01-06-2025

1001CJA106216250077 JA

PART-1 : PHYSICS

SECTION-I

1) Consider a system of n concentric conducting shells of radii r, 2r, 4r, 8r, ....., 2n–1r respectively. If
inner shell is earthed and Q charge is given on outer most shell, charge appears on inner most shell

is :-

(A) –Q

(B)

(C)

(D) +Q

2) Three 60W, 120V light bulbs are connected across a 120 V power source. If resistance of each
bulb does not change with current then find out total power delivered to the three bulbs.

(A) 180 W
(B) 20 W
(C) 40 W
(D) 60 W

3) Statement–1 : A sphere, a cube and a thin circular plate made of same material and of same
mass are initially heated to 200°C and kept same surrounding, the plate will cool at fastest rate.
and

Statement–2 : Rate of cooling . Surface area is maximum for circular plate.

(A) Statement–1 is True, Statement–2 is True; Statement–2 is a correct explanation for Statement–1
Statement–1 is True, Statement–2 is True; Statement–2 is not a 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) A thin conducting square loop of side L is placed in the first quadrant of the xy-plane with one of

the vertices at the origin. If a changing magnetic field is applied,

where β0 is a constant, then the magnitude of the induced electromotive force in the loop is :-

4
(A) 4β0L
4
(B) 3β0L
4
(C) 2β0L
4
(D) β0L

5) An aluminium ring B faces an electromagnet A. The current I through A can be altered. Then

which of the following statement is correct :-

(A) If I decreases A will repel B

(B) Whether I increases or decreases, B will not experience any force
(C) If I increases, A will repel B
(D) If I increases, A will attract B

6) A water cooler of storage capacity 120 litres can cool water at a constant rate of P watts. In a
closed circulation system (as shown schematically in the figure), the water from the cooler is used to
cool an external device that generates constantly 3 kW of heat (thermal load). The temperature of
water fed into the device cannot exceed 30°C and the entire stored 120 litres of water is initially
cooled to 10°C. The entire system is thermally insulated. The minimum value of P (in watts) for
which the device can be operated for 3 hours is: (Specific heat of water is 4.2 kJ kg–1 K–1 and the

density of water is 1000 kg m–3)

(A) 1600
(B) 2067
(C) 2533
(D) 3933

7) A capacitor is made of two square plates each of side 'a' making a very small angle α between

them, as shown in figure. The capacitance will be close to :

(A)

(B)

(C)

(D)

8) A particle of mass M and carrying charge Q is launched with initial speed v and at an angle of θ
relative to the horizontal direction. When it reaches the maximum height it enters a region of
uniform magnetic field. In the region it moves at constant velocity in the horizontal direction.
Determine the direction and strength of the magnetic field.

(A)
outward
inward
(B)

(C)
outward

(D)
inward

9) Consider a hollow pipe of infinite length and radius R. The pipe has uniform distribution of charge
on its surface. The pipe is rotating with angular velocity ω about its axis. The pipe is also moving
with velocity v along its length. There are two points P and Q as shown in the figure. Point P is inside
the pipe and point Q is outside the pipe. Then match the component of magnetic field in list-I with
description given in list-II.

List-I List-II

(P) Bx at point P (1) is zero

(Q) Bx at point Q (2) depends on ω

(R) By at point P (3) depends on v

(S) Bz at point Q (4) inversely proportional to distance from axis of the pipe

(5) None of these

(A) (P) - 1; (Q) - 1; (R) - 2; (S) - 3

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

10) Breaking stress of a wire is 10 N/m2. For a wire of same material but double the area of cross-
section, breaking stress will be

(A) 20 N/m2
(B) 40 N/m2

(C)

(D) 10 N/m2

11) A copper ball of mass 100 gm is at a temperature T. It is dropped in a copper calorimeter of

mass 100gm, filled with 170 gm of water at room temperature. Subsequently, the temperature of the
system is found to be 75°C. T is given by : (Given : room temperature = 30° C, specific heat of
copper = 0.1 cal/gm°C)

(A) 1250°C
(B) 825°C
(C) 800°C
(D) 885° C
12) The wall with a cavity consists of two layers of brick separated by a layer of air. All three layers
have the same thickness and the thermal conductivity of the brick is much greater than that of air.
The left layer is at a higher temperature than the right layer and steady state condition exists. Which
of the following graphs predicts correctly the variation of temperature T with distance d inside the
cavity?

