ii iit test08 pcm-main soln

7/24/2019 II IIT Test08 PCM-Main Soln

1/23

2IIT1516MT8PCMS 1

PHYSICS, CHEMISTRY AND MATHEMATICSClass II IIT-JEE Achiever 2015-2016

Solutions to Test 08 Main Date 07-11-2015

PART I: PHYSICSMultiple choice questions with one correct alternative

1. The deflection in an electric coil placed in a magnetic field is measured

using a lamp and scale arrangement. This arrangement consists of a

bright point source with a scale attached to it. A small mirror is

attached to the coil. Reflected beam from the mirror illuminates the

scale to produce bright spot. A schematic diagram of the arrangement

is shown in the figure.

When the coil rotates, the reflected beam reaches the scale at point P.

The angle of rotation of the coil is

(A) 30 (B) 15

(C) 60 (D) 45

Ans(B)

When mirror rotates, by an angle , the reflected beam rotates by 2.

tan (2)3

3

2= 30or = 15

2. If an object O is placed midway between two parallel mirrors facing each other and separated by 2b an

infinite number of images are formed. Find the distance of the nthimage formed by one of the mirrors

from the object O.

(A) 2(n 1)b

(B) 2nb

(C) nb

(D) (n 1)bAns(B)

I1is the image of O formed by mirror M1

I1acts as object for mirror M2and gives an image/1I .

For the mirror M1, I1is the first image and it is at a distance of 2b from O. I12is the second image and it is at a

distance 4b from O. Likewise the third image is at a distance = 2 3b= 6b. Generalizing this we see that the

nth

image is at a distance 2nb from O.

mirrorattachedto the coil

3 m

P

3 m

Scale

3 m

3 m2

I22

2bO/

b b b b 2b

I11 I21O

M1 M2

I12


2/23

2IIT1516MT8PCMS 2

3. A converging beam of light is incident on a convex mirror of focal length 8 cm. If the beam appear to

converge below the principal axis as shown in the figure then, the image position is (x and y co-ordinates

of the point O is 4 cm and 2 mm respectively)

(A) j4.0i8 cm

(B) cmj2.0j8 (C) cmj4.0i6 (D) cmj0.4i4 Ans(A)The point O acts a virtual object for the convex mirror. i.e., cm8f;cm4u

cm884

84V

image is formed in front of the mirror.

Thus x-coordinate is given by 8 cm.

The y-coordinate is given by

4

)8(

u

v

y

y

o

i yi= 2 (0.2) = 0.4 cm

4. Two transparent media A and B are separated by a plane boundary. The speed of light in medium A is

2.0 108ms1and in medium B, 2.5 108ms1. The critical angle for which a ray of light going from A

to B is total internally reflected is

(A) 11

sin2

(B) 12

sin5

(C) 14

sin5

(D) 11

sin3

Ans(C)

Refractive index of medium A with respect to medium B is8

8

2.5 10 5

2.0 10 4

For total internal reflection1 4

sin i 5 or1 4

i sin 5

Hence the correct choice is (C)

5. A vessel of depth t is half filled with oil of refractive index 1and the other half is filled with water of

refractive index 2. the apparent depth of the vessel when viewed from above is

(A) 1 2

1 2

2t

( )

(B) 1 2

1 2

2t ( )

(C) 1 2

1 2

t

2( )

(D) 1 2

1 2

t ( )

2

Ans(D)

The apparent depth is given by

1 2

1 2 1 2

t t t 1 1

A I 2

1 2

1 2

t( )

2

6. A diverging beam of light from a point source S having divergence angle

, falls symmetrically on a glass slab as shown in figure. The angles of

incidence for the two extreme rays are equal. If the thickness of the slab

is t and refractive is n, then the divergence angle of the emergent beam is

(A) zero (B)

(C)

1 1

sin n

(D)1 1

2sin n

Ox-axis

y-axis


3/23

2IIT1516MT8PCMS 3

Ans(B)

When a ray of light passes through a glass slab with parallel faces, it does not suffer any deviation ; it is

only displaced parallel to itself. Therefore, the direction of the beam remains unchanged after passing

through the glass slab. However, the rays are displaced slightly towards the outer side. Hence the

divergence angle of the emergent beam will be the same as that of the incident beam.7. A beam of light propagates through a medium 1 and falls onto another

medium 2, at an angle 1 as shown in figure. After that, it propagates in

medium 2 at an angle 2as shown. The lights wavelength in medium 1 is

1. What is the wavelength of light in medium 2?

(A) 1 12

sin

sin

(B) 2 1

1

sin

sin

(C) 1 12

cos

cos

(D) 2 1

1

cos

cos

Ans(B)

1sin = 2sin 2

1 2

1 2

c csin sin

v v

1 2

1 2

sin sin

f f

8. Light is incident on a glass block as shown in figure. If 1 is increased

slightly, what happen to 2?

