Mock IIT Advanced Test -...

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VMC/2013 1 Mock IIT Advanced Test-3/Paper-2

Mock IIT Advanced Test - 3/2013/Paper-2

12/05/2013

TEST CODE : ACEG M.M. : 240

02:00 PM - 05:00 PM 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.

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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 : 2 2H + I 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 4×10−4

and 3×10−3

respectively, calculate the value of KC. [Assume 1.11e 3≈ ]

(A) 4 (B) 0.4 (C) 3.6 (D) None of these

2. In a silver-silver chloride electrode, the concentration of chloride ions is 0.1 M. Calculate Ag AgCl Cl

E −/

assuming ( ) 10sp

AgClK = 10− and

Ag AgE 0.6 V+° =

/

(A) −0.049 V (B) 0.069 V (C) 0.01 V (D) None of these

SPACE FOR ROUGH WORK

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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 3NO− 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


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5. Consider following transformations :

I. 6 7XeF NaF Na [XeF ]+ −+ →

II. 5 4 62 PCl (s) [PCl ] [PCl ]+ −→

III. 3 2

2 6 2 2 5 3[Al(H O) ] H O [Al(H O) OH] H O+ + ++ → +

Possible transformations are :

(A) I, II, III (B) I, III (C) Ι, ΙΙ (D) II, III

6. 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 2 2Mn O(r r )+ −+

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


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7. Which of the following compounds will react with cyclopentanone to form an enamine?

(A) (B) (C) (D) All of above

8. End product of the following reaction is :

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


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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:

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−1

K−1

,

� Latent heat of vaporisation = 3.73 × 106 Jkg

−1

� Specific heat of steam = 1200 + 0.49T Jkg−1K-1

Which of the following statements are true? (Use : 373 473

0.2, 0.24300 373

ln ln = =

)

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

300 K through a pin hole is 4

35 times of rate of diffusion of O2 under identical conditions of P and T.

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) G° 5.5 kJ/mol∆ −≃ (D) Molecular weight of mixture = 70


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11. 2 4 5 4

Hydrolysis HClBorax (Na [B O (OH) ]) Pr oducts A + NaCl B C

∆ ∆→ → → →

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) 2 3HZn-Hg/HCl Br /RedP KCN O⊕ ∆

→ → → →

(B) 4 2 3 3 2NaBH Al O , O /H O(Oxidation)∆→ → →

(C) 2I + NaOH H⊕∆→ →→

(D) KMnO4/ H /+ ∆


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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 mZ 1 2A (t 10 day)= decays to give m 12

Z 6 B −− 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 ______________.


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

Anode : 2Pb(s) Pb (aq)+→

Cathode : 2

2PbO (s) Pb (aq)+→

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, +

2NO , NO+, CO

+, Br3O8


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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 Column 2

(A)

OH|

Ph C H COOH∆

− − →

(p) Cyclic

(B)

2Ph C H CH COOH|

OH

∆− − − →

(q) Exhibit cis-trans isomerism

(C)

2 2Ph C H CH CH COOH|

OH

∆− − − − →

(r) Can be optically active

(D) 2HO CH COOH

∆− − → (s) Lactone

(t) Dehydration


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Column 1 Column 2

(A)

(p) NaHCO3

(B)

(q) Na metal

(C)

(r) 2, 4-Dinitrophenyl hydrazine

(D)

(s) Lucas reagent

(t) NaOH


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PART - II (PHYSICS) 80 MARKS

SECTION - I


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 1 2andk k be the kinetic energy of the particle at the

bottom of the plane in two cases. Then :

(A) 1 2k k= (B) 1 2k k< (C) 1 2k k> (D) data insufficient

22. A sphere of radius 10 cm and density 500 kg/m3 is under water of density 1000 kg/ m

3. 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) 73 10 N× (B) 74 10 N×

(C) 72 10 N× (D) 75 10 N×


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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) 0

3

2B Towards corner G

(B) 03B Towards corner E

(C) 0

3

2B Towards corner H

(D) 03B 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) 0.54 J (B) 0.36 J (C) 0.18 J (D) 0.45 J

26. 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


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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

thereading of

reading of

C

A?

