mathematics problem sheet trigonometry – iv - jrs tutorials

JRS TUTORIALS Durgakund Varanasi 221005 Ph. No. 2311922, 2311777, 3290510

JRS TUTORIALS Mathematics Problem Sheet

Trigonometry – IV

1. Prove that, 28

7cos8

5cos8

3cos8

cos 2222

2. Prove that, 23

87sin

85sin

83sin

8sin 4444

3a. If ,2

3,43tan

xx find the value of 2

cosand2

sin xx

3b. If ,2

,41sin

xx find the values of

2tanand

2cos xx

3c. If ,2

3,31cos

xx find the values of 2

tanand2

sin xx

4. Prove that 2

tan1tan1

2sin12sin1

AA

AA 5. Show that 4

10cos3

10sin1

6. Prove that cosec A – 2cot 2A cos A = 2 sinA

7. Prove that AecAAA 2cos2cot4tancot 22

8. Prove that

A

AAAA

AA

4tan

sincossincos

2cos2sin1

9. Prove that

AAAA 2sin

4112cossincos 266

10. Prove that 23

3cos

3coscos 222

11. If ,15

prove that cos 2 cos 4 cos 8 cos 14 = 161

12. Prove that xxxxx 4sincos46sin4sin22sin 2

13. Prove that 12 2cos...2cos2coscos2

sin2sin nn

n

14. Show that 13)cos(sin4)cos(sin6)cos(sin3 6624 xxxxxx

15. Show that 01)cos(sin3)cos(sin2 4466 xxxx

16. Show that )(coscoscos2)(coscos 22 is independent of .


PS/Trig–IV/Page No. - 2

17. Prove that )20sin10(cos3)20sin10(cos4 33

18. Prove that 4sin)sincoscos(sin4 33

19. Prove that 4sin3)3cossin3sin(cos4 33

20. Prove that 3tan3)120tan()60tan(tan

21. Prove that 4 sin

3sin3

2sin3

sin

22. Prove that cot + cot (60 + ) + cot (120 + ) = 3 cot 3

23. Prove that )sin(cos)sin(cos)cos(cos)cos(cos84cos4cos

24. Prove that

42

3

tantan61tan4tan44tan

25. If ,135cosand

54sin prove that one value of

658

2cos

26. If ,tan21tan 22 prove that cos 2 = 1 + 2 cos 2

27. If and are acute angles and

2cos3

12cos32cos prove that tan = 2 tan

28. Prove that (a) 8

1512cos48sin 22 (b) 4 (sin24 + cos 6) = 3 + 15

29. Prove that (a) cot 6 cot 42 cot 66 cot 78 = 1 (b) tan12tan24tan48tan84 = 1

30. Prove that sin 165

54sin

53sin

52sin

5

31. If

cos1 cos costhatProve,

2tan

11

2tan

ee

ee

32. If ,41sinsinand

31coscos prove that

245

2cos

33. If ,2

tan2

tan2

prove that

cos35cos53cos

34. If cos = ,coscos1coscos

prove that one of the values of .

2tan

2tanis

2tan

35. If ,90 show that the maximum value of 21iscoscos .

36. Prove that tan tan (60 – ) tan (60 + ) = tan 3


PS/Trig–IV/Page No. - 3

37. If ,3

1sin,2

1cos show that 625or6252

cot2

tan

38. If and be two different roots of the equation a cos + b sin = c then prove that

(i) 푡푎푛 = (ii) 푐표푠(훼 + 훽) = (iii) 푡푎푛(훼 + 훽) =

39. If + = then show that + =( )

40. If 푐표푠(훼 + 훽) = and 푠푖푛(훼 − 훽) = , where , lie between 0 and 4 then prove

that tan(2훼) =

41. Prove that tan 20 tan 80 = 50tan3 .

42.

43.

44. (a) Prove that the value of 33

cos3cos5

lies between - 4 and 10.

(b) Find the maximum and the minimum values of 7 cos + 24sin .

(c) If ,4

cos4

sin1)(

f find the range of values of 푓(휃).