(A) 2also increases slightly

(B) 2is unchanged

(C) 2decreases slightly(D) 2changes abruptly, since the ray experiences total internal reflection

Ans(C)

As 1increases, also increases

So, (90 ) decreases and hence 2decreases

9. A uniformly charged thin spherical shell of radius R carries a uniform surface charge density . It is

made of two hemispherical shells, held together by pressing them with a force F, which is proportional to

(A) 2R2 (B) 2R

(C)2

R

(D)

2

2

R

Ans(A)

Electric pressure,2

E

0

p2

Repulsive force separating the two hemispheres2

2 2

0 0

2 R R2

Hence, pressing force 2 2F R

1

2

F F


4/23

2IIT1516MT8PCMS 4

10.A beam of light consisting of red, green and blue colours is incident on a right-angled prism, as shown in

the figure. The refractive indices of the material of the prism for the above red, green and blue

wavelengths are 1.39, 1.44 and 1.47 respectively. The prism will

(A) separate part of the red colour from the green and blue colours

(B) separate part of the blue colour from the red and green colours(C) separate all the three colours from one another

(D) not separate even partially any colour from the other two colours

Ans (A)

For total internal reflection

1 11.414

sin C sin 45

That is, for an angle of incidence 45, that color will suffer total internal

reflection for which the refractive index is less than 1.414

Therefore, red light will be total Internally reflected at interface AB,

whereas blue and green light will be refracted.

11.A given ray of light suffers minimum deviation in an equilateral prism P. Additional prisms Q and R of

identical shape and of the same material as P are now added, as shown in the figure. The ray will now

suffer

(A) greater deviation (B) no deviation

(C) same deviation as before (D) total internal reflection

Ans(C)

Since there will be no refraction from P to Q and then from Q to R (all being identical), the ray will now

have the same deviation as before the point N.

12.

In figure, ABC is the cross section of a right-angled prism and ACDE is the cross section of a glass slab.The value of so that light incident normally on the face AB does not cross AC is

(given sin1(3/5 = 37)

(A) 37

(B) < 37

(C) 53

(D) < 53

Ans(B)

A = 90r2= A = 90> c

cos > sin c

645

2 53

(cis critical angle)

1 4cos 375

13.The observer at O views two closely spaced spots on a vertical wall through an

angled glass slab as shown in figure. As seen by observer, the spots appear

(A) shifted upward, such that distance between them remains same

(B) shifted downward, such that distance between them remains same

(C) spaced farther apart(D) spaced closed together

45

A E

C DB

n

=3

/2

n

=6

/5

Slab

A

CB

r2

N

Q

P Rn

i = 45

A

B


5/23

2IIT1516MT8PCMS 5

Ans(A)

Emergent ray e will be parallel to incident ray i.

So both dots will appear to be shifted upward by same distance.

Hence, distance between them remains same.

14.A fish looks upward at an unobstructed overcast sky. What total angle does the sky appear to subtend?

(Take refractive index of water as 2 )

(A) 180 (B) 90 (C) 75 (D) 60

Ans(B)

The fish can observe the sky only if refraction takes place.

If TIR takes place, then image of sky cannot be observed.

i.e., i < icand ic= 45

So, angle subtended2

15.

An observer can see through a pin-hole the top end of a rod of height h, placed in a beaker as shown in

the figure. The beaker height is 3h and its radius h. When the beaker is filled with a liquid up to a height

2h, he can see the lower end of the rod. Then the refractive index of the liquid is

(A)5

2

(B)5

2

(C)3

2

(D) 32

Ans (B)sini 1

sin r n (i)

Since tan2h

r 12h

r = 45sinh

ih 5

sin1

i

5

1

1 55 n1n 2

2

16.A point object is placed at a distance of 20 cm from a thin planoconvex lens of focal

length 15 cm. The plane surface of the lens is now silvered. The image created by

the system is at

(A) 60 cm to the left of the system

(B) 60 cm to the right of the system

(C) 12 cm to the left of the system

(D) 12 cm to the right of the system

e

i

2h

r

h

3h

20 cm


6/23

2IIT1516MT8PCMS 6

Ans(C)

Refraction from lens1

1 1 1

v 20 15

v = 60 cm Positive direction

i.e., first image is formed at 60 cm to the right of lens system.Reflection from mirror

After reflection from the mirror, the second image will be formed at a distance of 60 cm to the left of

lens system

Refraction from lens

3

1 1 1

v 60 15 Positive direction

or v3= 12 cm

Therefore, the final image is formed at 12 cm to the left of the lens system.

17.The graph between object distance u and image distance v for a lens is given below. The focal length of

the lens is

(A) 5 0.1

(B) 5 0.05

(C) 0.5 0.1

(D) 0.5 0.05

Ans(B)

From the lens formula

1 1 1

f v u we have

1 1 1

f 10 10

or f = +5Further, u = 0.1

and v = 0.1( from the graph)

Now, differentiating the lens formula, we have

2 2 2

f v u

f v u

2

2 2

v uf f

v u

Substituting the values, we have

2

2 2

0.1 0.1f (5) 0.0510 10

f f = 5 0.05

18.A biconvex lens of focal length f forms a circular image of radius r of sun in focal plane. Then which

option is correct?

