**Physics**

**Section-A**

1. Two identical blocks A and B each of mass m resting on the smooth horizontal floor are connected by a light spring of natural length L and spring constant K. A third block C of mass m moving with a speed v along the line joining A and B collides elastically with A. The maximum compression in the spring is

(a)

(b)

(c)

(d)

2. A solid sphere of mass 2 kg radius 0.5m is rolling with an initial speed of 1 ms^{−}^{1} goes up an inclined plane which makes an angle of 300 with the horizontal plane, without slipping. How long will the sphere take to return to the starting point A?

(a) 0.80 s

(b) 0.60 s

(c) 0.52 s

(d) 0.57 s

3. If one mole of a polyatomic gas has two vibrational modes and β is the ratio of molar specific heats for polyatomic gas β=C_{p}/ C_{v} then the value of β is :

(a) 1.35

(b) 1.02

(c) 1.25

(d) 1.2

4. Two cells of emf 2E and E with internal resistance r_{1} and r_{2} respectively are connected in series to an external resistor R (see figure). The value of R, at which the potential difference across the terminals of the first cell becomes zero is

(a) r_{1} – r_{2}

(b) r_{1} + r_{2}

(c)

(d)

5. A sound wave of frequency 245 Hz travels with a speed of 300 ms^{−}^{1} along the positive x-axis. Each point of the medium moves to and fro through a total distance of 6 cm. What will be the mathematical expression of the travelling wave?

(a) Y (x, t) = 0.03 [ sin 5.1x − (0.2 x 10^{3})t]

(b) Y (x, t) = 0.06 [ sin 5.1x − (1.5 x 10^{3})t]

(c) Y (x, t) = 0.06 [ sin 0.8x − (0.5 x 10^{3})t]

(d) Y (x, t) = 0.03 [ sin 5.1x − (1.5 x 10^{3})t]

6. A carrier signal C(t) = 25 sin (2.512 × 10^{10}t) is amplitude modulated by a message signal m(t)= 5 sin (1.57 × 10^{8}t) and transmitted through an antenna. What will be the bandwidth of the modulated signal?

(a) 1987.5 MHz

(b) 2.01 GHz

(c) 50 MHz

(d) 8 GHz

7. Two particles A and B of equal masses are suspended from two massless springs of spring constants K_{1} and K_{2} If the maximum velocities during oscillations are equal, the ratio of the amplitude of A and B is :

(a)

(b)

(c)

(d)

8. Match List I with List II

List I | List II |

(a) Phase difference between current and voltage in a purely resistive AC circuit | (i) 𝜋/2, current leads voltage |

(b) Phase difference between current and voltage in a pure inductive AC circuit | (ii) zero |

(c) Phase difference between current and voltage in a pure capacitive AC circuit | (iii) 𝜋/2, current lags voltage |

(d) Phase difference between current and voltage in an LCR series circuit | (iv) |

Choose the most appropriate answer from the options given below :

(a) (a)-(ii), (b)-(iii), (c)-(iv), (d)-(i)

(b) (a)-(i), (b)-(iii), (c)-(iv), (d)-(ii)

(c) (a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)

(d) (a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)

9. A geostationary satellite is orbiting around an arbitrary planet ‘P’ at a height of 11R above the surface of ‘P’, R being the radius of ‘P’. The time period of another satellite in hours at a height of 2R from the surface of ‘P’ is _____. ‘P’ has a time period of rotation of 24 hours.

(a) 6/√2

(b) 3

(c) 6√2

(d) 5

10. The velocity of a particle is v=v_{0} + gt + Ft^{2}. Its position is x = 0 at t = 0; then its displacement after time (t = 1) is :

(a)

(b) v_{0} + 2g + 3f

(c) v_{0} + g + F

(d)

11. A block of mass 1 kg attached to a spring is made to oscillate with an initial amplitude of 12 cm. After 2 minutes the amplitude decreases to 6 cm. Determine the value of the damping constant for this motion. (take ln 2 = 0.693).

