Solution for Xyz

7/28/2019 Solution for Xyz

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XIII(XYZ) RT-5 [JEE MAINS] Page # 1

CHEMISTRY

PART-A

[SINGLE CORRECT CHOICE TYPE]

Q.1 to Q.27 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [3 Marks]

Q.1 Write the major product of the following reaction?

Me

N Et

Me

Me

OH

(A)

Me

(B)

Me

(C)

CH2

(D*)22 CHCH

For Z Batch

Q.1

OH OH

H

A

Product A is:-

(A)O

(B)

(C)

O

(D*)

O

Q.2 Which of the following is correct order of the o :

(A) [CoF6]2< [CoF

6]3< [Co(en)

3]3+

(B*) [Ru(H2O)

6]2+ < [Ru(ox)

3]3< [Ru(CN)

6]3

(C) [Fe(CN)6]4< [Fe(CN)

6]3< [Fe(ox)

3]3

(D) [Cr(H2O)

6]3+ < [Cr(CN)

6]3< [Cr(en)

3]3+

[Sol. o depends upon charge on metal ion as well as ligand strength.For Z - Batch

Q.2 Which of the following ores is not oxide ore?

(A) Tinstone (B) Bauxite (C) Limonite (D*) Carnallite

Q.3 pH of resultant solution when 700 mL of an aqueous solution of HCl (pH = 2.0) is mixed with 300 mL

of an aqueous solutionof NaOH (pH = 12.0) will be

(A) 5.2 (B*) 2.4 (C) 4.2 (D) 7.3

Q.4 In which of the following reaction carbene is formed as reaction intermediate:-

(A) Simmon-smith reaction (B*) Reimer-Teimann reaction(C) Hoffmann Bromamide (D) Curtius rearrangement

XIII-XYZ


2/25


CHEMISTRY

For Z Batch

Q.4 CH2

(i) BH3

(ii) H O /OH2 2

(A*) CH OH2 (B) O

(C)

CH3

OH

(D) CH2OH

OH

Q.5 Cu(s) + 2Ag+(0.1M) Cu2+(0.1M) + 2Ag(s)

In the above cell, the e.m.f. of cell increases

(A*) increase in concentration of Ag+ (B) increase in concentration of Cu2+

(C) decrease in concentration of Ag+ (D) increase inconcentration of Cu+

Q.6 Consider the following first order consecutive reactions : CBA 21 kk . The initial concentration

of A is 100 M and k1: k

2= 1.0 : 0.15. Given that k

1= 4.0 102 min1.

Nearly the time at which concentration of B will become maximum:

(A) 100 min (B) 15 min (C*) 56 min (D) 87 min.

For Z Batch

Q.6 Which of the following pair show negative deviation from Raoult's law.

(A) Ethanol + water (B) Cyclohexzane + ethanol

(C*) CHCl3

+ acetone (D) Methanol + CH2Cl

2

Q.7 Which of the following alkane cannot be prepared using wurtz and Kolbes electrolysis reaction(A*) Methane (B) Ethane (C) Butane (D) Hexane

For Z Batch

Q.7 2CH3MgBr

H)ii(

ONHCH)i( 23

Which of the following product can not be obtained in above reaction.

(A*) 33 CHNHOCH (B) 4CH

(C) 23 NHCH (D) OHCH3

Q.8 Which of the following d orbital of metal ion take part in hybridition of [Ni(en)2]2+

(A) 3dxy (B*) 3dx2-y2 (C) 3dz2 (D) 3dxz

[Sol. d-d splitting in square planar complex.

For Z BatchQ.8 Give the correct order of initials True (T) or False (F) for following statements :

(I) The number of electrons having l = 0 is 10 in Zn.

(II) The value of Zeffective

for 3d electrons of Cr and 3d electron of Mn is same as the number of

electrons in d sub shell of Cr and Mn are the same.

(III) Spin multiplicity of Fe is equal to that of Ni2+.

(IV) The value of l/n for the last incomming electron of the element having atomic number 57 is 0.4.

(A) T T T T (B) F T T T (C) T F T F (D*) F F F T


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CHEMISTRY

Q.9 Ag | Ag2SO

4(satd. sol.) || Ag+ (0.125 M) | Ag; K

spof Ag

2SO

4= 1.4 105;

o

Ag|AgE = +0.80 VV

Ecell

of the following cells at 25C is (Given log 7 = 0.84)

(A) 0.84 volt (B*) 0.084 volt (C) 0.036 volt (D) 0.36 volt

Q.10 How many species among them which are paramagnetic?

BaO2, KO

2, CaC

2, K

4[Fe(CN)

6], K

3[Co(OX)

3], [Cu(NH

3)4]SO

4, [Ni(NH

3)6]Cl

2, [Zn(NH

3)4]Cl

2,

NaNO3

(A*) 3 (B) 5 (C) 7 (D) 6

[Sol: KO2, [Cu(NH

3)4]SO

4, [Ni(NH

3)6]Cl

2]

For Z BatchQ.10 How many species are non polar among them?

