2013 Mocktest 3 Paper 2

Vidyamandir Classes

Mock IIT Advanced Test - 3/2013/Paper-2 12/05/2013 02:00 PM - 05:00 PM

M.M. : 240

TEST CODE : ACEG

TIME : 3.00 Hrs

1.

Blank spaces and blank pages are provided in this booklet for your rough work.

2.

Using a black ball point pen, darken the bubbles on the upper original sheet. Apply sufficient pressure so that the impression is created on the bottom sheet.

3.

Write your Name, Registration Number and the name of examination centre and sign with pen in the boxes provided on the right part of the ORS. Do not write any of this information anywhere else. Darken the appropriate bubble UNDER each digit of your Registration Number.

4.

The question paper consists of 3 parts (Chemistry, Physics and Mathematics). Each part consists of four sections.

5.

Section I contains 8 Straight objective type questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

6.

Section II contains 4 Multiple type questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.

7.

Section III contains 6 Integer (Subjective) type questions. Each question has an integer answer lying between 0 and 9.

8.

Section IV contains 2 Match the columns type questions. Each question contains statements given in 2 columns. Statements in the first column have to be matched with statements in the second column. The answers to these questions have to be appropriately bubbled in the answer sheet.

9.

For each question in Section I, you will be given 3 Marks if you have darkened only the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases, minus ONE (–1) marks (NEGATIVE MARKING) will be given.

10.

For each question in Section II, you will be given 4 Marks if you have darkened only the bubbles corresponding to the correct answers and zero mark if no bubble is darkened. There is NO Negative Marking.

11.

For each question in Section III, you will be given 4 Marks if you have darkened only the bubble corresponding to the correct answer and zero mark if no bubble is darkened. In all other cases, minus one (–1) marks (NEGATIVE MARKING) will be given.

12.

For each question in Section II, you will be given 8 Marks if you darken ALL the bubbles corresponding ONLY to the correct answer or given 2 Marks each for correct bubbling of answer in any row. No Negative mark will be given for an incorrectly bubbled answer.

VMC/2013

1

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

PART - I (CHEMISTRY)

80 MARKS SECTION - I STRAIGHT OBJECTIVE TYPE

This Section contains 8 Multiple Choice Questions. Each Question has 4 choices A, B, C & D, out of which Only One is Correct : 1.





⇀ For the reaction : H 2 + I2 ↽



2 HI (in equilibrium), the activation energy for the forward reaction is 1,71,828 Jmol−1 whereas for the reverse reaction is 1,80,142 Jmol−1. The presence of catalyst lowers the activation energy by 80 kJ mol−1. Assuming that the reactions are done at 27°C and the frequency factor for forward and backward reactions after addition of the catalyst are 4×10 respectively, calculate the value of KC. [Assume e1.11 ≈ 3 ] (A) 4 (B) 0.4 (C) 3.6

2.

(D)

−4

and 3×10

None of these

In a silver-silver chloride electrode, the concentration of chloride ions is 0.1 M. Calculate E Ag assuming ( K sp ) (A)

AgCl

−0.049 V

= 10

−10

(B)

−3

AgCl /Cl−

and E°Ag+ /Ag = 0.6 V 0.069 V

(C)

0.01 V

(D)

None of these

SPACE FOR ROUGH WORK

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2

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 3.

For a series of indicators, the colours and pH range over which colour change takes place are as follows: Indicator

Colour change over pH range

U

yellow to blue pH 0.0 to 1.6

V

Red to yellow pH 2.8 to 4.1

W

Red to yellow pH 4.2 to 5.8

X

Yellow to blue pH 6.0 to 7.7

Y

Colourless to red pH 8.2 to 10

Which of the following statements is correct? I. Indicator V could be used to find the equivalence point for 0.01 M acetic acid and 0.1 M ammonium hydroxide (ammonia solution) titration II. Indicator Y could be used to distinguish between 0.1 M HCl and 0.001 M NaOH solution in water III. Indicator X could be used to distinguish between two different solutions of ammonium chloride and sodium acetate IV. Indicator W could be suitable for use in determining the concentration of acetic acid in vinegar by base titration The correct choice is : (A) I, II (B) II, III (C) I, II, III (D) II, III, IV 4.

