testpaper2 (3)

CENTERS: MUMBAI / DELHI / AKOLA / KOLKATA / LUCKNOW / NASHIK / GOA

ANDHERI / BORIVALI / DADAR / CHEMBUR / THANE / MULUND/ NERUL / POWAI IIT – JEE - 2013 FULL TEST – 2 MARKS: 240 (ADVANCED PATTERN) PAPER - II

SECTION - I PHYSICS

PART I: Single Correct Answer Type This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. (+3, –1)

1. Three identical spheres each having a charge 2q and radius R are kept such that each touches the other two. Find the magnitude of the electric force on any sphere due to the other two.

(a) 2

20

3q4 R

(b) 2

20

3q2 R

(c) 2

20

3q8 R

(d) 2

20

3q4 R

2. In the shown arrangement of the meter bridge if AC corresponding to null deflection of

galvanometer is x, what would be its value if the radius of the wire AB is doubled?

(a) x (b) x4

(c) 4x (d) 2x

3. All the edges of a block with parallel faces are unequal. Its longest edge is thrice its shortest edge.

The ratio of the maximum to minimum resistance between parallel faces is (a) 3

(b) 6 (c) 9 (d) indeterminate unless the length of the third edge is specified

4. The ratio of the magnetic field at the centre of a current carrying coil of radius a to the field at a distance 3a on its axis is

(a) 20 10 (b) 2 10 (c) 10 10 (d) 10 5. A conducting circular loop of radius r carriers a constant current I. It is placed in a uniform magnetic

field 0B

such that 0B

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

(a) 02 IrB

(b) 0IrB

(c) 0IrB

(d) ZERO 6. A coil having N turns is wound tightly in the form of a spiral with inner and outer radii a and b

respectively. When a current I passes through the coil, the magnetic field at the centre will be

(a) 0NIb

(b) 02 NIa (c)

0

eNI blog

2 b a a

(d)

0

eNI blog

b a a

G

xA BC

2R1R


7. Two thin long parallel wires separated by a distance b are carrying a current I ampere each. The magnitude of force per unit length exerted by one wire on the other is

(a) 2

02

Ib (b)

20I

2 b

(c) 0I2 b

(d) 02

I2 b

8. A circular loop of radius 2m is kept in a magnetic field of strength 2T

(plane of loop is perpendicular to the direction of magnetic field.)

Resistance of the loop wire is 12 m

. A conductor of length 4m is

sliding with a speed 12 ms as shown in the figure. Find the instantaneous force acting on the rod:

Assuming the rod to have a negligible resistance, the instantaneous force acting on the rod is

(a) 8 N (b) 16 N (c) 32 N (d) 64 N

PART II: Multiple Correct Answer(s) Type This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) or (D) out of which ONE or MORE are correct. (+5, –2) 9. A milliammeter of range 10 mA and resistance 9 is joined in a circuit as shown. The

milliammeter gives full scale deflection for current I when A and B are used as its terminals. It gives a full scale deflection again for a current I ' when A and C are used as terminals. Then

0.1 0.9

9 ,10 mA

A B C

(a) I ' = 111 mA (b) I ' = 900 mA (c) I = 1 A (d) I = 1.1 A

10. A long, straight wire of radius R carriers a current distributed uniformly over its cross-section. The magnitude of the magnetic field is

(a) maximum at the axis of the wire (b) minimum at the axis of the wire (c) maximum at the surface of the wire (d) minimum at the surface of the wire 11. Current flows through a straight cylindrical conductor of radius r. The current is distributed

uniformly over its cross-section. The magnetic field at a distance x from the axis of the conductor has magnitude B

(a) B = 0 at the axis (b) B x for 0 x r

(c) 1B for x rx

(d) B is maximum for x = r

12. In the circuit shown, LRC

, the switch S is closed at time t = 0.