(A) r2f

(B) r2f2

(C) If lower half part is covered by black sheet, then area of the image is equal to r2/2

(D) If f is doubled, intensity will increase

Ans(B)

r = f tan

r f

r2f2

f

r

11

10

10 119 u

v


7/23

2IIT1516MT8PCMS 7

19.A light beam is traveling from Region I to Region IV (Refer figure). The refractive index in Regions

I, II, III and IV are 0 00n n

n , ,2 6

and 0n

8, respectively. The angle of incidence for which the beam just

misses entering Region IV is

Region I Region II Region III Region IV

0n

2 0

n

6 0

n

8

(A) 13

sin4

(B) 11

sin8

(C) 11

sin4

(D) 11

sin3

Ans(B)

00

nn sin sin

8 2

20.How will the image formed by a convex lens be affected if the central portion of the lens is wrapped in

black paper?

(A) No image will be formed

(B) Full image will be formed but it is less bright

(C) Full image will be formed but without the central portion

(D) Two images will be formed, one due to each exposed half.

Ans(B)

Only the light-gathering power is reduced

21.A converging lens has a focal length of 1m. The minimum distance between a real object and its real

image formed by this lens is

(A) 0.5 m (B) 1 m (C) 2 m (D) 4 mAns(D)

Minimum distance between an object and its real image is 4f

i.e., 4 1 m = 4 m22.Two thin lenses of focal lengths 20 cm and 25 cm are placed in contact. The effective power of the

combination is

(A) 45 D (B) 9 D (C)1

D9

(D) 6 D

Ans(B)

1 2

1 1 100 100P

f f 20 25 = (5 + 4) D = 9 D

23. In the part of a circuit shown in the figure, all capacitors

have a capacitance of 2 F. Then charge on

(A) Capacitor C1is zero

(B) Capacitor C2is zero

(C) Capacitor C3is zero

(D) Any capacitor cannot be determined

Ans(C)

Applying KVL to the part ABC, 1 3A C1 3

q qV 2 V

C C

n0

C2+

D 2V

3V

CA

B2V8V C3C1

+ +


8/23

2IIT1516MT8PCMS 8

1 3q q8 2 32 2

q1+ q3= 6 (1)

Applying KVL to the part ABD, 1 2A D1 2

q qV V

C C

1 2q q8 22 2

q1q2= 12 (2)

From KCL at B, we get q3= q1+ q2 (3)

Solving equation (1), (2) and (3) we get q1= 6C ; q2= 6C ; q3= 0

24.A cubic container is filled with a liquid whose refractive index increases linearly from top to bottom.

Which of the following, best represents the path of the light ray inside liquid?

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

Ans(A)

Due to variable , ray cannot be linear

Best answer is (A).

25.

A plano-concave lens is placed on a paper on which a flower is drawn. How far above its actual positiondoes the flower appear to be?

(A) 5 cm

(B) 12 cm

(C) 10 cm

(D) 15 cm

Ans(C)

2 1 2 1 1 3 / 2 1 3 / 2 v 10 cmv u R v 20 20

i.e., 10 cm above actual position of flower.

26.Two mutually perpendicular long conductors carrying currents i andi

3 lie

on x-y plane as shown. O is the origin. The locus of the points at which the

magnetic induction is zero is y x3

. Find

(A) 7 (B) 8

(C) 9 (D) 10

Ans(C)

Let at P(x, y), field is zero then 0 0i i / 3 i i

0 02 y 2 x y 3x

y = 3x = 9

R = 20 cm

20 cm

Paper

3

Air

i

i

3y

xO


9/23

2IIT1516MT8PCMS 9

27.The magnetic induction at the point O, if the wire carrying current

i = 4A has the shape shown in the figure, is 0 tesla

. Find .

Given R 2m . The linear parts of the wire are very long. Also

assume the wires to be having uniform resistance(A) 4

(B) 3

(C) 2

(D) 1

Ans(D)

1 21

2

i 1 ii R i

i 3 3

Also i1+ i2= 4

2

2 2

i

i 4 i 3A3 or i1= 1A

0 0 1AB BCDEi 3 i B j ; B k

4 R 8R

0 0 2 0EF BGEi i 3 B i , B k k

4 R 8R 8R

0 0 0neti 3 3 B i j k k

4 R 8R 8R

0 0 0

net

i 4B 2 2 1

4 R 4 2

Each of the following questions consists of a statementI and a StatementII. Examine

both of them and select one of the options using the following codes

(A) Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement-I.

(B) Statement-I and Statement-II are true, but Statement-II is not the correct explanation ofStatement -I

(C) Statement-I is true, but Statement -II is false

(D) Statement-I is false, but Statement -II is true

28.Statement-I: The formula connecting u, v and f for a spherical mirror is valid only for mirrors whose

sizes are very small compared to their radii of curvature.

Statement-II:Laws of reflection are strictly valid for plane surfaces, but not for large spherical surfaces.

Ans (C)

29.Statement-I:A charge q is placed at the centre of metallic shell as shown in figure. Electric field at

point P on the shell due to charge q is zero

Statement-II:Net electric field in a conductor under electrostatic conditions is zero.