The complex [Fe(H2O)5NO]2+ is formed in the brown ring test for nitrates when freshly prepared FeSO4 solution is added to aqueous solution of NO3− followed by addition of conc. H2SO4. Select correct statement about this complex. I. Colour change is due to charged transfer between NO and Fe II. It has iron in +1 oxidation state and NO as nitrosonium III. It has magnetic moment of 3.87 BM confirming three unpaired electrons in Fe IV. H2O in general is a weak field ligand but in the given complex it behaves as strong field ligand (A) I, II, III, IV (B) II, III (C) I, II, III (D) II, III, IV SPACE FOR ROUGH WORK

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3

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 5.

Consider following transformations : I.

XeF6 + NaF → Na + [XeF7 ]−

II.

2 PCl5 (s) → [PCl4 ]+ [PCl6 ]−

III.

→ [Al(H2O)5 OH]2+ + H3O+ [Al(H2O)6 ]3+ + H2O 

Possible transformations are : (A) I, II, III (B) 6.

I, III

(C)

Ι, ΙΙ

(D)

II, III

The structure of unit cell of perovskite – a salt of lanthanum (La), manganese (Mn) and oxygen, has Mn2+ at the each corner, oxide on every edge centre and a lanthanum ion at the body centre. Assuming that all ions are in contact with each other, which of the following statement are true ? I. Radius of oxide ion must be the least II. Radius of lanthanum ion must be the greatest III. Edge length of unit cell = 2 (r 2+ + r 2− ) Mn

O

IV. Charge on lanthanum ion is +4 The correct choice is : (A) II, III (B) I, II, III, IV

(C)

I, III, IV

(D)

I, II, IV

SPACE FOR ROUGH WORK

VMC/2013

4

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 7.

Which of the following compounds will react with cyclopentanone to form an enamine? (A)

8.

(B)

(C)

(D)

All of above

End product of the following reaction is :

(A)

(B)

(C)

(D)

SPACE FOR ROUGH WORK

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

SECTION - II MULTIPLE CORRECT ANSWERS TYPE This Section contains 4 Multiple Choice Questions. Each Question has 4 choices A, B, C & D, out of which one or More Choices may be Correct: 9.

1 kg of water is heated from 27°C to 200°C forming super heated steam under constant pressure. Assuming that under the given conditions :




Specific heat of water = 4200 Jkg−1K−1 , Latent heat of vaporisation = 3.73 × 106 Jkg−1 Specific heat of steam = 1200 + 0.49T Jkg−1K-1

 373   473  Which of the following statements are true? (Use : ln   = 0.2, ln   = 0.24 ) 300    373  (A) On increasing temperature entropy of substance increases linearly (B) Increase in entropy of water during heating (27°C to 100°C) is more than increase in entropy on conversion of same water to steam. (C) Total Entropy change on heating water from 27°C to 200°C is 11.17 kJ/K (D) The entropy change of water during heating from 27°C to 100°C at constant pressure is at 840 J/K 10.




⇀ 2A) , in one litre vessel of at The rate of effusion of an equilibrium mixture of A2 and A (A 2 ↽




4 times of rate of diffusion of O2 under identical conditions of P and T. 35 Assuming total equilibrium pressure of 10 units, which of the following are correct if atomic weight of A is 50 ? (A) Mole ratio of A/A2 = 1.5 (B) KP = 9 (C) (D) Molecular weight of mixture = 70 ∆G° ≃ − 5.5 kJ/mol

300 K through a pin hole is

SPACE FOR ROUGH WORK

VMC/2013

6

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 11.