Equal currents will flow through L and C at time 0t . Then (a) 0t RC (b) 0t RCIn 2

(c)

0

L In 2t

R (d) 0t LR

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x12ms

VS

C

RL

R

PART III: PARAGRAPH TYPE This section contains 3 multiple choice questions relating to ONE paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. (+4, –1)

PARAGRAPH – I

A person wants to roll a solid non-conducting spherical ball of mass m and radius r on a surface whose coefficient of static friction is . He placed the ball on the surface wrapped with n turns of closely packed conducting coils of negligible mass at the diameter. By some arrangement he makes a current I to pass through the coils either in the clockwise direction or in the anti-clockwise direction. A constant horizontal magnetic field B

is present throughout the space as shown in the figure.

Assume is sufficient enough to ensure pure rolling motion. Based on the facts provided, answer the following questions.

13. The maximum torque in the coil is (a) 2 ˆnIr B k (b) 2 ˆnIr B j (c) 2 ˆnIr B j (d) 2 ˆnIr B k 14. Angular acceleration of the ball after it has rotated through an angle o180 , is

(a) 5 nIB cos7 m

(b) 2 nIB cos5 m

(c) 7 nIB cos5 m

(d) 5 nIB cos2 m

15. The angular velocity of the ball when it has rotated through an angle is o180 , is

(a) 10 nIB sin7 m

(b) 5 nIB sin14 m

(c) 5 nIB cos14 m

(d) 5 nIB sin7 m

PART IV: Integer Answer Type

This section contains 4 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive) (+4,0) 16. In which branch of the circuit shown in figure, an 11 V battery be inserted so that it dissipates

minimum power. What will be the current, in ampere, through the 2 resistance for this position of the battery.

BI

y

x

17. A uniform disc of radius r and mass m is charged uniformly with the charge

q. This disc is placed flat on a rough horizontal surface having coefficient of friction . A uniform magnetic field is present in a circular region a r

but varying as 3kt as shown in figure. Find the time, in second after which the disc begins to rotate. (Given r = 1 m, m = 18 kg, q = 1C, 0.1 , K = 4,

2g 10 ms ) 18. A small conducting loop of radius a and resistance per unit length , is

pulled with velocity perpendicular to a long straight conductor carrying a current 0I . If a constant power P is dissipated in the loop, the

variation of velocity of the loop as a function of x is 2

20 0

x PI a

ka ,

where k is integer. {Given that x >> a}. 19. A long solenoid of radius 2 R contains another coaxial solenoid of

half the radius. The coils have the same number of turns per unit length and initially both carry no current. At a same instant, the currents in both solenoids starts increasing linearly with time. At any moment the current flowing in the inner coil is twice as large as that in the outer one and their directions are the same. Due to increasing currents, a charged are the same. Due to increasing currents, a charged particle, initially at rest between the solenoids, starts moving along a circular trajectory (see figure) of radius r kR . Then find k.

PART V: Matrix Match Type

This section contains 1 question. Each question contains statements given in two columns, which have to be matched. The statements in Column-I are labelled A, B, C and D, while the statements in Column - II are labelled p, q, r, s and t. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column- II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example: If the correct matches are A - p , s and t ; B - q and r ; C - p and q ; and D - s and t. (+8, 0) 20. Column 1 Column 2 (a) Uniformly charged ring (P) Electric field is present (b) A dielectric ring with uniformly

distributed charge is rotating with constant angular velocity.

(Q) Magnetic field is present

(c) A wire carrying constant current (R) Induced electric field is present (d) I = Io cos t (S) Magnetic moment is present.

PAPER – II (SOLUTION) 1. (d)

2 2

2 20 0

2q1 qF4 4 R2R

2 2netF F 2F cos60

2

net 2qF 3F 3

4 R

2

net 20

3qF4 R

2. (a)

The ratio ACCB

will remain uncharged.

3. (c)

maxmax

minR

A

0max 2

0

3R

0 Minimum length min

minmax

RA

0

min 20

R3

max

min

R 9R

4. (c)

20 0

centre axis 3 22 2

I IaB , B2a 2 a 9a

So, the desired ratio is

0

3 22

3 22

I2a 10 10 10Ia

2 10a

5. (d) Magnetic force on a current carrying loop in uniform magnetic field is zero. 6. (c)

The width b a is having N turns. So number of turns per unit length is Nnb a

n=fN

Consider a circular coil of radius x , radial thickness dx and if dN is the number of turns in it, then

