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12th Mains 23rd July Test Solution

101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232

2

Subject-PHYSICS Part : 1

SECTION-A

(NEXT THIRTY QUESTIONS: SINGLE CHOICE – ONLY ONE CORRECT ANSWER)

Q1 In the figure the electric lines on the right have twice the separation of those on the left. If a charge particle

takes time t to move a distance x in left region, then it will take time to travel the same distance in the right

side region is:

(a.)

2t

(b.) t (c.) t2 (d.) t2

Sol. (c)

Q2

A surface has the area vector .)ˆ3ˆ2( 2mjiA

The flux of an electric field through it if the field is

:ˆ4mViE

(a.) mV -8 (b.) mV -12 (c.) mV -20 (d.) Zero

Sol. (a)

mVjiiAE -8)ˆ3ˆ2(.ˆ4.

Q3 A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of larger

radius. Both the cylinders are initially electrically neutral (a.) A potential difference appears between the two cylinders when a charge density is given to the inner cylinder. (b.) A potential difference appears between two cylinders when a charge density is given to the outer cylinder. (c.) No potential difference appears between the two cylinders when a uniform line charge is kept along the axis of

the cylinders. (d.) No potential difference appears between the two cylinders when same charge density is given to both the

cylinders.



3

Sol. (a)

When charge is given to inner cylinder, an electric field will be in between the cylinders. So there is potential

difference between the cylinders.

Q4 Figure shows three paths along which we can move positively charged sphere A closer to positively charged

sphere B which is fixed in a plane. If 21 , WW and 3W are the work done along the three respective paths, then

(a.)

321 WWW (b.) 231 WWW (c.)

321 WWW (d.) 213 WWW

Sol. (c)

Work done, .)( qVVW ABAB In all the three cases, potential difference is same and so .321 WWW

Q5 Two small spheres each having the charge Q are suspended by insulating threads of length L from a hook.

This arrangement is taken in space where there is no gravitational effect, then the angle between the two

suspensions and the tension in each will be

(a.)

2

2

0 )2(41,180

LQ

(b.)

2

2

041,90

LQ

(c.)

2

2

0 241,180

LQ

(d.)

2

2

041,180

LQ

Sol. (a)

In the absence of gravitational force, the only force acts on the spheres is electrostatic repulsion and so the

angle between two suspension becomes .180 So force between the sphere .)2(4

12

2

0 LQF

Q6 In the electric field of a point charge ,q a certain charge is carried from point A to DCB ,, and .E Then the

work done

(a.) is least along the path AB (b.) is least along the path AD (c.) is zero along all the paths ADACAB ,, and AE (d.) is least along AE

Sol. (c)

The potential difference between either of EDCB ,,, and A is zero, and so work done along all the paths is

zero.

Q7 Four equal charges Q are placed at the four corners of a square of each side is .a Work done in removing a

charge Q from its centre to infinity is



4

(a.) 0 (b.)

aQ

0

2

42

(c.)

aQ

0

22

(d.)

aQ

0

2

2

Sol. (c)

Q8 The displacement of a charge Q in the electric field kejeieE ˆˆˆ

321 is .ˆˆˆ jbiar The work done is

(a.) )( 21 beaeQ (b.) 22

21 )()( beaeQ (c.) 22

21 )( baeeQ (d.) )()( 22

21 baeeQ

Sol. (a)

Q9 Three charges qQ , and q are placed at the vertices of a right angled isosceles triangle as shown. The net

electrostatic energy of the configuration is zero if Q is equal to

(a.)

2 q

(b.)

222 q

(c.) q2 (d.) q

Sol. (b)

Q10 The spatial distribution of the electric field due to charges ),( BA is shown in figure. Which one of the

following statements is correct?



5

(a.) A is +ve and B – ve and |||| BA (b.) A is –ve and B + ve; |||| BA

(c.) Both are +ve but BA (d.) Both are –ve but BA

Sol. (a)

Spacing between the lines of field increases from A to ,B and so magnitude of charge at B will be less than .A

Q11 A square surface of side L meters is in the plane of the paper. A uniform electric field

E (volt/m), also in the

plane of the paper, is limited only to the lower half of the square surface, (see figure). The electric flux in SI

units associated with the surface is

(a.) Zero (b.) 2EL (c.)

