IIT STS4 Questions Solutions

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IIT-JEE 2008 STS IV/PCM/P(I)/QNS

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IIT-JOINT ENTRANCE EXAMINATION, 2008 PAPER I PHYSICS

−

CHEMISTRY − MATHEMATICS QUESTION PAPER CODE

Time: 3 Hours

4

Maximum Marks: 243

Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose. INSTRUCTIONS: R O T A. General A L I 1. This booklet is your Question Paper containing 66 questions. The booklet has 24 pages. G I V N 2. This question paper CODE is printed on the right hand top corner of this sheet. I E H 3. This quest io ion paper cont ai ains 1 blank page for your rough work. No additional sheets will be T provided for rough work. M O R F 4. Blan Blank k pape papers, rs, clipb clipboa oards rds,, log log tabl tables es,, slide slide rule rules, s, calcu calcula lator tors, s, cellu cellula larr phon phones es,, page pagers rs and and elec electro troni nicc S gadgets in any form are not allowed t o be carried i nside the examination hall. N O I T 5. Fill in the t he boxes provided below on this page and also writ e your Name and Enrollment No. in the C U space provided on the back page (page no. 24) of this booklet. R T S 6. This booklet also c ontains the answer sheet (i.e., a machine gradable response sheet) ORS. N I T I 7. DO NOT TAMPER WITH/MUTILATE THE ORS OR THE BOOKLET. A W A B. Filling the ORS , T E 8. On the lower part of the ORS, write i n ink, your name in box L1, your Enrollment No. in box L2 and L K Name of the Centre in box L3. Do not write these anywhere else. else . O O B 9. Put your signature in ink in box L4 on the ORS. S I H T N O C. Question paper format: Read the instructions pri nted on the back page (page no. 24) of this booklet. S L A D. Marking scheme: Read the instructions on t he back page (page no. 24) of this booklet. E S E H Enrollment No.: T Name: . K A E R B I have read all the instructions I have have verified all the informations T filled in by the Candidate. abide by them. O and shall abide N O D

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2

PAR PART A : PHYS HYSICS SECTION I Straight Objective Type This section conta ntains 9 mult ultipl iple choice ques uestion ions num numbered 1 to 9. Each question has four choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

1. A point moving with uniform acceleration in a straight line and describes th

distances S and S′ in the m

(A)

S  S′

(B)

mn

S  S′

and n

th

(C)

mn

seconds. What is its acceleration? S mn

(D)

S′ mn

cal dis distance an astronaut can jump on earth surfac face is 0.5 m. 2. The maximum vertical Estimate the vertical dis distance nce through which he can jump ump on moon, whic hich has has a mean density

2 3

(A) 1.5 m

that of earth and radius one quarter that of earth.

(B) 6 m

(C) 7.5 m

(D) 3 m

nstant voltage V = 25 volts is maintained 3. A const between points A and B of the circuit. Find the the magn magnit itude ude of the the curr curren entt flowi flowing ng thr throu ough gh CD if R = 1 oh ohm, R = 2 ohm, R = 3 ohm 1

2

3

and R = 4 ohm. 4

(A) 2 A (B) 0.8 A (C) 1 A (D) 1.2 A SPACE FOR ROUGH WORK



























3

4. If the number of turns per unit length of a coil of a solenoid is doubled, the self

inductance of solenoid will (A) remain the same

(B) be halved

(C) be doubled

(D) become four times

5. A radio adioac acttive ive subst ubstan ancce disi disint nteg egrrate ates to

1 64

of its initial value in 60 seconds.

What is the half -life of the substance? (A) 5 s

(B) 10 s

(C) 30 s

(D) 20 s

6. The fri friction coeffi fficient between flo floor and board of mass m on which a man of 



mass M stan stands ds is µ. Find the maximum for force that the man can exert on the rope so that the board does not slip. 

(A)

(B)

(C)



µ Mm g 1µ

µ Mm g 1µ

µ Mg µ

(D) µ (M + m) g

and cros crosss-sectio sectional nal area area A is fitted on to a wooden disc 7. A steel ring of radius r and 







of radiu adiuss R (R > r). If Young s modulus is Y, what is the force with which the steel ring is expanded? 

(A)

AYR r





(B)

AY R  r r

(C)

Y R  r A

r

(D)

SPACE FOR ROUGH WORK

Yr AR



























4

nductting ing sphe pheres having char harges of oppo pposite sign attract each 8. Two identical conduc other with a force 0.108 N, when separated by 0.5 m. The spheres which are connected by a conducting wire is removed. Now they repel each other with a force 0.036 N. Find the charges.

(A) 3 µC and 1 µC

(B) 5 µC and 2 µC

(C) 5 µC and 4 µC

(D) 3 µC and 2 µC

9. The initial pressure and volume of a given mass of gas for which γ = 1.5 is P and V. It is slowly compressed to one half of its initial volume. It is then rapidly compressed to volume V/4. Find the final pressure. (A) 4P

(B) 2P

(C) 5.64P

(D) 1.5P

SECTION II Assertion and Reason Type This section contains 4 questions numbered 10 to 13. Each question contains STATEMENT 1 (Assertion) and STATEMENT 2 (Reason). Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

(A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation for Statement 1. (B) Statement 1 is True, Statement 2 is True; Statement 2 is not a correct explanation for Statement 1. (C) Statement 1 is True, Statement 2 is False. (D) Statement 1 is False, Statement 2 is True. 10. Stateme Statement nt 1: All simple harmonic motions are periodic bu t all periodic motions are not necessarily simple harmonic. because Statement 2:

The necessary condition for S.H.M is that the restoring force must be pro proportion ional to the dis displa placement and dir directed towards the mean position.




