E

F

R

R

R

NdxdNb a

If dB is the field due to this element at the centre, then

0NIdxdB

2 b a x

b0

ea

NI bB dB log2 b a a

7. (b) Force per unit length between two wires carrying currents 1I and 2I at distance r is given by

0 1 2I IF2 r

Here, 2I I I and r b

2

0IF2 b

8. (d)

Halfeq

B vI whereRR

2

Half2R 2 4

2 4 2

I 8A2

So, if mF is the magnetic force, then mF BI mF 2 8 4 64 N 9. (a,c) gI 10mA 0.01A A B g gV V I I 0.1A I 9.9

gI 0.1 I 10

0.01A 10I 1A

0.1

Similarly g gI ' I 0.9 I 9.1

gI ' 0.9 I 10 1I ' A 111mA9

10. (b,c) 11. (a,b,c,d) 12. (b,c)

LRC

2 LRC

LRCR

c L say

Since 1 1LL 0

VI I 1 e 1 eR

and

1

cdq VI edt R

For L CI I , we have t t

1 e e

t

1 2e e

t log 2 ln 2

Lt ln 2 RC ln 2 ln 2R

13. (a) B

, where

is the dipolement

2 ˆ ˆnI r j Bi

2 ˆnIr B k

14. (a) cmBcos fr I ……(1)

Where cmI is the moment of inertia of the ball about the centre of mass. (Do not confuse cmI with the current I) and cmf ma mr ….(2)

5 nIB cos7 m

15. (a)

d 5 nIB cosd 7 m

2 5 nIB sin

2 7 m

10 nIB sin7 m

16. (1) 17. (2) 18. (8) 19. (2) 20. a P ;b P,Q,S ;c Q ;d Q,R

B f

r


SECTION - II CHEMISTRY

PART I: Single Correct Answer Type This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. (+3, –1)

1. During electrophilic substitution reaction of the compound

A B C , the ring that

is most likely to undergo electrophilic substitution is (a) Ring A (b) Ring B (c) Ring A and C (d) All the three rings 2. Consider the following statements, (1) CH3 group is o,pdirecting group due to hyperconjugation. (2) CCl3 is meta directing group due to reverse hyperconjugation. (3) CCl3 group is meta directing group due to mesomeric effect. (4) CH = CH2 group is o,pdirecting group of these statements (a) (1) and (2) are correct (b) (1) and (3) are correct (c) (1), (2) and (4) are correct (d) (3) and (4) are correct 3. The time required for electroplating of 39 g of chromium from a Cr(III) solution, using an average

current of 30 ampere with 75% efficiency, is (a) 1.68 hours (b) 2.68 hours (c) 5.4 hours (d) 10.8 hours 4. At pH = pKin + 1 (where Kin= dissociation constant of the indicator), the ratio [In]/ [HIn] in the

solution is

110 . For this pH, the percentage dissociation of the indicator is

(a) 91 (b) 90 (c) 10 (d) none of these 5. Whichof the following species is iso-structural with XeF4? (a)

5TeF (b) 3I (c)

4BrF (d) XeO3

6. Addition polymers include 1. polyamide 2. polyethylene 3. polyester (a) 1 only (b) 2 only (c) 2 and 3 only (d) 1,2 and 3 7. The major product of the following reaction is

OH OH H+ ?

O O CHO

(a) (b) (c) (d) CH2


8. In how many elements does the last electron have the quantum numbers of n = 4 and l = 1?

(a) 4 (b) 6 (c) 8 (d) 10

PART II: Multiple Correct Answer(s) Type This section contains 4 multiple choice questions. Each question has four choices (A), (B), (C) or (D) out of which ONE or MORE are correct. (5, –2) 9. 2 2H O is prepared

(a)by the electrolysis of 50% 2 4H SO

(b)by the electrolysis of 4 4NH HSO

(c)by the auto-oxidation of 2-ethyl anthraquinol (d) by adding hydrated 2BaO to conc. 2 4H SO at 0200 C 10. Pick out the wrong statement. An electrochemical cell stops working only when (a) electrode potential of the two half cells becomes equal. (b) whole of the metal used as cathode is consumed. (c) whole of the metal used as anode is consumed. (d) molar concentrations in the two half cells becomes equal. 11. Which of the following do not change the value of equilibrium constant for a reaction ? (a) addition of catalyst (b) increase in temperature (c) increase in pressure (d) removal of one of the products 12. Which of the following statement(s) is/are correct : (a) Hydrolysis of ethyl acetate in acid medium is a first order reaction. (pseudo 1st order) (b) The unit of rate constant of 3rd order reaction is litre2mol1 sec1. (c) The rate of a chemical change is always directly proportional to concentration. (d) The rate constant of a chemical reaction is altered by a catalyst.