(A)

(B)

(C)

(D)

13) A uniform elastic string has length a1 when tension T1 and length a2 when tension is T2. The
amount of work done in stretching it from its natural length to a length (a1 + a2) is :

(A)

(B)

(C)

(D)
14) A Helmholtz coil has a pair of loops, each with N turns and radius R. They are placed coaxially at
distance R and the same current I flows through the loops in the same direction. The magnitude of
magnetic field at P, midway between the centres A and C, is given by [Refer to figure given below]:

(A)

(B)

(C)

(D)

15) Statement–I: A charged particle can never move along a magnetic field line in absence of any
other force.
Statement–II : Force due to magnetic field is given by

(A) Statement–I is true, Statement–II is true ; Statement–II is correct explanation for Statement–I.
Statement–I is true, Statement–II is true ; Statement–II is NOT a correct explanation for
(B)
statement–I
(C) Statement–I is true, Statement–II is false
(D) Statement–I is false, Statement–II is true

16) A long infinite current carrying wire bent in the shape as shown in figure. The magnetic

induction at point O is :–

(A)

(B)

(C)

(D)

17) An infinitely long hollow conducting cylinder with inner radius R/2 and outer radius R carries a
uniform current density along its length. The magnitude of the magnetic field, as a function of
the radial distance r from the axis is best represented by

(A)

(B)

(C)

(D)

18) A uniform magnetic field of 0.4 T acts perpendicular to a circular copper disc 20 cm in radius.
The disc is having a uniform angular velocity of 10 π rad s–1 about an axis through its centre and
perpendicular to the disc. What is the protential difference developed between the axis of the disc
and the rim ? (π = 3.14)

(A) 0.0628 V
(B) 0.5024 V
(C) 0.2512 V
(D) 0.1256 V

19) Figure shows three regions of magnetic field, each of area A, and in each region magnitude of
magnetic field decreases at a constant rate α. If is induced electric field then value of line integral
 along the given loop is equal to

(A) αA
(B) –αA
(C) 3αA
(D) –3αA

20) Two inductors L1 and L2 are connected in parallel and a time varying current i flows as shown.

The ratio of currents i1/i2 at any time t is

(A) L1/L2
(B) L2/L1

(C)

(D)

SECTION-II

1) A hollow conducting cone (mass m, radius R, height h) has a current I flowing uniformly along its
curved surface area. This cone is kept on a very rough horizontal surface and a uniform magnetic

field exists in space. The maximum value of I so that the cone does not topple about point P

is given by , then n is
2)

Three identical rods are joined with each other at point P as shown. Lateral surfaces of each rod is
insulated. If temperature of their ends are 90°C, 90°C and 0°C then temperature of junction P is

x°C. Find value of .

3) A steel rod with Y = 2.0 × 1011 Nm–2 and α = 10–5 °C–1 of length 4 m and area of cross-section 10
cm2 is heated from 0° C to 400°C without being allowed to extend. The tension produced in the rod
is x × 105 N where the value of x is .............

4) A rectangular loop PQRS is formed by joining the corners of cube of side 20 cm as shown in
figure. If I = 8 A and then the magnitude of torque on loop is P × 10–3 N-m. Find the value

of P.

5) A loop of area 0.04 m2 with 100 turns is kept in a magnetic field perpendicular to plane of coil
depending on time as B = 4 sin 2t. What is value of maximum emf induced in the loop ? (B is in tesla
and give your answer in S.I. units)

PART-2 : CHEMISTRY

SECTION-I

1) Statement-I : The equilibrium of A(g) ⇌ B(g) + C(g) is not affected by changing the volume.
Statement-II : Kc for the reaction does not depend on volume of the container.

