.36V and –2V. From these
values, it is evident that Sn2+ would be oxidised first, followed by Ag at anode. At cathode,
either Na+ can get reduced to Na or Sn2+ to Sn or H+ to H2. The reduction potential value for

Na+ is highly negative while for Sn2+ | Sn is –0.14V and for

is –0.413V. Thus Sn2+ will get reduced at cathode followed by H+.

21)

(A) The molecularity of an elementary reaction indicates how many reactant species take part
in the step. Thus in unimolecular reaction, only one molecule participates in the step and in
bimolecular reaction, two molecules participate in the step.
(B) The rate law of an elementary relation can be predicted by simply seeing the stoichiometry
of reaction. Thus, for the reaction 2A + B → C, the rate law cannot be written as rate =
k[A]2[B]
(C) The slowest elementary step in sequence of the reactions governs the overall rate of
formation of product. This step is called rate determining step.
(D) A rate law is often derived from a proposed mechanism by imposing the steady state
approximation or assuming that there is a pre-equilibrium.

22)

(A) Chemisorption does not change into physisorption at higher pressure.

(D) If is irreversible.

23) After sufficient time osmotic pressure of all solution will become same.
as T is same i.e. molar concentration should be same, for this ratio of volume should be the
same as that of ratio moles.
Urea : KCl : AlCl3
20 m moles 30 m moles 40 m moles
Total volume (300 ml) should be divided in 2 : 3 : 4

24)

2N2O5(s) CCl4 4NO2(g) + O2(g)

t=0 a
t=t a–x 2x x/2
t=∞ a–a 2a a/2

t=t .... (1)

t=∞ ....(2)

(2) - (1) ∝ v ∞ – vt

(Q)
(A) → Q
(B) C12H22O11 + H2O → C6H12O6 + C6H12O6
Sucrose Glucose Fructose
t=0 a x x
t=t a–x a a
t=∞ a–0
(i) 2x ∝ rt
(ii) 2a ∝ r∞
So 2(a – x) ∝ r∞ – rt

(B) → (T) "C"

(C) CH3COOC2H5 + H2O HCl CH3COOH + C2H5OH
t=0 a – –
t=t a–x x x
t=∞ a–a a a
t=0 eq HCl = Eq NaOH
C ∝ v0 (1)
t=t Eq HCl + eq CH3COOH = Eq NaOH
C + x ∝ vt (2)
t=∞ Eq HCl + Eq CH3COOH = Eq NaOH
C + a ∝ v∞ (3)

from eq. (C → S)
(D) 2H2O → 2H2O + O2
t=0 a
t=t a–x
Eq. H2O2 = Eq. KMnO4
t=0 a ∝ v0
t=t a–x ∝ vt

(D → R)

25) Theoretical

26)

Option (P) :
On adding AgNO3 solution to KCl solution precipitation of AgCl will occur due to which Cl–
already present will be replaced by NO3– ions. So conductance of solution will decrease till
equivalence point. After complete precipitation of AgCl, further added AgNO3 will increase the
number of ions in resulting solution so conductance will increase.
Option (Q) :
On adding KCl solution to AgNO3 solution precipitation of AgCl will occur due to which already
present Ag+ ions will be replaced by K+ ions in solution. So conductance of solution will
increase. After complete precipitation of AgCl further added KCl will increase the number of
ions in resulting solution so conductance will increase further.
Option (R):
On adding HCl solution to NaOH solution, OH– will be replaced by Cl– ions so conductance of
solution decreases. After complete neutralisation further added HCl will increase number of
ions in the solution. So conductance will increase futher.
Option (S):
On adding CH3COOH solution to NaOH solution OH– will be replaced by CH3COO– ions, so
conductance of solution decreases. After complete neutralisation further added CH3COOH will
remain undissociated because it is a weak acid and there is also common ion effect on acetate
ions. So number of ions in solution will remain almost constant therefore conductance of
solution will remain constant.

27)

R = K1[A]x
K' = K[H+]y

= K[10–2]y

= K[10–3]y
⇒y=2
As half life is independent of concentration of A
Hence x × 1.

28) CO2 escaped out : 853.5 –851.3 = 2.2 gm.

= 3.4 × 103 × = 6.786

29)

nA = 10 nA An (s)
nB = 20 10 mole ---

10-x mole mole

After 10 min, number of moles of A, B and another solute added
= (10 –x ) + 20 + 1 = (31 – x) mole

vapour pressure of solution =

450 =

279 – 9x = 60 – 6x + 200 = 260 – 6x

i.e. 0.30 m–1

30) n = 4(3 – 0) = 12
ΔGº = – = – 12 × 96500 × 1.5 = – 1737000 J
ΔGº = ΔHº – TΔSº
 or, –1737 = 2(– 840) –
or, – 57 = – 0.3 ΔSº
or, ΔSº = 190 Jk–1
or,
or, 2 × 280 – 4 × 10 – = 190
or, = 330

or,

31)

Explanation: The degree of dissociation of the NH₄OH solution is to be found.