5

11. Stateme Statement nt 1:

When When ultr ultras ason onic ic wave wavess trave ravell from from air air to wat water, er, it bend bendss tow owar ards ds the normal to the air water surface. surf ace.

because Statement 2: 12. Stateme Statement nt 1:

The speed of a sound wave is greater in water than in air. The specific fic hea heat capacity of a solid is diff diffe erent when the solid is heated at (i) constant volume and (ii) at constant pressure.

because Statement 2:

When solid is hea heated at const nstant pre pressure, re, it does expan pand a litt ittle and some heat is required for doing the mechanical work associated with this expansion.


A dentist uses a convex mirror to examine a small cavity in the tooth.

because Statement 2: A convex mirror forms only diminished virtual images. SECTION III Linked Comprehension Type This section contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

Paragraph for Questions 14 to 16 The vibration of a string of length 60 cm fixed at both ends are represented by the equation y = 4 sin

πx 15

cos 96 π t , where x and y are in cm and t in sec.

14. What is the maximum displacement of a point at x = 5 cm? (A) 3.46 cm

(B) 4.8 cm

(C) 2.96 cm

(D) 5.2 cm
























































7

SECTION IV Matrix-Match Type This section contains 3 questions. Each question contains statements given in two columns which have to be matched. Statements (A, B, C, D) in Colu olumn I have to be matched with statements (p, q, r, s) in Colu olumn II. The answers to these questions have to be appropriately bubbled as illustrated in the following example. If the correct matches are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then the correctly bubbled 4 × 4 matrix should be as follows:

2 0.

p

q

r

s

A

p

q

r

s

B

p

q

r

s

C

p

q

r

s

D

p

q

r

s

Column I (A) Magnetic force on a charged particle of charge q

Column II (p)

µ Ni 0

L moving in a magnetic field with velocity v perpendicular to it

(B) Magnetic field induction due to an infinitely

(q)

µ Ni 0

2R long straight conductor at a distance R

(C) Magnetic field induction at the centre of a circular loop of radius R having N turns

(r) Bqv

µ i (D) Magnetic field induction due to a long solenoid

(s)

0

2π R



























8

21. Column I gives the various defects of human eye and Column II lists the laser that are used to correct them. Column I

Column II

(A) Myopia

(p) Concave lens

(B) Hypermetrobia

(q) Cylindrical lens

(C) Presbyopia

(r)

Bifocal lens

(D) Astigmatism

(s)

Convex lens

22. Column I gives list of common units and Column II gives the dimensional formula of these units to measure the physical quantities. Column I

Column II 2

−3

2

−2

(A) Pascal

(p) ML T

(B) Becquerel

(q) ML T

(C) Watt

(r)

T

(D) Joule

(s)

ML

−1

−1

−2

T



























9

P A R T B : C H E M IS IS T R Y SECTION I S t r a i g h t O b je je c t i v e T y p e

T h is is s e ct io ion con t ai a i n s 9 m u ltlt ip ip le le ch oi oi ce q u es es t io ion s n u mb m b e r ed ed 2 3 t o 3 1 . E a ch ch q u es es t io i on h a s 4 choices choices (A), (A), (B), (B), (C) (C) an d (D), ou ou t of which O NLY ONE is cor cor r ect. 2 3 . A m et et a l cr ys ys t a llll i ze s i n t o t w woo cu bi b i c p ha h a se s e s F CC CC a n d B C C w h os os e u n itit ce l l l e n gt gt h s ar e 3.5 and 3.0 Å respectively respectively.. The ra tio of of the den sities of of FCC an d BCC ar e (A) (A ) 1.26 (B ) 1.0 (C ) 1.8 (D ) 2.2 1. 0 1. 8 2. 2 n +

n +

n 3+

2 4 . T h e s t an a n da d a rd r d r ed ed u ct ct io ion p ot en en ti t i a ls ls of t hr h r ee ee ca ti t i on s A 1 , B 2 a n d C + 0.52, − 2.8 2.8 an d − 1.5 V r espectively. espectively. The ord ord er of th eir r edu cing powers is (A) (A )

A

n 1+ n 2+

>

B

n 2+ n 3+

>

C

n 3+

(B )

A

n 1+

>

C

n 3+

>

B

n 2+

n 1+

(D ) can not be pr edicted edicted B > C > A 2 5 . What is tru e regarding adsorpt adsorpt ion ion ph enomenon? enomenon? (A) (A ) Ph ysical ysical a dsorpt dsorpt ion ion is en doth doth erm ic but chem ical ical a dsorpt dsorpt ion ion is exoth exoth erm ic. ic. (B ) Reverse of th e above. (C ) Both are endothermic. (D ) Both Both ar e exo exoth erm ic. ic. 2 6 . H ow ma ny t ypes of of car bon bon -car bon bon bonds are pr esent esent in but yne -1? (A) (A ) 2 (B ) 3 (C ) 1 (D ) 4 2 7 . Following is an organic reaction, X Y C 6H 5S O 2 OH → C 6H 5 S O 2 Cl → C 6H 5 S O 2 N H 2 (C )