(A) Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I.
 Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for
(B)
statement-I
(C) Statement-I is true, Statement-II is false
(D) Statement-I is false, Statement-II is true

2) A gas behaving ideally was allowed to expand reversibly and adiabatically from 1
litre to 32 litre. It's initial temperature was 327° C. The molar enthalpy change for the process is :-

(A) –1125 R
(B) – 575 R
(C) –1575 R
(D) –525 R

3) One mole of an ideal gas expanded from 1 litre to 4 litre against constant pressure of 2 atm and
constant temperature 300 K, then heat absorbed by gas is : (1 lit-atm = 100 J)

(A) 600 J
(B) 800 J
(C) –600 J
(D) –800 J

4) A hypothetical reaction , A ® 2B, proceed through following sequence of steps -

A → C; ΔH = q1
C → D; ΔH = q2

D → B; ΔH = q3
The heat of reaction A → 2B is :

(A) q1 – q2 + 2q3
(B) q1+ q2 – 2q3
(C) q1 + q2 + 2q3
(D) q1 + 2q2– 2q3

5) The reaction Cis-X trans-X is 1st order in both direction. At 25ºC the equlibrium constant
is 0.10 and the rate constant Kf = 3 × 10–4 sec–1. In an experiment starting with the pure cis form,
how long would it take for half of the equilibrium amount of the trans isomer to be formed -
(ln2 = 0.7)

(A) 150 sec.

(B) 200 sec.
(C) 240 sec.
(D) 210 sec.

6)
A reaction takes place in various steps. The rate constant for first, second, third and fifth steps are
k1, k2, k3 and k5 respectively. The overall rate constant is given by

If activation energy are 40, 60, 50 and 10 kJ/mol. respectively, the overall energy of activation
(kJ/mol.) is

(A) 10
(B) 20
(C) 25
(D) 40

7) If a ground water contains H2S at concentration of 2 mg/l, determine the pressure of H2S in head
space of a closed tank containing the ground water at 20°C. Given that for H2S, Henry's constant is
equal to 6.8 × 103 bar at 20°C.

(A) 720 Pa
(B) 77 × 102 Pa
(C) 553 Pa
(D) 55 × 102 Pa

8) Find major product of following reaction sequence is :

(A)

(B)

(C)
(D)

9) For the reaction sequence given below, the correct statement(s) is(are)

(In the options, X is any atom other than carbon and hydrogen, and it is different in P, Q and R)

(A) C–X bond length in P, Q and R follows the order Q > R > P.
(B) C–X bond enthalpy in P, Q and R follows the order R > P > Q.
(C) Relative reactivity toward SN2 reaction in P, Q and R follows the order P > R > Q.
pKa value of the conjugate acids of the leaving groups in P, Q and R follows the order R > Q >
(D)
P.

10) Consider the reaction given below

Number of monochlorinated product obtained if X and if product undergo fractional distilation and
number of fractions obtained is Y then find out 2X – Y

(A) 18
(B) 15
(C) 21
(D) None of these

11)
Product (D) will be :

(A)
(B)

(C)

(D)

12) Which of the following reaction is correctly matched with major product ?

(A)

(B)

(C)

(D)

13)

List - I List - II
(Reagent) (Major product)

(P) (1) 1º ROH

(Q) (2) 2º ROH

MeMgX +
 (R) (3) Cyclic product
MeMgX +

Major product contains

(S) (4)
chiral carbon
MeMgX +

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

14)

correct option?

(A)
(P) is (Phthalic acid)

(B)

(Q) is (Phthalic anhydride)

(C) Both (A) and (B) correct
(D) Both (A) and (B) incorrect

15) Read the following statements:

(I) In the electrolytic reduction of alumina dissolved in fused cryolite (Na3AlF6) mixed with fluorspar
(CaF2), cryolite and fluorspar are used to decrease the M.P. and increase the conductivity of the
electrolyte
(II) Leaching is a method to dissolve Al2O3 from red bauxite.
(III) The conversion of PbS into Pb requires roasting and self reduction.
(IV) Sodium is extracted by the electrolytic-reduction of aqueous sodium chloride.
Pick the correct set of statements.

(A) I, II
(B) I, III, IV
(C) II, III, IV
(D) I, II, III

16) Which statements is/are correct about sodium nitroprusside test?

(I) This test is used for detection of S2– anion.
(II) H2S also gives positive test.
(III) Formation of Na2[Fe(H2O)5NOS] complex confirm the presence of S2– anion.
(IV) Iron has +2 oxidation state in sodiumthionitroprusside complex.

(A) I & II
(B) I, II & III
(C) I & IV
(D) I, II, III & IV

17) XeFn+ mH2O → A(Contain Xe) + Other Products

Regarding above reaction, which of the following option is correct :

(A) When n = 6 and m = 2, the hybridisation of Xe in A is sp3d2.

