JEE-ADVANCED_Part Test 1 paper - 2013

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JEE-ADVANCED(2013)

PART TEST - 1 (Paper I)

Time: 3 Hours Maximum Marks: 210

Syllabus Covered

Physics : Unit and dimension, Error Analysis, Kinematics, Laws of Motion and friction, Work, Energy and Power

Chemistry : Some Basic Concept, Atomic Structure, Chemical Bonding and Molecular Structure, States of Matter (Gaseous & liquid) Mathematics : Functions, Limits, Continuity and Differentiability, Application of Derivatives, Indefinite Integration

Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose.

INSTRUCTIONS A. General:

1. This booklet is your Question Paper containing 60 questions.

2. The Test ID is printed on the right hand top corner of this booklet. This should be entered on the OMR sheet.

3. Blank papers, clipboards, log tables, slide rules, calculators, cellular phones, pagers, and electronic gadgets in any form are not allowed to be carried inside the examination hall.

4. The answer sheet, a machine-readable Optical mark recognition sheet (OMR Sheet), is provided separately. 5. DO NOT TAMPER WITH / MUTILATE THE OMR OR THE BOOKLET.

6. Do not break the seals of the question-paper booklet before being instructed to do so by the invigilators.

B. Question paper format:

7. The question paper consists of 3 parts (Part I: Physics, Part II: Chemistry, Part III: Mathematics). Each part has 3 sections.

8. Section I contains 10 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), for its answer, out of which only one is correct

9. Section II contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), for its answer, out of which one or more is/are correct.

10. Section III contains 5 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9. The answer will have to be appropriately darkened in the OMR as per the instructions given at the beginning of the section.

C. Marking Scheme:

11. For each question in Section I, you will be awarded 3 marks if you darken the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In case of bubbling of incorrect answer, minus one (–1) mark will be awarded. 12. For each question in Section II, you will be awarded 4 marks if you darken the bubble(s) corresponding to the correct

choice(s) for the answer and zero mark if no bubble is darkened. No negative marks will be awarded in this Section. 13. For each question in Section III, you will be awarded 4 marks if you darken the bubble corresponding to the correct answer

and zero mark if no bubble is darkened. No negative marks will be awarded for in this Section.

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PHYSICS

SECTION – I

Single Correct Choice Type

This section contains 10 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct.

1. A block of mass 4 kg is pressed against the wall by a force of 80 N

as shown in the figure. Determine the value of friction force and

block’s acceleration [take g = 10 m/s2, µs = 0.2, µk = 0.15]

(A) 8 N, 0 m/s2 (B) 32 N, 6 m/s2

(C) 8 N, 6 m/s2 (D) 32 N, 2 m/s2 37

o

80 N 4 kg

2. If the acceleration of wedge in the shown arrangement is ‘a’

towards left then at this instant acceleration of the block would be, (assume all surfaces to be frictionless)

(A) 4a (B)

(

17−8 cosα

)

a (C) 17a (D) 17 cos a

2 α

3. A racing car driver drives his car on a flat circular track of radius 25/3 m and coefficient of

friction 0.5. The magnitude of its tangential acceleration at an instant when car starts slipping at a speed of 5 m/s is

(A) 2 m/s2 (B) 3 m/s2 (C) 4 m/s2 (D) 1 m/s2

4. An object of mass 10 kg is launched from the ground at t = 0, at an angle of 37º above the

horizontal with a speed of 30 m/s. At some time after its launch an explosion splits the projectile into two pieces. One piece of mass 4 kg is observed at (105, 43 m) at t = 2 s. Find the location of

second piece at t = 2 s ? [Take g = 10 m/s2]

(A) (10, 2) (B) (48, 16)

(C) (10, –2) (D) Information is insufficient

5. Two masses mA and mB moving with velocities vA and vB in opposite directions collide elastically.

After that the masses mA and mB move with velocity vB and vA respectively. The ratio (mA / mB ) is