Hydrolysis

HCl

∆

∆

→ B  →C Borax (Na 2 [B 4 O 5 (OH) 4 ]) → Pr oducts → A + NaCl 

Which of the following is(are) true? (A) 5 moles of water are required to hydrolyse 1 mole of borax. (B) 2 moles of HCl are required to neutralise the products of hydrolysis of 1 mole of borax (C) A is orthoboric acid, B is metaboric acid and C is boric oxide (D) 4 moles of each A, B and C are formed if 1 mole of borax is used in the above reaction 12.

The following conversion reaction can be carried out by using reaction sequences :

(A)

H3⊕O∆ Zn-Hg/HCl Br2/RedP KCN  →  → →  →

(B)

4 2 3 3 2  →   →   →

(C)

2 →  →  →

(D)

NaBH

∆

Al O ,∆

I + NaOH

O /H O(Oxidation)

H⊕

KMnO4/ H + / ∆ SPACE FOR ROUGH WORK

VMC/2013

7

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

SECTION - III SUBJECTIVE TYPE This Section contains 6 Subjective Questions. Each question has an integer answer between 0 and 9. Fill the answer bubbles in OMR Sheet appropriately and CAREFULLY. [Please note that an answer ‘5’ should be filled as ‘5’ in the OMR sheet] 13.

A solution of palmitic acid (M = 256 g mol-1) in benzene contains 6.4 g of acid per litre. This solution on pouring on water surface forms a monomolecular level of palmitic acid as benzene gets evaporated. If 600 cm2 area of water surface is to be covered by a monolayer, the volume of the platimic acid solution needed is x × 10−5 litres? Area covered by one molecule of palmitic acid is 0.2 nm2. The value of x is __________.

14.

Following radial probability distribution curve will be valid for how many of the given orbitals : 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f, 6s, 6p, 7s, 7p

15.

A radioactive isotope

ZA

m

(t1 2 = 10 day) decays to give

Z−6 B

m −12

stable atom along with α -particles.

If m grams of A are taken and kept in a sealed tube, then Helium collected in 20 days at STP is x × 101 litres. Integer closest to x is ______________.

SPACE FOR ROUGH WORK

VMC/2013

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 16.

H2SO4 is used in a battery having the below given reactions :

Anode :

→ Pb 2 + (aq) Pb(s) 

→ Pb 2 + (aq) PbO 2 (s)  Cathode : An arrangement is made in the battery such that the same acid solution is used at both cathode and anode. During the discharge of this battery the density of a 2 L 40% H2SO4 solution by weight fell from 1.225 to 0.98 and 20% by weight solution was obtained. Charge leaving anode in Farads is ________.

17.

A unit of a silicate mineral has three SiO4 tetrahedra (that are connected to each other by Si − O − Si links) and it also contains Ca2+ and Cu2+ ions in 1 : 1 molar ratio. Total number of atoms of Cu, Si and Ca per unit of mineral is _______.

18.

How many of the species are paramagnetic? N2O, NO, NO2, KO2, Na2O2, O2, NO +2 , NO+, CO+, Br3O8

SPACE FOR ROUGH WORK

VMC/2013

9

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

SECTION - IV MATRIX MATCH TYPE This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. Statements in Column I are labelled as (A), (B), (C) & (D) whereas statements in Column II are labeled as p, q, r, s & t. The answers to these questions have to be appropriately bubbled. More than one choice from Column II can be matched with Column I. 19.

MATCH THE COLUMN : Column 1

(A) (B)

OH | Ph − C H − COOH

Column 2 (p) Cyclic

∆

 → ∆

Ph − C H − CH 2 − COOH | OH

(q) Exhibit cis-trans isomerism

 →

(C)

Ph − C H − CH 2 − CH 2 − COOH | OH

(D)

HO − CH 2 − COOH

∆

 →

∆

 →

(r)

Can be optically active

(s)

Lactone

(t)

Dehydration

SPACE FOR ROUGH WORK

VMC/2013

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 20.