(B) When n = 2 and m = 1, then other products has a diatomic colored gas.
(C) When n = 6 and m = 3, then Electron pair structure of A is tetrahedral.
(D) When n = 6 and m = 1, the hybridisation of Xe in A is sp3d.

18)
Select the INCORRECT statements for compound X, Y and Z.

(A) Colours observed yellow, orange & blue respectively.

(B) O.N. of Cr is +6 in all three
3+
(C) [Z] decomposes rapidly in aqueous solution into Cr & O2
(D) [X] is paramagnetic

19) In which process redox change is not observed :-

(A) P4 + conc. HNO3 →

(B) Fe2O3 + dil. HNO3 →
(C) I2 + conc. HNO3 →
+
(D) NaNO2 + K2Cr2O7 + H →

20) Assertion (A) : CO forms weak bonds to Lewis acid such as BF3. In contrast, CO forms strong
bonds to transition metals.
Reason (R) : In both case the bonding pattern is different that is and

(A) Assertion and Reason both are true and the Reason is a correct explanation of Assertion.
(B) Assertion and Reason both are true but the Reason is NOT a correct explanation of Assertion.
(C) Assertion is true but the Reason is false.
(D) Assertion is false but the Reason is true.

SECTION-II
1) Consider the reaction N2O4(g) ⇌ 2NO2(g). The temperature at which KC = 20.4 and KP = 600.1,
is_____K. (Round off to the Nearest Integer).
[Assume all gases are ideal and R = 0.0831 L bar K–1 mol–1]

2) Based on Heisenberg's uncertainty principle, the uncertainty in the velocity of the electron to be
found within an atomic nucleus of diameter 10–15 m is ______× 109 ms–1 (nearest integer)
[Given : mass of electron = 9.1 × 10–31 kg, Plank's constant (h) = 6.626 × 10–34 Js] (Value of π = 3.14)

3) Total number of compounds which show positive Tollen's as well as Fehling's test -

(a) (b) (c) (d)

(e) (f) (g)

(h) (i)

4) If X is the number of basic oxides among CrO3 , MnO , Bi2O3 , Fe2O3 , V2O5 , Ga2O3 , SnO
Then find total stereo isomers form by complex [Fe(NH3)x(H2O)3]

5) If non existing species among CℓF7 , , , XeH4 , NOF3 are 'x' then find value of 'y' in
silicate mineral Hemimorphite Zny(OH)x[Si2O7].

PART-3 : MATHEMATICS

SECTION-I

1) The value of

(A) 1
(B)

(C)
(D)

2) Let x = secθ – cosθ and y = sec3θ – cos3θ then the value of at x = 0 is

(A) 0
(B) 2
(C) 4
(D) 9

3) Assertion A : π4 > 4π

Reason R : The function y = xx is decreasing

(A) Both A and R are correct and R is the correct explanation of A

(B) Both A and R are correct but R is NOT the correct explanation of A
(C) A is correct but R is not correct
(D) A is not correct but R is correct

4) A function g(x) is defined as and f'(x) is an increasing function

then g(x) is increasing in the interval ?

(A)

(B)

(C) (–1, 1)

(D)

5) Let then (λ1 + λ2) is equal

to (where 'C' is the constant of integration)

(A) 0
(B) 1
(C) 2
(D) 3

6) The circle x2 + y2 = 4 cuts the line joining the points A(1, 0) and B(3, 4) in two points P and Q. Let

and . Then α and β are roots of the quadratic equation :

(A) 3x2 + 2x – 21 = 0
(B) 3x2 + 2x +21 = 0
(C) 2x2 + 3x – 21 = 0
(D) 3x2 – 16x + 21 = 0

7) The integral equals (where 'c' is constant of integration)

(A)

(B)

(C)

(D)

8) The range of values of λ (λ > 0) such that the angle θ between the pair of tangents drawn from (λ,

0) to the circle x2 + y2 = 4 lies in is :

(A)

(B)

(C) (1, 2)
(D) None of these

9) Let I = and J = . Then for any arbitrary constant C match the

following List-I with List-II

List-I List-II

(A) I (I)
tan–1 +c

(B) J + I (II)
ln +c

(C) J–1 (III)

(D) J (IV)

(A) (A) – IV, (B) – I, (C) – II, (D) – III

(B) (A) – III, (B) – IV, (C) – I, (D) – II
(C) (A) – I, (B) – III, (C) – IV, (D) – II
(D) (A) – III, (B) – I, (C) – II, (D) – IV

10) Let . If the value of is mπ, then the

value of m is

(A) 100
(B) 98
(C) 99
(D) 117

11) If f : R → R is a continuous function satisfying f(2x) – f(x) = x ∀ x ∈ R and =

P(x) then P(x) is

(A) a constant function

(B) a linear polynomial in x
(C) a quadratic polynomial in x
(D) a cubic polynomial in x

12) Let which one of the following statements is correct?