The r eagents u sed fo for t he st eps X an d Y respectively respectively ar e (A) (A ) P Cl , SOCl , N H N H (B ) P Cl or S OCl , N H 5 2 2 2 5 2 3 (C )

P 4H 10 , N H 3

(D )

HCl, NaN H 2

are



























10

2 8 . An or g a n i c com p ou n d of m ol e cu l a r for m u l a C H 5

10

O

2

g i v e s t he h e fol l ow i n g

properties. (i) It evo evolves lves eff effervesc ervescenc encee with NaH CO solution. 3

(ii) (ii) On chlorin chlorin at ion, ion, gives gives only only one one m onochloro nochloro derivat ive. ive. (iii) iii) Its sodium sodium salt when fused with with soda soda lime giv gives es neobuta neobuta ne. (iv) iv) Its calc calcium ium salt when distilled distilled giv gives es di-t er -but yl ketone. The str uctur e of of the origina origina l co compoun d is (A) (A ) (CH (C H ) C − COOH 3 3

(B ) CH

3

− CH 2 − CH 2 − CH 2 − COOH

(C )

(D )

2 9 . Pentaamminenitrito -N -cobalt (III) cat cat ion ion is (A) (A ) [Co(ONO)(NH ) ]

2+

3 5

(B ) [Co(NO )(NH ) ] 2

2+

3 5

(C ) [Co(ONO)(NH ) ]

3+

3 5

(D ) [Co(NO )(NH ) ] 2

3+

3 5

3 0 . An e x a m pl pl e of a n ox y g e n con t ai a i n in i n g com po p ou n d w h ic ich d oe s n ot ot g i v e ox y g e n g a s on strong heating is (A) (A ) (N H ) Cr O 4 2

(C ) KClO

2

7

(B ) KMnO (D ) P b O

4



























11

3 1 . When When H Cl and HNO ar e mixed in th e ra tio 3:1 3:1 by vo volum e, the pr oducts obta obta ined 3

are (A) (A ) Cl

(B ) NOCl

2

(C ) H O

(D ) all th e above above

2

SECTION II Assertion

-

R e a s o n Ty p e

T h i s s e ct i on con t a i n s 4 q u e s t i on s n u m b e r e d 3 2 t o 3 5 . E a ch q u e s t i on con t a i n s S T AT E M E N T 1 (As s e r t i on ) a n d S T AT E M E N T 2 (R e a s on ). ). E a ch q u e s t i on h a s 4 choices choices (A), (A), (B), (B), (C) a n d (D), ou ou t of wh ich ON LY ON E is cor cor r ect. (A) (A ) S t a t e m e n t 1 i s T r u e , S t a t e m e n t 2 i s T r u e ; S t a t e m e n t 2 i s a c o r r e c t explana explana tion tion for Stat ement 1. (B ) S t a t e m e n t 1 i s T r u e , S t a t e m e n t 2 i s T r u e ; S t a t e m e n t 2 i s n o t a cor r ec ect explana explana tion tion for Stat ement 1. (C ) Sta tement 1 is True, Stat ement 2 is False. False. (D ) Sta tement 1 is False, Sta tement 2 is True. 3 2. 2 . S t a t e m e n t 1: 1:

Dibora Dibora ne on complete meth ylation ylation gives gives B H (CH ) . 2

2

3 4

because Statement 2: 3 3. 3 . S t a t e m e n t 1: 1:

Dibo Dibora ne str ucture sh ows two three -cent re bonds. L i n k ag a g e i s om er er is i s m i s n ot ot p os os s ib ib l e i n coor di d i a n ti t i on com po p ou n ds ds −

when th e ligand ligand present in it is OH ion. because Statement 2: 3 4. 4 . S t a t e m e n t 1: 1:

−

OH ion ion is a base. RMgX is a Lewis base.

because Statement 2: 3 5. 5 . S t a t e m e n t 1: 1:

It a tt acks on on electr electr on r ich ich a toms. The conjugate base of H S O



4

in a queous solution solution is H S O .


H S O is a st rong acid acid in a queous solut solut ion. ion. 2

4

2

4





























12

SECTION III L i n k e d C o m p re re h e n s i o n Ty p e

T h is is s e ct io ion con ta t a in in s t wo wo p a ra r a gr gr ap a p h s. s. B a se se d u po p on e a ch ch p a ra r a gr gr ap a p h , 3 m ul u l t ip ip l e ch oi oi ce q u es es t io ion s h av a v e t o b e a ns n s w e r ed ed . E ac a ch q u es es t io ion h as a s 4 ch oi oi ce s (A), (B ), (C ) a nd nd (D), (D), ou ou t of which ON LY ONE is corr corr ect. ect. P a r a g r a p h f o r Qu Qu e s t i o n s 3 6 t o 3 8