MATCH THE COLUMN : Column 1

Column 2

(p) NaHCO3

(A)

(q)

(B)

(r) (C)

(s)

(D)

(t)

Na metal

2, 4-Dinitrophenyl hydrazine

Lucas reagent

NaOH

SPACE FOR ROUGH WORK

VMC/2013

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

PART - II (PHYSICS)

80 MARKS SECTION - I STRAIGHT OBJECTIVE TYPE

This Section contains 8 Multiple Choice Questions. Each Question has 4 choices A, B, C & D, out of which Only One is Correct : 21.

A particle is released from the top of two inclined rough surface of height ‘h’ each. The angle of inclination of the two planes are 30° and 60° respectively. All other factors (e.g. coefficient of friction, mass of block etc.) are same in both the cases. Let k1 and k2 be the kinetic energy of the particle at the bottom of the plane in two cases. Then : (A) (B) k1 = k2 k1 < k2

(C)

k1 > k2

(D)

data insufficient

22.

A sphere of radius 10 cm and density 500 kg/m3 is under water of density 1000 kg/ m3. Viscosity of water is 1.0 centipoise. If g = 9.8 m/s2 find, the velocity of the sphere at the instant where its acceleration is 9.8 m/s2 upward is: (A) 9 m/s (B) 10 m/s (C) 11 m/s (D) 12 m/s

23.

The figure shows a conical container of half-apex angle 37° filled with certain quantities of kerosene and water. The force exerted by the water on the kerosene is approximately. (Take atmospheric pressure = 105 Pa)

(A)

3 × 107 N

(B)

4 × 107 N

(C)

2 × 107 N

(D)

5 × 107 N

SPACE FOR ROUGH WORK

VMC/2013

12

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 24.

Current I is flowing along the path ABCD, along the four edges of the cube (figure-a), creates a magnetic field in the centre of the cube of B0. Find the magnetic field B created at the center of the cube by a current I flowing along the path of the six edges ABCGHEA (figure-b)

(A)

3 B0 Towards corner G 2

(B)

3B0 Towards corner E

(C)

3 B0 Towards corner H 2

(D)

3B0 Towards corner F

Paragraph for Question 25 - 26 Two tennis balls of mass 60gm are attached with a massless rubber thread, and held in the vertical position as shown in the figure. In this position length of the rubber thread is 40 cm and it is unstretched. The upper ball is slowly raised vertically upward, until the lower ball just becomes unsupported by the ground. At this time the length of the thread is 1m. The rubber thread exerts a force which is proportional to its extension.

25.

How much work is done by external agent while the upper ball was raised?

(A) 26.

0.54 J

(B)

0.36 J

(C)

0.18 J

(D)

0.45 J

If we release the upper ball at this position at what speed will it hit the lower one?

(A)

4 m /s

(B)

18 m / s

(C)

30 m / s

(D)

26 m / s

SPACE FOR ROUGH WORK

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes Paragraph for Question 27 - 28 A circuit is shown below :

27.

If A is an ideal ammeter, B an ideal Battery of voltage V, and C an ideal voltmeter, what will be reading of C the ? reading of A R (D) 0 2 If B is an inductor of inductance L, A an ideal battery of voltage V and C an ideal battery of voltage 2V each connected so that the positive terminal is facing left, what is voltage across B as soon as the circuit is connected: V 3V (A) (B) V (C) (D) 0 2 2

(A)

28.

R

(B)

2R

(C)

SPACE FOR ROUGH WORK

VMC/2013

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

SECTION - II MULTIPLE CORRECT ANSWERS TYPE This Section contains 4 Multiple Choice Questions. Each Question has 4 choices A, B, C & D, out of which one or More Choices may be Correct: 29.

30.