(A) f(x) is continuous for all x when a = 5

(B) f(x) must be discontinuous for all x, when a = 5

(C)
f(x) is continuous for all x = 2nπ – tan–1 , n ∈ I when a = 5

(D)
f(x) is continuous for all x = 2nπ – tan–1 , n ∈ I when a = 5

13) Let a and b ∈ R and f: R → R be defined by f (x) = . Then which of the

following is INCORRECT?

(A) differentiable at x = 0 if a = 1 and b = 0

(B) differentiable at x = π if a = 1 and b = 1
(C) non differentiable at x = 0 if a = 1 and b = 1
(D) differentiable at x = 2π if a = 1 and b ∈ R

14) Let f(x) = sin3 x + λ sin2 x for – < x < and λ > 0. The range of values of λ for which f(x) has
exactly one minimum and one maximum is
(A)

(B) 0 < λ < 1

(C) 1 < λ < 2
(D) 1 < λ < 3/2

15) Statement-1 : Tangents at two distinct points of cubic polynomial cannot coincide.
Statement-2: If P(x) is a polynomial of degree n(n ≥ 2), then P'(x) = k cannot hold for n or more
distinct values of x.

(A) Both the statments are true and Statement 2 is correct explanation of Statement-1.
(B) Both the Statements are true and Statement-2 is not the correct explanation of Statement 1.
(C) Statement 1 is true and Statment 2 is false.
(D) Statement – 1 is false and Statement 2 is true.

16) Let a, b, c ∈ R satisfy the equations a + [b] + {c} = 3⋅1, {a} + b + [c] = 4⋅3 and [a] + {b} + c =
5⋅4 (where [⋅] denotes the greatest integer function and {⋅} is fractional part), then 6a + b – 3c is

(A) 0
(B) 2
(C) 4
(D) 6.3

17) Let f : R → R is a function satisfying the property f(2x + 3) + f(2x + 7) = 2, x ∈ R, then the
period of f(x) is

(A) 2
(B) 4
(C) 6
(D) 3

18) A point on the hypotenuse of a right angle triangle is at distance 64 and 27 from the sides of
triangle, then minimum length of the hypotenuse is -

(A) 175
(B) 75
(C) 125
(D) 225

19) Let line L is normal to the curve . If slope of line L is , then sum of x intercept
and y intercept of line L is -

(A)

(B)
(C)

(D)

20) Number of solutions of the equation x3 + 2x2 + 5x + 2cosx = 0 in [0, 2π], is

(A) 1
(B) 2
(C) 3
(D) 0

SECTION-II

1) Let f : R → R be a function defined by f(x) = x3 – 3x2 – 9x + 27.

If f3(x) – 3f2(x) – 9f(x) + 27 < f(x3 – 4x2 – 3x + 19) for x ∈ (a, b) ∪ (c, d), then (b – a) + (c + d) is
(where a < b < c < d)

2) Let , , and A = , B = tan α . tan β

+ tan β . tan γ + tan γ . tan α then A + B is

3) If , then 2 (λ + µ) is equal to :-

4) If the length of the largest interval in which the function f(x) = sin–1|sin αx| + cos–1(cos αx) is
constant is 2π, then the value of 20α is______ where (α > 0)

5) For a polynomial P(x) with real coefficient, let np denote the number of distinct real roots of P(x).
Suppose M is the set of polynomials with real coefficients defined by
M = {(x – 2)2(x – 4)2 (α7x7 + α6x6 + α5x5 + α4x4 + α3x3 + α2x2 + α1x + α0) ;
Where α is ∈ R and α7, α0 > 0}
For a polynomial g, let g' and g'' denote its first and second order derivatives, respectively. Then the
minimum possible value of (ng + ng' + ng''), where g ∈ M, is
 ANSWER KEYS

PART-1 : PHYSICS

SECTION-I

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

SECTION-II

Q. 21 22 23 24 25
A. 7 6 8 128 32

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 A C D C A C B C B B B C D C C D B A

SECTION-II

Q. 46 47 48 49 50
A. 354 58 5 2 4

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. B D C B C D B A D C B C C A A C B C D D

SECTION-II

Q. 71 72 73 74 75
A. 10 2 3 10 10
 SOLUTIONS

PART-1 : PHYSICS

1) Charge on inner most shell is .