(A) R (B) 2R (C) 2

R (D) 0

28. 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:

(A) V

2 (B) V (C)

3

2

V (D) 0


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SECTION - II




29. 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 :

(A) The number of waves striking the surface per second is ( )c v

fc

+

(B) The wavelength of reflected wave is ( )( )

c c v

f c v

−

+

(C) The frequency of the reflected wave is ( )( )

c vf

c v

+

−

(D) The number of beats heard by a stationary listener to the left of the reflecting surface is vf

c v−

30. 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


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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 Fµ ; and the phase constant has a magnitude of 60 0. ° (Take 2 10π = )

(A) Here current leads the voltage in phase (B) Here voltage leads the current in phase

(C) Resistance of circuit is 80

Ω3

π (D) At resonance 42

ω 103

= × rad/sec


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SECTION - III

SUBJECTIVE TYPE




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 : π ε π volt 0 5 1 ρ 1Ω 0 25Ω 100B T , . m L m / m R . m gm= = , = , = , = , = , =ℓ ]

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. 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).

[Given : 86 10σ −= × W/m2 and length of filament is

10

3cm

π]

37. The magnetic flux through each of five faces of a neutral playing dice is given by B NWbΦ = ± 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.


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SECTION - IV

MATRIX MATCH TYPE





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) 3v S∝ (p) P = constant

(B) 2v t∝ (q) P v∝

(C) 2v S∝ (r) F = constant

(D) v t∝ (s) 1F

v∝

(t) P t∝


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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.


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PART - III (MATHEMATICS) 80 MARKS

SECTION - I


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 1 1

2 3

x ycos cos θ− −+ = then the value of 2 29 12 4x xy cos yθ− + is equal to 2 2n sin , n Nθ ∈ , and complete set

of values of x for which ( ) ( )2 21 1 0cos x sin x− −− > is satisfied, is [p, q)

41. If [ ]∈ p, qα , then possible value of ‘α ’ is :

(A) 3n − (B) 4n − (C) 5n − (D) 6n −

42. If α is as obtained above, then number of solutions of ( ) ( )2 221

q p xsin sin x e

−− = in [ ]2 2,π π− is :

(A) 8 nα + − (B) 10 nα + − (C) 10n α− − (D) n α−

43. A circle 2 2 22 2x y gx gy a+ + + = intersects the hyperbola 2xy c= in four points ( ) ( )1 1 2 2P x , y , Q x , y ,

( )3 3R x , y and ( )4 4S x , y . 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 :

1x

g

∑ + 1 2

2

x x

a

∑ + 1 2 3

1 2 3

x x x

y y y

∑∑

+ 2OG

OM=

(A) 0 (B) 4 (C) 6 (D) 8

44. Let ( )21

xf x

x=

+ and ( )

( )2

1

1g x

x x=

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

1x

e= and x π= , and A2 is the area of region bounded between y = g(x), x-axis and between

1x

π= and

1x

e= . Then 1 2A A− equals to :

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

2nℓ (D)

13

11nℓ


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45. If 1 2 3z , z , z be 3 complex numbers such that 1 2z z≠ and

1 2 3

2 3 1

3 1 2

0

z z z

z z z

z z z

= then

2

3 1

2 1

z zarg

z z

− = −

(A) 2

1

zarg

z

(B) 3

2

zarg

z

(C) 1

3

zarg

z

(D) 2

π

46. If a , b��

are perpendicular vectors, then the projection of the vector ( )a ba b

l m na b a b

× + + ×

��

�� along

the angle bisector of the vectors a��

and b��

may be given as :

(A) 2 2

2 2 2

l m

l m n

+

+ + (B)

2 2 2l m n+ + (C) 2 2

2 2 2

l m

l m n

+

+ + (D)

2

l m+

47. Let ( )( )( )

0 1

1 1

→∞

− > ≠

= +

=

n n

n nn

x sin xlim , if x , x

f x x sin x

, if x

. Then at x = 1

(A) f is continuous

(B) f has removable discontinuity (i.e. ( )1x

lim f x→

exists, but this limit is different from f (1))