45. If a right angle be divided into three parts , , prove that

tantantantan1tan

46. If + = 60 prove that 43coscoscoscos 22

47. prove aaaaaa

aaecaa 22

2222

2222 sincos2cossin1cossin

sincos1

cossec1

48. prove )sec(coscottan2)sec(cot)cos(tan 22 BAecBAABBecA

49. Prove that : to n terms

= .

50. Show that

Answers: 3(a) sin =√

, cos = −√

3(b) cos = √ , tan = √√

3(c) sin = , tan = −√2

44(b) -25, 25 44(c) [-1, 3]

3 cos 4 cos2 4cos8 sin8.

1 cos2x cos 4x cos 6x 4 cos x cos 2x cos 3x

tan tan 2 tan2 tan3 tan3 tan4 ...

n 1 cot tann 1

tan9 . tan27 . tan45 . tan 63 tan81 1

PHYSICS

JRS Tutorials, Durgakund, Varanasi-221005, Ph No.(0542) 2311922, 2311777

JRS TUTORIALSVECTOR (11th IIT)

DPP-01

1. A physical quantity which has a direction :

(A) cannot be a vector (B) must be a vector (C) must be a scalar (D) may be a vector

2. The magnitude of a vector cannot be :

(A) positive (B) unity (C) negative (D) zero

3. The forces, each numerically equal to 5 N, are acting as shown in the Figure. Find the anglebetween forces?(A) 90°

(B) 120°

(C) 0°

(D) 60°

4. For the figure shown.

(A) A B C

(B) B C A

(C) C A B

(D) A B C 0

5. Six vectors, a through f

have the mangitudes and directions indicated in the figure.Which of

the following statements is true ?

`

(A) b + c = f

(B) d + c = f

(C) d + e = f

(D) b + e = f

6. A man moves 10 m in a direction 30° East of North.The displacement of man is [assuming east

as positive X axis and north as positive Y axis]

(A) ˆ ˆs 5 3 i 5 j m

(B) ˆ ˆs 5i 5 3 j m

(C) ˆ ˆs 5 3i 5 j m

(D) ˆ ˆs 5 i 5 3 j m

PHYSICS


7. A person pushes a box kept on a horizontal surface with force of 100 N. In unit vector notationforce F

can be expressed as :

(A) 100 ˆ ˆ(i j)

(B) 100 ˆ ˆ(i j)

45°

y

xF

(C) 250 ˆ ˆ(i j)

(D) 50 2 ˆ ˆ(i j)

8. Just after firing, a bullet is found to move at an angle of 37° to horizontal. Its acceleration is 10m/s2 downwards. Find the component of acceleration in the direction of the velocity.

(A) – 6 m/s2

(B) – 4 m/s2

(C) – 8 m/s2

(D) – 5 m/s2

9. A particle moves along a path ABCD as shown in the figure. Then the magnitude of netdisplacement of the particle from position A to D is :(A) 10 m

(B) 25 m(C) 9 m(D) 27 m

10. Find the resultant of three vectors OA

, OB

and OC

and each of magnitude R

(A) 2R

(B) R(1 + 2 )

(D) R 2

(D) R( 2 – 1)

11. Find resultant force as vector in ˆ ˆai bj format.

(A) ˆ ˆ225 i 300 j

(B) ˆ ˆ475 i 900 j

(C) ˆ ˆ475 i 600 j

(D) ˆ ˆ475 i 900 j12. An insect crawls 10 m towards east, turns to its right, crawls 8 m, and again turns to its right,

Now crawling a distance of 2 m it turns to its right and stop after moving 2 m more. Find its netdisplacement.(A) 10 m , 37° S of E (B) 10 m , 37° E of S(C) 5 m , 37° S of E (D) 5 m , 37° W of N

PHYSICS


13. A sail boat sails 2 km due East, 5 km 37o South of East and finally an unknown displacement .If the final displacement of the boat from the starting point is 6 km due East, the third displace-ment is ______(A) 3 km North (B) 3 km South (C) 4 km South (D) 4 km East