W e k no n ow t ha h a t, t , i n a ch em em ic ica l r ea ea ct ct io ion on ly ly e l e ct ro r on s (e x t ra r a n uc u cl e a r p a rt r t ) of t he he a t om t a k e p a r t , w h il il e t h e n uc u cl e u s of t h e a t om r e m a i n s u n af a ffe ct e d . H ow e v e r , t h e r ev e v e r se s e r ea e a ct ct io i on s (w h er e r e on ly l y n u cl cl e i of a t om om s t a k e p a rt r t i n r ea e a ct ct io i on s ) a r e a llss o p os os s ib ib l e . S u ch ch r ea ea ct ct io ion s i n w h ic ich n u cl cl e u s of a n a to t om i t se s e l f u n de de r go goe s s p on ta t a n eo eou s ch a ng n g e or i n t er er a ct ct w it it h ot h er e r n u cl cl e i of l i g h t er e r p ar a r t ic icl e s , r es e s ul u l t in i n g i n t h e for m a ti t i on of n ew e w n u cl cl e i an d one one or more lighter lighter par ticles ticles are called nuclear nuclear rea ctions. 3 6 . Which Which is n ot a postulat e of th eory eory of rad ioactivi ioactivity? ty? (A) (A ) Radioactivity Radioactivity is due to th e un sta bility bility of th e nu cleus. (B ) R a di di oa ct ct iv iv i t y i s n ot ot a ffe ct e d b y ch em em i ca l com b i n at a t i on b u t a ff ffe ct e d b y temperature. (C ) If n/p n/p ra tio is is higher t ha n 1.6, 1.6, β-par ticles ticles are emitt ed. (D ) If n/p rat io is less tha n 1, positr positr on em ission ission t ak es place. place. 3 7 . T h e h a lf lf - l i fe p er er io i od of a r a di d i oa ct ct iv iv e s ub u b s ta t a n ce ce i s 1 2 . 2 d a ys ys . T h e t im im e t a k en en t o fa llll th e am oun t t o 20% of the init ial co concentr at ion ion is (A) (A ) 8.45 days

(B ) 28.3 days

(C ) 40.5 days

3 8 . Which Which a mong th e fo following llowing is used in r adiother apy?

(D ) 100 days



























13

Thu s acetoac acetoacetic etic ester exists as ta ut omers .

3 9 . Which Which one of th e co compoun ds does not not sh ow ta ut omer ism?

(A) (A ) CH COCH COCH 3

(C)

2

3

(B )

(D) C H CO C H 6

5

6

5

4 0 . Which Which is n ot a correct rrect stat ement r egardin egardin g tau tomerism? tomerism? (A) (A ) Nitr o compoun ds show ta ut omer ism. (B ) The ta ut omer ic form form s ar e chem chem ically ically distinct distinct en tities. (C ) Tautomers can can not be separ separ ated. (D ) Polar pr otic solve solvent nt like water ten ds to stabilize th e keto fo form . 4 1 . Tau tomerism can be distin distin guished from from th e resona resona nce, as



























14

SECTION IV Matrix -M a t c h T y p e

T h is i s s ec ect io ion con t a in in s 3 q u es e s ti t i on s . E a ch ch q u es e s ti t i on con t a in in s s ta t a t em em en en t s g i v e n i n t wo wo l u mn m n I h av col u mn m n s w h ic ich h av a v e t o b e m at a t ch ch ed ed . S t at a t em em en en ts t s (A, B , C , D ) i n C o lu a ve t o b e l u mn m n I I . T h e a ns m a tc t ch ed ed w i t h s t a te t e m en en ts t s (p , q , r , s ) i n C o lu n s w e r s t o t he h e s e q u es es t io ion s ha ve to be appr opria tely bubbled as illustr at ed in th e fo following llowing example. I f t he h e cor re r e ct m at a t ch ch es es a re r e A-p , A-s , B -q, B -r , C -p , C -q a n d D -s , t h en en t h e cor r ec ect ly ly bubbled 4 × 4 ma tr ix should be as follo follows: ws:

42.

p

q

r

s

A

p

q

r

s

B

p

q

r

s

C

p

q

r

s

D

p

q

r

s

Co l u m n I ( A)

(B )

− COCl

Co l u m n II (p )

[Ag(NH ) ]

(q )

− CH O

3 2

2+



























15

43.

Co l u m n I

Co l u m n II

2

(A) (A ) s p hybridisation 3

(B ) s p hybridisation 3

(C ) s p d hybridisat ion ion

(p ) SnCl (q ) bent

(r ) linear

(D ) Thr ee lone lone pair s of elec electr tr ons (s ) XeF 44.

Co l u m n I (A) (A ) K N O 4

(p ) CO (q ) N

2

(C ) (N H ) Cr O 4 2

(D ) CaCO

3

2

Co l u m n II

3

(B ) N H N O

2

2

7

(r ) O

2

2 2

(s ) Oxidises Fe

2+



























16

PART PAR T C : MATH MATHEM EMATICS ATICS SECTION I Straight Objective Type This section contains 9 multiple choice questions numbered 45 to 53. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. 2

45. A solution x of [x] − 2 [x] − 15 > 0, where [ ⋅] is the greatest integer function is (A) − 1.2

(B) 4.2

(C) 5.3 2

46. If p ∈ {0, 1, 2, ... 9} such that x − px +

(D) 8.3 1

(2p + 3) = 0, then the probability that

4 the roots of the above equation are real is

(A)

3 10

(B)

1 5

(C)

2 5

(D)

7 10

47. If the eccentric angles α, β of two points on the ellipse

x

2

2



y

2 2

= 1 of

a b eccentricity e differ by a constant, then the chord joining these two points touches another ellipse of eccentricity

(A) e

(B) e/2

(C) e/3

(D) e/4





























17

50. If f(x) =

sin π x

1  x f(4) + f(− 4) is

(A) 3



π 2

x



,where [⋅] denotes the greatest integer function, then

(B) 4 th

(C) 0

th

(D) 1

th

51. If the (p + 1) , (q + 1) , (r + 1) terms of an A.P. are in G.P., p, q, r are in H.P., then the ratio of the common difference to the first term of the A.P. is (A)