(a) 3.3 × 10^{2} kg s^{−}^{1}

(b) 5.7 × 10^{−}^{3} kg s^{−1}

(c) 1.16 × 10^{−}^{2} kg s^{−1}

(d) 0.69 × 10^{2} kg s^{−1}

12. An object is located 2 km beneath the surface of the water. If the fractional compression ∆V/V is 1.36%, the ratio of hydraulic stress to the corresponding hydraulic strain will be ……..

[Given: density of water is 1000 kgm^{–3} and g= 9.8 ms^{–2}]

(a) 2.26 ×10^{9} Nm^{−}^{2}

(b) 1.96 ×10^{9} Nm^{−}^{2}

(c) 1.44 ×10^{7} Nm^{−}^{2}

(d) 1.44 ×10^{9} Nm^{−}^{2}

13. Two identical photocathodes receive the light of frequencies f_{1} and f_{2} If the velocities of the photo-electrons coming out are v_{1} and v_{2} respectively, then

(a)

(b)

(c)

(d)

14. The atomic hydrogen emits a line spectrum consisting of various series. Which series of hydrogen atomic spectra lie in the visible region?

(a) Balmer series

(b) Lyman series

(c) Brackett series

(d) Paschen series

15. Which one of the following will be the output of the given circuit?

(a) NAND Gate

(b) AND Gate

(c) XOR Gate

(d) NOR Gate

16. The four arms of a Wheatstone bridge have resistances as shown in the figure. A galvanometer of 15 resistance is connected across BD. Calculate the current through the galvanometer when a potential difference of 10 V is maintained across AC.

(a) 4.87 mA

(b) 4.87 μA

(c) 2.44 μA

(d) 2.44 mA

17. Which one is the correct option depicting the two different thermodynamic processes?

(a) (c) and (d)

(b) (b) and (c)

(c) (a) only

(d) (c) and (a)

18. A hairpin-like shape as shown in the figure is made by bending a long current-carrying wire. What is the magnitude of a magnetic field at point P which lies on the centre of the semicircle?

(a)

(b)

(c)

(d)

19. A rubber ball is released from a height of 5 m above the floor. It bounces back repeatedly, always rising to (81/100) of the height through which it falls. Find the average speed of the ball. (Take g=10 ms^{−}^{2})

(a) 2.50 ms^{−}^{1}

(b) 3.50 ms^{−}^{1}

(c) 3.0 ms^{−}^{1}

(d) 2.0 ms^{−}^{1}

20. What happens to the inductive reactance and the current in a purely inductive circuit if the frequency is halved?

(a) Both, including reactance and current, will be doubled

(b) Both, inductive reactance and current will be halved

(c) Inductive reactance will be halved and current will be doubled

(d) Inductive reactance will be doubled and current will be halved

**Section-B**

21. The electric field intensity produced by the radiation coming from a 100 W bulb at a distance of 3 m is E. The electric field intensity produced by the radiation coming from 60 W at the same distance is Where the value of x = ________

22. The image of an object placed in the air formed by a convex refracting surface is at a distance of 10 m behind the surface. The image is real and is at 2nd/3 of the distance of the object from the surface. The wavelength of light inside the surface is 2/3 times the wavelength in air, The radius of the curved surface is x/13 m. The value of ‘x’ is _____

23. A 2μF capacitor C_{1} is first charged to a potential difference of 10V using a battery. Then the battery is removed and the capacitor is connected to an uncharged capacitor C_{2} of 8 μF. The charge in C_{2} on equilibrium condition is ________ μC. (Round off to the Nearest Integer)

24. A particle of mass m moves in a circular orbit in a central potential field U(r) =U_{0r4}. If Bohr’s quantization conditions are applied, radii of possible orbital r_{n} vary with n^{1/α}, where α is _____

25. The electric field in a region is given by with The flux of this field through a rectangular surface are 0.4 m^{2} parallel to Y-Z plane is _______Nm^{2}C^{−}^{1}.