PART III: PARAGRAPH TYPE This section contains 3 multiple choice questions relating to three paragraphs with two questions on each paragraph. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. (+4, –1)

4

4

NaIO.2

KMnO.1 [Q]

H.2

NaOH.1 [R] HBr.2

excessMgBrCH.1 3 [S] NaH C11H12O2 C11H12O2 C13H19BrO

O


13. The product [Q] is :

(a)

OH

OH

(b)

OH

O

(c)

CHO

CHO

(d)

O

OH

14. The structural formula of intermediate product [R] is :

(a)

O

CHO (b)

O

OH

(c)

O

O

(d)

O

O

15. The structural formula of intermediate product [S] is :

(a)

Br

OH (b)

Br

OH

(c)

OH

Br (d)

OH

Br

PART IV: Integer Answer Type

This section contains 4 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive) (+4, 0)

16. In an adsorption experiment, a graph between log (x/m) versus log p was found to be linear with a slope of 1.0. The intercept on the log axis was found to be 0.3010. What will be the amount of gas adsorbed per gram of charcoal under a pressure of 0.5 atmospheres?

17. The reaction 2O3 3O2 is assigned the following mechanism

3 2O O O

O3 + O 2O2 (slow). The overall order of the reaction


18. The bond order of O2–2 will be.

19. At 400 K, the root mean square (rms) speed of a gas X (molecular weight = 40) is equal to the most probable speed of gas Y at 60 K. The molecular weight of the gas Y is


This section contains 1 question. Each question contains statements given in two columns, which have to be matched. The statements in Column-I are labelled A, B, C and D, while the statements in Column - II are labelled p, q, r, s and t. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column- II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example: If the correct matches are A - p , s and t ; B - q and r ; C - p and q ; and D - s and t. (+8, 0) 20. Column – I (Compounds/class Column – II (Structural features) of compounds) (A) ortho silicates (p) d-d transition (B) Aluminium chloride (anhydrous) (q) Intramolecular H-bonding (C) CuSO45H2O (r) sp3 – p overlap (D) KMnO4 (s) p – p or p – d overlap


PAPER – II (SOLUTION) 1. Reaction occurs on the middle ring since it is bonded to two activating phenyl groups

(with +R effect). (b) 2. (c) 3. The electroplating reaction would be Cr3+ + 3e Cr Let the current is passed for t hours.

5231

965003600t

1007530

= 39

t = 2.68 hours (b)

4. % of [In] = 100110

10100]HIn[]In[

]In[

91%

(a) 5. For XeF4,

62

482N

Structure is octahedral with 2 lone pairs, i.e. square planar. For

5TeF ,

62

1562N

But it has one lone pair so it is not iso-structural with XeF4. For

3I ,

52

1272N

For 4BrF ,

62

1472N

It is iso-structural with XeF4 as it also has two lone pairs. For XeO3,

428

2N

(c) 6. b 7. b 8. b 9. (a,b,c) 10. (b), (c), (d) 11. a, c, d Equilibrium constant depends upon mode of writing the system, stoichiometry of the system and

temperature. 12. a, d The unit of rate constant is given by : (Concentration)(1n) (Time)1 Hence for third order reaction unit will be : L2mole2sec1 13. c


CHO

CHO

[Q]

4KMnO OH

OH

4NaIO

14. d

NaOH CHO

CHO COO

CH2OH

HCOOH

CH2OH tionesterifica

rramoleculaint O

O

[R] 15. a

O

O

OH.2

ExcessMgBrCH.1

3

3 HBr OH

OH

Br OH

[S]

NaH O

Br Br O

Br OH

[S] 16. 1

1 1.0 log K 0.3010 p 0.5n

1/nx p .km

1/1x 2 (0.5) 1m

17. 1 18. 1 1s2*1s22s2*2s22px2 2py2 = 2pz2*2py2 = * 2pz2

B order = 10 8 12

19. 4

rms mp3RT 2RTU UM M

From questions 13R 400 2R 60 4 g mol40 M

M = 4.