A sound wave of frequency f travels horizontally to the right. It is reflected from a large vertical plane surface moving to left with a speed v. The speed of sound in medium is c :

(c + v)

(A)

The number of waves striking the surface per second is f

(B)

The wavelength of reflected wave is

(C)

The frequency of the reflected wave is f

(D)

The number of beats heard by a stationary listener to the left of the reflecting surface is

c

c (c − v)

f (c + v)

(c + v) (c − v) vf c−v

The plates of a parallel-plate capacitor are separated by a solid dielectric. This capacitor and a resistor are connected in series across the terminals of a battery. Now the plates of the capacitor are pulled slightly farther apart. When equilibrium is restored in the circuit, (A) The potential difference across the plates has increased (B) The energy stored in the capacitor has decreased (C) The capacitance of the capacitor has increased (D) The battery would have gained energy

SPACE FOR ROUGH WORK

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 31.

Choose the correct statement(s) (A) Electrons in a conductor have no motion in the absence of a potential difference across it (B) Two identical metallic spheres of exactly equal masses are taken. One is given a positive charge Q coulombs and the other an equal negative charge. Their masses after charging are different (C) A line of force in an electric field is the path traced by a unit positive charge, free to move in that field (D) The energy of a charged conductor is stored partly inside the conductor and partly outside the conductor

32.

A series RLC circuit is driven by a generator at frequency 1000 Hz. The inductance is 90.0 mH; capacitance is 0.500 µ F ; and the phase constant has a magnitude of 60.0° (Take π 2 = 10 )

(A)

Here current leads the voltage in phase (B)

(C)

Resistance of circuit is

80π 3

Ω

(D)

Here voltage leads the current in phase

At resonance ω =

2 ×104 rad/sec 3

SPACE FOR ROUGH WORK

VMC/2013

16

Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

SECTION - III SUBJECTIVE TYPE This Section contains 6 Subjective Questions. Each question has an integer answer between 0 and 9. Fill the answer bubbles in OMR Sheet appropriately and CAREFULLY. [Please note that an answer ‘5’ should be filled as ‘5’ in the OMR sheet] 33.

Side rail of length 2L are fixed on a horizontal plane at a distance ℓ from each other. These ends are connected by two identical ideal batteries with emf E by resistanceless wires (see figure). On the rails is a rod of mass m, which may slide along them. The entire system is placed in a uniform vertical magnetic field B. Assuming that the resistance of the rod is R and the resistance per unit length of each of the rails equal to ρ , find the period of small oscillations (in sec.) arising from shifting the rod from the equilibrium along the rails. Neglect friction, internal resistance of batteries and induced emf in the rod. [Take : B = πT , ε = π volt, ℓ = 0.5m, L = 1m, ρ = 1Ω / m, R = 0.25Ω, m = 100 gm ]

34.

A uniform rod of length ℓ = 1m is free to move and rotate in gravity-free space. When an impulse is given to one end of the rod, perpendicular to its length, its centre of mass moves with velocity v = 1 m/s. What will be its angular velocity (in rad/s).

35.

A positively charged particle starts at rest 25cm from a second positively charged particle which is held stationary throughout the experiment. The first particle is released and accelerates directly away from the second particle. When the first particle has moved 25cm, it has reached a velocity of 10 2 m/s. If the maximum velocity (in m/s) that the first particle will reach is 10V0, find V0 ?

36.

37.

The filament of an incandescent lamp of power 64W is made of Tungsten. The operation temperature of the lamp is 2000K. Consider the filament a black body and find its radius (in mm). 10 [Given : σ = 6 × 10−8 W/m2 and length of filament is cm ] 3π The magnetic flux through each of five faces of a neutral playing dice is given by Φ B = ± N Wb where

N (= 1 to 5) is the number of spots on the face. The flux is positive (out-ward) for N even and negative (inward) for N odd. What is the flux through the sixth face of the die? 38.

