Part Test for IIT Mains 2016

8/9/2019 Part Test for IIT Mains 2016

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PAPER-1(PHASE TEST-1)

Time : 3Hours Maximum Marks : 264

READ THE INSTRUCTIONS CAREFULLY

GENERAL :

1.

This sealed booklet is your question paper. Do not break the seal till you are instructed to do so.

2.

The question paper CODE is printed on the left hand top corner of this sheet and the right hand top

corner of the back cover of this booklet.

3.

Use the Optical Response Sheet (ORS) provided separately for answering the questions.4.

The ORS CODE is printed on its left part as well as the right part. Insure that both these codes are

identical and same as that on the question paper booklet. If not, contact the invigilator.

5.

Blank spaces are provided within this booklet for rough work.

6.

Write your name and roll number in the space provided on the back cover of this booklet.

7.

After breaking the seal of the booklet, verify that the booklet contains 32 pages and that all the 60

questions along with the options are legible.

QUESTION PAPER FORMAT AND MARKING SCHEME:

8.

The question paper has three parts: Physics, Chemistry and Mathematics. Each part has three

sections.

9.

Carefully read the instructions given at the beginning of each section.

10.

Section 1 contains 8 questions. The answer to each question is a single digit integer ranging from 0

to 9 (both inclusive).

Marking scheme: +4 for correct answer and 0 in all other cases.

11.

Section 2 contains 10 multiple choice question with one or more than one correct option.

Marking scheme: +4 for correct answer, 0 if not attempted and -2 in all other cases.

12. Section 3 contains 2 “match the following” type questions and you will have to match the entries in

column I with entries in column II.

Marking scheme: for each entry in column I, +2 for correct answer, 0 if not attempted and -1 in all

other cases.

Candidate’s Name


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Mathematics

PART I: MATHEMATICS

SECTION 1(Maximum Marks: 32)

This section contains EIGHT questions.

The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both

inclusive For each question, darken the bubble corresponding to the correct integer in the ORS

Marking Scheme:

+4 if the bubble corresponding to the answer is darkened

0 In all other cases

1. The number of positive integers in the range of the function f ( x) =2

2

tan 4 tan 9

1 tan

x x

x

2. Let : 0,4 0, f be defined by 1cos cos f x x .The number of points 0,4 x

satisfying the equation 1010

x f x is

3. The number of integral values of k for which the equation 7cos 5sin 2 1 x x k has a

solution

4. The number of integer values of m for which 3 2 3 11 f x x mx x is invertible

5. The maximum value of 2

2

3 9 17

3 9 7

x x f x

x x

is 5 1k then the value of k is

6. If x,y,z are non-zero real numbers are 0

3111

1211

111

z z z

y y

x

then

z y x

111

7. If ,222,42 z y x z y x 4 z y x are inconsistent then

8. If 0

sincoscos

cossincos

coscossin

x x x

x x x

x x x

44

x then tan x =

Space for rough work


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Mathematics


This section contains TEN questions.

Each question has FOUR option(s) (A), (B), (C) & (D). ONE OR MORE THAN ONE of these

four options is(are) correct

For each question, darken the bubble corresponding to the correct option(s) in the ORS

Marking Scheme:

+4 if only the bubble(s) corresponding to the all corresponding to the correct option(s)

is(are) darkened

0 If none of the bubbles darkened

-2 In all other cases

9. Which of the following is/are periodic?

(a) 1

0

x is rational f x

x is irrational

(b) ; 2 2 1

1 ; 2 1 2 22

x x n x n

f xn x n

Where [.] denotes the greatest integer function, nZ

(c) [2 / ]

1 , x

f x

where [.] denotes the greatest integer function.

(d) 3 tan / 2 , f x x x x Where [.] denotes the greatest integer function, and a is a

rational number.

10. : 1, ln sin 2 cos 2 f R and f x x x ; (Where [.] denotes the greatestinteger function). Then,

(a) f x has range Z (b) f x is periodic with fundamental period π/4.

(c) f x is invertible in [0 , π/4]. (d) f x is into function.



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Mathematics

11. Let 1sgn cot tan / 2 f x x x , Where [x] denotes the greatest integer function lessthan or equal to x. Then which of the following alternatives is/are true?

(a) f x is a many-one function but not an even function. (b) f x is a periodic function.

(c) f x is a bounded function. (d)The graph of f x remains above the x-axis.

12. Which of the following functions have the graph symmetrical about the origin?

(a) .1

x y f x given by f x f y f

xy

(b) 2 2 1 1 f x given by f x f y f x x y x

(c) , f x given by f x y f x f y x y R (d)None of these.

13.