02 2/ EL (d.) 2/2EL

Sol. (a)

.090cos EA

Q12 Two thin wire rings each having a radius R are placed at a distance d apart with their axes coinciding. The

charges on the two rings are Q and .Q The potential difference between the centres of the two rings is

(a.) Zero (b.)

220

114 dRR

Q

(c.) 204/ dQR (d.)

220

112 dRR

Q

Sol. (d)

Q13 Two concentric shells are having radii R and ,2R charges 1q and 2q and potentials V3 and V respectively.

Now outer shell is earthed, (a.) Potential difference between shell increases (b.) Potential difference between shell decreases (c.) Potential of inner sphere will become V2 (d.) Potential of inner sphere will become V5.2

Sol. (c)



6

Previously, if 1V and 2V are the potentials of inner and outer shells due to own charge respectively, then

VVV 321

VVV 31 or VV 21

When outer shell is earthed, its potential becomes zero, and so potential of inner becomes VVV 202,1

Q14 In an isolated parallel plate capacitor of capacitance ,C the four surface have charges 321 ,, QQQ and 4Q as

shown. The potential difference between the plates is

(a.)

CQQQQ

24321

(b.)

CQQ

232

(c.)

CQQ

232

(d.)

CQQ

241

Sol. (c)

Q15 For the circuit shown, which of the following statements is true?

(a.) With 1S closed, VVVV 20,15 21 (b.) With 3S closed, VVV 2521

(c.) With 1S and 3S closed 021 VV (d.) With 1S and 3S closed, VVVV 20,30 21

Sol. (d)

Charges on the capacitors will redistribute only when all the three switches will be closed.

Q16 Plates of area A are arranged as shown. The distance between each plate is ,d the net capacitance is

(a.)

dA0

(b.)

dA07

(c.)

dA06

(d.)

dA05

Sol. (a)



7

The equivalent circuit is shown in figure

Thus .0

dACCab

Q17 What is equivalent time constant of following RC circuit?

(a.) RC5.1 (b.) RC3 (c.) RC2 (d.)

2RC

Sol. (a)

The effective circuit is shown in figure.

The effective resistance across C is

Q18 In the circuit shown in figure, find the steady state charge on capacitor .1C

(a.) C2 (b.) C3 (c.) C4 (d.) Zero

Sol. (c)

At steady state, the current in the resistors are shown in figure.

If q be the charge on each of the capacitor, then 06

323

8 qq

Cq 4

Q19 The figure here shows conduction electrons moving leftward through a wire. Select the quantity/(ies) which

have direction rightward:

(a.)

dvi, (b.) ji, (c.) dvEj ,, (d.) ji, and E

Sol. (d)



8

The direction of i and j are opposite of the direction of motion of electron, and the direction of electrons are

opposite to that of .

E

Q20 Figure shows a rectangular solid conductor of edge lengths LL 2, and .3L A certain potential difference V is

to be applied between pairs of opposite faces of the conductor as shown in figure: left-right, top-bottom and

front-back. In which pair current is maximum

(a.) left-right (b.) top-bottom (c.) front-back (d.) equal in all

Sol. (b)

As ,ALR

and so it is least along top-bottom. Current will be maximum.

Q21 For the shown circuit the effective resistance between the points A and B will be

(a.) R2 (b.) R4 (c.) R (d.) 2/R

Sol. (c)

Q22 A meter bridge is set up as shown, to determine an unknown resistance X using a standard 10 ohm resistor.

The galvanometer shows null point when tapping-key is at 52 cm mark. The end corrections are 1 cm and 2

cm respectively for the ends A and .B The determined value of X is



9

(a.) 2.10 ohm (b.) 6.10 ohm (c.) 8.10 ohm (d.) 1.11 ohm

Sol. (b)

At null point

Q23 A torch bulb rated as VW 5.1,5.4 is connected as shown in the figure. The e.m.f. of the cell needed to make

the bulb glow at full intensity is

(a.) V5.4 (b.) V5.1 (c.) V67.2 (d.) V5.13

Sol. (d)

Q24 In the circuit shown below, the cell has an e.m.f. of V10 and internal resistance of 1 ohm. The other

resistances are shown in the figure. The potential difference BA VV is



10

(a.) V6 (b.) V4 (c.) V2 (d.) V2

Sol. (d)

)()( 00 ABBA VVVVVV

4121

.2V

Q25 The equivalent resistance between A and B is

(a.)