20. (A) – (r), (s), (B) – (r), (C) – (p), (q), (D) – (r), (s)

CENTERS : MUMBAI / AKOLA / NASHIK / PUNE / DELHI / KOLKATA / LUCKNOW / GOA

PART I : Single Correct Answer Type

This section contains 8 multiple choice questions. Each question has four choices (A),(B), (C) and (D) out of which ONLY ONE is correct. (+3, –1)

1 If the sum of the roots of the quadratic equation 2ax bx c 0, abc 0 is equal to sum

of the squares of their reciprocals, then a b c, ,c a b

, are in

(A) arithmetic progression (A.P.) (B) geometric progression (G.P.)(C) harmonic progression (H.P.) (D) none of these

2 If three unequal positive real numbers a,b,c are in G.P. and b c,c a,a b are in H.P. thenthe value of a b c is independent of(A) a (B) b (C) c (D) none of these

3 Suppose a, b 0 and 1 2 3x , x , x are three roots of x a x b b ab a x a x b

such that

1 2 3x x x and 1 2 3x x x c. Then a, b, c are in(A) A.P. (B) G.P. (C) H.P. (D) none of these

4 Four dice are rolled. The number of possible outcomes in which at least one die shows 2 is(A) 1296 (C) 625(C) 671 (D) none of these

5 The number of signals that can be generated by using 6 differently coloured flags, when anynumber of them may be hoisted at a time is(A) 1956 (B) 1957(C) 1958 (D) 1959

6 If mx occurs in the expansion of 2n

2

1xx

, then the coefficient of mx is

(A)

2n !m! 2n m !

(B) 2n !3!3!2n m !

(C)

2n !2n m 4n m

! !3 3

(D) none of these

7 The angle at which the curve Kxy Ke intersects the y-axis is :

(A) 1 2tan k (B) 1 2cot k (C) 1 4sec 1 k (D) none

8 If a variable tangent to the curve 2 3x y c makes intercepts a, b on x and y axis respectively, then

the value of 2a b is

(A) 327c (B) 3

4 c27

(C) 3

4 c27

(D) 3

4 c9

SECTION III : MATHEMATICS


PART II : Multiple Correct Answer(s) Type

This section contains 4 multiple choice questions. Each question has four choices (A),(B), (C) and (D) out of which ONE or MORE are correct. (+5, –2)

9 The real value of for which the expression, 1 icos1 2i cos

is a real number is

(A) 2n2

(B) 2n2

(C) 2n2

(D) none of these

10 If cos cos cos sin sin sin 0 , then

(A) cos 2 cos 2 cos 2 0

(B) sin 2 sin 2 sin 2 0

(C) cos cos cos 0

(D) sin sin sin 0

11 If n objects are arranged in a row, then the number of ways of selecting three of theseobjects so that no two of them are next to each other is

(A) n 2 n 3 n 46

(B) n 23C

(C) n 3 n 33 2C C (D) none of these

12 If the third term in the expansion of 105log xx x is 610 , then x can be

(A) 1310

(B) 10 (C) 5210

(D) 210

PART III : Paragraph Type

This section contains 3 multiple choice questions relating to three paragraphs with twoquestions on each paragraph. Each question has four choices (A), (B), (C) and (D) out ofwhich ONLY ONE is correct. (+4, –1)

In a sequence of 4n 1 terms the first 2n 1 terms are in A.P. whose common differ

ence is 2 and the last 2n 1 terms are in G.P. whose common ratio is 0.5. If the middleterms of the AP and GP are equal then