PCl3F

2, PCl

2F

3, XeOF

4, XeO

2F

2, XeF

4, CH

2Cl

2, SO

3, CHClCCHCl

(A*) 3 (B) 4 (C) 2 (D) 5

Q.11HBr

(A)

the major product of the following reaction

(A*)

Br

(B)

Br

(C)

Br

(D)

Br

For Z Batch

Q.11 Which of the following will give haloform reaction?

(A*)32223 CHCHCCHCCHCH

||||OO

(B)33 CHCOCCH

||||OO

(C) OHCHCHCH 223 (D)

OH|

CHCHCCH|CH

323

3

Q.12 For the reaction : B .3H2O(s) B. H

2O (s) + 2H

2O(g) ; K

P= 9 atm2

A 24.63 litre flask contains 1 mole of B. H2O(s) at 300 K. How many moles of H2O(g) should be addedto the flask at the given temperature to drive the backward reaction for completion.

(A) 3 moles (B*) 5 moles (C) 8 moles (D) 2 moles


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CHEMISTRY

Q.13

Br

CH3 CH CH3

(i) PPh3

(ii) Bu Li

O

(iii)

(A)

Product A in the following reaction

(A)

CH CH3

(B)

CH2

(C*)

C CH3

CH3

(D) 323 CHCHCH

For Z Batch

Q.13 PhMgBr +

O

O

(A)

OH

O

(B*) (C)

O

OH

Ph (D)

OH

OH

Q.14 For the reversible reaction: AK1

K2B, K

1= 0.02 min1 and K

2= 0.03 min1. The reaction is

started with A only. The half life of such reaction is defined as the time in which half of the equilibrium

amount of B is formed. What is the half life of reaction?

(A) 34.65 min (B) 23.10 min (C*) 13.86 min (D) Infinite

Q.15 Which of the following species has higher electrons in t2g/t

2set of orbitals.

(A) [Cr(en)3]3+ (B*) [Ni(H

2O)

6]2+ (C) [Fe(NH

3)6]2+ (D) [NiCl

4]2

[Sol. A : Cr3+ = t2g3 eg0

B : Ni2+ = t2g6 eg2

C : Fe2+ = t2g4 eg2

D : Ni2+ = e4 t24 (for tetrahedral geometry 'g' is not used )


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CHEMISTRY

For Z Batch

Q.15 If n, l, m & s are 2, 1, 0 &2

1 respectively for the 7th electron of oxygen atom. Which set of magnetic

and spin quantum numbers is correct for its 8th electron :

(A) m = 0 (only), s =

2

1 (B) m =1 or +1 (only), s =

2

1

(C*) m = 0 or +1 or 1, s =2

1 (D) m =1 or +1 (only), s =

2

1

Q.16 COOHCH3 Q

OHCH)iv(

,OAg)iii(

NCH)ii(

SoCl)i(

3

2

22

2

product Q in the following reaction

(A) COOHCHCH 23 (B) COClCHCH 23

(C*) 323 COOCHCHCH (D) COOHCH3

For Z Batch

Q.16Cl

Cl+ 23NHCH

Product of above reaction is

(A)

N

H

(B*)

N

(C)

N

H

(D)

N

H

Q.17 The standard free energy of formation of MgO and CO at temperature 1000C and 2000C are given

below (they refer to the reaction involving one mole of oxygen at one atmosphere pressure).

The free energy change to the reaction.

2Mg + O2 2MgO G

1000C=941 kJ/mol

G2000C

=341 kJ/mol

2C + O2 2CO G

1000C=439 kJ/mol

G2000C=

628 kJ/mol

The following reaction is feasible at :

2MgO + 2C 2Mg + 2CO

(A) At 1000C temperature (B*) At 2000C temperature

(C) Both (A) and (B) (D) None of these

Q.18 Which of the following are incorrect order of the mentoned property in increasing order?

(A) First ionization potentialNa


6/25


CHEMISTRY

Q.19 A hydrogenation reaction is carried out at 500 K. If the same reaction is carried out in the presence of a

catalyst at the same rate with same frequency factor, the temperature required is 400 K. What is the

activation energy of the reaction, if the catalyst lowers the activation energy barrier by 16 kJ/mol?

(A) 100 kJ/mol (B*) 80 kJ/mol (C) 60 kJ/mol (D) 50 kJ/mol

Q.20 Which of the following carbocation will not undergo rearrangement?

(A) (B)

Et

Me

(C) (D*)

For Z Batch

Q.20 Which of the following will not give ether as major product?

(A)

3

33

3

CH

|BrCHNaOCCH

|CH

(B) BrCHONa 3

(C*)

3

33

3

CH|

ONaCHBrCCH|

CH

(D) OAgClCHCH 223 (dry)

Q.21 Which of the following central atom has sp3d2 hybridization .