The half-life of substance X is 45 years, and it decomposes to substance Y. A sample from a meteorite was taken which contained 2% of X and 14% of Y by quantity of substance. If substance Y is not normally found on a meteorite, and the approximate age of the meteorite is 15m years find m.

SPACE FOR ROUGH WORK

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

SECTION - IV MATRIX MATCH TYPE This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. Statements in Column I are labelled as (A), (B), (C) & (D) whereas statements in Column II are labeled as p, q, r, s & t. The answers to these questions have to be appropriately bubbled. More than one choice from Column II can be matched with Column I. 39.

A particle is moving on a straight line. It is initially at rest. v = instantaneous velocity P = instantaneous power S = displacement F = force t = time Match the possible expression of the quantities in column 1 with the situation in column 2.

Column 1

Column 2

(A)

v3 ∝ S

(p)

P = constant

(B)

v2 ∝ t

(q)

P∝v

(C)

v2 ∝ S

(r)

F = constant

(D)

v∝t

(s)

F∝

(t)

P∝t

1 v

SPACE FOR ROUGH WORK

VMC/2013

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 40.

MATCH THE COLUMN : Column 1

Column 2

(A)

A circular conducting loop rotating about its axis in a uniform constant magnetic field perpendicular to its plane.

(p)

Net induced emf in the loop is nonzero

(B)

A circular conducting loop moving along its plane in pure translation in a uniform constant magnetic field perpendicular to its plane.

(q)

Net induced emf is zero but a small part of the loop may have some emf induced across it.

(C)

A circular conducting loop placed in a magnetic field perpendicular to its plane which is decreasing with time.

(r)

Induced emf in any small part of the loop is zero.

(D)

A square conducting loop rotating about its side in a uniform constant magnetic field perpendicular to its axis. Consider a time when the plane of the loop is parallel to the magnetic field.

(s)

Induced electric field outside the conductor is zero.

(t)

Current flows in the loop.

SPACE FOR ROUGH WORK

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

PART - III (MATHEMATICS)

80 MARKS SECTION - I STRAIGHT OBJECTIVE TYPE

This Section contains 8 Multiple Choice Questions. Each Question has 4 choices A, B, C & D, out of which Only One is Correct :

Paragraph for Q.41 - 42 If cos −1

x y + cos −1 = θ then the value of 9 x 2 − 12 xy cos θ + 4 y 2 is equal to n 2 sin 2 θ , n ∈ N , and complete set 2 3

(

of values of x for which cos −1 x

41.

) − ( sin x ) 2

−1

2

> 0 is satisfied, is [p, q)

If α ∈ [ p, q ] , then possible value of ‘ α ’ is : (A) n−3 (B) n−4

(C)

n−5

n−6

(D)

(

2 q 2 − p2

)x

in [ −2π , 2π ] is : n −α

42.

If α is as obtained above, then number of solutions of sin −1 ( sin x ) = e (A) (B) (C) n − α − 10 (D) α +8−n α + 10 − n

43.

A circle x 2 + y 2 + 2 gx + 2 gy = a 2 intersects the hyperbola xy = c 2 in four points P ( x1 , y1 ) , Q ( x2 , y2 ) ,

R ( x3 , y3 ) and S ( x4 , y4 ) . If O is the centre of the hyperbola, C is the centre of the circle and G is the centroid of the quadrilateral PQRS and M is the mid-point of OC, then : ∑ x1 x2 ∑ x1 2OG ∑ x1 x2 x3 + + + = 2 ∑ y1 y2 y3 g OM a (A) 44.

0

Let f ( x ) =

(B) x

1+ x

2

and g ( x ) =

4

(C)

(

1

x 1 + x2

)

6

(D)

8

. If A1 be the area of region bounded between y = f (x), x-axis,

1 1 and x = π , and A2 is the area of region bounded between y = g(x), x-axis and between x = and e π 1 x = . Then A1 − A2 equals to : e 3 13 (A) 0 (B) 1 (C) (D) ℓn ℓn 2 11 x=

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes

45.