321 RRR (b.) 2R (c.)

142

42 RRR

RR

(d.)

41

41

RRRR

Sol. (d)

The effective circuit is shown in figure

41

41

RRRRRAB

Q26 Two infinitely long conducting wires carry currents, as shown in figure. If the horizontal wire carrying current

produces a magnetic field B at the point ),0,,2( ddP then the resultant magnetic field at this point is

(a.) kB ˆ2 (b.) kB ˆ2 (c.) kB ˆ2 (d.) Zero

Sol. (a)

Given .)2(.2

0

dB

Magnetic field due to A4 wire, .2

)4(2

' 0 Bd

B



11

So, resultant magnetic field BBB 2 along – ve z -axis.

Q27 A vertical straight conductor carries a current vertically upwards. A point P lies to the east of it at a small

distance and another point Q lies to the west at the same distance. The magnetic field at P is

(a.) greater than at Q (b.) Same as at Q

(c.) less than at Q (d.) greater or less than at Q depending upon the strength of the current

Sol. (b)

As the distance of the points P and Q is equal from the wire and so magnetic field is also equal.

Q28 The magnetic induction at the centre O in the figure shown is

(a.)

21

0 114 RR

i

(b.)

21

0 114 RR

i

(c.) )(

4 210 RRi

(d.)

)(4 210 RRi

Sol. (a)

Q29 A part of a long wire carrying a current i is bent into a circle of radius r as shown in figure. The net magnetic

field at the centre O of the circular loop is

(a.)

ri

40

(b.)

ri

20

(c.) )1(

20

ri

(d.)

)1(2

0

ri

Sol. (c)

,2

.2

00

ri

riB

out of plane of paper.

Q30 Wires 1 and 2 carrying currents 1i and 2i respectively are inclined at an angle to each other. What is the

force on a small element d of wire 2 at a distance of r from wire 1 (as shown in figure) due to the magnetic



12

field of wire 1

(a.)

tan2 21

0 diir

(b.)

sin2 21

0 diir

(c.)

cos2 21

0 diir

(d.)

sin4 21

0 diir

Sol. (c)



13

Subject-CHEMISTRY Part : 2

SECTION-A


Q1 If the same type of atoms are packed in hexagonal closest packing (HCP) and cubical closest packing (CCP)

separately, then

(a.) Density of HCP will be greater than CCP (b.) Density of HCP will be smaller than CCP (c.) Density of HCP will be equal to CCP (d.) Density of HCP may be equal or greater than or smaller than CCP depending upon the temperature of the system

Sol. (c)

Q2 In the closest packing of atoms, there are

(a.) One tetrahedral void and two octahedral voids per atom (b.) Two tetrahedral void and one octahedral voids per atom (c.) Two of each tetrahedral and octahedral void (d.) Three of each tetrahedral and octahedral void

Sol. (b)

Q3 In spinel structure, oxides ions are cubical-closest packed, whereas 1/8th of tetrahedral holes are occupied by

cations 2A and 1/2 of octahedral holes are occupied by cations 3B ions. The general formula of the

compound having spinel structure is (a.)

42BOA (b.) 42OAB (c.) OBA 42 (d.) OBA 24

Sol. (b)

Q4 In an antiferromagnetic material

(a.) All the magnetic moment vectors are aligned in one direction (b.) Half of the magnetic moment vectors point in one direction and the rest in opposite direction (c.) All the magnetic moment vectors are randomly oriented (d.) Is characterized by a very large magnetic moment

Sol. (b)

Q5 The mole fraction of a solute in carbon tetrachloride is .235.0 The molality of the solution is about

(a.) 1kgmol2.0 (b.) 1kgmol5.1 (c.) 1kgmol0.1 (d.) 1kgmol5.0

Sol. (a)



14

Q6 Which of the following behaviors is true for an ideal binary liquid solution?