(A) 1 (B) A B A B − + v v v v (C) (mA+m ) / mB A (D) vA/v B

6. Two particles are projected from the same point on ground simultaneously with speeds 20 m/s

and 20 / 3 m/s at angles 30° and 60° with the horizontal in the same direction. The maximum

distance between them till both of them strike the ground approximately:

(g = 10 m/s2)

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7. Two bodies of mass m and 4m are attached with string as shown in the figure. The body of mass m hanging from a string of length

l is executing oscillations of angular amplitude θo while the other

body is at rest. The minimum coefficient of friction between the mass 4m and the horizontal surface should be

(A) 2 – cos 0 3 θ       (B) 2 cos 2 0 2 θ      

(C) 1 – cos 0 2 θ       (D) 0 3 – 2cos 4 θ      

8. With what minimum speed v must a small ball should be pushed

inside a smooth vertical tube from a height h so that it may reach the top of the tube? Radius of the tube is R:

(A) 2 (g h+2 )R (B) 5

2R

(C) g R(5 – 2 )h (D) 2 (2 – )g R h

9.

A thin rod AB is moving in a vertical plane. At a certain instant when the rod is inclined

at 30° to the horizontal the point A is moving horizontally with 3m/s while B is moving in

the vertical direction. Then velocity of B is

A B VB 30° 3 m/s (A)

1 m/s

3

(B) 3 m/s (C) 3 3 m/s (D) 3 m/s 2

10.

The mass and linear momentum in two systems of units are related as under:

2 1

m

y

x

m

_











=

;

P

₁

=

(

x

2

y

)

P

₂

. All the symbols with subscript (1) belong to one system and

all the symbols with subscript (2) belong to the other system. x and y are dimensionless

constants.

U and

₁

U are potential energies in two systems of units. Then

₂

(A)

U

₁

=

(

x

2

y

3

)

U

₂ (B)

U

₁

=

(

x

3

y

3

)

U

₂

(C)

U

₁

=

(

x

2

y

)

U

₂ (D)

U

₁

=

(

x

2

y

2

)

U

₂

SECTION – II

Multiple Correct Choice Type

This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct.

4m m θ0 _l v h R

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11. Which of the following pairs have the same dimensions ?

(A) h

e and magnetic flux (B)

h

e and electric flux

(C) electric flux and

0

ε q

(D) electric flux and µ0I

12. A simple pendulum of length L and mass (bob) M is oscillating in a plane about a vertical line

between angular limits –φ and +φ. For an angular displacement θ (|θ| < φ), the tension in the string

and the velocity of the bob are T and V respectively. The following relations hold good under the above conditions: (A) T cos θ = Mg (B) T – mg cos θ = 2 MV L

(C) the magnitude of the tangential acceleration of the bob |aT| = g sin θ

(D) T = Mg cos θ

13. A particle of mass m moves on the x-axis as follows: it starts from rest at t = 0 from the point

x = 0, and comes to rest at t = 1 at the point x = 1. No other information is available about its

motion at intermediate times (0 < t < 1). If α denotes the instantaneous acceleration of the

particle, then:

(A) α cannot remain positive for all t in the interval 0 ≤ t ≤ 1

(B) |α| cannot exceed 2 at any point in its path

(C) |α| must be ≥ 4 at some point of points in its path

(D) α must change sign during the motion, but no other assertion can be made with the

information given

14. Two resistances R1 = (3.0 ± 0.1) Ω and R2 = (6.0 ± 0.2) Ω are to be joined together.

(A) The maximum resistance obtainable is (9.0 ± 0.3) Ω

(B) The maximum resistance obtainable is (9.0 ± 0.2) Ω

(C) The minimum resistance obtainable is (2.0 ± 0.3) Ω

(D) The minimum resistance obtainable is (2.0 ± 0.2) Ω

15. A particle is projected in air with an initial velocity u directed at an angle θ above horizontal.

After time t0 the velocity of the particle is directed at angle φ above horizontal.