Ans (D)

OD

C

i = 4A

B

i = 4A

i1

Ei2 G

F

A

q

P


10/23

2IIT1516MT8PCMS 10

30.Statement-I:Three identical long solenoids P, Q and R are connected to each other as shown in the

figure. The ratio of magnetic fields at the centres of the solenoids is 2 : 1 : 1

Statement-II:The magnetic field within a long solenoid is given by B = 0ni. This implies B i

Ans (A)

PART II: CHEMISTRYMultiple choice questions with one correct alternative

1. A crystal plane intercepts the three crystallographic axes at the multiples of unit distance2

3, 2 and 1.

What are the Miller indices of the plane?

(A) 3, 2, 1 (B) 2, 4, 2 (C) 6, 4, 2 (D) 4, 3, 6

Ans(D)

x y z

Intercepts on crystallographic axes2

3 2 1

Reciprocals of intercepts3

2

2

1 1

Clear the fractions (by multiplying by (LCM) 4 3 6

Miller indices of the plane (4 3 6)

2.

Which of the following aqueous solutions has osmotic pressure nearest to that of an equimolar solutionof K4[Fe(CN)6]?

(A) Na2SO4 (B) BaCl2 (C) Al2(SO4)3 (D) C12H12O11

Ans(C)

Al2(SO4)3gives 5 ions on dissociation in water. K4[Fe(CN)6] also gives 5 ions in solution. Since their

molar concentrations in aqueous solution are the same, their osmotic pressures are also the same.

3. Which one of the following pairs of solution can we expect to be isotonic at the same temperature?

(A) 0.1 M urea and 0.1 M NaCl (B) 0.1 M urea and 0.2 M MgCl2

(C) 0.1 M NaCl and 0.1 M Na2SO4 (D) 0.1 M Ca(NO3)2and 0.1 M Na2SO4

Ans(D)

Both Ca(NO3)2 and Na2SO4give 3 particles each on ionization. Their molar concentrations are same.

They form isotonic solutions.

4.

Which of the following solutions has the highest boiling point?

(A) 1% solution of C6H12O6 (B) 1 M NaCl

(C) 1 M BaCl2solution (D) 1 M (NH2)2CO solution

Ans(C)

(A) and (D) are non-electrolytes. (B) is A+Btype of electrolyte, which gives two particles per molecule

on ionization. However, (C) is AB2 type of electrolyte which gives 3 particles per molecule on

ionisation. Thus, BaCl2solution will show the highest boiling point.

i i

Q

R

P


11/23

2IIT1516MT8PCMS 11

5. The molecular weight of benzoic acid in benzene as determined by depression in freezing point method

corresponds to

(A) ionization of benzoic acid (B) solvation of benzoic acid

(C) dimerisation of benzoic acid (D) polymerization of benzoic acid

Ans(C)Benzoic acid undergoes dimerisation in benzene as the solvent. Thus, the molecular weight of it

determined by freezing point depression method would be nearly double the true molecular weight.

6. The osmotic pressure of an aqueous solution containing 30.0 g of a newly-isolated protein in one litre of

solution is 12.7 mm Hg at 25C. The approximate molecular weight of protein in g/mol is

(A) 440 (B) 4400 (C) 44000 (D) 444000

Ans(C)

V = nRT RTMV

W

Molecular mass1

molg440001

760

7.12

2980821.030

V

RTWM

7. Which of the following elements does not occur in the native state?

(A) S (B) Si (C) Hg (D) Cu

Ans(B)

8. Which one among the following set is not an oxide ore/mineral?

(A) Cassiterite and Zincite (B) Rutile and Cuprite

(C) Corundum and Pyrolusite (D) Argentite and Cinnabar

Ans(D)

9. Which of the following does not contain copper in it?

(A) Malachite (B) Azurite (C) Pyrite (D) Cuprite.

Ans (C)

10.

Octahedral voids in a fcc lattice are located as follows:

(A) 6 at edge centres and 8 along body diagonals (B) 12 at edge centres and 1 at body centre

(C) 6 along body diagonal and 6 at edge centres (D) At all the edge centres only.

Ans(B)

In an fcc lattice, 12 octahedral voids are located at edge centres and one at the body centre.

11.Experimentally it was found that a metal oxide has formula M0.98O1. Metal M, is present as M2+and M3+

in its oxide. Fraction of the metal which exists as M3+would be

(A) 6.05 % (B) 5.08 % (C) 7.01 % (D) 4.08 %

Ans(D)

Let the number of M3+ions is x, then the number of M2+ions will be (0.98 x). Net charge of oxide will

be zero.

+2(0.98 x) + 3x + (2)1 = 0

1.96 + x 2 = 0

x = 0.04

Percentage of %08.410098.0

04.0M3


12/23

2IIT1516MT8PCMS 12

12.A mineral AB2crystallises in ccp lattice, with A atoms occupying the lattice points. The CN of A is 8

and that of B is 4. What percentage of the tetrahedral sites is occupied by B atoms?