(A) XeF2

(B*) ICl4 (C) TeCl

4(D) I

3+

Q.22 When a plot of log Keq vs 1/T is obtained, then slope of the line is

0.5 Kelvin. Where Keq is equilibriumconstant for the reaction, hence H for the reactions is

(A) 9.57 KJ/mol (B*) 9.57 J/mol (C) 95.7 KJ/mol (D) 95.7 J/mol

Q.23 Which of the following will be the major product for the reaction written below:-

OH

OHHO

H

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

O

OO


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CHEMISTRY

For Z Batch

Q.23

CH3

Br2

H O2

(A)

OH

Br

H

CH3

(B)

OH

Br

H

CH3

(C)

OH

Br

H

CH3

(D*) OH

Br

H

CH3

Q.24 If electron excite from ground state to 6 shell in hydrogen atom. then if

a = total visible line b = lyman spectral line

c = balmer spectral line d = I.R. spectral line

then report your answer as abcd.

(A) 5456 (B) 5556 (C) 4556 (D*) 4546

Q.25 Which of the following is incorrect order about the mention property

Order Property

(A) Na < Cl < F Electronegativity

(B) Be < Mg < Na Metallic character

(C) O < F < Cl | HEG

|

(D*) Cl2O

7> Al

2O

3> Na

2O Basic nature of oxide

Q.26 Which of the following will give only three mono chloroderivatives?

(A) n-pentane (B) Iso-octane (C) Ethane (D*) n-butane

For Z Batch

Q.26 Decreasing order of rate of reaction of molecules towards electrophilic addition reaction is:-

(a)

CH=CH2

NO2

(b)

CH=CH2

NH2

(c)

CH=CH2

CH3

(d)

CH=CH2

OH

(A) d > c > b > a (B) a > c > d > b (C*) b > d > c > a (D) c > b > d > a

Q.27 For an orbital, 2122r/k krkrrekr 1 select the correct statement.

(A) Orbital is 5p

(B*) Orbital is 5d

(C) Angular wavefuntion of orbital does not depend on ,

(D) Curve of2

r vs r has maximum value at origin.


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CHEMISTRY

[PARAGRAPH TYPE]

Q.28 to Q.30 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [3 Marks]

[3 question 3 marks]Paragraph for question nos. 28 to 30

To obtain stability elements get combined to each other and forms compounds of ionic nature andcovalent nature etc. The melting and boiling point of compound depends upon structrue as well as force

of attraction between the constituiting particles.

Q.28 Which of the following have the highest melting point?

(A*) covalent crystal (B) Ionic compound (C) Metallic solid (D) Covalent solid

Q.29 Which of the following is least soluable in water?

(A) LiCl (B*) LiF (C) LiBr (D) LiI

Q.30 Which of the following species has highest bond angle?

(A) BF3

(B) SO2

(C) SO3

(D*) XeF2


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XIII (XYZ) RT-5 [JEE Mains] Page # 1

PHYSICS

XIII-XYZ REVIEW TEST-5 DATE : 16.12.2012

JEE MAINS

PART-A


Q.1 to Q.27 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.

Q.1 From the homogeneous square plate we cut a triangle (Figure). Side of the square is a and the apex of

the triangle is at the center of the square. Distance from the center of the square to the center of mass of

the remainder of the plate is:

(A*)9a (B)

3a (C)

4a (D)

6a

[Sol. xc

=

4

mm

3

a

4

m0m

=

4

m312

ma

=9

a

3

a

3

2

2

a

a/2m/4 ]

Q.2 The strength of the magnetic field at a point at a perpendicular distance R near a long straight thin currentcarrying wire is B. The field at a perpendicular distance R/2 from wire will be

(A) B/2 (B) B/4 (C*) 2B (D) 4B

Q.3 Three identical point charges are at the vertices of an equilateral triangle with side a. The Electric field

generated by a individual charge at a distance a is equal to E0. Find the value of electric field intensity at

a point at the middle of one of the sides of the triangle:

(A)2

E0(B)

3

E2 0(C)

4

E3 0(D*)

3

E4 0

[Sol. E0

= 2akq

60

a

E

q

q q

2

3a

E = 2

2

a3

kq

=

3

4E

0]


10/25


PHYSICS

Q.4 A long wire is bent into shape ABCDE as shown in the figure, with BCD being a semicircle with centre

O and radius R meter. A current of I flows through it in the direction ABCDE. Then themagnetic induction at the point O is

(A*)

R4

1

R2

10 (B)

R4

1

R2

10 (C)

R4

I0(D)

R

I0

Q.5 A uniform disc of mass M and radius R is rolling without slipping with angular speedon a horizontalplane. The magnitude of angular momentum of the disc about the origin O is

(A) 2MR

2

1(B) MR2 (C*)

2MR2

3(D) 2MR2

Q.6 A sphere of radius R carries charge such that its volume charge density is proportional to the square of

the distance from the centre. What is the ratio of the magnitude of the electric field at a distance 2R from

the centre to the magnitude of the electric field at a distance of R/2 from the centre (i.e.2/RrR2r

E/E )?