(A) At time t0 magnitude of vertical component of velocity of the particle is u cos θ cot φ

(B) At time t0 magnitude of rate of change of speed of the particle is g sin φ

(C) the radius of curvature of the trajectory at P is

φ

θ

3 2 2

cos

g

u

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SECTION – III

Integer Answer Type

This section contains 5 questions. The answer to each of the questions is a single digit integer, ranging from 0 to 9. The appropriate bubbles below the respective question numbers in the ORS have to be darkened. For example, if the correct answers to question numbers X, Y, and Z (say) are 6, 0 and 9, respectively, then the correct darkening of bubbles will look like the following:

16. A particle is moving along the path given by 6

6

c

y= t (where c is a positive constant). The

relation between the acceleration (a) and the velocity (v) of the particle at t = 5 sec is a = kv, what is the value of k ?

17. A 10 H.P. motor pumps out water from a well of depth 20m and fills a water tank of volume

22380 litres at a height of 10m from the ground. The running time of the motor to fill the empty

water tank is _________ minutes (g = 10ms–2)

18. The chain of length 6m and coefficient of friction 1/2. The maximum length of chain which can

be held outside of table without sliding _________ m.

19. The number of significant figures in all the given numbers 25.12, 2009, 4.156 and 1.217 × 10–4 is ___

20. A light string fixed at one end to a clamp on ground passes over a

fixed pulley and hangs at the other side. It makes an angle of 30° with the ground. A monkey of mass 5 kg climbs up the rope. The clamp can tolerate a vertical force of 40 N only. The maximum acceleration in upward direction with which the monkey can climb

safely is ______m/s2 (Neglect friction and take g = 10 m/s2)

CHEMISTRY

SECTION – I

Single Correct Choice Type

21. If a real gas obey following equation P(V – nb) = nRT, at very low pressure, then the intercept

and slope of graph between d

p Vs P are respectively. (A)

( )

2 M RT MR , T b (B)

( )

2 M Mb , RT _RT − (C)

( )

2 Mb M , RT RT − (D)

( )

2 RT b , M M RT − 30° a

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22. The number of S-S bonds in sulphur trioxide trimer S3O9 is

(A) Three (B) Two (C) One (D) Zero

23. The correct order of bond length of Si–O, P–O, S–O and Cl–O in SiO4₄−, PO₄3−, SO2₄− and

4

ClO− is

(A) ClO₄− > SO2₄− > PO3₄− > SiO4₄− (B) PO3₄− > SiO4₄− > SO2₄− > ClO₄−

(C) SiO4₄− > PO3₄− > SO2₄− > ClO₄− (D) SO2₄− > SiO4₄− > PO3₄− > ClO₄−

24. Calculate de-Broglie wavelength of an electron travelling at 1% of the speed of light

(A) 2.73 × 10–24 m (B) 2.42 × 10–10 m

(C) 242.2 × 1010 m (D) None of these

25. A sample of H2O2 solution labelled as 33.6 volume has density 264 gL

–1

. Mark the correct option representing concentration of same solution in other units. (Assume that solution contains only H2O and H2O2).

(A) Mole fraction of H2O2 in the solution = 0.20

(B) % w/V = 62% (C) Molarity (M) = 6 M

(D) Molality (m) = 1000m

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26. The density of O2 is 16 at NTP. At what temperature its density will be 14? Consider that the

pressure remain constant at

(A) 50°C (B) 39°C (C) 57°C (D) 43°C

27. The velocity of an electron in an excited state of hydrogen atom is 1.093 10× 6m s/ . The

circumference of the orbit in which the electron is revolving is.

(A) 3.32 10× −10m (B) 6.64 10× −10m (C) 13.3 10× −10m (D) 13.28 10 m× −8

28. MnO2 on ignition converts into Mn3O4. A sampleof pyrolusite having 75% MnO2, 20% inert

impuries and rest water is ignited in air to constant weight. What is the percentage of Mn in the ignited sample?