Let the function : f R be defined by 2 sin f x x x for x R . Then f is(a) One-to-one onto. (b)One-to-one but not onto.

(c)Onto but not one to one. (d)Neither one-to-one nor onto.

14. If 2sin cos , 1 f x x x g x x , then g f x is invertible in the domain

(a) 0,2

(b) ,4 4

(c) ,2 2

(d) 0,



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Mathematics

15. If ‘A’ is an odd order sequence matrix such that A2 + A + 2I = 0 then which of the following

is/are true

(a)A is non-singular (b) A is symmetric (c) A cannot be skew symmetric (d) I A A 2

11

16. If AB = A and BA = B, then (a) B = I (b) A = I (c) A

2 = A (d) B

2 = B

17. Let

21

21

21

x x x

x x x

x x x

f exdxcxbxax 2345 then :

(a) f = 1 (b) e = 3 (c) a + c = 1 (d) b + d = 1

18. If x f and x g are functions such that y f x g y g x f y x f .. , then

f g f

f g f

f g f

is independent of

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



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Mathematics


This section contains TWO questions.

Each question contains two columns, Column I & Column II.

Column I has four entries (A), (B), (C) & (D).

Column II has five entries (P), (Q), (R), (S) & (T).

Match the entries in Column I with the entries in Column II. One or more entries in Column I may match with one or more entries in column II.

The ORS contains a 4 X 5 matrix whose layout will be similar to the one shown below:

(A)

(B)

(C)

(D)

For each entries in Column I, darken the bubbles of all the matching entries. for example if entry (A) in

Column I matches with entries (Q), (R), (T) then darken these three bubbles in the ORS. Similarly, for the

entries (B), (C) & (D).

Marking Scheme:

+2 if only the bubble(s) corresponding to the all corresponding to the correct option(s) is(are)

darkened




19. Column I Column II

A : 0, 0,1

x f f x

x

P One-one ,onto

B 1

: 0 f R R f x x x

Q One-one , not onto

C 1

: 0 f R R f x x

x

R Onto, not one-one

D 1

: 0 f R R f x x x

S Neither one-one nor onto


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Mathematics

20. If

122

212

221

A and 3B =A then match column – I with column – II

Column I Column II

A Adj A is P Orthogonal matrix

B B is Q Involutory matrix

C BT is R Cofactor matrix of A

D B – 1

S - 3 A



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PART II : PHYSICS





Marking Scheme:



1. Three charges are placed at the vertices of a triangle. A dipole moment is

placed at O at shown. Take . The force exerted by charges

on dipole is

. Find .

2. The electric field in a region is radially outward with magnitude . Calculate the value

of charge contained in sphere of radius centered at origin. If

3. A capacitor stores when connected across a battery. When the gap between the plates

is filled with a dielectric, a charge flows through the battery. The dielectric constant

of material is…



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4. If the following combination of capacitor are connected with 4V battery then charge on

second 1μF capacitor (in μC) is…

5. Suppose one wishes to construct a capacitor using circular discs. If the sepration

between the disc is kept at , what would be approximate radius of disc(in km)

6. A wire is bent in the form of a regular hexagon and a total charge q is distributed uniformly

on it. What is the electric field at center?

7. A charge of is given to one plate of a parallel plate capacitor of capacitance and a

charge of is given to other plate. The potential difference between the plates (in Volts)

is…

8. Two identical particles, each having a charge of and mass of , are kept at a

seperation of and released. What should be speed of each particle (in x/s) when

the seperation becomes large



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For each question, darken the bubble corresponding to the correct option(s) in the ORS

Marking Scheme:


is(are) darkened



9. A thin walled spherical conducting shell S of radius R is given a charge Q. The same amount

of charge is also placed at its center C. Select the correct statements from following:

(a) charge density on outer surface of S is

(b) Electric field is zero at all points inside S(c) At a point just outside S, Electeric field is double the field at a point just inside S

(d) Electric field is inversely propotional to square of its distance from C at points outside

and inside the conductor.

10. An electric charge is placed at point (1,2,4). At the point (3,2,1), the electric

(a) field will increase by factor K if the space between the points is filled with a dielectric of

dielectric constant K.

(b) field will be along y-axis

(c) potential will be approx 49.9 V

(d) field will hae no y-component



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11. Electric potential in a region of space is given by: , where V is in volts

, x is in meters and A,B,C are constants.Then,

(a) E varies linearly with x (b) Unit of E is newton coulomb-1

(c) E is in negatie x direction (d) E is constant in this region

12. A conducting spherical shell of radius R has a charge. Then

(a) the charge is uniformly distributed over its surface in absence of external fields.