(A) 1 (B*) 2 (C) 4 (D) 8

[Sol. = r2 [where is a constant]Charge in the shell (element)

dq = (4r2dr) = (4)r4dr

Charge enclosed in sphere of radius r, q = 4 r

0

4drr =5

r4 5

By Gauss's theorem,

at r = R/2 (Er = R/2

)

2

2

R4 =

0

5

5

)2/R(4

Er = R/2 =0

3

40

R

Total charge enclosed, Q =5

R45

Er = 2R

=0

5

5

R4

2)R2(4

1

=

0

3

20

R

2/Rr

R2r

E

E

= 2 ]


11/25


PHYSICS

Q.7 A coil of inductance 8.4 mH and resistance 6 is connected to a 12 V battery at time t = 0. The currentin the coil is 1 A at the time (approximately).

(A) 500 s (B) 20 s (C) 35 ms (D*) 1 ms

Q.8 A proton enters in a uniform electric and magnetic fields E

and B

respectively. Velocity of proton is v

.

All the three vectors are mutually perpendicular. The proton is deflected along positive x-axis when

either of the fields or both are switched on simultaneously. Which of the following statement isINCORRECT?

(A) v

may be along positive y-axis (B) E

must be along positive x-axis

(C) B

may be along positive z-axis (D*) B

must be along negative y-axis

Q.9 A conducting circular loop of radius r carries a constant current i. It is placed in a uniform magnetic field

0B

such that0B

is perpendicular to the plane of the loop. The magnetic force acting on the loop is

(A) ir0B

(B) 2ir0B

(C*) zero (D) ir0B

Q.10 Eight field lines emerge from a closed surface surrounding an isolated point charge. Would this fact

change if a second identical charge were brought to a point just outside the surface?

(A) The number of lines would change but the shape of the lines remains the same.

(B*) The number of lines would remain the same but the shape of the lines change.

(C) The number of lines as well as the shape of the lines remains the same.

(D) The number of lines as well as the shape of the lines would change.

[Sol. The second charge is kept just outside surface.

Flux will not change but another carge influences the shape. ]

Q.11 A voltmeter connected in series with a resistance R1

to a circuit indicates a voltage V1= 198V. When a

series resistor R2 = 2R1 is used, the voltmeter indicates a voltage V2 = 180V. If the resistance of the

voltmeter is Rv

= 900 is, then the applied voltage across A and B is

V

A

B

R1

198V

A

B

R 2R2 = 1

180VV

(A) 210 V (B) 200 V (C*) 220 V (D) 240 V

Q.12 Figure shows a uniform yo-yo placed on a rough surface friction is sufficient for pure rolling. After forceF acts pure rolling begins.

F y

x

Statement-1: Angular acceleration of yo-yo is in clockwise sense.

Statement-2: Torque due to F about centre of mass must be larger than the torque due to friction.

(A*) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

(C) Statement-1 is true, statement-2 is false.

(D) Statement-1 is false, statement-2 is true.


12/25


PHYSICS

Q.13 A 2.0 V potentiometer is used to determine the internal resistance of a 1.5 V cell. The balance point of

the cell in the open circuit is 75 cm. When a resistor of 10 is connected across the cell, the balancepoint shifts to 60 cm. The internal resistance of the cell is

(A) 1.5 (B*) 2.5 (C) 3.5 (D) 4.5

Q.14 A uniform disk, a thin hoop (ring), and a uniform sphere, all with the same mass and same outer radius,

are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotationaxis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the

rims, as shown. Rank the objects according to their angular momentum after a given time t, least to

greatest.

Disk Hoop Sphere

F F F

(A*) all tie (B) disk, hoop, sphere

(C) hoop, disk, sphere (D) hoop, sphere, disk

[Sol.RJL

= tLdtL

Same for all ]

Q.15 A straight very long wire of diameter 0.5 mm carrying a uniform current 2A is replaced by another wire

of diameter 1mm carrying the same current. The strength of magnetic field at a distance 2m away from

the centre axis of wire is

(A) half of the previous value (B) twice the previous value

(C*) unchanged (D) quarter of its previous value.

Q.16 Statement -1 :An electric dipole placed in the electric field of a point charge can never experience zero

resultant force.

Statement -2 : Electric field of a point charge is non uniform.

(A*) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

(C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.

[Sol. Newton's third law. ]

Q.17 For the circuit arrangement shown in fig., in the steady state condition charge on the capacitor is

12V 2

4

6

2F

(A) 12 C (B) 14 C (C) 20 C (D*) 18C


13/25


PHYSICS

Q.18 A students connects an ammeter A and a voltmeter V to measure a resistance R as shown in fig. If the

voltmeter reads 20V and the ammeter reads 4A, then R is

(A) equal to 5(B) greater than 5

+

+

20V

4A

V

AR

(C*) less than 5(D) greater or less than 5 depending upon the direction of current.