(a.) Plot of totalp versus Ay (mole fraction of A in vapour phase) is linear

(b.) Plot of totalp versus By is linear

(c.) Plot of total/1 p versus Ay (or By ) is linear

(d.) Plot of total/1 p versus Ay (or By ) is nonlinear

Sol. (c)

Q7 The vapour pressure of pure benzene and toluene are 160 Torr and 60 Torr, respectively. The mole fraction of

benzene in vapour phase in contact with equimolar solution of benzene and toluene would be

(a.) 50.0 (b.) 84.0 (c.) 73.0 (d.) 27.0

Sol. (c)

Q8 Arrange the following ( 1.0 molal) solution in the increasing order of freezing point.

(I) Soln of 243 )(POCa ; (II) Soln of 42SONa ; (III) Soln of glucose

(a.) I > II > III (b.) I > III > I (c.) III > II > I (d.) III > I > II

Sol. (c)

Larger the sum of stoichiometric number of ions of the molecule, larger ions it produces in solution, and thus

larger decrease in freezing point. Consequently, the freezing point is the lowest.

Q9 A solution of sucrose in g100 of water boils at C.100.25 If this solution is boiled till its temperature is

C,101 the mass of water evaporated from the solution is

(a.) g25 (b.) g50 (c.) g75 (d.) g85

Sol. (c)

Q10 A solution of sodium sulphate in water is electrolysed using inert electrodes. The respective products at the

cathode and anode are

(a.) 22 O,H (b.)

22 H,O (c.) Na,O2 (d.) 22 SO,O



15

Sol. (a)

OH2 is reduced in preference to Na and OH2 is oxidized in preference to .SO24

Q11 The time required to coat a metal surface of 2cm80 with mm0.005 thick layer of silver (density

3cmg10.5 ) with the passage of A3 current through a silver nitrate solution is

(a.) s125 (b.) s130 (c.) s135 (d.) s140

Sol. (a)

s0.125)molg(108A)(3

(1))molC(96500g)10.50.005(80MI

vF)(VMIvmF

t 1

1ee

Q12 The conductivity of M0.001 crotonic acid is .cmS3.83 1 If ,molcmS126(NaCl)Λ 12m

12m molcmS426(HCl)Λ and m (sodium salt of crotonic acid) = ,molcmS83 12 then the ionization

constant of crotonic acid is

(a.) M10 3 (b.) M10 5 (c.) M10 7 (d.) M10 8

Sol. (c)

Q13 The standard potentials at K298 are

V,0.6762Zn)|(ZnE 2 V,0.740Cr)|(CrE 3

00.0Pt)|H|(HE 2 and V0.770Pt)|Fe,(FeE 23

Which is the strongest reducing agent? (a.) (s)Zn (b.) (s)Cr (c.) (g)H2 (d.) (aq)Fe2

Sol. (a)

The smaller the value of reduction potential, lesser the reduction tendency of the ion and larger the oxidation

tendency of the corresponding metal. Larger oxidation tendency implies stronger reducing agent.

Q14 Given V0.337Cu)|(CuE 2 and V.0.136Sn)|(SnE 2 Which of the statements is correct?

(a.) 2Cu ions can be reduced by (g)H2 (b.) Cu can be oxidized by H

(c.) 2Sn can reduced by (g)H2 (d.) Cu can reduce 2Sn

Sol. (a) 2Cu has larger tendency to reduce than .H 2Sn has lesser tendency to reduce than .H



16

Hence, 2Cu ion can be reduced by (g).H2

Q15 When toluene is treated with 2Cl at high temperature without the use of any catalyst, the product obtain is

(a.) Only o chlorotoluene (b.) Only m chlorotoluene

(c.) Only p chlorotoluene (d.) A mixture of ortho–and para–chlorotoluenes

Sol. (d)

At low temperature in the presence of a catalyst ),(FeCl3 electrophilic substitution at benzene ring takes place.

Q16 Which of the following orders regarding the boiling points of the given alkyl chloride is correct?