26. A body of mass 1 kg rests on a horizontal floor with which it has a coefficient of static friction 1/√3. It is desired to make the body move by applying the minimum possible force F N. The value of F will be __________. (Round off to the Nearest Integer)

[Take g = 10 ms^{−}^{2}]

27. Seawater at a frequency f = 9 × 10^{2} Hz, has permittivity ∈ = 80∈_{0} and resistivity ρ = 0.25 Ωm. Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternating voltage source V(t) = V_{0} sin(2πft). Then the conduction current density becomes 10x times the displacement current density after time t= 1800. The value of x is __________.

(Given : )

28. The disc of mass M with uniform surface mass density σ is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position x is _____(Round off to the Nearest Integer) [a is an area as shown in the figure)

29. Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes 0.01 cm^{3} of oleic acid per cm^{3} of the solution. Then you make a thin film of this solution (monomolecular thickness) of area 4 cm2 by considering 100 spherical drops of radius Then the thickness of the oleic acid layer will be x × 10^{−}^{14} m where x is _________.

30. A boy of mass 4 kg is standing on a piece of wood having mass 5 kg. If the coefficient of friction between the wood and the floor is 0.5, the maximum force that the boy can exert on the rope so that the piece of wood does not move from its place is ________N. (Round off to the Nearest Integer)

[Take g = 10 ms^{−}^{2}]

**Chemistry**

**Section-A**

1. Match List-I with List-II.

**List-I** **List-II**

Chemical Compound Used as

(a) Sucralose (i) Synthetic detergent

(b) Glyceryl ester of stearic acid (ii) Artificial sweetener

(c) Sodium benzoate (iii) Antiseptic

(d) Bithionol (iv) Food preservative

Choose the correct match:

(a) (a)-(i), (b)-(ii), (c)-(iv), (d)-(iii)

(b) (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)

(c) (a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)

(d) (a)-(iii), (b)-(ii), (c)-(iv), (d)-(i)

2. In the below reactions, enzyme A and enzyme B respectively are:

(a) Invertase and Amylase

(b) Amylase and Invertase

(c) Invertase and Zymase

(d) Zymase and Invertase

3. The correct pair(s) of the ambident nucleophiles is (are):

(A) AgCN/KCN

(B) RCOOAg/RCOOK

(C) AgNO_{2}/KNO_{2}

(D) AgI/KI

(a) (A) and (C) only

(b) (B) only

(c) (B) and (C) only

(d) (A) only

4. During which of the following processes, does entropy decrease?

(A) Freezing of water to ice at 0°C

(B) Freezing of water to ice at -10°C

(C) N_{2}(g) + 3H_{2}(g) →2NH_{3}(g)

(D) Adsorption of CO(g) on lead surface.

(E) Dissolution of NaCI in water

(a) (A), (B), (C) and (D) only

(b) (A), (C) and (E) only

(c) (A) and (E) only

(d) (B) and (C) only

5. Match List-I with List-II:

**List – I List – II**

(a) [Co(NH_{3})_{6}] [Cr(CN)_{6}] (i) Linkage isomerism

(b) [Co(NH_{3})_{3} (NO_{2})_{3}] (ii) Solvate isomerism

(c) [Cr(H_{2}O)_{6}]Cl_{3} (iii) Coordination isomerism

(d) cis-[CrCl_{2}(ox)_{2}]^{3}^{−} (iv) Optical isomerism

Choose the correct answer from the options given below:

(a) (a)-(iii), (b)-(i), (c)-(ii), (d)-(iv)

(b) (a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)

(c) (a)-(ii), (b)-(i), (c)-(iii), (d)-(iv)

(d) (a)-(iv), (b)-(ii), (c)-(iii), (d)-(i)

6. The common positive oxidation states for an element with atomic number 24, are

(a) +1 and +3

(b) +1 to +6

(c) +1 and +3 to +6

(d) +2 to +6

7. The set of elements that differ in a mutual relationship from those of the other sets is:

(a) Be – Al

(b) Li – Na

(c) B – Si

(d) Li – Mg

8. Given below are two statements:

Statement I: 2-methyl butane on oxidation with KMnO_{4} gives 2-methyl butane-2-ol.

Statement II: n-alkanes can be easily oxidized to corresponding alcohols with KMnO_{4}.