Q.19 A block of mass m is lying on a wedge having inclination angle= tan1(1/5). Wedge is moving with aconstant acceleration a = 2 m/s2. The minimum value of coefficient of friction , so that m remains

stationary w.r.t. to wedge is,

a = 2 m/s2

m

(A)92 (B*)

125 (C)

51 (D)

52

[Sol. mg sin a + ma cos < N

2

mg

maN + ma sin = mg cos N = mg cos ma sin

>

sinmacosmg

cosmasinmg

=26

1226

510

26

52

26

110

= 48

20

= 12

5

]

Q.20 In the fig. shown, a charge q moving with a velocity v along the x-axis enters into a region of uniform

magnetic field. The minimum value of v so that the charge q is able to enter the region

x > b.

b

a

q

y

xv

(A)m

qBb(B)

m

qBa(C*)

m

)ab(qB (D)

m2

)ab(qB


14/25


PHYSICS

Q.21 In the circuit shown in fig., L = 10H; R = 5E = 15V. The switch S is closed at t = 0. At t = 2s, thecurrent in the circuit is

E S

L R

(A*) Ae

1

13

(B) Ae

1

13 4

(C) Ae

1

3

(D) Ae

1

3 4

Q.22 A ring having charge Q is concentric with a uniformly charged solid sphere of total charge Q. Ring has

radius 2R and sphere has radius R. The potential at common centre is

04

1K

R

2R

(A)R2

kQ3(B)

R2

kQ(C*)

R

kQ2(D)

R

kQ

[Sol. V =R2

kQ3+

R2

kQ=

R

kQ2]

Q.23 A semi-circular conducting ring abc of radius R moves with constant

speed v in a plane perpendicular to uniform magnetic field B as shown

in fig. Identify the correct statement.

(A)Va

Vc = BRv (B) Vb

Vc = BRv(C) V

aV

b= 0 (D*) all the above

Q.24 An insulating rod of length l carries a charge q uniformly distributed on it. The rod is pivoted at one of its

ends and is rotated at a frequency f about a fixed perpendicular axis. The magnetic moment of the rod is

(A)12

qf2

l(B)

2

qf2

l(C)

6

qf2

l(D*)

3

qf2

l

Q.25 The ring B is coaxial with a solenoid A as shown in fig. As the the switch S is closed at t = 0, the ring B

(A) is attracted towards A

A B

S

(B*) is repelled by A

(C) is initially repelled and then attracted.

(D) is initially attracted and then repelled.

Q.26 A solenoid (having uniform number of turns per unit length) of inductance L and resistance R is cut into

two parts in the ratio 1: , which are then connected in parallel. The combination is then connected to acell of emf E. The time constant of the circuit is (neglect mutual induction)

(A*)R

L(B)

R)1(

L

(C) RL

1

(D)R

L1


15/25


PHYSICS

6Q.27 A pendulum bob is released from horizontal position shown. At an angular position, its total accelerationmakes an angle with string, is -

aT

(A) 0 (B) tan1 (2 tan ) (C*) tan1

2

tan(D)

2

[Sol. mgl cos =2

1mv2

aT at

l

2v

= 2g cos

mat= mg sin

tan f =R

t

a

a=

cosg2

sing=

2

tan ]

[PARAGRAPH TYPE]


Paragraph for question nos. 28 to 30

A uniform conducting ring of masskg and radius 1m is kept on smooth horizontal table.A uniform but

time varying magnetic field B = jti2 Tesla is present in the region. (Where t is time in seconds).

Resistance of ring is 2. Then, (g = 10m/s2)

Q.28 Net magnetic force (in Newton) on conducting ring as function of time is

(A) 22t (B) 22t2 (C) 22t3 (D*) zeroQ.29 Time (in second) at which ring start toppling is

(A*)

10(B)

20

(C)5

(D)25

Q.30 Heat generated (in kJ) through the ring till the instant when ring start toppling is

(A)31

(B)2

(C*)32

(D)1


16/25

XIII (XYZ) RT-5 [Mains] Page # 1

MATHEMATICS

PART-A


Q.1 to Q.27 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [27 3= 81]

Q.1 The terms a1, a

2, a

3(in that order) form an increasing arithmetic sequence whose sum is 18.

If a1

+ 1, a2, a

3+ 2 form a geometric sequence (in that order), then sum of all possible values of a

1is

(A*) 2 (B) 11 (C) 13 (D) 9[Sol. a d, a, a + d A.P.

3a = 18 a = 6Now 7 d, 6, 8 + d G.P. 36 = (7 d) (8 + d) d = 4 or 5 (rejected) a

1= 2 S = 2. Ans.]

Q.2 The value of definite integral

2

0

dxx4sinx3sinx2sinxsin is equal to

(A)4

(B)

8

(C*)

16

(D)

32

[Sol. Let I =

2

0

dxx4sinx3sinx2sinxsin .......(1)

Apply King property, we get

I = 2

0

dxx4sinx3cosx2sinxcos ......(2)

(1) + (2)

2I =

2

0

dxx4sinx2sinx2cos 4I =

2

0

2dxx4sin 8I =

2

0

dx)x8cos1(

I =2

0

x8sin8

1x

8

1

=

0

28

1=

16

. Ans.]