2 q

52. If in ∆ ABC

(B) 

x =tan

y = tan

2

(C)

q B



C

tan

2 C



2

2 r

A 2

A

tan

B 2

(D) 

2 p



























18

(A) Statement 1 is True, statement 2 is True; statement 2 is a correct

explanation for statement 1. (B) Statement 1 is True, statement 2 is True; statement 2 is not a correct

explanation for statement 1. (C) Statement 1 is True, statement 2 is False. (D) Statement 1 is False, statement 2 is True. 54. Stateme Statement nt 1:

c

If P(A ∪ B) = 5/6, P(A ) = 1/4, P(B) = 1/3, then A and B are independent events.

because Statement 2: 55. Stateme Statement nt 1:

If P(A ∩ B) = P(A) ⋅ P(B), then A and B are independent. Tangents are drawn from the points on 2x + y = 1 to the circle 2

2

x + y = 4. Then the chords of contact passes through (8, 4). because Statement 2:

Forr diff Fo differ eren entt valu values es of λ ax + by + c + λ (a′x + b′y + c′) = 0 passes



























19

SECTION III Linked Comprehension Type This section contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

Paragraph for Question Nos. 58 to 60 If f(x) = 0 be an n

th

degree equation in x, then

S = sum of the roots =  coeff. of x

n  1

1

coeff. of x

n

S = sum of the products of the roots taken 2 by 2 = coeff. of x

n  2

2

coeff. of x

n



























20

4

3

2

60. If x − 8x + ax + bx + 16 = 0 have equal roots, then the ordered pair (a, b) is (A) (24, 32)

(B) (24, − 32)

(C) (8, 24)

(D) (24, 8)

Paragraph for Question Nos. 61 to 63 If Lt

f x = 1 and Lt

x→a

g x =

∞

, then Lt

f x

g x

Lt

f x

is e x → a

x→a

x→a 1

61. The value of Lt

a

x



x



c

x

x

is

3

x→0

(A) abc

b

3

(B) abc

(C) abc 1

(D) a + b + c



1 g x



























21

matched with statements (p, q, r, s) in Colu Colum mn II. The answers to these questions have to be appropriately bubbled as illustrated in the following example. If the correct matches are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then the correctly bubbled 4 × 4 matrix should be as follows:

64.

p

q

r

s

A

p

q

r

s

B

p

q

r

s

C

p

q

r

s

D

p

q

r

s

Column I (A) PQ and RS are parallel tangents of a circle

Column II (p) 1





























22

65.

Column I

Column II

(A) If A (a, b, c) be any point on the plane

(p) 2

2

7 2

2

3x + 2y + z = 7, then the least value of a + b + c is ¯ v, ¯ w ¯ be such that | u ¯ | = 1, | v ¯ | = 2 and (B) Let u, ¯ is equal ¯ | = 3. Also the projection of v ¯ on u |w

(q)

1 3



























23




























Name: .

Enrollment No.:

Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose.

C. Question paper format: 10. The question paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part has 4 sections.



























1 ®

®

IIT-JEE 2008 STS IV/PCM/P(I)/SOLNS



























2

At moon’s surface, acceleration due to gravity = g′ . With the same velocity he can jump a distance, 2

v = 2g′ s g

(

2

= g)



























































































































































































































































































































































































































































































































































































































































































































































17

PART C : MATHEMATICS SECTION I Straight Objective Type This section contains 9 multiple choice questions numbered 45 to 53. Each question has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. 2

2

2

45. The value of x satisfying the equation cot (x + y) + tan (x + y) = 1 + 2x − x is (A) 1

(B) 2

46. The remainder when 3 (A) 3

(D) − 1

(C) 3 2006

2006

+7

(B) 5

+ 2007 is divided by 29 is

(C) 6

(D) 7

47. If f{x + f(y)} = f(x) + y for all x, y ∈ R and f(0) = 1, then the value of f(10) is (A) 10

(B) 11

(C) 9

(D) 1

48. If the product of the slopes of any two normals out of three normals drawn from 2

(h, k) to y = 4ax is 2, then the locus of (h, k) is

(A) a straight line

(B) a circle

(C) a parabola

(D) an ellipse

49. If π < α <

3π 2

4

value of 4 sin α

(A) 2 + 4 sin α (B) 2 − 4 sin α



2

sin 2 α



4cos

(C) 2

2

π

4

α 

2

is

(D) 2 + 4 cos α 2

50. Given that x ∈[0, 1] and y ∈ [0, 1] let E1 be the event of (x, y) satisfying y ≤ x 2

and E be the event of (x, y) satisfying x ≤ y then 2

(A) P (E ∩ E ) = 1 1 2 3

(B) P (E ∩ E ) = 1 1 2 2

(C) E and E are exhaustive

(D) E and E are independent

1

2

1

2




























18

2

2

51. The real value of k such that a root of (k − 2) m − (3k − 4) m + k = 0 is the slope of a line parallel to yy -axis is (A) ± 1

(C) ± 2

(B) ± 2

(D) ± 3

2

52. The value of

∫e

x  x

d x x

where [ ] represents the greatest integer

0

function is 2

2

(A) (e + 1) (e − 1)

(B) (e − 1) (e + 1)

(C) e − 1

(D) e + 1

53. The sum of cot (A)

π

4

2

−1

−1

3 + cot

(B)

π

−1

7 + cot

(C) cot

2

−1

13 + cot

1

5 7

−1

21 + cot

31 is

(D) cot

1

7 5

SECTION II Assertion and Reason Type This section contains 4 questions numbered 54 to 57. Each question contains STATEMENT 1 (Assertion) and STATEMENT 2 (Reason). Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

(A) Statement 1 is True, statement 2 is True; statement 2 is a correct explanation for statement 1. (B) Statement 1 is True, statement 2 is True; statement 2 is not a correct explanation for statement 1. (C) Statement 1 is True, statement 2 is False. (D) Statement 1 is False, statement 2 is True.




