13 Middle term of the sequence is

(A) n 1

n

n 24 1

(B) nn 2 (C) n 1

n

n 22 1

(D) none of these

14 First term of the sequence is

(A) n

n

4n 2n 22 1

(B) n

n

2n n 22 1

(C) n

n

4n 2n 22 1

(D) n

n

2n n 22 1


15 Middle term of the G.P. is

(A) n

n

22 1

(B) n

n2 1 (C)

n

n

n 22 1

(D) n

2n2 1

PART IV : Integer Answer Type

This section contains 4 questions. The answer to each question is a single digit integer,ranging from 0 to 9 (both inclusive). (4, 0)

16 If roots of the equation 2x 10cx 11d 0 are a, b and these of 2x 10ax 11b 0 are c, d then

find the value of a b c d

605

(a, b, c and d are distinct numbers)

17 If , are two distinct real roots of the equation 3ax x 1 a 0 , 1a 1, 0 , none of which

is equal to unity, then the value of

3 2

1 x1x

1 x x alim

e 1 x 1

is a k

.

Find the value of kl.

18 If the variable line 3x 4y k 0 lies between the circles 2 2x y 2x 2y 1 0 and2 2x y 16x 2y 61 0 without intersecting or touching either circle, then the range of k is

(a, b) where a, b I. Find the value of b a .

19 If 2f x ln x x 2 ; R R and

g x x 1 ; 1, 2 1, 2 , where {x} denotes fractional part of x.

If the domain and range of f g x are a, b and c, d respectively a b, c d , then find the

value of b da c



This section contains 1 question. Each question contains statements given in two columns, which have tobe matched. The statements in Column-I are labelled A, B, C and D, while the statements in Column -II are labelled p, q, r, s and t. Any given statement in Column-I can have correct matching with ONE ORMORE statement(s) in Column- II. The appropriate bubbles corresponding to the answers to thesequestions have to be darkened as illustrated in the following example: If the correct matches are A - p ,s and t ; B - q and r ; C - p and q ; and D - s and t. (+8, 0)

20 Column I Column II

(A) The period of sin xe sin x3

where {x} denotes the fractional (P)2

part of x

(B) The period of 6 6 2sin x cos x cot x sin 3x (Q) 1

(C) The period f x min tan2x, cot4x (R) 6

(D) The period of 3 x2 sin x where {} denotes the (S) 2fractional part and [ ] denotes the integral part

PAPER - II (SOLUTION)1

Sol 22 2

2 2 2 2 2 2

21 1a

2

2 2 2 22

2 2 2 2

2

22 2

b cb b ac a b b ab bca a

ca c c c a aca

2 2 2 22 a b ca c ab bc

b c a [ dividing by abc ]

, ,c a b

a b c are in A.P.. , ,a b c

c a b are in H.P..

2Sol As , ,a b c are in G.P., 2b ac and , ,b c c a a b are in H.P..

2 1 1 a c

c a b c a b b c a b

22 b c a b a c

2 222 ab ac b bc a c a c

2 222 2ab b bc a c a c

2 2 22 0b a c a c a c a c

2 2 2b a c ac a c b

3 3a c b ac which is not independent of ,a b and .c

3Sol The given equation can be writtern as

x a b a x b

b x a x b a

2 22 2x a b a x b

b x a x b a

x a b x a b x b a x b a

b x a x b a

0x a b a x b x a b b x b a x a

2 2 2 2 0x a b a x b ax ab b x a bx ab

2 2 2 0x a b a b x a b x

2 2 0x x a b a b x a b

0,x x a b or 2 2a b

xa b

Since 2 2 2 since , 0b ab

a b a b a ba b a b

We take 2 2

1 2 3, & 0.a bx a b x x

a b

since 1 2 3x x x c we get2 2a b

a b ca b

2 2 2 2a b a b abc c

a b a b

, ,a b c are in H.P..

4Sol The total number of possible outcomes is 46 . The nfumber of possible outcomes in which

does not appear on any die is 45 , so that tens digit of 10! 11! ... 49! is zero. Therefore,the number of possible outcomes in which at least one die shows a 2 is

4 46 5 1296 625 671.

5Sol When one flag is used, the number of signals that can be generated is 6

1P . When two flags

are used, the number of signals that can be generated is 62P . When three flages are used, the

number of signals that can be generated is 63P , and so on. Hence, the number of different

signals that can be generated is6 6 6 6 6 6

1 2 3 4 5 6 6 30 120 360 720 720 1956P P P P P P .