Q.3 If the angle between the vectors

kj)x(cosia

and kxsin43j)x(cosi)xsinx(sinb 2

is obtuse and x

2,0 , then the exhaustive set of values of 'x' is equal to

(A) x

6

,0 (B*) x

2

,

6

(C) x

3

,

6

(D) x

2

,

3


17/25


MATHEMATICS

[Sol. Given, kj)x(cosia

and kxsin43j)x(cosi)xsinx(sinb 2

Now, ba

< 0 (sin2x sin x) (1 sin2x) + (3 4 sin x) < 0

2

1< sin x < 1 x

2,

6Ans.]

Q.4 The curve represented by the differential equation xdy

ydx = ydy, intersects the y-axis at A(0, 1) andthe line y = e at (a, b), then the value of (a + b) is equal to

(A*) 0 (B) 1 (C) 2 (D) 3

[Sol. 2y

xdyydx = 2y

ydy

y

xd =

y

dy Cyn

y

x l

As, A(0, 1) lies on it, so 0 + 0 = C C = 0

0yny

x l .........(1)

Put y = e in (1), we get

0ene

x l x = e

(a = e, b = e)

Hence, (a + b) = 0. Ans.]

Q.5 Let f (x) =

2x,1x5

2x,dtt15

x

0,

then f(x)

(A) has maxima but no minima. (B) has minima but no maxima.

(C*) is continuous but non-differentiable at x = 2. (D) is continuous and differentiable at x = 2.

[Sol. f(x) =

2x,1x5

2x,dt)1t(5dt)t1(5

1

0

x

1=

2x,1x5

2x,2

xx41

2

Now, verify alternative. Ans. ]

Q.6 Consider a matrix A = [aij]3 3

, where aij

=

evenijif,i

oddijif,ji2 .

If bij

is cofactor of aij

in matrix A and dij

=

3

1k

jkikba , then the value of 3 ij)d(det is equal to

(A) 4 (B) 24 (C*) 48 (D) 24


18/25


MATHEMATICS

[Sol. Clearly,A =

092

444

210

| A | = 48

dij =

3

1kjkikba =

jiif|,A|

jiif,0

ijd =A00

0A0

00A

=3

A 3 ij)d(det = A = 48. Ans.]

Q.7 The solution for x of the equation

x

22 1tt

dt=

2

, is

(A) 2 (B) 2 (C) 2 (D*) 2

[Sol.

x

22 1tt

dt=

2

x

2

1 tsec =2

4xsec 1

=

2

sec1x =

4

3

x = 2 Ans.]

Q.8 Let f (x) =

2x0,)5x3(

0x2,)x1(2

3

2

Which one of the following statement is correct?

(A) f (x) is continuous at each point on [2, 2].

(B) f (x) is differentiable for all 'x' in (2, 2).

(C) Rolle's Theorem is applicable for f(x) in [2, 2 ].

(D*) There exists at least one c (2, 2) for which f '(c) = 0.[Sol. Clearly,

y = f (x) is discontinuous at x = 0 (A) is incorrect.y = f (x) is not differentiable at x = 1 (B) is incorrect.f (2) = f (2) (C) is incorrect

But f '(x) = 0 has a root x =35 (D) is correct ]

Q.9 Let B1, C

1and D

1are points on AB, AC and AD of the parallelogram ABCD, such that

ABkAB 11 ,

ACkAC 21 and

ADkAD31

where k1, k

2and k

3are scalar, then if B

1, C

1, D

1are collinear then

k1, 2k

2, k

3are in

(A) A.P. (B) G.P. (C*) H.P. (D) A.G.P.


19/25


MATHEMATICS

[Sol:

Vector equation of B1D

1 bkdkbkr 131

Satisfy dbkr 2

bkdkbkdbk 1322

k2

= k1 k

1

k2

= k3

k2

= k1 k

13

2

kk

k2k

3= k

1k

3 k

1k

2

k1, 2k

2, k

3are in H.P. ]

Q.10 Let f : [0, 1] [1, 3] be a function defined by f(x) = x3 + x + 1, then the value of

3

1

1 dx)x(f is

(A*)4

5(B) 1 (C)

4

3(D) 3

[Sol. We know thatIf f : [a, b] [c, d]

then, b

a

dx)x(f +

d

c

1dx)x(f = bd ac

1

0

dx)x(f +

3

1

1 dx)x(f = 3

3

1

1 dx)x(f = 1

0

dx)x(f3 = 3 4

7=

4

5. Ans.]

Q.11 If equation of tangent to the curve y = 2x

e

where it crosses the y-axis isb

y

a

x = 1,

then (a b) is equal to

(A) 3 (B) 2 (C) 2 (D*) 3

[Sol. At x = 0, y = 1, Also, y '(0) =2

1

Equation of tangent is (y + 1) =2

1(x 0)

1

y

2

x

= 1 (a b) = 2 (1) = 3. Ans.]