(b) distribution of charge over its surface will be non-uniform if an external electric field is present in space.

(c) potential at every point of the shell must be same

(d) the electric field inside the sphere will be equal to zero only when no external field exists.

13. Consider two identical charges placed 2d distance apart along x-axis. The equilbirium of a

positive test charge placed at O midway between them is

(a) stable for displacement along x-axis (b) neutral

(c) unstable for displacement along y-axis (d) stable for displacement along y-axis

14. Two identical parallel plate capacitor are connected in one case in parallel and in the other

case in series. In each case the plates of one capacitor are brought closer by a distance and

the plates of other are moved apart by same distance . Then

(a) total capacitance of first system increases

(b) total capacitance of first system decreases

(c) total capacitance of second system increases

(d) total capacitance of second system decreases



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15. A charged particle having a positive charge q approaches a grounded metal spheres of radius

R with a constant small speed v as shown in figure. Then select the correct alternative

(a) as charge q draws nearer to the sphere, a current flows into the ground

(b) as charge q draws nearer to the sphere, a current flows out of the ground

(c) as charge q draws nearer to the sphere, magnitude of current in wire increases

(d) as charge q draws nearer to the sphere, magnitude of current in wire decreases

16. Which of the following is true? (a) The electric field inside a closed conductor is always zero.

(b) The electric field at the surface of a conductor is perpendicular to it.

(c) The potential at the surface of a conductor is constant.

(d) The electric field lines never forms a closed loop. 17. A isolated conducting sphere

(a) Can be used as an energy storing device.

( b) The electric field inside it would be zero if it doesn’t have any charge within its body.

(c)The electric field at the surface of the charge won’t be continuous/

(d) Its capacitance decreases as the radius increases.

18.

A small conducting sphere of radius mounted on an insulated handle and having a positive

charge q is inserted through a hole in the wall of a hollow conducting sphere of inner radius

and outer radius .The hollow sphere is supported on an insulated stand and is initially

uncharged. The small sphere is placed at the center of the hollow sphere. Neglect any effect

of the hole. Which of the following statement(/s) will be true for this system?

(a) No work will be done in carrying a charge from inner conductor to outer conductor

(b) The electric field at a point in the region between the spheres at a distance r from the

center is equal to

(c) The electric field at a point outside the sphere at a distance r from the center is equal to

(d) The potential of the inner sphere with respect to the outer sphere is given by



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Match the entries in Column I with the entries in Column II.

One or more entries in Column I may match with one or more entries in column II.


(A)

(B)

(C)

(D)

For each entries in Column I, darken the bubbles of all the matching entries. for example if entry (A) in

Column I matches with entries (Q), (R), (T) then darken these three bubbles in the ORS. Similarly, for the


Marking Scheme:

+2 if only the bubble(s) corresponding to the all corresponding to the correct option(s) is(are) darkened

0 If none of the bubbles darkened-1 In all other cases


19.

Columns I Columns II

A Zero potential PConcentric circles with centres onaxis of a dipole

B Zero electric field QAt a fixed distance d from a point

charge

CConstant electric field

RInside a uniformly charged

conducting spherical shell

DConstant electric potential

SIn the vicinity of an infinitely

uniformly charged large sheet

20.


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Column-I Column-I I

AThe electric potential at surfaceincreases

PA positive charge q is put inthe cavity of a solid sphere

B The electric flux at surface increases QA negative charge q is placed

at the body center of a cube

C The electric potential at surfacedecreases

R Two spheres of same radius but different surface chargedensity are connected.

D The electric flux at surface decreases SA positive charge is kept at

the surface of a conductor


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Chemistry


PART I : PART III: CHEMISTRY





Marking Scheme:



1. A metal (Atomic mass = 75 g mol -1 ) crystallizes in cubic lattice, the edge length of unit cell

being 5Å. If the density of metal is 2 h cm-3 , then find the radius of metal atom (nA = 6 ×

1023). If radius of metal atom is X pm. Then find the value of [

]. Where [ ] denotes thegreatest integer value.

2. KCL crystallizes in the same type of lattice as does NaCl. Given that

= 0.5 and

=

0.7. Calculate the ratio of the side of the unit cell for KCl to that of NaCl. If value of ratio is

a.bcd ( where a, b, c & d are numbers. For eg. if ratio= 2.987 then c=8 ) then, find the value

of c.

3. The edge length of unit cell of LiCl having rock salt type lattice is 5014 Å. If Li + ions

precisely fit into the octahedral void of clased packed structure of Cl- ions, if the ionic radius

of Cl- ions is X pm. Then find the value of [

]. Where [ ] denotes the greatest integervalue.