14. Two horizontal forces of magnitudes 10 N & P N act on a particle. The force of magnitude 10 Nacts due west & the force of magnitude P N acts on a bearing of 30° east of north as shown infigure. The resultant of these two force acts due north. Find the magnitude of this resultant.(A) 310 N

(B) 20 N(C)10 N(D) 10 / 3 N

15. As shown in figure, find the magnitude of the unknown forces X and Y if sum of all forces iszero.(A) 5N, 5N(B) 5N, 10N(C) 10N, 5N(D) 10N, 10N

16. Three forces acting on a body are shown in the figure. To have the resultant force only along the

y-direction, the magnitude of the minimum additional force needed is:

(A) 0.5N (B) 1.5N (C) 3 N

4(D) 3N

INTEGER TYPE QUESTIONS

17. The rectangular components of a vector are (2, 2). The corresponding rectangular components

of another vector are (1, 3 ). Find one third of the angle between the two vectors in degree.

18. A vector B

which has a magnitude 8.0 is added to a vector A

which lie along the x-axis. Thesum of these two vectors is a third vector which lie along the y-axis and has a magnitude that is

twice the magnitude of A

. The magnitude of A

is x5 then find x .

PHYSICS


MATCH THE COLUMN19. Column-I show vector diagram relating three vectors a, b

and c . Match the vector equation in

column-II, with vector diagram in column-I

Column-I Column-II

(I)a

bc(P) 0)cb(a

(II)a

cb (Q) acb

(III)a b

c(R) cba

(IV)b

ca (S) cba

PASSAGEThree forces of 3N, 2N and 1N act on a particle as shown in the figure. Calculate the

20. Net force along the x-axis.

(A) 6 N (B) 3 N (C) 32 N (D)

32

N

21. Net force along the y-axis.

(A) 6 N (B) 3 N (C) 32 N (D)

32

N

22. Single additional force required to keep the body in equilibrium.

(A)3 3ˆ ˆF i j

2 2

N (B)3 3ˆ ˆF i j

2 2

N

(C)3 3ˆ ˆF i j2 2

N (D)3 3ˆ ˆF i j2 2

N

PHYSICS


ANSWER KEY

VECTOR DPP-01

1. D 2. C 3. B 4. C 5. C 6. D

7. C 8. A 9. D 10. B 11. D 12. A

13. A 14. A 15. B 16. A 17. 5 18. 8

19. (I)-R ; (II)-S ; (III)-P ; (IV)-Q 20. C 21. D 22. A


PHYSICS

JRS TUTORIALSELECTROSTATICS (12-IIT)

DPP-04 : Coulomb’s Law

1. An infinite number of charges, each of charge 1 C, are placed on the x-axis with co-ordinates

x = 1, 2, 4, 8, .... . If 1 C is kept at the origin, then what is the net force acting on 1 C charge

(A) 9000 N (B) 12000 N (C) 24000 N (D) 36000 N2. Given are four arrangements of three fixed electric charges. In each arrangement, a point labeled

P is also identified — test charge, +q, is placed at point P. All of the charges are the samemagnitude, Q, but they can be either positive or negative as indicated. The charges and point Pall lie on a straight line. The distances between adjacent items, either between two charges orbetween a charge and point P, are all the same.

I. II.

III. IV.

Correct order of choices in a decreasing order of magnitude of force on P is(A) II > I > III > IV (B) I > II > III > IV (C) II > I > IV > III (D) III > IV > I > II

3. Four positive point charges are arranged as shown in the accompanying diagram. The forcebetween charges 1 and 3 is 6.0 N; the force between charges 2 and 3 is 5.0 N; and the forcebetween charges 3 and 4 is 3.0 N. The magnitude of the total force on charge 3 is most nearly

(A) 6.3 N

(B) 8.0 N

2 3 4

1

(C) 10 N

(D) 11 N4. A point charge +Q is placed at the centroid of an equilateral triangle. When a second charge +Q

is placed at a vertex of the triangle, the magnitude of the electrostatic force on the central charge

is 4N. What is the magnitude of the net force on the central charge when a third charge +Q is

placed at another vertex of the triangle?