(A) 25 % (B) 12.5 % (C) 50 % (D) 100 %

Ans(D)

4 spheres/unit cell 8 tetrahedral sites all occupied by B atoms.

13.Choose the correct statement

(A) Schottky and Frenkel defects are intrinsic defects but are non-thermodynamic defects

(B) In Schottky defect, entropy decreases whereas in Frenkel defect entropy increases

(C) Schottky defect is a stoichiometric defect whereas the Frenkel defect is a non-stoichiometric defect

(D) The same crystalline solid may have both the Schottky and Frenkel defects.

Ans(D)

14.KBr crystal structure has the CN ratio of 6 : 6 at normal temperature and pressure. At high pressure, the

CN ratio

(A) remains the same (B) changes to 6 : 8

(C) changes to 8 : 6 (D) changes to 8 : 8

Ans(D)

When KBr crystal is subjected to high pressure (~5000 atm) the structure transforms to a CsCl-like

structure with 8 : 8 coordination. Electron rich anions are particularly susceptible to compression.

Reverse can be done by )NaClCsCl( heating to 760 K

15. In diamond, the co-ordination number of carbon is

(A) 4 and its unit cell has 8 carbon atoms

(B) 4 and its unit cell has 6 carbon atoms

(C) 6 and its unit cell has 4 carbon atoms

(D) 4 and its unit cell has 4 carbon atoms

Ans(A)

Crystal structure of diamond is similar to that of ZnS.

Thus, CN of carbon is 4.

Since carbon atoms are in place of both Zn2+and S2ions, the number of carbon atoms per unit cell is 8.

16.The density of an ionic solid belonging to cubic system, with the unit cell dimension of 654 pm is

2.75 g/cm3. If the molecular mass of the solid is 119, which of the following statements is true? The

solid belongs to

(A) fcc system with coordination number = 6 (B) scc system with coordination number = 4

(C) bcc system with coordination number = 8 (D) none of these

Ans(A)

Density3

A

ZM

a N

654 pm = 654 1012m = 654 1010cm

10 3 23

Z 1192.75 or Z 3.89 4

(654 10 ) 6.02 10

Since the number of particles per unit cell is 4, the crystalline solid belongs to fcc type.


13/23

2IIT1516MT8PCMS 13

O

H

O

H* *

*

O O

O O

OO

O

O

O2

Ca +2

Ca 2 + 2CaCO3

O

Cl

O

H

R

I

O

II

O

HIII

O O

HIV

O

H

O H3O+

X +OH

O Y + O

H3O+

17.Number of optically active aldehydes possible with molecular formula C5H8O is

(A) 2 (B) 4 (C) 6 (D) 8

Ans(C)

(2 + 4) = 6

18.A mixture of calcium acetate and calcium but-2-enoate is heated in absences of air. The cross product

formed in the ration is

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

Ans(D)

19.Consider the following reaction

The reagent (R) of the reaction is

(A) H2/Zn (B) H2/p.d.BaSO4 (C) CH3CdCH3 (D) ZnHg/HCl

Ans(B)

20.The order of boiling point of following compounds is

(A) IV < II < III < I (B) IV < III < II < I (C) I < III < II < IV (D) I < III < IV < II

Ans (A)

21.The carbonyl compound fails to responds to NaHSO3test is

(A) (B) CCl3CHO (C) ClCH2CHO (D) CH3CH2COCH3

Ans (B)

22.Consider the following reactions

The reagent is used to distinguish between compounds (X) and (Y) is/are

(A) (NaOH + I2) (B) [Ag(NH3)2]OH (C) Schiffs reagent (D) all of theseAns (D)


14/23

2IIT1516MT8PCMS 14

H

OO

H H

O

O

H

O

Br

O(R)

Na ;HO

OH/H+ ; H3O+/heat

Na, H3O+

Na ; H3O+

HOOH/H+

; NaHO

OH/H+ ;

OH

OH

TsCl/heat

TsOH X

HBr(excess)

Peroxide Y

(i) KCN/alc.

excess

(ii) Ca(OH)2/heatZ

O

O

O

O O

O

O

H

OH

O

H

N

O

H

HO

23.Which of the following compounds does not reduce Fehlings solution?(A) CHOHC 56 (B) CHOCH3 (C) OHCOCHCH 23 (D) HCOOH

Ans(A)

Arylaldehydes does not reduce Fehlings reagent.

24.

The reactivity of following aldehydes towards nucleophilic addition reaction is

I II III IV

(A) II > III > I > IV (B) III > IV > II > I (C) II > I > IV > III (D) IV > III > I > II

Ans(A)

Greater the steric hindrance at -position, lesser the reactivity.

25.The correct sequence of reagent(s) used in the following conversion are

(A) (B)

(C) (D)

Ans(A)

26.Consider the following sequence of reactions

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

Ans (D)

27.The compound/s which fails to respond Cannizzaros reaction is/are

(A) (B) (C) (D) all of these

Ans(D)

Each of the following questions consists of a statementI and a StatementII. Examine



(B) Statement-I and Statement-II are true, but Statement-II is not the correct explanation of

Statement-I

(C) Statement-I is true, but Statement-II is false(D) Statement-I is false, but Statement-II is true


15/23

2IIT1516MT8PCMS 15

28.Statement-I:In the thermite reduction, Al powder is chosen as the reducing agent.