(A) 24.6% (B) 37% (C) 55.24% (D) 74.5%

29. A 10.00 g mixture of Cu2S and CuS was treated with 200.0 mL of 0.7500 M MnO4

−_{in acid}

solution, producing SO2, Cu

2+

and Mn2+. The SO2 was boiled off, and the excess MnO4

−_was

titrated with 175.0 mL of 1.000 M Fe2+ solution. The percent CuS in the original mixture is:

(A) 78% (B) 64% (C) 58% (D) 52%

30. An electron whose total energy is −2.42 × 10−19 J given up a photon as its energy fall to

−21.76 × 10−19 J. What is the wavelength of the emitted light?

(A) 806 Å (B) 1226 Å (C) 2812 Å (D) 5408 Å

SECTION – II

Multiple Correct Choice Type

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31. Which of the following statement(s) is (are) correct?

(A) The electronic configuration of Cr is [Ar] 3d5 4s1. (Atomic number of Cr = 24)

(B) The magnetic quantum number may have a negative value

(C) In silver atom, 23 electrons have a spin of one type and 24 of the opposite type. (Atomic number of Ag = 47)

(D) The oxidation state of nitrogen in HN3 is –3

32. In I₃−

(A) I2 is Lewis acid and I− is Lewis base (B) I2 is Lewis base and I− is Lewis acid

(C) I₃− itself behaves as amphoteric anion (D) I₃− is linear

33. Which of the following is correct for critical temperature

(A) It is the highest temperature at which liquid and vapour can coexist

(B) Beyond the critical temperature, there is no distinction between the two phases and a gas cannot be liquefied by compression

(C) At critical temperature (TC) the surface tension of the system is zero

(D) At critical temperature the gas and the liquid phases have different critical densities

34. H2S acts only as a reducing agent while SO2 can act both as a reducing and oxidizing agent

because

(A) S in H2S has –2 oxidation state

(B) S in SO2 has oxidation state + 4

(C) Hydrogen in H2S more +ve than oxygen

(D) Oxygen is more –ve in SO2

35. For radial probability curves, which of the following is/are correct?

(A) The number of maxima in 2s orbital are two

(B) The number of spherical or radical nodes is equal to n− − 1

(C) The number of angular nodes are " "

(D) 3d has 3 angular nodes2_Z

SECTION – III

Integer Answer Type

36. A gas is present at a pressure of 2 atm. What should be the increase in pressure (in atm) that the

volume of the gas can be decreased to 1/4th of the initial value if the temperature is maintained

constant ?

37. 100 ml of 0.1M HCl is mixed with 100 ml 0.2 M H2SO4 so molarity of 50 ml NaOH for complete

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38. The uncertainty in the location of circulating electron is equal to its de Broglie wavelength. The minimum percent error in its measurement of velocity under this circumstance will be approximately _______

39.

Al and KClO

3

react together to form Al

2

O

3

according to

2KClO

3 →

2KCl + 3O

2

4Al + 3O

2 →

2Al

2

O

3

4 mol of KClO

3

(50% pure) on reaction with excess of Al forms Al

2

O

3

________ mol

40. The bond order of CN− is _______

MATHEMATICS

SECTION – I

Single Correct Choice Type

41. Which of following functions is derivable at x = 0;

(A) sin |x| + cos |x| (B) |x| + cos |x|

(C) |x| + sin |x| (D) |cos x| + cos |x|

42. If f(x) = 4 x2 1

sin x sin x

− +

− , then the domain of f(x) is

(A) [−2, 0] (B) (0, 2] (C) [−2, 2] (D) [−2, 0) 43. Let f(x) = n 1 x sin , x 0 x 0 , x 0  ≠    ₌ 

, then f(x) is continuous, but not differentiable at x = 0, if

(A) n ∈ (0, 1] (B) n ∈ [1, ∞)

(C) n ∈ (–∞, 0) (D) n = 0

44. If the equation x5 – 10a3 x2 +b4 x +c5 = 0 has three equal roots, then

(A) 2b2 –10 a3b2+c5 = 0 (B) 6a5 + c5 = 0 (C) 2c5 –10 a3b2+ b4c5 = 0 (D) b4 = 15 a5 45. I = x dx 1 e+