19 π ⁄4


∫  π ⁄4

1  cos 2x 1  cos 2x

π

dx =

4

because

a Statement 2:

a

∫ f x dx = 2∫ f x dx if f(x) is even a

0

= 0 if f(x) is odd 55. Stateme Statement nt 1:

n

C

n

n  r

 3⋅ C

n

n  r  1

=

 3⋅ C (n + 3)

C

n

n  r  2

 C

n  r  3

r



n

n

C + C r

r−1

=

(n + 1)

C

r 2

2

The angle of intersection of the curves x = 5 − 4y and y = x at (1, 1) is

π

2


The angle of intersection of two curves at a point of intersection is 1

= tan

m  m 1

2

1 m m 1

where m

1

an d m

2

tangents to the curves at the point. SPACE FOR ROUGH WORK

2

are the slopes of the



























20

2


If one root of the equation x + ax + 12 = 0 is 4 while the equation 2 x + ax + b = 0 has equal roots, then the value of b is equal to 49 4

because 2

2

If the roots of ax + bx + c = 0 are equal, then b − 4ac = 0.

Statement 2:

SECTION III Linked Comprehension Type This section contains 2 paragraphs. Based upon each paragraph, 3 multiple choice questions have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

Paragraph for Questions 58 to 60 Let S = 0

.. . ( 1 )

L=0

... (2)

be the equations respectively conic and a line. Then the combined equation of the straight line ines joinin ning the origin to the poi points of intersection of (1) (1) and (2) (2) can be got by homogenising (1) with (2) 2

2

58. The lines joining the origin to the points of intersection of y = mx + c and x + y = 1 2

are at right angles then c is 2

(A) m + 1

ngent to 59. A tange

(B) x

2 

m

2



2 y

1

(C)

m

2



3

1

(D)

m

2



1

4

2 =

1 meets its director circle at P and Q. Then the product

16 9 of the slopes of CP and CQ where C is the centre of the ellipse is

(A) 

9 16

(B)

9 16

(C)

16 9

(D) − 1




























21

2

2

60. The line x + y = 1 meets x − y = 4 at P and Q. Angle between CP and CQ where C is the centre of the hyperbola is 

(A) tan

1

1



(B) tan

2

1

1



(C) tan

3

1

1

4

(D)

π

4

Paragraph for Questions 61 to 63 sin

−1

−1

x + cos

x=

π

if − 1 ≤ x ≤ 1

2 tan

−1

x + tan

−1

y = tan

−1

x 1

= π + tan −1

tan

x − tan

−1

y = tan

−1



(A) tan

1

48 79



π

12



13



(B) tan

62. The value of tan (A)

1

−1

63. Let f(x) = sin

−1

1 

1

79

−1

y xy

is 

(C) tan

1

46 79



(D) tan

3 is (C)

4

x + cos

49

if x > 0, y > 0 and xy > 1

xy



50

47 −1



y



3π

(D)

4

5π 4

π

x. Then

2

is equal to

(B) f (− 2)

1 1 λ



π

(A) f (− 3) (C) f

1

2 + tan

(B) 

4

cos

x

−1

x

if x > 0, y > 0 and xy < 1

xy



1 61. The value of cos

y



4

, λ ∈ R

2

(D) f (λ − 2λ + 3), λ ∈ R


1

36 79





























22

SECTION IV MATRIX-MATCH TYPE This his sect sectio ion n cont contai ains ns 3 que questio stions ns.. Each ach ques questi tion on cont contai ains ns stat tateme ements nts give given n in two columns which have to be matched. Statements (A, B, C, D) in Colum lumn I have to be matched with statements (p, q, r, s) in Colu Colum mn II. The answers to these questions have to be appropriately bubbled as illustrated in the following example. If the correct matches are A-p, A-s, B-q, B-r, C-p, C-q and D-s, then hen the correctly bubbled 4 × 4 matrix should be as follows: p

q

r

s

A

p

q

r

s

B

p

q

r

s

C

p

q

r

s

D

p

q

r

s

Column I 64 .