3) For same mass surface area in maximum for thin circular plate and rest parameter depends
on material.

4)
Area element,

Magnetic flux,

Thus magnitude of e.m.f.

5) According to Lenz' law.

6) P × 3 × 60 × 60 + 120 × 4.2 × 103 × 20 = 3 × 103 × 3 × 60 × 60

P = 2066.67 = 2067 watt

7) Assume small element dx at a distance x from left end

Capacitance for small element dx is
dC = ;C=

8) 2vcosθB = Mg

10) Breaking stress depends on material not on area.

11) Heat given = Heat taken

(100) (0.1)(T – 75) = (100)(0.1)(45) + (170)(1)(45)
10(T – 75) = 450 + 7650 = 8100
T – 75 = 810
T = 885 °C

12)

13) Let ℓ is natural length and K is spring force constant

T1 = K (a1 - ℓ) ......(i)
T2 = K (a2 - ℓ) ......(ii)
From (i) and (ii)

The amount of work done

14)

Here & a = R, N turns

16) (Due to semi–infinite);

17)

M.F. inside will be zero

M.F. in between B ∝
M.F. outside B ∝ 1/r

18)

B = 0.4 T
r = 20 cm
ω = 10π rad/s

= 0.2512 V

20)

L1I1 = L2I2

21)

23)

Thermal force F = Ay∝ΔT

F = (10 × 10–4) (2 × 1011) (10–5)(400)
F = 8 × 105 N ⇒ x = 8
 24) As we known
.... (i)

Here

25)

As
f = BA = 4 Sin2t (0.04)
ε = –100 × 4 cos2t (2).(0.04)
= –32 cos2t
∴ εmax = 32

PART-2 : CHEMISTRY

26) Theory based.

30)

Kf + Kb =

t= ln
t = 210 sec.

32) Henry's law,

Pi = KH.Xi; where;
Pi = partial pressure of gas above the liquid
Xi = mole fraction of gas in the liquid
KH = Henry's constant
substituting values in the expression; we get

⇒
= 7.2 × 10–3 bar = 720 Pa
 39)

42) XeF6 + H2O → XeOF4 + 2HF

XeF6 + 2H2O → XeO2F2 + 4HF
XeF6 + 3H2O → XeO3 + 6HF
2XeF2 + 2H2O → 2Xe(g) + 4HF(g) + O2(g)

46) N2O4(g) ⇌ 2NO2(g) ; Δng = 2 – 1 = 1

Now, Kp = Kc . (RT)Δng
or, 600.1 = 20.4 × (0.0831 × T)1
∴ T = 353.99 K = 354K

47)

48)

b, c, d, e, g

PART-3 : MATHEMATICS

51)
by using standard limit

52) x = secθ – cosθ

= secθ tanθ + sinθ = tanθ(secθ + cosθ)

also y = sec3θ – cos3θ
 = 3sec2θ secθ⋅tanθ + 3cos2θ⋅sinθ
= 3 tanθ (sec3θ + cos3θ)

53) Let y =

decreasing If x > e

Now, for y = xx

⇒ y = exℓux

Increasing

54)

If, x > 0 ⇒

If, x < 0 ⇒

55)
∴ λ1 = 1, λ2 = 1, λ3 = 1 & λ4 = 0

56) The equation of the line joining A(1, 0) and B(3, 4) is y = 2x

– 2. This cuts the circle x2 + y2 = 4 at Q(0, –2) and P .

We have BQ = , QA = , BP = and PA =

∴ and
2
∴ α, β are roots of the equation x – x(α + β) + αβ = 0

i.e., or 3x2 + 2x – 21 = 0

57) x5 + x + 1 ≡ (x2 + x + 1) (x3 – x2 + 1)

Thus 3x4 + 2x3 – 2x + 1 ≡ x3 – x2 + 1 + (3x2 – 2x)(x2 + x + 1)

58) We have

i.e. .