Statement-II: 2 2 33

2Al O Al O ; G2

for this reaction is highly negative.

Ans(A)

29.

Statement-I:Choice of flux depends upon the nature of impurities within the ore.Statement-II:Flux is added from an external source, to remove the gangue in the form of a fusible slag.

Ans (A)

30.Statement-I: The reactivity of ethanal towards aldol reaction is greater than propanal.

Statement-II:Addition of a few drops of NaOH solution to a mixture of ethanal and propanal gives six

optically active cross aldol products.

Ans (B)

PART III: MATHEMATICSMultiple choice questions with one correct alternative

1. The function f(x) = | sin x | + 2| cos x | + g() has period2

if is

(A) 2 (B) 1 (C) 3 (D) none of these

Ans(B)

Since period of | sin x | | cos x | , 12

2. If S is the set of all real x satisfying1

1x1 e 0,

then S =

(A) (, 0) (1, ) (B) (, ) (C) (, 0] [1, ) (D) none of these

Ans(A)

By data,1 1

1 1x x1 e 0 e 1

1 x 11 0 0 x ( , 0) (1, )

x x

3. The range of the function f : [0, 1] R defined by f(x) = x3x2+ 4x + 2 sin1x is

(A) [, 2, 0] (B) [2, 3] (C) [0, 4 + ] (D) (0, 2 + ]

Ans(C)

By data, f(x) = x3x2+ 4x + 2 sin1x2

2

2f (x) 3x 2x 4 0, x [0, 1]

1 x

f(x) is an increasing function.

Range = [f(0), f(1)] = [0, 4 + ]

4. The domain of f (x) x 4 2 (x 5) x 4 2 (x 5) is

(A) [5, ) (B) (, 2] (C) [5, ) (D) none of these

Ans (C)

Here, f (x) (x 5) 1 2 x 5 (x 5) 1 2 x 5

x 5 1 x 5 1

x 5


16/23

2IIT1516MT8PCMS 16

5. If 0 < a < b, then 1

n n n

nlim b a

(A) e (B) a (C) b (D) none of these

Ans (C)

Consider

1/nn

n n 1/nn n

alim (b a ) lim b. 1b

n

na 1

.b nb

n a

n

alim b 1b

b . e0= b

6.

2 2

3x 0 2

x y (y x)lim

8xy 4x 8xy

(A)1

4

(B)1

2

(C)1

2 2

(D) none of these

Ans (D)

2 2 2 2 2

3 3x 0 x 02 2

x y (y x) x y y x 2xylim . lim.

8xy 4x 8xy 8xy 4x 8xy

3/ 2

3x 0 3/ 2

x 2y xlim .

x 2 2y x 2 2y

3 3

2y 2y 1

128y2 2y 2 2y 4 2y

7.n

1 2

nr 1

3lim cot r

4

(A) 0 (B) tan11 (C) tan12 (D) none of these

Ans(C)n

1 2

nr 1

3lim cot r

4

n n1 1

n n2r 1 r 1

1 1r r

1 2 2lim tan lim tan

1 1 1r 1 1 r r

4 2 2

n1 1

nr 1

1 1lim tan r tan r

2 2

1 1 1 13 1 5 3tan tan tan tan . . . to2 2 2 2

1 1 11 1tan cot tan 22 2 2

8. Let f be a function satisfying f(x + y) = f(x) + f(y) and f(x) = x 2.g(x) x and y where g(x) is a

continuous function, then f (x) is equal to

(A) g (x) (B) g(0) (C) g(0) + g(x) (D) 0

Ans (D)


17/23

2IIT1516MT8PCMS 17

h 0 h 0

f (x h) f (x) f (x) f (h) f (x)f (x) lim lim (by data)

h h

2

h 0 h 0

f (h) h .g(h)lim lim 0

h h

9.

If

[cos x], x 1

f(x) | x 2 |, 1 x 2

, where [x] denotes the integral part of x, then f(x) is

(A) continuous and non-differential at x = 1 and x = 1

(B) continuous and differentiable at x = 0

(C) discontinuous at1

x2

(D) continuous but not differentiable at x = 2

Ans(C)

By data,

3 11, x

2 2

10, x 02

1, x 0f(x)

10, 0 x

21

1, x 12

2 x, 1 x 2

We can check from the above definition

that f(x) is discontinuous at1

x2

10. If f (x) = g(x) and g(x) = f(x) x and f(2) = 4 = f (2), then f 2(19) + g2(19) =


Ans(B)

By data, f (x) = g(x) f (x) = g (x) = f(x)

2 .f (x) . f (x) = 2.f(x) . f (x)

2f (x).f (x) 2.f (x).f (x)

2 2f (x) [f (x)] c . . .