6Sol The general term in the expansion of the given expression is

2 2 2 2 31 2

1 .r

n n r n n rr r rT C x C x

x

Putting 12 3 ,i.e., 2 ,3

n r m r n m the coefficient of mx is

2

1 23

2 ! 2 !1 1 1 12 2 ! 2 ! 4 ! 2 !3 3 3 3

n

n m

n nC

n n m n m n m n m

7

[Hint:dydx x

0

= k2 tan = k2 cot

2

= k2

2

= cot1 k2 = sin1

1

1 4 k B ]

8[ Hint : x2y = c3

x2dxdy

+ 2xy = 0 dxdy

= xy2

equation of tangent at (x,y)

Y – y = )xX(xy2

Y = 0, gives , X = 2x3

= a

and X = 0 , gives , Y = 3y = b

Now a2b = y3.4x9 2

= 32 c427yx

427

(C) ]

9Sol We have

2

4

1 2cos 3cos1 cos 1 2 cos1 cos1 2 cos 1 2 cos 1 2 cos 1 4cos

ii ii

i i i

Thus

1 cos1 2 cos

i

i

is a real number if cos 0

22

n , where n is an integer..

10Sol Let

cos sin , cos sin , & cos sina i b i c i We have

cos cos cos sin sin sin 0 0 0a b c i i

Also 1 1 11 1 1a b c

a b c

cos cos cos sin sin sini

[ using De Moiver’s theorem ]0 0 0i

0bc ca ab

cos sin cos sin cos sin 0i i i

cos cos cos 0

and sin sin sin 0

Also 20 0a b c a b c

2 2 2 2 0a b c ab ca ab

2 2 2 0 0a b c bc ca ab

cos 2 sin 2 cos 2 sin 2 cos 2 sin 2 0i i i [ using De Moiver’s theorem ]

cos 2 cos 2 cos 2 0

and sin 2 sin 2 sin 2 0 .

11Sol Let 0x be the number of objects to the left of the first object chosen, 1x the number of

objects between the first and the second, 2x the number of objects between the second

and the third and 3x the number of objects to the right of the third object. We have

0 3 1 2, 0, 1x x x x and 0 1 2 3 3x x x x n ...(1)The number of solution of (1)= coefficient of 3ny in

2 2 2 3 2 31 ... 1 ... ... ...y y y y y y y y y y

= coefficient of 3ny in 42 2 31 ...y y y y

= coefficient of 5ny in 41 y

= coefficient of 5ny in 4 5 2 6 31 2 31 ...C y C y C y

5 3 25 3

2 3 46

n nn

n n nC C

Also 3 3 23 2 3

n n nC C C

12Sol The third term in the expansion of the given expression is

10 10 1025 5 2 3

3 2log log 3 2log10 10x x xT C x x x x x

Since 63 10 ,T we therefore get

10 106 53 2log 3 2log10 10 10x xx x

Taking logarithms and putting 10log ,y x we get

510 10 10 103 2 log log log 10 5log 10 5x x

2 22 3 5 0 2 2 5 5 0y y y y y

2 1 5 1 0 2 5 1 0.y y y y y

That is, 52

y or 1, so that 5210x

or 10.

Sol 13 to 15Let 2n 1t x

then, n 1 3n 1t t

n

1x 2n x.

2

n 1

n

n 2x

2 1

first term x 2 2n

n 1 n n n

n n n

n 2 2n 2 4n 2 4n 4n 2n 24n

2 1 2 1 2 1

3n 1 n n

1 2nt x

2 2 1

16 [Ans. 1210][Sol. As a + b = 10c and c + d = 10a

ab = – 11d, cd = – 11b ac = 121 and (b + d) = 9(a + c)

a2 – 10ac – 11d = 0c2 – 10ac – 11b = 0

a2 + c2 – 20ac – 11(b + d) = 0

(a + c)2 – 22(121) – 11 × 9(a + c) = 0 (a + c) = 121 or – 22 (rejected) a + b + c + d = 1210 Ans. ]