(A) zero (B) 4 N (C) 4 3N (D) 8 N5. Three identical spheres each having a charge q and radius R, are kept in such a way that

each touches the other two. The magnitude of the net electric force on any sphere is

(A) 2

0

314 4

qR

(B) 2

0

212 4

qR

(C) 2

0

214 4

qR

(D) 2

0

312 4

qR


PHYSICS6. A charge Q is placed at each of the opposite corners of a square. A charge q is placed at

each of the other two corners. If the net electrical force on Q is zero, then Q/q equals

(A) – 22 (B) – 1 (C) 22 (D) –21 [AIEEE-2009]

7. Given figure shows an arrangement of six charged particles. The net electrostatic force F acting

on charge +q at the origin due to other charges is

(A) 20

2

a4q6

(B) zero (C) 20

2

a2q7

(D)

3

23

a4q

20

2

8. The magnitude of electric force on 2 C charge placed at the centre O of two equilateraltriangles each of side 10 cm, as shown in figure is P. If charge A, B, C, D, E & F are 2 C,2 C, 2 C, -2 C, - 2C, - 2 C respectively, then P is:(A) 21.6 N

(B) 64.8 N OA

B

C

D

E

F

(C) 0

(D) 43.2 N9. Through the exact centre of a hydrogen molecule, an -particle passes rapidly, moving on

a line perpendicular to the internuclear axis. The distance between the two hydrogen nucleiis b. The maximum force experienced by the -particle is

(A) 2

20

43 3

eb (B)

2

20

83

eb (C)

2

20

83 3

eb (D)

2

20

43

eb

10. Which of the following graphs best represents the force acting on a charged particle kept at

distance x from the centre of a square and on the axis of the square whose corners haveequal charges.

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


PHYSICS11. Three charges +Q1, +Q2 and q are placed on a straight line such that q is somewhere in between

+Q1 and +Q2. If this system of charges is in equilibrium, what should be the magnitude and sign

of charge q ?

(A) ve,)QQ(

QQ2

21

21 (B) ve,

2QQ 21

(C) ve,)QQ(

QQ2

21

21 (D) ve,

2QQ 21

12. Three charges + 4q, -q and +4q are kept on a straight line at position (0, 0, 0), (a, 0, 0) and(2a, 0, 0) respectively. Considering that they are free to move along the x-axis only(A) all the charges are in stable equilibrium(B) all the charges are in unstable equilibrium(C) only the middle charge is in stable equilibrium(D) only middle charge is in unstable equilibrium

13. A mass particle (mass = m and charge = q) is placed bewteen two point charges of charge

q separtion between these two charge is 2L. The frequency of oscillation of mass particle,

if it is displaced for a small distance along the line joining the charges–

(A) 30Lm

12q

(B) 30Lm

42q

(C) 30Lm4

12q

(D)2q

30mL161

14. Two positive point charges are fixed at some distance apart. A third negative charge ' q ' is placedat the centre of the line joining the two charges. Then the number of lines along which ' q ' canperform SHM for small displacements is(A) 1 (B) 2 (C) 3 (D)

15. Four point positive charges are held at the vertices of a square in a horizontal plane. Theirmasses are 1 kg, 2 kg, 3 kg & 4 kg. Another point positive charge of mass 10 kg is kept onthe axis of the square. The weight of this fifth charge is balanced by the electrostatic forcedue to those four charges. If the four charges on the vertices are released such that they canfreely move in any direction (vertical, horizontal etc) then the acceleration of the centre ofmass of the four charges immediately after the release is: (Use g = 10 m/s2)

(A) 10 m/s2 (B) 20 m/s2 (C) zero (D) 10 m/s2 16. Under the influence of the Coulomb field of charge +Q, a charge –q is moving around it in an

elliptical orbit.Find out the correct statement(s). [JEE -2009](A) The angular momentum of the charge –q is constant.(B) The linear momentum of the charge –q is constant.(C) The angular velocity of the charge – q is constant.(D) The linear speed of the charge –q is constant.