Column II

(A) If 3 is a root of ax 2 + bx + c = 0 and a, b, c are in AP, then the other root is (B) The value of tan 9 ° − tan 27° − tan 63° + tan 81° is 

(C) The value of sin

π

1





sin

1

π 

2 2 where [ ] denotes greatest integer function is x

(D) If f x

=

cos





1

x





1

(p)

4

(q)



(r)

0

(s)



2 5

1 x 1 x

then f ′(2) is

7 5



























23

Column I

Column II 2

65.(A) The value of

∫

1  x

2

dx is

(p) 0

2 π ⁄4

(B) The value of

∫

x

2

e sec e

 π ⁄4

2x

x

dx is

(q) 4

1

π

(C) The value of

dx

∫

0 1  5 1

(D) The value of

dx

∫ 0

cos cos x

is

x 

1  x

2

(r)

is

(s)

Column I

π

2

(p) 5 3

(B) If z , z , z are the complex numbers of 2

4

Column II

66.(A) The greatest value of |z + 1| if |z + 4| ≤ 7 is 1

π

3

(q) 2 z

the vertices of an equilateral triangle, 1 1 1   then is z z z z z z 1

2

2

3

3

1

(C) The area of the parallelogram whose diagonals represent the vectors

(r) 10

3 i  j  2 k and i  3 j  4 k is (D) x, y, z are distinct real numbers such that (s) 0 the scalar triple product xa  y b  zc × x b  yc  za ⋅ xc  ya  z b = 0 where a, b, c are nonnon-coplanar vectors then the value of x + y + z is

1



























24




























25




























Name: .

Enrollment No.:

Please read the instructions carefully. You are allotted 5 minutes specifically for this purpose.

C. Question paper format: 10. The question paper consists of 3 parts (Physics, Chemistry and Mathematics). Each part has 4 sections. 11. Section I con contain tainss 9 mult multip iple le cho choice ice ques questi tio ons. Each questio stion n has 4 choice oicess (A), (A), (B), (B), (C) (C) and (D), (D), out out of which only one is correct. correct . 12. Section II contai contains ns 4 quest question ions. s. Each Each questi question on contai contains ns STAT STATEM EMENT ENT-1 (Assert (Assertion ion)) and and STAT STATEM EMENT ENT2 (Reason). Bubble Bubble (A) (A)

if both both the statements statements are are TRUE TRUE and and STATEM STATEMENT ENT-2 is the correct explanation of STATEMENT-1.

Bubble Bubble (B) (B)

if both both the statements statements are are TRUE TRUE but but STATEME STATEMENT NT-2 is not the correct explanation of STATEMENT-1.

Bubb Bubble le (C) (C)

if STAT STATEM EMEN ENT T-1 is TRUE and STATEMENT-2 is FALSE.

Bubb Bubble le (D) if STAT STATEM EMEN ENT T-1 is FALSE and STATEMENT-2 is TRUE. 13. 13. Sect Sectio ion n III III cont contai ains ns 2 para paragr grap aphs hs.. Based Based upon upon each each para paragr grap aph, h, 3 mult multip iple le choi choice ce quest questio ions ns have have to be answered. Each question has 4 choices (A), (B), (C) and (D), out of which only one is correct. correct. 14. 14. Sect Sectio ion n IV contains 3 questions. Each question contains statements given in 2 columns. Statements in the first column have to be matched with statements in the second column. The answers to thes these e ques questi tion onss have ave to be appr approp opri riat atel ely y bubb bubble led d in the the ORS ORS as per per the the inst instru ruct ctio ions ns giv given at the beginning of the section. D. Marking scheme: 15. For each question in Sect ection I , you will be awarded 3 marks if you darken only the bubble corr corres espo pond ndin ing g to the the corr correc ectt answe answerr and and zero zero mark mark if no bubble is dark darke ened. ed. In all oth other case cases, s, minus one (−1) mark will be awarded. 16. For each question in Sec Section ion II, II, you will be awarded 3 marks if you darken only the bubble corr corres espo pond ndin ing g to the the corr correc ectt answe answerr and and zero zero mark mark if no bubble is dark darke ened. ed. In all oth other case cases, s, minus one (−1) mark will be awarded. 17. For each question in Sect Sectio ion n III III, you will be awarded 4 marks if you darken only the bubble corr corres espo pond ndin ing g to the the corr correc ectt answe answerr and and zero zero mark mark if no bubble is dark darke ened. ed. In all oth other case cases, s, minus one (−1) mark will be awarded. 18. For each questi stion in Sect Sectio ion n IV you will be awarded 6 marks if you darken ALL the bubbles





























1

®

®

IIT JEE 2008 STS IV/PCM/P(II)/SOLNS -

BRILLIANT S 

HOME BASED FULL SYLLABUS SIMULATOR TEST SERIES -

FOR OUR STUDENTS TOWARDS

IIT JOINT ENTRANCE EXAMINATION, 2008 -

PAPER II

-

SOLUTIONS

PHYSICS − CHEMISTRY − MATHEMATICS

PART A: PHYSICS SECTION I 1. (B) The rock rocke et is subjecte cted to a constant force rce (the fuel uel burns rns at const nstant rate) but since its total mass is decre creasing with time, the acce ccelerat ration keeps on



























2 4. (C) The relative refractive index of the pair of media

=

=

Speed of light in media 1 Speed of light in media 2 2.2

×

10

2.4

×

10

∴ θc = sin

8 8

=

11 12

11

−1

12

5. (D) The elect lectri ricc fie field E is

V l

and V = curr curre ent × resista resistance. nce. Since Since

l

and and curr curren entt are are

the same for all the three wires, the electric fields are proportional to the resistances of the three wires. 6. (A) We have

mv

2

= qvB

R R= R=

qB 2mE

=

qB mv

q m

⋅

B 2E

For α particles, mα = 4Mp and qα = 2qp Hence for same R, E α = Ep 7. (C) No work is done by the charge AB



























3

Wave velocity = n λ y π

∴ 2πny0 = 4nλ or λ =

o

2

9. (A) In the first case, the equivalent capacity = C + 2C

= 3C The charge stored = 3CV coulomb In the second case, the equivalent capacity = kC+ 2C = C (k + 2)