(a.) CCl)(CHCHClCHCHCHCl)(CHCH 33323323

(b.) CCl)(CHCHClCHCHCHCl)(CHCH 33323323

(c.) CCl)(CHCHClCHCHCHCl)(CHCH 33323323

(d.) CCl)(CHCHClCHCHCHCl)(CHCH 33323333

Sol. (a)

Boiling point decreases with increasing branching of the alkyl group.

Q17 Which of the following is an example of Sandmeyer reaction?

(a.)

(b.)

(c.)

(d.)

Sol. (d)

The major product is 2-butene ).CHCHCH(CH 33

Q18 The order of reactivity for the 2SN reaction involving replacement of X by a nucleophile in alkyl halide is

(a.) 21CH3 (b.) 21CH3 (c.) 21CH3 (d.) 21CH3

Sol. (b)

The rate decreases with increasing bulk of alkyl group.

Q19 In the reaction of p chlorotoluene with 2KNH in liquor ammonia, the major product is

(a.) o-toluidine (b.) m-toluidine (c.) p-toluidine (d.) p-chloroaniline

Sol. (b)

The reaction proceeds via elimination-addition mechanism. The major product is m-toluidine



17

Q20 The compound which gives the most stable carbonium ion on dehydration is

(a.) |

CH

OHCHCHCH

3

23

(b.)

3

3

3

CH|

OHCCH

CH|

(c.) OHCHCHCHCH 2223 (d.) |

OH

CHCHCHCH 323

Sol. (b)

3 carbonium ion is most stable.

Q21 The reaction of ROH with R’MgX produces

(a.) RH (b.) R’H (c.) R – R (d.) R’ – R’

Sol. (b)

HR'ROMgXMgXR'ROH

Q22 Which of the following statements is correct?

(a.) The reaction OHROH HOHRO lies far towards the left side

(b.) The reaction SHR'NaRONaSR'ROH is feasible

(c.) ,RS in a protic solvent, acts as a stronger nucleophile than RO

(d.) The bond angle HOR in methanol is smaller than that of HSR in methanethiol

Sol. (c)

(a) The differences in acidities and basicities are slight, all species are present at equilibrium.

(b) ROH is a weaker acid than SHR' and also SR' is a weaker base than .RO Hence, the reaction will

not be feasible as the reactants contain weaker acid and weaker base as compared to the products.

(c) Because of the bigger size of S, the negative charge on it is more easily polarised which makes it easier for

the nucleophilic attack. The nucleophilic nature of RO is also diminished in a protic solvent because

there will be more stronger hydrogen bonding due to the large negative charge density on the smaller sized

O atom.

(d) The bond angles are 109 and 96 for HOC and H,SC respectively. The oxygen involves 3sp

hydrid orbitals while sulphur involves p orbitals for bonding with C and H atoms.

Q23

The reaction is known as (a.) Wurtz reaction (b.) Riemer-Tiemann reaction (c.) Schotten-Baumann reaction (d.) Kolbe reaction

Sol. (d)

Q24 The reaction of 32CHPhOCH with one equivalent of HI produces



18

(a.) OHCHCHPhI 23 (b.) 33CHCHPhI (c.)

3266 CHICHHC (d.) 32CHICHPhOH

Sol. (d)

Aryl ethers do not cleave on the aromatic side, but only on the alkyl side.

Q25

The reaction of with OHCH3 in (i) acid ,H and (ii) base ,OCH3 respectively, give

(a.) OH)CHC(OCH)(CH 2323 and 3223 OCHCH(OH)CH)(CH

(b.) OH)CHC(OCH)(CH 2323 and OH)CHC(OCH)(CH 2323

(c.) 32323 OCH)CHC(OCH)(CH and OHC(OH)CH)(CH 223

(d.) OHC(OH)CH)(CH 223 and 32323 OCH)CHC(OCH)(CH

Sol. (a)

There occurs cleavage of ether linkage in oxiranes (cyclic three membered ring including oxygen). In acid-

catalyzed reaction, nucleophile 3OCH from OHCH3 attaches to the more substituted carbon where as in

base-catalyzed reaction, it is attached to the less substituted carbon.

Q26 Which reagent on condensation with benzaldehyde product phenylhydrazone?

(a.) OHNH2 . (b.) 256 NHHC (c.)

22 NNHCONHH (d.) 256 NHNHHC

Sol. (d)

Phenylhydrazine )NHNHH(C 256 produces phenylhydrazone.