Given Data:

A. Molar conductivity of NH₄Cl at infinite dilution (λ°NH₄Cl) = 150 Ω⁻¹ cm² mol⁻¹
B. Molar conductivity of OH⁻ at infinite dilution (λ°OH⁻) = 200 Ω⁻¹ cm² mol⁻¹
C. Molar conductivity of Cl⁻ at infinite dilution (λ°Cl⁻) = 75 Ω⁻¹ cm² mol⁻¹
D. Molar conductivity of 0.01 M NH₄OH solution (λ) = 22 Ω⁻¹ cm² mol⁻¹

Concept: kohlrausch's Law and its application

Calculation:

A. Molar conductivity of NH₄OH at infinite dilution (λ°NH₄OH):

Using Kohlrausch's law:
λ°NH₄OH = λ°NH₄⁺ + λ°OH⁻
λ°NH₄OH = (λ°NH₄Cl - λ°Cl⁻) + λ°OH⁻
λ°NH₄OH = (150 Ω⁻¹ cm² mol⁻¹ - 75 Ω⁻¹ cm² mol⁻¹) + 200 Ω⁻¹ cm² mol⁻¹
λ°NH₄OH = 275 Ω⁻¹ cm² mol⁻¹
B. Degree of dissociation (α):
α = λ / λ°
α = (22 Ω⁻¹ cm² mol⁻¹) / (275 Ω⁻¹ cm² mol⁻¹)
α = 0.08

Final Answer: 0.08

32) For maximum packing efficiency in 2D, hexagonal arrangement will be used.
 Radius of Marble = 5 mm
Area of Marble = pr2 = p(5 mm)2
There will be some gaps left over while keeping the marbles

Total half of marble covers area of an equilateral triangle of side (2r).

Area covered by 1 marble =

Total area to be covered = a2 = (8r)2

No. of marbles = = 18.47

Hence, we can at maximum put centre of 18 marbles in the given area.

MATHS

33) Diameter of the inscribed circle of triangle OCF is 2.

Radius = 1

Centre of the circle is (1, 1) and equation of CF is

Now, length of perpendicular from (1, 1) on CF =1 (radius)

On squaring both sides, then

Also,

Hence, (AB) (CD) = (2a) (2b) = 13 x 5 = 65

34)
h'(x) = f (x)

h(x) = g'(x) h'(x) =

g"(x) = f(x)

∴ Apply IBP

⇒ 3+g =5⇒ g =2
2 2
Let p(x) = k(x – 2) (x – 3) + 3
As, p(1) = 4k + 3 = 7
∴ k = 1 ⇒ f(x) = (x – 2)2 (x – 3)2 + 3
Now, p'(x) = 2(x – 2)(x – 3)(2x – 5)
⇒ p (x) has local maximum at x =

∴ =

35)

I=

=
I = I 1 + I2 …… {Let}
For I1, let tan x cos θ – sin θ = t2

For I2, let

I = I 1 + I2

=
=
Comparing
AB =

36)

...(1)
 ...(1)

in (1) put x = 0 f(0) = –1

37)

⇒
⇒ 3–2a – 4.3–a + 3 ≥ 0
⇒ (3–a – 1) (3–a – 3) ≥ 0
⇒ 3–a ≥ 3 or 3–a ≤ 1
⇒ a ≤ – 1 or a ≥ 0
∴ a ∈ (–∞, – 1] ∪ [0, ∞)

38) +

where &

⇒ ⇒ = critical point

graph of g'(x) =
39)

Here tan2θ = and ...(2)

...(1)

solving (1) & (2)

is a hyperbola

,
foci ≡ (±2, 0)

equation of directrix ⇒

length of L.R.

40) We know

Now consider

= 1 + 2k . 2 + 4k2
Put

put x = 4, –4 = 2 + c

∴ c = –6

x=4

41)
F(1) = F(0) = 1

42) (P) The equation of tangents to hyperbola having slope m are y = mx

Now, (Given)

(Q) As, and are eccentricities of a hyperbola and its conjugate hyperbola, so
Length of chord =

43) H = xy = c2, S ≡ (x – α)2 + (y – β)2 = r2 let

S1 = t12 + t22 + t32 + t42 =

44)
The equation of first tangent
is t1y = x + at12
P(0,at1) & R(–at12, 0)
For second tangent
Q(0, at2) & S(–at22,0)
For concyclic points (OP) (OQ) = (OR) (OS)
D t1t2 = 1
Point x = at1t2
⇒ x=a
⇒ λ1 = 1

45)
as

Therefore, sum of all coefficients =

46) y = 1 – x2
Consider point
0 < x0 ≤ 1
equation of tangent at P is

intersection with x-axis at

⇒
intersection with y-axis at

area of ΔOAB Δ =

= ⇒

changes sign from + ve to – ve

0
at x = So point of minimum ⇒
47)

Let
Let

or

So,

or

So, [I] = 5.

48)

Let

=
∴ k2 = 64 ⇒ k = 8