17 Ans. 1Sol. Roots of equation ax3 + x – 1 – a = 0 are 1, ,

= – 1 and = aa1

1xlim

)1x)(1e(axx)a1(

x1

23

=

1xlim

)1x)(1e(]aaxx)a1)[(1x(

x1

2

=

1xlim

)1e(aaxxa

x1

2

=

1xlim

x1

xxx1(a 2

=

1xlim

x1)x1)(x1(a

=

1xlim

a(1 – x) =

)(a

k = 1 and = 1 and so k = 1

18 [Ans. 6]

19 [Ans. 4]Sol. f(x) = n (x2 – x + 2) : R+ R

g(x) = {x} + 1 [1, 2] [1, 2]Domain of f(g(x)) = {x : x [1, 2] and {x} + 1 (0, )]

= [x : 1 x 2 and – 1 < {x}]= [1, 2] a = 1, b = 2

x [1, 2] {x} + 1 [1, 2) 1 g(x) < 2

g(x)2 – g(x) + 2 = g(x)2 – g(x) + 41 +

47 =

2

21)x(g

+

47

2 (g(x))2 – g(x) + 2 < 4 range of f(g(x)) = [n2, n4) c = n 2, d = n 4

ab

+ cd

= 2 + 2 = 4

20Sol A R, B S,C P, D Q

(A) The period of sin{x}e is 1 as the period of x x x is 1 and the period of sin x3

is 2 6

3

period of the given function is LCM of 1 and 6 6.

(B) The period of 6 6sin x cos x is 2 , period of 2cot x is and the period of sin 3x is 2

3

Hence 2 , LCM of all the three is the period of the gieven function.

(C) The period of tan 2x is 2

and that of cot 4x is 4

. the required is 2

(D) sin x 0 period of 3 {x}2 sin x is period of x 1.


ANDHERI / BORIVALI / DADAR / CHEMBUR / THANE / MULUND/ NERUL / POWAI FULL TEST – 2 (ADVANCED PATTERN)

PAPER – I (ANSWER KEY)

PHYSICS 1. A 2. A 3. C OR D 4. B 5. C 6. A 7. A 8. D 9. A, B, C, D 10. B, C, D 11. A, B, D 12. A, B, C, D 13. B 14. A 15. C 16. (9) 17. (16) 18. (1) 19. (24) 20. a S ; b P, R ;c R ;d Q,S

CHEMISTRY

1. (A) 2. (D) 3. (A) 4. (C) 5. (B) 6. (C) 7. (D) 8. (B) 9. (A,B,D) 10. (A,B,C) 11. (A,C,D) 12. (A, B, C, D) 13. (D) 14. (B) 15. (B) 16. (6) 17. (4) 18. (1) 19. (3) 20. (A – Q, S), (B – Q, S), (C – P, R, T), (D –S)

MATHS

1. D 2. A 3. A 4. A 5. C 6. D 7. A,C 8. A,D 9. A,B,C,D 10. B,C 11. A 12. B 13. A 14. B 15. 14 16. 3 17. 3 18. 6 19. 0 20. A Q;B P,Q;C R; D P


PAPER – II

PHYSICS 1. D 2. A 3. C 4. C 5. D 6. C 7. B 8. D 9. A, C 10. B, C 11. A, B, C, D 12. B, C 13. A 14. A 15. A 16. (1) 17. (2) 18. (8) 19. (2) 20. a P ;b P,Q,S ;c Q ;d Q,R

CHEMISTRY

1. (B) 2. (C) 3. (B) 4. (A) 5. (C) 6. (B) 7. (B) 8. (B) 9. (A,B,C,D) 10. (B, C, D) 11. (A, C, D) 12. (A, D) 13. (C) 14. (D) 15. (A) 16. (1) 17. (1) 18. (1) 19. (4) 20. (A) – (R), (S), (B) – (R), (C) – (P), (Q), (D) – (R), (S)

MATHS

1. C 2. D 3. C 4. C 5. A 6. C 7. B 8. C 9. A,B,C 10. A,B,C,D 11. A,B,C 12. B,C 13. C 14. C 15. D 16. 1210 17. 1 18. 6 19. 4 20. A R;B S;C P,S;D Q, R

testpaper2 (3)

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