Q27 The Cannizzaro reaction is not given by

(a.) Trimethylactaldehyde (b.) Benzaldehyde (c.) Acetaldehyde (d.) Formaldehyde

Sol. (c)

CHOCH3 contains α hydrogen atom. It will not show Cannizzaro reaction.

Q28 In the Cannizzaro ,PhCOOHCHPhCHO2Ph 22OH the slowest step is

(a.) The attack of OH at the carbonyl group (b.) The transfer of hydride to the carbonyl group

(c.) The abstraction of proton from the carboxylic acid (d.) The deprotonation of OHCHPh 2

Sol. (b)

Transfer of hydride to carbonyl group is the slowest step.

Q29 Which of the following tests distinguishes acetophenone from propionaldehyde?

(a.) Baeyer test (b.) Iodoform test (c.) Tollens test (d.) Lucas test

Sol. (b)

563 HCOCCH does not react with .NaHSO3

Q30 The reduction potential of hydrogen electrode will be positive if



19

(a.) M1][Hatm,1)p(H2 (b.) M2][Hatm,1)p(H2

(c.) M1][Hatm,2)p(H2 (d.) M1.5][Hatm,2.5)p(H2

Sol. (b)

][Hp

lnF

RTE1/2H

PtHH2

2

The combination atm1)p(H2 and M2][H will make PtHH 2

E positive.



20

Subject-MATHEMATICS Part : 3

SECTION-A


Q1

If 03111

1211111

cccbb

a where 0,0,0 cba then 111 cba is

(a.) 4 (b.) 3 (c.) 2 (d.) 1

Sol. (b)

Take cba ,, common from 321 ,, RRR respectively.

311111

12111

1111

ccc

bbb

aaaabc

Apply 3211 RRRR

ccc

bbbcbaabc

131111

11211111

1113

Now apply

233 CCC and 122 CCC & expand

011132

cbaabc

As 0,0,0 cba

3111 cba

Q2 The system of equation 1822,2245,2254 zyxzyxzyx is

(a.) Inconsistent (b.) Consistent and determinate (c.) Consistent but dependent (d.) Has infinite solution



21

Sol. (a)

The system is ,BAX where

122

,,822245254

Bzyx

XA

0822245254

||

A

1A does not exis. We find ..adj A

Cofactor are 9,18,18,18,36,36,18,36,36

91818183636

183636C

91818183636183636

.adj TCA

We have 063126126

)adj(

BA

Thus .0|| A But .0)adj( BA Therefore the system has no solution.

Q3

If )(sin)(sin)(sin)(sin)(sin)(sin)(cos)(cos)(cos

)(

xxxxxx

xf and ,5)2( f then

20

1

)(r

rf equals

(a.) 5 (b.) 10 (c.) 100 (d.) 50

Sol. (c)

Clearly )(0)(' xfxf constant

But 5)2( f

5)( xf

1002055)(20

1

20

1

rr

rf

Q4 If A is square matrix of order 22 such that

1021;A2 BI and ,ABAC then 2009)(C



22

(a.) AA

10220091

(b.)

AA

10

220091 (c.) AA

2009021

(d.)

AA

20090220091

Sol. (a)

Q5 Which of the following is correct

(a.) If A and B are square matrices of order 3 such that ,3||,1|| BA then the determinant of AB3 is equal

to 27

(b.) If A is an invertible matrix, then )(det 1A is equal to )(det A

(c.) If A and B are matrices of the same order, then 222 2)( BABABA is possible if IAB :

(d.) 111)( BAAB

Sol. (d)

(a) We have |||||| BAAB

Also for a square matrix of order ||||,3 3 AkkA because each element of the matrix A is multiplied by

k and hence in this case we will have 3k common

81)3()1(27||||3|3| 3 BAAB

(b) Since A is invertible, therefore 1A exists and 11 AA

)(det)(det 1 IAA 1)(det)(det 1 AA )(det

1)(det 1

AA

(c) )()()( 2 BABABA

22 BBAABA



23

22 2 BABA if .BAAB

Q6 Let

000

A and .50)( 50

dcba

AIA Then the value of dcba is

(a.) 2 (b.) 1 (c.) 4 (d.) 3

Sol. (a)

We have, 2,,2 kOAOA k

Thus,

AIIA 50)( 50

AIIA 50)( 50

1,0,0,1 dcba

Q7 If ,1sin1sin1sin 111

bax then the value of x is

(a.)