Choose the correct option:

(a) Both statement I and statement II are incorrect

(b) Statement I is correct but statement II is incorrect

(c) Both statement I and statement II are correct

(d) Statement I is incorrect but statement II is correct

9. Amongst the following, the linear species is:

(a) N_{3}^{−}

(b) Cl_{2}O

(c) O_{3}

(d) NO_{2}

10. For the coagulation of a negative sol, the species below, that has the highest flocculating power is:

(a) SO_{4}^{2}^{−}

(b) Na^{+}

(c) Ba^{2+}

(d) PO_{4}^{3}

11. The functional groups that are responsible for the ion-exchange property of cation and anion exchange resins, respectively, are:

(a) –SO_{3}H and –COOH

(b) –SO_{3}H and –NH_{2}

(c) –NH_{2} and –SO_{3}H

(d) –NH_{2} and –COOH

12. Choose the correct statement regarding the formation of carbocations A and B given.

(a) Carbocation A is more stable and formed relatively at a faster rate

(b) Carbocation B is more stable and formed relatively at a faster rate

(c) Carbocation A is more stable and formed relatively at a slow rate

(d) Carbocation B is more stable and formed relatively at a slow rate

13. In the below reaction, the structural formula of (A), “X” and “Y” respectively are:

14. Fructose is an example of:

(a) Heptose

(b) Aldohexose

(c) Pyranose

(d) Ketohexose

15. Which of the following statement(s) is (are) the incorrect reason for eutrophication?

(A) excess usage of fertilisers

(B) excess usage of detergents

(C) dense plant population in water bodies

(D) lack of nutrients in water bodies that prevent plant growth

Choose the most appropriate answer from the option given below:

(a) (D) only

(b) (C) only

(c) (B) and (D) only

(d) (A) only

16. Primary, secondary and tertiary amines can be separated using:

(a) Para-Toluene sulphonyl chloride

(b) Chloroform and KOH

(c) Acetyl amide

(d) Benzene sulphonic acid

17. Match List-I with List-II

**List-I List-II**

(a) Haematite (i) Al_{2}O_{3}. xH_{2}O

(b) Bauxite (ii) Fe_{2}O_{3}

(c) Magnetite (iii) CuCO_{3}. Cu(OH)_{2}

(d) Malachite (iv) Fe_{3}O_{4}

Choose the correct answer from the options given below:

(a) (a)-(ii), (b)-(iii), (c)-(i), (d)-(iv)

(b) (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)

(c) (a)-(i), (b)-(iii), (c)-(ii), (d)-(iv)

(d) (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)

18. The set that represents the pair of neutral oxides of nitrogen is:

(a) NO and N_{2}O

(b) NO and NO_{2}

(c) N_{2}O and NO_{2}

(d) N_{2}O and N_{2}O_{3}

19. Nitrogen can be estimated by Kjeldahl’s method for which of the following compound?

20. One of the by-products formed during the recovery of NH_{3} from Solvay process is:

(a) NaHCO_{3}

(b) Ca(OH)_{2}

(c) CaCl_{2}

(d) NH_{4}Cl

**SECTION – B**

21. The reaction 2A + B_{2}→ 2AB is an elementary reaction. For a certain quantity of reactants, if the volume of the reaction vessel is reduced by a factor of 3, the rate of the reaction increases by a factor of ……………. (Round off to the Nearest Integer).

22. In the ground state of atomic Fe (Z = 26), the spin-only magnetic moment is ……………..x 10^{−}^{1} (Round off to the Nearest Integer).

[Given: √3 = 1.73, √2 = 1.41]

23. Consider the below-given reaction. The percentage yield of an amide product is ………………. (Round off to the Nearest Integer). [Given: Atomic mass: C: 12.0 u, H : 1.0 u, N : 14.0, O : 16.0 u, Cl : 35.5 u]

24. On complete reaction of FeCl_{3} with oxalic acid in an aqueous solution containing KOH, resulted in the formation of product A. The secondary valency of Fe in the product A is …………..(Round off to the Nearest Integer)

25. Consider the reaction N_{2}O_{4} (g) ⇌ 2NO_{2} (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}]

26. A KCl solution of conductivity 0.14 S m-1 shows a resistance of 4.19 Ω in a conductivity cell. If the same cell is filled with an HCl solution, the resistance drops 1.03 Ω. The conductivity of the HCl solution is ……………….. × 10^{−}^{2} S m^{−}^{1}. (Round off to the Nearest Integer).