Q.12 If the equations x2 + px + 3 = 0, x2 + qx + 5 = 0 and x2 + (p + q)x + 24 = 0 have a common negative

root, then the value of (p + q), is

(A*) 10 (B) 12 (C) 14 (D) 16


20/25


MATHEMATICS

[Sol. We have,

x2 + px + 3 = 0 .........(1)

x2 + qx + 5 = 0 .........(2)

and x2 + (p + q) x + 24 = 0 .........(2)

(1) + (2) (3) x2 = 16 x = 4, 4So, the common negative root is 4

Put x = 4 in (1), we get p =4

19 and put x = 4 in (2), we get q=421 .

Hence, (p + q) =4

19+

4

21= 10. Ans.]

Q.13 The maximum value of f(x) = x3x2 + x for all x A where A = {x | f(x) 0}, is(A) 3 (B) 2 (C) 1 (D*) 0

[Sol. f(x) 0 x (x2x + 1) 0 x 0Also, f '(x) = 3x22x + 1 > 0 x R and f(0) = 0So, f

max.(x = 0) = 0. x 0. Ans.]

Q.14 The true set of values of x

2,

2satisfying 51xsin4 is equal to

(A)

10

3,

10(B*)

10

3,

10(C)

5,

5(D)

3,

3

2

[Sol. | 4 sin x 1 | < 5

4

51xsin

4

51

x

10

3,

10Ans.]

Q.15

x

x

20xdt)t1(n

x1Lim l is equal to

(A*) 0 (B) 1 (C) 1 (D) 2

[Sol.2

x

x

0x x

dt)t1(n

Lim

l

0

0=

x2

)x1(n)x1(nLim

0x

ll=

2

1(1 1) = 0. Ans.]

Q.16 The number of value(s) of x satisfying the equation

07xn5xn3xn14 23xlnlog2 lll , is

(A) 0 (B) 1 (C*) 2 (D) 3

[Sol. (lnx)21 + (lnx 1) (ln2x 2lnx 7) = 0

(ln x 1) [(ln x + 1) + (ln2x 2 ln x 7)] = 0

ln x = 1; ln x = 2; ln x = 3But ln x = 1 & ln x = 3 are acceptable only

x = e & x = e3. Ans.]

Q.17 In triangle ABC, if = a2(b c)2 then tan A is equal to[Note :All symbols used have usual meaning in triangle ABC.]

(A)15

7(B*)

15

8(C)

15

11(D)

15

13


21/25


MATHEMATICS

[Sol. Given, = (a + b c) (a + c b) = 2(s c) 2(s b) ......(1)

Also, tan2

A=

)as(s

)cs()bs(

=

)cs()bs(=

4

1(Using (1))

So, tan A =

2

Atan1

2

Atan2

2 =2

4

11

4

12

= 15

8

. Ans.]

Q.18 Let f(x) = 13ax a sin 5x 7x sin9x. If f(x) is a strictly increasing function for all x R andhas no critical points, then the minimum integral value of a is

(A) 2 (B*) 3 (C) 4 (D) 5

[Sol. f(x) = 13 ax a sin 5x 7x sin 9x

f '(x) = 13 a 5a cos 5x 7 9 cos 9x > 0 8a 16 + 5a(1cos5x) + 9(1 cos9x) > 0 x 8a 16 > 0

a > 72 a (2, ) aleast integer

= 3. Ans.]

Q.19 A function f is differentiable in [0, 10] and f(0) = 4 and f(10) =1. If g(x) =2x

)x(f

and c (0, 10) then

g' (c) is must be equal to

(A)10

5(B)

24

5(C)

10

5(D*)

24

5

[Sol. We have g(x) =

2x

)x(f

g(0) =20

)0(f

=

2

4= 2 and g(10) =

210

)10(f

=

12

1

Now, apply LMVT to y = g(x) in x [0, 10] , there exist some c (0, 10)

such that g'(c) =010

)0(g)10(g

g '(c) =

10

212

1

=1210

25

=

24

5. Ans.]

Q.20 If the equation 2|3x| = p, where p is a constant integer has exactly three distinct solutions, then

the number of integral values of p, is

(A) 0 (B*) 1 (C) 2 (C) 4

[Sol. The following sequence of graphs gives a progression that leads directly to the answer.

(3,0)

y=| x + 3 |

y

x

(3,2)

y=|x + 3| 2

y

x

(3,2)

y= |x + 3| 2| |

y

x ]


22/25


MATHEMATICS

Q.21 If f(x) = (x2013 + 1) | x24x 5 | + sin | x | + cos ( | x 1 | ), then f(x) is not differentiable at

(A*) 2 points (B) 3 points (C) 4 points (D) zero point

[Sol. f(x) = (x5 + 1) )1x()5x( + sin | x | + cos (x 1).

f(x) is non-differentiable at two points viz. x = 0, 5. Ans.]

Q.22 If the system of equations

x + y + z = 0x + y + z = 0x + y + z = 0

has solution (x0, y

0, z

0) where x

0y

0z

00, then number of distinct real values of equals

(A) 0 (B) 1 (C*) 2 (D) more than 2

[Sol. Put = 0

11

11

11

= 0

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

( 1)2 ( + 2) = 0 = 2, 1. Ans.]