11 22 baab

(b.)

11 22 baab

(c.)

112

22 baab

(d.)

112

22 baab

Sol. (a)

Let

ba1sin;1sin 11 then

x1sin 1

)(sin1sinsin 1

x

ab

aab

bbabax

111.11111sincoscossin1 22

22

11 22

ba

abx

Q8 The inverse of

xxxf

11tan)( 1 is:

If given invertible

(a.)

yyyy

cottancottan

(b.)

yyyy

cottancottan

(c.)

yyyy

tancottancot

(d.)

yyyy

tancottancot

Sol. (c)

Given, xxy

11tan 1

yxx tan

11

yxx 2tan

11

yx

x2tan

111



24

yyyy

yyx

tancottancot

tan1tan1

2

2

Q9 The value of

sincoscot

sin1costan 11

xxx

is

(a.) 2 (b.) (c.) 2/ (d.) Independent of

Sol. (b)

Q10

Range of the function Rxxxxf

,3

1tan)(2

1 is

(a.)

2,

6

(b.)

3,

6

(c.)

4,

6

(d.) ),(

Sol. (c)



25

Q11 If ,2500...941)( Rxxxxxxf then all the values of x where )(xf has

minimum values lies in

(a.) )700,600( (b.) )678,576( (c.) )678,625( (d.) )200,100(

Sol. (c)

Q12

nnn 2tan

21....

2tan

21

2tan

21tanlim 22

(a.)

1

(b.)

2cot21

(c.) 2cot2 (d.)

tan1

Sol. (b)



26

Q13

Let ,}1)1{()1(

)(1

0

n

rn xrrxxLtxf then

(a.) )(xf is continuous but not differentiable at 0x (b.) )(xf is both continuous and differentiable at 0x

(c.) )(xf is neither continuous nor differentiable at 0x (d.) )(xf is a periodic function

Sol. (c)

1111 xrrx

xtr

111

111

xrrx

rxxr

1

0 111

n

rrn nx

tS

0,1 x

0,0 x

1

11nx

LtSLtnnn

Thus,

0,00,1

xx

xf

10

xfLtx

and 00 f

Hence xf is neither continuous nor differentiable at 0x

Clearly xf is not a periodic function.

Q14

2

11

1 1)cos(sinlim

xx

x

equals

(a.) 0 (b.) 1 (c.)

21

(d.)

1

Sol. (c)



27

Q15

))2ln(cos()sin(lim 2

2

0 xxx

x is equal to

(a.) 2 (b.) 2 (c.) 1 (d.) 1

Sol. (b)

Q16 In order that the function xxxf cot)1()( is continuous at )0(,0 fx must be defined as

(a.) 0)0( f (b.) ef )0( (c.) ef /1)0( (d.) 1)0( f

Sol. (b)

For continuity actual value must be equal to limiting value

A = x

xx cot

0)1(lim

log A = )1(logcotlim0

xxx



28

= x

xx tan

)1(loglim0

form00

= 1sec

11

lim 20

x

xx

(By L’ Hospital Rule)

log A = 1 A = e1 = e.

For )0(f must be defined as .)0( ef

Hence (b) is the correct answer.

Q17

If

2

2)441(logsinlog.

)2(sin1

)(222

xk

xxx

xxx

xf is continuous at ,2

x then k

(a.)

161

(b.)

321

(c.)

641

(d.)