If

z1 , z2 , z3

  z − z 2  arg   3 1   =   z2 − z1     z  (A) arg  2   z1 

46.

z  arg  3   z2 

(B)

(C)

and

z  arg  1   z3 

(D)  a  a

   If a , b are perpendicular vectors, then the projection of the vector  l     the angle bisector of the vectors a and b may be given as :

l 2 + m2

(A)

2

2

l +m +n

2

l 2 + m2

l 2 + m2 + n 2 (C)

(B)

( ) ( )

 x − sin x  lim Let f ( x ) =  n →∞ x n + sin x n  1  (A) f is continuous n

47.

z1 ≠ z2

be 3 complex numbers such that

2

2

l +m +n

2

z1

z2

z3

z2 z3

z3 z1

z1 z2

=0

then

π

2    a ×b  b + m  + n    along b a×b  

(D)

(

)

l+m 2

n

,

x > 0, x ≠ 1

if ,

. Then at x = 1

x =1

if

(B)

f has removable discontinuity (i.e. lim f ( x ) exists, but this limit is different from f (1))

(C)

f has finite (jump) discontinuity (i.e., f (1+) and f 1−

x →1

( )

both exist finitely, but they are

different)

(D) 48.

f has infinite or oscillatory discontinuity (for e.g. like sin

Tangent is drawn to hyperbola

x2 8

−

1 x

at x = 0 and tan x at x =

π 2

)

 π = 1 at 2 2 sec θ , tan θ ; θ ∈ 0,  . The value of θ such that  2 1

y2

(

)

sum of intercepts on axes made by this tangent is maximum is :

(A)

π 3

(B)

π

(C)

4

π 6

(D)

None of these

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SECTION - II MULTIPLE CORRECT ANSWERS TYPE This Section contains 4 Multiple Choice Questions. Each Question has 4 choices A, B, C & D, out of which one or More Choices may be Correct: 49.

All chords of the curve 3 x 2 − y 2 − 2 x + 4 y = 0 which subtend a right angle at the origin pass through:

(A) (B)

(1, − 2 ) the point of intersection of the lines y + 2 x = 0 and x = 1

(C)

the vertex of the parabola x 2 − 2 x − 4 y − 7 = 0

centre of the circle x 2 + y 2 + 2 x − 4 y − 4 = 0    Let a = 4ˆi + 3 ˆj and b be two vectors perpendicular to each other in xy-plane. The vectors c in the   same plane having projections 1 and 2 along a and b are :

(D)

50.

(A)

2 11 − ˆi + ˆj 3 2

(B)

2ˆi − ˆj

(C)

2 11 − ˆi + ˆj 5 5

(D)

2 ˆ 11 ˆ i+ j 3 2

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 51.

Which of the following statements is(are) correct ? (A) Three coins are tossed once. At least two of them must land the same way. No mater whether they land heads or tails, the third coin is equally likely to land either the same way or oppositely. So, the chance that all the three coins land the same way is 1/2. (B) Let 0 < P(B) < 1 and P(A/B) = P(A/BC). Then A and B are independent. (C) Suppose an urn contains w white and b black balls and a ball is drawn from it and is replaced along with d additional balls of the same colour. Now a second ball is drawn from it. The probability that the second drawn ball is white is independent of the value of d. (D) A, B, C simultaneously satisfy P (ABC) = P (A) P (B) P (C)

( ) ( ) P ( ABC ) = P ( A) P ( B ) P ( C ) P ( ABC ) = P ( A) P ( B ) P ( C )

P ABC = P ( A) P ( B ) P C

Then A, B, C are independent. 52.

tan 142

1° =2+ 2 − µ − λ 2

(A)

µ =3

(B)

λ =5

(C)

( µ + λ )1 2 = 3

(D)

( µ + λ )1 2 = 2

2

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Vidyamandir Classes

SECTION - III SUBJECTIVE TYPE This Section contains 6 Subjective Questions. Each question has an integer answer between 0 and 9. Fill the answer bubbles in OMR Sheet appropriately and CAREFULLY. [Please note that an answer ‘5’ should be filled as ‘5’ in the OMR sheet] 53.