Q.23 Let f : (1, 1) R be a differentiable function with f(0) = 1 and f '(0) = 1. Let g(x) = 2)2)x(f2(f ,then g'(0) equals

(A*) 4 (B) 0 (C) 2 (D) 4

[Sol. g '(x) = 2f 2)x(f2 f ' 2)x(f2 2 f '(x)Put x = 0, we get

g '(0) = 2 (1) 1 2 1 = 4. Ans.]

Q.24 Let f(x) = (1 + b2

)x2

+ 2bx + 1 and let m(b) be the minimum value of f(x). As b varies,the range of m (b) is

(A) [0, 1] (B) 01

2,

(C)

1

21,

(D*) (0, 1]

[Sol.

(a) f(x) = (1 + b2) x2 + 2bx + 1

Minimum value of f(x) whould occur when x =)b1(2

b22

= 2b1

b

Minimum value of f(x) = m (b)

m(b) = (1 + b2) 22

2

b1

)b(

2

2

b1

b2

+ 1

m(b) = 2

2

b1

b

+ 1 = 2b1

1

as and varies m(b) would vary from (0, 1] as and varies from (, ).]

Q.25 Let f (x) =

0xfor,1

2|x|0for|,x|. Then at x = 0, f has

(A*) a local maximum (B) no local maximum

(C) a local minimum (D) no extremum.


23/25


MATHEMATICS

[Sol.

(0,1)

(0,0)

y

x

Graph of y = f(x)

]

Q.26 Let a1, a

2,...., a

10be in A.P. and h

1, h

2, ....., h

10be in H.P. If a

1= h

1= 2 and a

10= h

10= 3

then a4h

7is equal to

(A) 2 (B) 3 (C) 5 (D*) 6

[Sol. a1, a

2,...., a

10, are in A.P. [TN, S&P] done

h1, h

2, ....., h

10are in H.P.

1021

h

1.....,,

h

1,

h

1are in A.P..

a10

= a1

+ 9d

3 + 2 + 9d

d =9

1

and D9h

1

h

1

110

; D92

1

3

1 ;

6

1= 9D; D =

54

1

a4

= a1

+ 3d =3

7

3

12

D6h

1

h

1

17

54

16

2

1

h

1

7

9

1

2

1

h

1

7

18

7

h

1

7

h7

=7

18; a

4h

7=

7

18

3

7 = 6 Ans.]

Q.27 The point (4, 1) undergoes the following transformation successively

(i) reflection about the line y = x

(ii) translation through a distance 2 units along the positive direction of xaxis

(iii) rotation through an angle4

about origin in anticlockwise direction.

(iv) reflection about the line x = 0

The final position of a given point is

(A)

2

7,

2

1(B)

2

7,

2

1(C*)

2

7,

2

1(D)

2

7,

2

1

[Sol: Ref. of (4, 1) in y = x is (1, 4)

Translation of 2 units along +ve direction of x-axis (3, 4)

Slope of line joining origin & new position after rotation


24/25


MATHEMATICS

tan ( + 45) =45tantan1

45tantan

=

3

41

13

4

=

1

7

= 7

New position5

25/7

0y

25/1

0x

x =

5 25

1

; y = 5 25

7

2

7,

2

1; Image of

2

7,

2

1in y-axis

2

7,

2

1ans.]

[PARAGRAPH TYPE]


Paragraph for question nos. 28 to 30

Let f(x) be a cubic polynomial which has local maximum at x =1 and f '(x) has a local minimum at

x = 1. If f(1) = 10 and f(0) = 5.Q.28 The distance between two horizontal tangents is

(A) 12 (B) 22 (C*) 32 (D) 10

Q.29 Sum of all non-negative integers which lies between the roots of f(x) = 0

(A) 9 (B*) 10 (C) 12 (D) 13

Q.30 If x1, x

2, x

3are the real roots of f(x) = 0 then [x

1]2 + [x

2]2 + [x

3]2 is equal to

[Note : [ k ] denotes the greatest integer less than or equal to k. ]

(A) 20 (B) 21 (C) 22 (D*) 25

[Sol. Let f "(x) = k (x 1)

f '(x) = k

x2

x2

+ C

f '(1) = 0

1

2

1k C

C2

k3 C

2

k3

f '(x) = k

x2

x2

2k3

Integrate f (x) = k

x

2

k3

2

x

6

x 23

f(0) = 5

(1,10)

(3,

22)

3 2 1

x1 x2

1 2 3 4 5

x30

5 =

f(1) = 10 k = 6 f(x) = x33x2 9x + 5 f(x) = 3x26x 9

= 3(x22x 3)3(x 3) (x+ 1) = 0


25/25


MATHEMATICS

x = 1 & x = 3

f(1) = 10 point of maximum

& f(3) = 22 point of minimum

Distance between horizontal tangents |10 (22)| = 32

[x1] = 3

[x2] = 0

[x3] = 4[x1]2 + [x

2]2 + [x

3]2 = 32 + 02 + 42 = 25. Ans.]

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Solution for Xyz