∫

is equal to (A) loge x x 1 e e  ₊     + c (B) loge x x e 1 e     +  + c (C) loge (e x ) (ex +1) + c (D) loge (e 2x + 1) + c

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46. Let f(x) = min { n (tan x), n(cot x)}. _{Which of the following statement is incorrect?}

(A) f(x) is continuous for x ∈ 0,

2 π

     

(B) Lagrange’s mean value theorem is applicable on f(x) for x ∈ ,

8 4 π π

     

(C) Rolle’s theorem is not applicable on f(x) for x ∈ ,3

4 8

π π

     

(D) Rolle’s theorem is applicable on f(x) for x ∈ ,3

8 8

π π

     

47. The number of solutions of [sin x] + |cos x| = 1, in 2π≤ x ≤ 4π, [where [.] = greatest integer

function], is (A) 3 (B) 4 (C) 5 (D) 6 48.

(

)

2 2 x 0

sin cos (tan(sin x))

lim x → π is equal to (A) π (B) 4 π _(C) 2 π _{(D) none of these} 49.

(

)

2 1 x 0 tan sec x lim x sin x− → π is equal to

(A) – π (B) π (C) 0 (D) none of these

50. dx ₂

sin x cos x =

∫

(A) log |cotx

2| – secx + c (B) log tan

x 2 + cosec x + c (C) log | tanx| 2 + secx + c (D) log x | cot | 2 – cosecx + c

SECTION – II

Multiple Correct Choice Type

51. Let f(x + y) = f(x) + f(y) + 2xy – 1 ∀ x , y ∈ R. If f(x) is differentiable and f ′(0) = sinφ, then

(A) f(x) < 0 ∀ x ∈ R (B) f(x) > 0 ∀ x ∈ R

(C) f(x) ≥ 3

4 ∀ x ∈ R (D) –1 ≤ f(x) ≤ 1 ∀ x ∈ R

52. f(x) is a real valued function, satisfying f(x + y) + f(x – y) = 2f(x).f(y) , x,∀ y∈R. Then

(A) f(x) is an even function (B) f(x) is even if f(0) = 1

(C) f(x) is odd if f(0) = 0 (D) f(x) is even if f(0) = 0

53. If f(x) = (sin2x –1)n (2+ cos2x), n ∈N; then x= π/2 is a point of

(A) local maximum, if n is odd (B) local minimum, if n is odd

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54. If ₂₂ dx₇ n(z )6 9z2 2z3 18z c x (x 6)   = λ_ + − − _+ −

∫

_{, then} (A) 1 54432 λ = (B) 7 7 x 6 z x  ₋  =    (C) 1 24432 λ = (D) 6 6 x 7 z x  ₋  =   

55. The interval in which

y

=

x

3 increases more rapidly than

y

=

6 x

2

+

15 x

+

5

is

(A) (−∞ −, 1) (B) (5, )∞ (C) ( 1, 5)− (D) (0, )∞

SECTION – III

Integer Answer Type

56. If the value of

(

)

5 2 2 3 x x tan 1 / ( x ) 3x 7 lim x 7 x 8 →−∞ π + + + + is equal to – K /π, then K = ?

57. Let f : R → Q be a continuous function ∀ x ∈ R and f (5) = 5, find the value of f (3).

58. If f(x) is a twice differentiable function such that f(a) = 0, f(b) = 2, f(c) = – 1, f(d) = 2,

f(e) = 0, where a < b < c < d < e, then find the minimum number of zeroes of g(x) = (f′(x))2 + f″(x) f(x) in the interval [a, e].

59. Let f(x) = (x – 1)2 (x – 2)3 ex. Then f(x) has point of inflexion at x = _____________

60. Normal to the curve y = (1 + x)y + sin–1(sin2x) at x = 0 is x + y = a. Find the value of a.

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