281

Sol. (c)

For )(xf to be continuous at ,2

x we must have



29

Q18 If ]),[]([.1)( xxxxf then ][ is GIF

(a.) )(xf is continuous and differentiable at 1x (b.) )(xf is discontinuous at 1x

(c.) )(xf is continuous at 2x (d.) )(xf is continuous but non-differentiable at 1x

Sol. (d)

Q19 The 2),1(),1(max)( xxxf is, where ),( x (a.) discontinuous at all points (b.) differentiable at all points (c.) differentiable at all points except 1 and 1 (d.) continuous at all points except 1 and 1

Sol. (c)

We draw the graph of xyxy 1,1 and 2y

}2,1,1{.max)( xxxf

1,1)( xxxf

21,2 x

2,1 xx

From graph it is clear that )(xf is continuous at all x and differentiable at all x except 1x and 1x Q20

If

1

1,0)( 2

2

xcaxbxxbbax

xf then )(xf is continuous and differentiable at 1x if

(a.) bac 2,0 (b.) Rcba , (c.) 0, cba (d.) 0, cba

Sol. (a)



30

Q21 If )(),( xgxf be twice differentiable functions on ]2,0[ satisfying 4)1(',2)1('),(")(" gfxgxf

and ,9)2(,3)2( gf then )()( xgxf at 4x equals

(a.) 0 (b.) 10 (c.) 8 (d.) 2

Sol. (b)

Let )()()( xgxfxh

)(')(')(' xgxfxh

0)(")(")(" xgxfxh

cxh )(' … (1)

kcxxh )( … (2)

From (1)

242)1(')1(')1(' gfhc

Also 693)2()2()2( gfh

From (2),

kch 2)2(

k 46 2k

Hence 22)( xxh

10242)4( h

Q22 The differential coefficient of xeca1

10 coslog

is

(a.)

axxxec

a xec

1021

log

log1

1cos

1cos10

(b.)

axxxec

a xec

1021

log

log1

1cos

1cos10

(c.)

10log1

1cos 21

log 1cos10

axxxec

a xec

(d.)

10log1

1cos 21

log 1cos10

axxxec

a xec

Sol. (b)

Let

xecay

1cos10log

axecy elog.cosloglog 110

aexec ee log.log.coslog 101

axece 101 log.coslog

Differentiating w.r. to x , we get



31

1

1cos

1.log12110

xxxeca

dxdy

y

xecxxay

dxdy

1210 cos1.

11log

xecxxaa xec

1210

log

cos1

log.

1cos10

Q23 If )(.)(.)()( zfyfxfzyxf for all zyx ,, and ,3)0(',4)2( ff then )2('f equals

(a.) 12 (b.) 9 (c.) 16 (d.) 6

Sol. (a)

Q24 Let the function )(xfy be given by ,7205 35 tttx and ,31834 23 ttty where

).2,2(t Then )(' xf at 1t is

(a.)

25

(b.)

52

(c.)

57

(d.)

75

Sol. (b)

Q25 The maximum slope of the curve 2723 23 xxxy is

(a.) 5 (b.) 5 (c.)

51

(d.)

51

Sol. (a)



32

Q26

Let ,1,23

10,2

sin)(

xx

xxxf

then

(a.) )(xf has local maxima at 1x (b.) )(xf has local minima at 1x

(c.) )(xf does not have any local extrema at 1x (d.) )(xf has a global minima at 1x

Sol. (a)

Q27 Consider the statements (i) xxn )1( in ),0( (ii) )1( xnx in ),0( (iii) xx tan in )2/,0( (iv) xx tan in )2/,0( Among above statements

(a.) (i) and (iii) are correct (b.) (i) and (iv) are correct (c.) (ii) and (iii) are correct (d.) (ii) and (iv) are correct

Sol. (a)



33

Q28 Let .41,|43|)( 2 xxxxf Then

(a.) )(xf is monotonically increasing in

23,1

(b.) )(xf is monotonically increasing in

4,

23

(c.) The minimum value of )(xf is

41

(d.) The maximum value of )(xf is 2

Sol. (a)

Q29 If Rxxf 0)(" and )6()2()( 22 xfxfxg then

(a.) )(xg is an increasing in ]0,2[ (b.) )(xg is an increasing in ),2[

(c.) )(xg has a local minima at 2x (d.) )(xg has a local maxima at 2x

Sol. (d)

Q30 The probability of a man hitting a target is .

41

If he fires n times such that the probability of his hitting the

target at least once is greater than ,32

then the value of n is

(a.) 3 (b.) 4 (c.) 5 (d.) 2

Sol. (b)

Let p is the probability of hit and q is the probability of miss



34

4n

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