54.

55.

 x2 , if x ≤ −1  Let a function f be defined as f ( x ) =  x − 1 , if x > −1  2  x +1 Then the number of critical point(s) on the graph of this function is_________.             d b c  a + d c a  b + d a b  c           If  a b c  = d = 6 , then find 6

πx

+ 16 − x 2 + x + log 2 ( x ( x − 2 ) ) . If p be the sum of all possible integers in the 2 domain of function f (x) and q be the sum of all possible integers in the range of function f ( x ) , then Let f ( x ) = sin

q q  p  −  p  = k then [k] is (where [.] and {.} represent greatest integer and fractional part function     respectively). 1

56.

If I1 =

dx

π 4

∫ e x (1 + x )

and I 2 =

0

57.

2

θ

∫ ( 2 − tan2 θ ) 0

sin θ dθ

cos θ 3

If G is the greatest and L is the least value of

(G + L − 5) 58.

etan

2

, then

eI1 is equal to : I2

z + 2i , where i = −1 , and 1 ≤ z − 1 ≤ 3 then

is____.

Let F ( x ) = f ( x ) g ( x ) h ( x ) for all real x, where f (x), g(x) and h(x) are differentiable functions. At some point x0,

F ′ ( x0 ) = 21F ( x0 ) , f ′ ( x0 ) = 4 f ( x0 ) g ′ ( x0 ) = −7 g ( x0 )

and

h ′ ( x0 ) = kh ( x0 ) . Then k/4 is equal to____.

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Vidyamandir Classes

SECTION - IV MATRIX MATCH TYPE This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. Statements in Column I are labelled as (A), (B), (C) & (D) whereas statements in Column II are labeled as p, q, r, s & t. The answers to these questions have to be appropriately bubbled. More than one choice from Column II can be matched with Column I. 59.

MATCH THE COLUMN :

Column 1 (A)

Column 2

If exactly two real common tangents can be drawn to the circles x 2 + y 2 − 2 x − 2 y = 0 and x 2 + y 2 − 8 x − 8 y + 6λ = 0 for λ ∈ Z (p) then the greatest possible value of λ equals

2

2π

(B)

∫ (| sin x | + | cos x |) dx equals

(q)

3

(r)

1

(s)

16

(t)

0

−2π

(C)

The slope of a curve at (x, y) is −

x+y x

and it passes through the

1  points  3, −  and ( λ , 1) then one of the values of λ is 6 

(D)

( λ , 6, 2 ) x +1

=

is a point on the plane passing through the line

y −1

=

z+3

and parallel to the line of intersection of 2 −1 1 the planes x − y − 5 z = 6 and 3 x + 5 y + 3 z = 4. Then λ is

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Mock IIT Advanced Test-3/Paper-2

Vidyamandir Classes 60.

MATCH THE COLUMN :

Column 1 (A)

(B) (C)

2z − i

z +1 z −i =

Column 2

= 1 , locus of point ‘z’ is

(1 + i ) z + (1 − i ) z + 1 4

then locus of ‘z’ is

If the points (1, 2, 3) and (2, 1, 0) lie on the opposite sides of the plane 2 x + 3 y − 2 z = k , then the

(p)

4

(q)

−1

(r)

A circle

(s)

An ellipse

(t)

A rectangular hyperbola

number of integral values of k is

(D)

If ( x1 , y1 , z1 ) is the image of the point (1, 2, 3) about the plane x + y + z = 0 then z1 is

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Vidyamandir Classes SPACE FOR ROUGH WORK




End of Mock IIT Advanced Test - 3/Paper-2/2013   

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Mock IIT Advanced Test-3/Paper-2