∴ potential =

charge capacity

=

3CV C k



2

=

3V k



2

SECTION II 10. (D) App Appllying ying the laws laws of cons conse erva rvation of line linea ar momentum ntum and kine kinettic energ nergy, y,

we find v2 − v1 = − (u2 − u1). Thus the relative velocity is unchanged in magnitude but gets reversed in direction. 11. (A) To obse observe rve beat beats, s, the the diff differ eren ence ce betw betwee een n two two inte interf rfer erin ing g freq freque uenci ncies es must must be

less than 10 to 16 Hz. In visible light, waves have very high frequencie cies. So beats are not observed due to persistence of vision. 12. (A) Since the mass and materia rial are same, the volumes must also be same. For

same volume the surface area of plate is greatest and for sphere it is least. 13. (A) Angle of refraction refraction r is given by sin r = sin i . Since angle angle i = 0, 0, sin r = 0 or r = 0



























4 15. (D) Let the echo be heard when train P is at distance x

from hill. Time taken by train to reach P =

dis distanc tance e SP velocit city of train

1  x

=

40

... (1)

Time taken taken by echo to reach P =

=

dist distan ance ce SH

dis distanc tance e PH



v  u

v  u

1

x



1200  40

1200  40

Comparing (1) and (2), 1  x

1

=

40

x



1240 x=

29 31

1160

km

16. (A) Frequency heard by driver

v  u  v n″ =

v  u

o

⋅ n′

... (2)



























5

Initial angular speed = 2π ×

5

=

60

π

rad/sec

6

270.4

Initial angular moment J = I ω =

×

3

π 6

= 15 π J-sec 18. (A) When each boy moves 0.6 m towards centre, the new distance of boy from centre = 0.7 m

∴ M.I. of system I ′ = M l 12 =

2



∑ mr ′

10 × 2.6

2

2

 2 × 25 × 0.7

12

2

2

= 30.13 kg -m

If ω′ is the angular speed then according to principle of case, value of angular momentum J = Iω = I′ ω′

ω′ =

Iω I′

π

=

15π



30.13

π 2

1

revolution ⁄





























6

PART B : CHEMISTRY

SECTION I

23. (A)

k1

A+B

Slow

→ C D

k2

k3

Rate = k3 [C], but

k k

1

=

2

Substituting for [C], k k Rate =

1

k

3

[A] [B]

C A B



























7 27. (D)

Formation of an additional pπ-pπ bond due to lateral overlap of vacant 2p-orbital of B and one filled 2p-orbital of F. Since any one of the three fluorine atoms can take part in back bonding, the structure of BF3 is a resonance hybrid of the following canonical forms:

28. (B) In compounds of type ECl 3, the angle is in the order B > P > As > Bi.



























8 SECTION II

32. (B) Benzene gets stabilised because of resonance. 33. (A) In p-nitrotoluene, both group moments act in the same directi ction, while in

p-chloronitrobenzene they act in the opposite direction.

34. (D) H − N − H bond angle in NH3 is greather her than han H − As − H bond angle in

AsH3. With ith the decr decre ease in elect lectro rone nega gattivit ivity y of the elem lements nts, the size of the elect lectro ron n clo cloud incr increa easses from rom N to P to As. This his caus cause es grea greatter dis distort ortion ion in pyra pyram midal idal stru struct ctur ure e of the the mole molecu cule le.. Th The e bond ond angl angle e appr approa oach ches es 90° and the 2

participation participation of s electrons in hybridisation becomes less important. H

N

H 106° 45′ ; H

P

H 94° ; H

As

H 91° 48′



























9 =

2

0.5

1.5 p

K p =

×

p

×

0.238 = 0.159 atm

2 NO

2

N O 2

=

0.159

2

= 0.319

0.0793

4

38. (C) When hen more NO2 is added dded at equi equili libr briu ium, m, acco accord rdin ing g to Le-Chatelie Chatelierr principle, principle,

the rea reacti ction procee ceeds in the rev reverse rse direction. i.e., concentrat ration of N2O4 increases. But K p does not change. 39. (A) − CH3 group roup exert xertss + I effe ffect. ct. Farthe rtherr the − CH3 group group from from − COOH COOH grou group, p,

greater is the acid strength.



























10 +

+

+

+

2Ag + 2KCl → 2AgCl + 2K 2Ag + H2S → Ag2S ↓ + 2H Black 2+

(D) Cu

+

+ H2S → CuS ↓ + 2H Black

44. (A) − (p); (B) − (p), (s); (C) − (q); (D) − (q), (r)

(A)

2, 3-pentadiene, non-superimposable mirror images.





























11

PART C : MATHEMA MATHEMATIC TICS S SECTION I 45. (A) LHS ≥ 2, 2



























12

 h t1t2 + t2t3 + t3t1 = 2a a

... (2)

t1 t2 t3 = k a

... (3)

If the product of the slopes of the normals at t and t



























13



























14 51. (C) Slope of the line parallel to y -axis is infinity. ∴ 2

k −2=0 2

2

coefficient m = 0

k =2



























15

Similarly, t3 = tan t4 = tan t5 = tan

−1

−1

−1

−1

4 − tan

−1

5 − tan

−1

6 − tan

3 4 5















































































































































































































IIT STS4 Questions Solutions

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