4. A metal crystallizes into two cubic phases, face centered cubic(fcc) and body centeredcubic(bcc), whose unit cell lengths are 3 and 6 Å respectively. Calculate the ratio of

densities of bcc and fcc.


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Chemistry


5. 29.2% (w/w) HCl stock solution has a density of 1.25 g mL-1. The molecular weight of HClis 36.5 g mol-1. The volume(mL) of stock solution required to prepare a 200 mL solution of

0.4 M HCl is:

6. 2.71 g of sulphur in 72 g naphthalene shows a freezing point lowered by 1.015 oC. K f ’ for

C10H8 is 6.9 K mol-1 kg. How many atoms of S are present in a molecule of sulphur?

7. A metal M of molar mass 96 g mol -1 reacts with fluorine to form a salt that can berepresented as MFx. In order to determine x, a 9.18 g of the sample of the salt is dissolved in

100 g of water and its boiling point was determined to be 374.38 K. What is the chemical

formula of the salt? Given K b (water) = 0.512 kg mol-1. Assume complete dissociation of salt.

8. An element X of atomic mass 25.0 exists as X4 in benzene to the extent of 100%. When

10.30 g of saturated solution of X in benzene is added to 20.0 g of benzene, the depression in

freezing point of the resulting solution is 0.51 K. If K f for benzene is 5.1 K kg mol-1, the

solubility of X(in g) in 100g of benzene will be


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Chemistry






For each question, darken the bubble corresponding to the correct option(s) in the ORS Marking Scheme:


is(are) darkened


9. Let Benzene and naphthalene form an ideal solution at room temperature. For this process,the true statement(s) is(are)(a) ΔG is positive (b) ΔSsystem is positive

(c) ΔSSurroundings=0 (d) ΔH=0

10. Which of the following is/are correct statements(s) about the Raoult’s law applied to a

solution of non-volatile solute?

(a) Vapour pressure of solution is proportional to mole fraction of solute(b) V.P. of solution is proportional to the mole fraction of solvent

(c) Relative lowering of V.P.= mole fraction of solute

(d) Lowering of vapour pressure is proportional to the mole fraction of solute.

11. Which pair(s) of liquids on mixing are expected to show no net volume change and no heat

effect(a)

Acetone and ethanol

(b) Chlorobenzene and bromobenzene

(c) Chloroform and benzene

(d) N-butyl chloride and n-butyl bromide


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Chemistry


12. Composition of an azeotrope(a) Is independent of external pressure because it is a compound

(b) Alters on changing the external pressure as it is not a compound

(c) Remains unchanged during distillation at a constant external pressure

(d) Fluctuates even at constant pressure

13.

Colligative properties are similar in that they all(a) Were discovered in college laboratories(b) Are due to solvent molecules linked together

(c) Have no harmful side effects

(d) Depend on the concentration os solute particles in solution

14. In benzene the intermolecular forces is/are:

(a) Dipole-dipole attractions

(b) Hydrogen bonding

(c) Dispersion forces

(d)

ϭ and π bonding

15. Which of the following(s) have non-stoichiometric defect?

(a) Frenkel defect

(b) Schottky defect

(c) Metal excess defect(d)

Electronic defect

16. Select the correct statement(s):

(a)

Schottky defect is shown by CsCl(b)

Frenkel defect is shown by ZnS.

(c) Hcp and ccp structures have the same coordination number 12

(d) On increasing pressure, coordination number of CsCl decreases to that of NaCl


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20/21

Chemistry







Match the entries in Column I with the entries in Column II. One or more entries in Column I may match with one or more entries in column II.


(A)

(B)

(C)

(D)

For each entries in Column I, darken the bubbles of all the matching entries. for example if entry (A) inColumn I matches with entries (Q), (R), (T) then darken these three bubbles in the ORS. Similarly, for the


Marking Scheme:

+2 if only the bubble(s) corresponding to the all corresponding to the correct option(s) is(are)darkened




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Chemistry


19. Match the following at constant pressure of 1 atm.

Column- I Column-II

(a) Mixture of two immicible liquid. (P) Composition dependent

(b)

Solution of two miscible liquids (q) Constant, independent of relativefollowing Roult’s law amount of the liquids.

(c) Maximum B. point azeotropes (r) Constant along with composition

(d) Minimum B. point azeotropes (s) Constant, composition of

solution and vapour being

identical

20. Match the following column

Column I Column II

Structure Packing efficiency(a) atoms are present at corners as well as on edge centre (i) /6

(b) atoms are present at corners as well as on face centre (ii) /8

(c) atoms are present at corners as well as on body centre (iii) /12

(d) atoms are present only at corners (iv) /6

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Part Test for IIT Mains 2016