At x = 2, 16 = 16 + c c = 32

By (1), [f (x)]2+ [f(x)]2= 32

[g(x)]2+ [f(x)]2= 32

g2(19) + f 2(19) = 32

11. If P(x) is a polynomial such that P(x2+ 1) = [P(x)]2+ 1 and P(0) = 0, then P(0) is equal to


Ans(C)

By data, P(x2+ 1) = [P (x)]2+ 1

P(x) = x P(x) = 1 P(0) = 1


18/23

2IIT1516MT8PCMS 18

12.The chord joining the points x = p and x = q on the curve y = ax 2+ bx + c is parallel to the tangent at the

point on the curve whose abscissa is

(A)p q

2

(B)

p q

2

(C)

pq

2 (D) none of these

Ans(A)

By data, at x = p, y = ap2+ bp + c

and at x = q, y = aq2+ bq + c

Slope of tangent2 2aq bq c (ap bp c)

(q p)

= a(q + p) + b

Also, if this is to be true at (x, y)

dy p q 2ax b a(q p) b x

dx 2

13.The angle at which the two curves x33xy2= 4 and 3x2y y3= 4 intersect, is

(A)6 (B)

4 (C)

2 (D) 0

Ans(C)

Consider x33xy2= 4 3x23y23x. 2yy= 02 2 2 2

1

y x x yy m

2xy 2xy

and

2 3 2 2dy dy3x y y 4 3x . 6xy 3y . 0dx dx

22 2

dy 2xym

dx x y

2 2

1 2 2 2

(x y ) 2xym m 1

2xy (x y )

the two curves cut at right angles

14.Let y = x2. ex. then the interval in which y increase w.r.t. x is

(A) (, ) (B) (2, 0) (C) (2, ) (D) (0, 2)

Ans(D)

Here, x 2 xdy

2xe x edx

xdy 0 xe (2 x) 0

dx

But ex> 0 x x(2 x) > 0 x(x 2) < 0 0 < x < 2

15.The greatest value of the function f(x) = 2 sin x + sin 2x on the interval3

0,2

is

(A)3 3

2 (B) 3 (C)

3

2 (D) none of these

Ans(A)

By data, f (x) = 2 cos x + 2 cos 2x3x x

f (x) 0 cos x cos2x 0 2cos cos 02 2

3x (2n 1) x (2n 1)or , n Z

2 2 2 2


19/23

2IIT1516MT8PCMS 19

(2n 1)x or x (2n 1) , n Z

3

3x , in 0,

3 2

Also, f (x) 2sin x 2sin 2x 0 at x 3

The greatest value2

f 2.sin sin3 3 3

2 3 3 3 3

2 2 2

16.N characters of information are held on magnetic tape in batches of x characters each. The batch

processing time is + x2seconds, , being constants. The optional value of x for fast processing is

(A)

(B)

(C)

(D)

Ans(C)

Each batch hasN

xcharacters.

Let 2N

P x N xx x

2

dPN

dx x

Both for maximum and minimum,2

dP0

dx x

x

Also,2

2 3

d p 2N 0 at x

dx x p

When P is least, x

17. If 21

x log 1 dx f (x).log(x 1) x .g(x) L.x c.,x

then

(A)2x

f(x)2

(B) g(x) = log x (C) L = 1 (D) none of these

Ans(D)2 2

2

1 1 x 1 1 xx log 1 dx log 1 . .dx

1x x 2 x 21x

2 1x log 1xx

dx2 2(1 x)

2 1x log 11 (1 x) 1x

dx2 2 (1 x)

2 1 xx . log1 1x

x log(1 x)2 2 2

2 2

(x 1) x log x 1log(1 x) x c2 2 2


20/23

2IIT1516MT8PCMS 20

18.

3

2 4 4

1dx

x x 1

(A)

1

4

4

11 c

x

(B) 4 1/4(x 1) c (C)

1

4

4

11 c

x

(D) none of these

Ans (D)

3/ 4

2 4

4

dxG.I.

1x x 1

x

3/ 4

5

4

dx

1x 1

x

Put 4 51 4

1 t dx dtx x

1/41/4

3/ 4 4

dt 1 t 1. 1 c

14(t) 4 x

4

19. If n Z,(n 1)

2

n

f (x)dx n ,

then the value of4

2

f(x)dx is


Ans(C)

By data,n 1

2

n

f (x)dx n

4 1 0 1 2 3 4

2 2 1 0 1 2 3

f (x)dx f (x)dx f (x)dx f (x)dx f (x)dx f (x)dx f (x)dx

2 2 2 2 2 2( 2) ( 1) 0 1 2 3 = 4 + 1 + 1 + 4 + 9 = 19

20.Let f (x) min x 1, (1 x) . Then area bounded by f(x) and x-axis is

(A)1

6 (B)

5

6 (C)

7

6 (D)

11

6

Ans(C)

Here, f (x) min. x 1, 1 x

x 1, 1 x 0

0 x 11 x,

0 1

1 0

R.A. (x 1) dx 1 x .dx

7sq. units

6

21.A spherical raindrop evaporates at a rate proportional to its surface area at any instant t. The D.E.

giving the rate of change of the radius of the raindrop is

(A)2

2

d r2r 0

dt (B)