But sin ⇒

⇒ .

59) Given I = and J =

J= =
∴ I+J=
Put ex = t ⇒ ex dx = dt

∴ I+J= =

Put t – = y ⇒ dy = dt

∴ I+J= tan–1

I+J= ....(1)
Similarly,

J–I= ℓn +c ....(2)
Equation (1) – (2) gives

I= +C
Equation (1) + (2) gives

J= +C

60)

61)

Write the functional equation as f (x) – f

⇒
---------------------------
---------------------------

Adding up we obtain

f(x) – f
Hence,

62)

f(x) is continuous where 3 sin x + a2 – 10a + 30 = 4 cos x

or a2 – 10a + 30 = 4 cos x – 3 sin x
LHS ≥ 5, RHS ≤ 5
⇒ (a – 5)2 + 5 = 4 cos x – 3 sin x

⇒ a = 5, x = 2nπ – tan–1 ,n∈I

63)

f(x) = aex sin x + b cos x

It is differentiable everywhere

64)

f (x) = sin3 x + λ sin2 x

f' (x) = 3 sin2 x cos x + 2 λ sin x cos x
for exactly one minimum and one maximum

f' (x) = 0 at two points in

therefore f'(x) = 0 ⇒ cos x = 0 or, sin x (3 sin x + 2 λ) = 0 ⇒ sin x = or sin x = 0, cos x ≠

0

but λ > 0 ⇒ 0 < λ <

65)

Let A (a, P(a)), B(b, P(b)), then slope of AB = P'(a) = P'(b) from LMVT where P'(c) =
slope of AB

66) On adding all equation a + b + c = 6.4

Now subtracting one by one all three equation, we get a = 2, b = 1.3, c = 3.1

67) f(2x + 3) + f(2x + 7) = 2 ..... (1)

Replace x by x + 1, f(2x + 5) + f(2x + 9) = 2 ..... (2)
Now replace x by x + 2, f(2x + 7) + f(2x + 11) = 2 ..... (3)
from (1) – (3) we get f(2x + 3) – f(2x + 11) = 0
f(2x + 3) = f(2x + 11) ⇒ T = 4.
68)

at tanθ =

and at tanθ =

69)

Slope of Normal =

∴y= +1=3+1=4

Point P(3, 4) and Normal L : y – 4 = (x – 3)

X intercept = Y intercept =

70) f(x) = x3 + 2x2 + 5x + 2cosx

f ′ (x) > 0 ∀ x ∈ R
f(0) = 2, f(2π) > 0
∴ No real roots in [0, 2π]

71)
2
f'(x) = 3(x – 2x – 3) = 3(x + 1)(x – 3)
Case-I f'(x) > 0 i.e. f(x) is increasing the x > 3 or x < –1
f(f(x) < f(x3 – 4x2 – 3x +19) ⇒ f (x) < x3 – 4x2 – 3x +19
⇒ x2 – 6x + 8 < 0 ⇒ x ∈ (2,4)
∴ x ∈ (3, 4)
Case-ll f'(x) < 0 i.e. f(x) is decreasing then –1 < x < 3
f(x) > x3 – 4x2 – 3x + 19
⇒ x2 – 6x + 8 > 0 ⇒ (x – 2)(x – 4) > 0 ⇒ x < 2 or x > 4
∴ x ∈ g (–1, 2)
∴ x ∈ (–1, 2) ∪ (3, 4)
∴ b – a + c + d = 2 + 1 + 3 + 4 = 10

72) α + β + γ =

⇒ and hence A =
and
73)

Now,

⇒ I = ex

⇒ µ = 1,

74) Let g(x) = sin–1|sinx|+cos–1(cosx) =

g(x) is periodic with period 2π and is constant in the interval ,n∈I

Now f(x) = g(αx)

∴ f(x) is constant in

75) g(x) = 0
⇒ x = 2, x = 4, x = β (7th degree poly)
(β < 0)
g(2) = g(4) = g(β) = 0
g'(r1) = 0 for r ∈ (2, 4)
g'(r2) = 0 for r ∈ (β, 2)
also g'(2) = g'(4) = 0
so g'(x) = 0 at x = 2, 4, r1, r2
⇒ g''(x) = 0 at atleast three points.