(C) f has finite (jump) discontinuity (i.e., f (1+) and ( )1f − both exist finitely, but they are

different)

(D) f has infinite or oscillatory discontinuity (for e.g. like 1

sinx

at x = 0 and tan x at 2

xπ

= )

48. Tangent is drawn to hyperbola

2 2

18 1

x y− = at ( )2 2 , ; 0,

2sec tan

πθ θ θ

∈

. The value of θ such that

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

(A) 3

π (B)

4

π (C)

6

π (D) None of these


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SECTION - II




49. All chords of the curve 2 23 2 4 0x y x y− − + = which subtend a right angle at the origin pass through:

(A) ( )1 2, −

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

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

(D) centre of the circle 2 2 2 4 4 0x y x y+ + − − =

50. Let 4 3= +��

ˆ â i j 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 :

(A) 2 11

3 2− +ˆ î j (B) 2 −ˆ î j (C)

2 11

5 5− +ˆ î j (D)

2 11

3 2+ˆ î j


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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. 1

142 2 22

tan µ λ°

= + − −

(A) 3µ = (B) 5λ = (C) ( )1 23µ λ+ = (D) ( )1 2

2 2µ λ+ =


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SECTION - III

SUBJECTIVE TYPE




53. Let a function f be defined as ( )2

2

if 1

1if 1

1

≤ −

= −> − +

x , x

f x x, x

x

Then the number of critical point(s) on the graph of this function is_________.

54. If 6a b c d = = ��

, then find 6

+ + �� d b c a d c a b d a b c

55. Let ( ) ( )( )2216 2

2

xf x sin x x log x x

π= + − + + − . If p be the sum of all possible integers in the

domain of function f (x) and q be the sum of all possible integers in the range of function ( )f x , then

q qk

p p

− =

then [k] is (where [.] and {.} represent greatest integer and fractional part function

respectively).

56. If ( )

1

1

01x

dxI

e x=

+∫ and

( )24

2 320

2

tane sin dI

costan

π θ θ θ

θθ=

−∫ , then 1

2

eI

I is equal to :

57. If G is the greatest and L is the least value of 2z i+ , where 1i = − , and 1 1 3z≤ − ≤ then

( )25G L+ − is____.

58. 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,

( ) ( ) ( ) ( )0 0 0 021 4F x F x , f x f x= =′ ′

( ) ( )0 07g x g x= −′ and ( ) ( )0 0h x kh x=′ . Then k/4 is equal to____.


Vidyamandir Classes


SECTION - IV

MATRIX MATCH TYPE






Column 1 Column 2

(A)

If exactly two real common tangents can be drawn to the circles 2 2 2 22 2 0 and 8 8 6 0 for+ − − = + − − + = ∈x y x y x y x y Zλ λ

then the greatest possible value of λ equals

(p) 2

(B) ( )2

2−

+∫ | sin x | | cos x | dx

π

π

equals (q) 3

(C)

The slope of a curve at (x, y) is +

−x y

xand it passes through the

points ( )1

3 and 16

−

, ,λ then one of the values of λ is

(r) 1

(D)

( )6 2, ,λ is a point on the plane passing through the line

1 1 3

2 1 1

+ − += =

−

x y z and parallel to the line of intersection of

the planes 5 6 and 3 5 3 4 Then− − = + + =x y z x y z . λ is

(s) 16

(t) 0


Vidyamandir Classes



Column 1 Column 2

(A) 2

11

z i

z

−=

+, locus of point ‘z’ is (p) 4

(B) ( ) ( )1 1 1

4

+ + − +− =

i z i zz i then locus of ‘z’ is (q) 1−

(C)

If the points (1, 2, 3) and (2, 1, 0) lie on the opposite

sides of the plane 2 3 2x y z k+ − = , then the

number of integral values of k is

(r) A circle

(D)

If ( )1 1 1x , y , z is the image of the point (1, 2, 3)

about the plane 0x y z+ + = then z1 is (s) An ellipse

(t) A rectangular hyperbola


Vidyamandir Classes



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