2

2

d r3r 0

dt (C)

2

2

d r0

dt (D) none of these

Ans(C)

We know that volume 34V r3


21/23

2IIT1516MT8PCMS 21

2 2dv 4 dr dr .3r . 4 r .dt 3 dt dt

(1)

By data, 2 2dv dr

4 r 4 r .k dt dt

(2)

(1) and (2),

2 2dr dr

4 r . 4 r .k k dt dt 2

2

d r0

dt

22.Equation of a circle passing through the origin and belonging to the co-axial system, of which the

limiting points are (1, 2) and (4, 3) is

(A) x2+ y22x + 4y = 0 (B) x2+ y28x 6y = 0

(C) 2x2+ 2y2x 7y = 0 (D) x2+ y26x 10y = 0

Ans(C)

Clearly, the point circles of the system are

(x 1)2

+ (y 2)2

= 0 x2

+ y2

2x 4y + 5 = 0and (x 4)2+ (y 3)2= 0 x2+ y28x 6y + 25 = 0

Equation of R.A. is 6x + 2y 20 = 0 3x + y 10 = 0

Equation of any member of the family will be

x2+ y22x 4y + 5 + (3x + y 10) = 0 (1)

If it passes through 0(0, 0)1

5 ( 10) 02

By (1), R.E. will be

2 2 1x y 2x 4y 5 (3x y 10) 02

2x2+ 2y2x 7y = 0

23.

Given that q2pr < 0, p > 0, the value of

p q px qy

q r qx ry

px qy qx ry 0

is

(A) zero (B) positive (C) negative (D) q2+ pr

Ans(C)

By data, 13 3 1 22 2

p q 0

G.D. q r 0 C C xC yC

px qy qx ry (px 2qxy xy )

= (px2+ 2qxy + ry2) (pr q2)

= (px2+ 2qxy + ry2) (q2pr)

2 2 2 21 (px qy) y (pr q ) (q pr) 0p

24. If At4+ Bt3+ Ct2+ Dt + E =

2t 3t t 1 t 3

t 1 2 t t 3

t 3 t 4 3t

, then E =

(A) 33 (B) 39 (C) 27 (D) 24

Ans(A)

Put t = 0 on either side to get


22/23

2IIT1516MT8PCMS 22

0 1 3

E 1 2 3 9 3(10) 39

3 4 0

25. If a and b

are unit vectors and is the angle between them, thena b

2

(A) sin2

(B) sin (C) 2 sin (D) sin 2

Ans(A)

w.k.t. a b | a | . | b | cos cos

22

2 2a b a b | a | | b | 2.a.b

2 4 4

21 1 2cos 1 cos sin4 2 2

a b sin2 2

26. If the angle between the linesx 1 y 1 z 2

1 2 2

and the plane 2x y .z 4 0 is such that

1sin ,

3 then =

(A)3

4 (B)

4

3

(C)

5

3 (D)

3

5

Ans(C)

We know that the angle between the line and the normal to the plane is1.2 2( 1) 2

cos2 3 5

where is the angle between the line and the plane.

2 1sin by data

33 5

52 5 4 5

3

27.The distance between the parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is

(A)3

2 (B)

5

2 (C)

7

2 (D)

9

2

Ans(C)

Clearly, required distance

5( 8)

24 1 4

21 7

6 2

Each of the following questions consists of a StatementI and a StatementII. Examine



(B) Statement-I and Statement-II are true, but Statement-II is not the correct explanation of

Statement-I

(C) Statement-I is true, but Statement-II is false

(D) Statement-I is false, but Statement-II is true


23/23

2IIT1516MT8PCMS 23

28.Suppose1 0 0 2

A and B0 1 2 0

. Let X be a 2 2 matrix such that X AX B

Statement I: X is non-singular and det.(X) = 2

Statement II: X is asingular matrix.

Ans (C)If X = 0, then XAX = 0 B = 0, a contradiction

X 0, let | X | = a | X | = a

Also, | XA X | = | B | (a) (1) (a) = 4 a2= 4 a = 2

As a = | X | 0, X cannot be a singular matrix.29.Statement I:If a and c

are non-collinear, then the lines r 6a c (2c a)

and r a c (a 3c)

are coplanar.

Statement II: There exist and so that the two values of r

become same.

Ans (A)

If the lines have a common point, then and such that

6 = 1 + and 1 + 2= 1 + 3

= 3, = 2

30.Statement I:The point A (3, 1, 6) is the mirror image of B (1, 3, 4) in the plane x y + z = 5

Statement II: The plane x y + z = 5 bisects the line joining A(3, 1, 6) and B(1, 3, 4).

Ans (B)

Direction ratios of AB are 2, 2, 2 or 1, 1, 1

AB is normal to x y + z = 6

Also, midpoint of AB = (2, 2, 5) lies on x y + z = 5

The plane bisects AB statement 2 is true.

As the plane is the perpendicular bisector of the segment AB,

A is the mirror image of B and statement I is also true but doesnot follow from statement II alone

* * *

ii iit test08 pcm-main soln

Documents