27. A 1 molal K_{4}Fe(CN)_{6} solution has a degree of dissociation of 0.4. Its boiling point is equal to that of another solution which contains 18.1 weight per cent of a nonelectrolyte solute A. The molar mass of A is …………… g/mol. (Round off to the Nearest Integer).

28. The number of chlorine atoms in 20 mL of chlorine gas at STP is ………10^{21}. (Round off to the Nearest Integer).

[Assume chlorine is an ideal gas at STP

R=0.083 L bar mol^{−}^{1} K^{−}^{1}, NA = 6.023 × 10^{23}]

29. KBr is doped with 10^{−}^{5} mole per cent of SrBr_{2}. The number of cationic vacancies in 1 g of KBr crystal is ……… 10^{14}. (Round off to the Nearest Integer).

[Atomic Mass: K = 39.1 u, Br = 79.9 u, N_{A} = 6.023 × 10^{23}]

30 The total number of C–C sigma bond/s in mesityl oxide (C_{6}H_{10}O) is …… (Round off to the Nearest Integer).

**Mathematics**

**Section-A**

1. If the Boolean expression (p ∧ q) ⍟ (p ⊗ q) is a tautology, then ⍟ and ⊗ are respectively given by :

(a) ∧, →

(b) →, →

(c) ∨, →

(d) ∧, ∨

2. Let the tangent to the circle x^{2} + y^{2} = 25 at the point R (3, 4) meet the x-axis and y-axis at points P and Q, respectively. If r is the radius of the circle passing through the origin O and having a centre at the incentre of the triangle OPQ, then r^{2} is equal to:

(a) 625/72

(b) 585/66

(c) 125/72

(d) 529/64

3. Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with a probability of occurrence of 0 at even places be 1/2 and probability of occurrence of 0 at the odd place be 1/3. Then the probability that ‘10’ is followed by ‘01’ is equal to :

(a) 1/6

(b) 1/18

(c) 1/9

(d) 1/3

4. The number of solutions of the equation x + 2 tan x = π/2 in the interval [0, 2π] is :

(a) 5

(b) 2

(c) 4

(d) 3

5. If the equation of the plane passing through the mirror image of a point (2, 3, 1) with respect to line and containing the line is ɑx + βy + γz = 24, then, then ɑ + β + γ is equal to:

(a) 21

(b) 19

(c) 18

(d) 20

6. Consider the function f : R → R defined by Then f is

(a) monotonic on (0, ∞) only

(b) Not monotonic on (–∞, 0) and (0, ∞)

(c) monotonic on (–∞, 0) only

(d) monotonic on (–∞, 0) ⋃ (0, ∞)

7. Let O be the origin. Let x, y ∈ R, x > 0, be such that |PQ| = √20 and the vector is perpendicular to If z ∈ R, is coplanar with then the value of x^{2} + y^{2} + z^{2} is equal to:

(a) 2

(b) 9

(c) 1

(d) 7

8. Let L be a tangent line to the parabola y^{2} = 4x – 20 at (6, 2). If L is also a tangent to the ellipse then the value of b is equal to:

(a) 20

(b) 14

(c) 16

(d) 11

9. Let f : R → R be defined as f (x) = e^{–x} sin x. If F : [0,1] R→ is a differentiable function such that then the value of lies in the interval

(a) [330/360, 331/360]

(b) [327/360, 329/360]

(c) [331/360, 334/360]

(d) [335/360, 336/360]

10. If x, y, z are in arithmetic progression with common difference d, x ≠ 3d, and the determinant of the matrix is zero, then the value of k^{2} is :

(a) 6

(b) 36

(c) 72

(d) 12

11. If the integral where ɑ, β, γ are integers and [x] denotes the greatest integer less than or equal to x, then the value of ɑ + β + γ is equal to:

(a) 20

(b) 0

(c) 25

(d) 10

12. Let y = y (x) be the solution of the differential equation cos x (3 sin x + cos x + 3) dy = (1 + y sin x (3 sin x + cos x + 3)) dx, 0 ≤ x ≤ π/2, y(0) = 0. Then, y (π/3) is equal to :

(a)

(b)

(c)

(d)

13. The value of the limit is equal to:

(a) −1/2

(b) −1/4

(c) 0

(d) 1/4

14. If the curve y = y(x) is the solution of the differential equation 2 (x^{2} + x^{5/4}) dy − y (x + x^{1/4}) dx = 2x^{9/4 }dx, x > 0 which passes through the point then the value of y(16) is equal to:

(a)

(b)

(c)

(d)

15. Let S_{1}, S_{2} and S_{3} be three sets defined as S_{1} = {z ∈ C : |z – 1| ≤ √2}, S_{2} = {z ∈ C : Re ((1 – i) z) ≥ 1}, S_{3} = {z ∈ C : Im (z) ≤ 1}. Then the set S_{1} ⋂ S_{2} ⋂ S_{3}

(a) has infinitely many elements

(b) has exactly two elements

(c) has exactly three elements

(d) is a singleton

16. If the sides AB, BC, and CA of a triangle ABC have, 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices is equal to:

(a) 360

(b) 240

(c) 333

(d) 364

17. The value of where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to:

(a) 0

(b) r

(c) r/2

(d) 2r

18. The value of The value of is equal to:

(a) 1124

(b) 924

(c) 1324

(d) 1024

19. Two tangents are drawn from a point P to the circle x^{2} + y^{2} – 2x – 4y + 4 = 0, such that the angle between these tangents is If the centre of the circle is denoted by C and these tangents touch the circle at points A and B, then the ratio of the areas of △PAB and △CAB is :

(a) 11 : 4

(b) 9 : 4

(c) 2 : 1

(d) 3 : 1

20. The number of solutions of the equation and [x] denotes the greatest integer less than or equal to x, is:

(a) 0

(b) 2

(c) 4

(d) Infinite

**Section-B**

21. Let the coefficients of third, fourth and fifth terms in the expansion of be in the ratio 12:8:3. Then the term independent of x in the expansion is equal to ………

22. Let such that AB = B and a + d = 2021, then the value of ad-bc is equal to …………….

23. Let f : [–1,1] → R be defined as f (x) = ax^{2} + bx + c for all x ∈ [–1, 1], where a, b, c ∈ R such that f (–1) = 2, f’ (–1) = 1 and for x ∈ [–1, 1] the maximum value of f’’ (x) is 1/2. If f (x) ≤ ɑ, x ∈ [–1, 1], then the least value of ɑ is equal to …………… .

24. Let where n ∈ If (20)I_{10} = αI_{9} + βI_{B}, for natural numbers α and β, then α – β equal to ………………… .

25. Let f : [−3, 1] → R be given as

If the area bounded by y = f(x) and x-axis is A, then the value of 6A is equal to …………… .

26. Let be a vector in the plane containing vectors If the vector is perpendicular to and its projection on then the value of is equal to ……………… .

27. Consider a set of 3n numbers having variance 4. In this set, the mean of the first 2n numbers is 6 and the mean of the remaining n numbers is 3. A new set is constructed by adding 1 into each of the first 2n numbers and subtracting 1 from each of the remaining n numbers. If the variance of the new set is k, then 9k is equal to ………………..

28. If 1, log_{10}(4^{x} – 2) and are in arithmetic progression for a real number x, then the value of the determinant is equal to:

29. Let P be an arbitrary point having the sum of the squares of the distances from the planes x + y + z = 0, lx – nz = 0 and x – 2y + z = 0, equal to 9. If the locus of the point P is x^{2} + y^{2} +z^{2} = 9, then the value of l – n is equal to ………………

30. Let tan α, tan β and tan γ; be the slopes of three-line segment OA, OB and OC, respectively, where O is the origin. If the circumcentre of ∆ABC coincides with the origin and its orthocentre lies on the y-axis, then the value of is equal to

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