PHYSICS17. A ring of radius R is made out of a thin metallic wire of area of cross section A. The ring has a

uniform charge Q distributed on it. A charge q0 is placed at the centre of the ring. If Y is theyoung’s modulus for the material of the ring and R is the change in the radius of the ring, then

(A) R = RAY4Qq

0

0

(B) RAY4QqR

0

0

(C) RAY8

QqR 20

0

(D) RAY8

QqR0

20

18. Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap

film of side ‘ ’. The surface tension of the soap film is . The system of charges and planar

film are in equilibrium, and = k1/N2q

, where ’k’ is a constant. Then N is

(A) 3 (B) 2 (C) 4 (D) 6 [JEE-2011]

MULTIPLE CORRECT ANSWERS19. Three charged particles are in equilibrium under their electrostatic forces only

(A) The particles must be collinear.(B) All the charges cannot have the same magnitude.(C) All the charges cannot have the same sign.(D) The equilibrium is unstable.

20. Two equal negative charges –q each are fixed at the points (0, a) and (0, -a) on the y-axis .Apositive charge Q is released from rest at the point (2a, 0) on the x-axis. The charge Q will:(A) At origin velocity of particle is maximum.(B) Execute simple harmonic motion about the origin(C) Move to infinity(D) Execute oscillatory but not simple harmonic motion.

21. A negative point charge placed at the point A is

(A) in stable equilibrium along x-axis

(B) in unstable equilibrium along y-axis

(C) in stable equilibrium along y-axis

(D) in unstable equilibrium along x-axis22. Two identical charges +Q are kept fixed some distance apart. A small particle P with charge q is

placed midway between them. If P is given a small displacement , it will undergo simpleharmonic motion if(A) q is positive and is along the line joining the charges.(B) q is positive and is perpendicular to the line joining the charges.(C) q is negative and is perpendicular to the line joining the charges.(D) q is negative and is along the line joining the charges.


PHYSICSParagraph

Three charged particles each of +Q are fixed at the corners of anequilateral triangle of side ‘a’. A fourth particle of charge –q andmass m is placed at a point on the line passing through centroid of triangle and perpendicular to the plane of triangle at a distance xfrom the centre of triangle

23. Force on the fourth particle is

(A) 2/3220 )ax3(

Qx394

1 (B) 2/322

0 )ax3(Qx33

41

(C) 2/3220 )ax2(

Qx224

1 (D) 2/322

0 )ax2(Qx24

41

24. Value of x for which the force is maximum is

(A) 3a

(B) 2a

(C) 6a

(D) 5a

MATCH THE COLUMN25. Four charge Q1,Q2,Q3, and Q4,of same magnitude are fixed along the x axis at x = –2a –a, +a

and +2a, respectively. A positive charge q is placed on the positive y axis at a distance b > 0.Four options of the signs of these charges are given in List-I . The direction of the forces on thecharge q is given in List- II Match List-1 with List-II and select the correct answer using thecode given below the lists [JEE (ADV)-2014]

List-I List-IIP. Q1,Q2,Q3, Q4, all positive 1. +xQ. Q1,Q2 positive Q3,Q4 negative 2. –xR. Q1,Q4 positive Q2, Q3 negative 3. +yS. Q1,Q3 positive Q2, Q4 negative 4. –yCode :(A) P-3, Q-1, R-4,S-2 (B) P-4, Q-2, R-3, S-1(C) P-3, Q-1, R-2,S-4 (D) P-4, Q-2, R-1, S-3


PHYSICS26. In the following Fig. charges, each +q, are fixed at L and M. O is the mid point of distance LM.

X- and Y –axes are as shown. Consider the situations given in column I and match them with

the information in column II

Column I Column –II(A) Let us place a charge +q at O, displace it (P) force on the charge is zero slightly along X-axis and release. Assume that it is allowed to move only along X-axis. At position O,(B) Place a charge –q at O, Displace it slightly (Q) potential energy of the system along X-axis and release. Assume that it is maximum is allowed to move only along X-axis. At position O,(C) Place a charge +q at O. Displace it slightly (R) potential energy of the system is along Y-axis and release. Assume that it is minimum allowed to move only along Y-axis. At position O,(D) Place a charge –q at O. Displace it slightly (S) the charge is in equilibrium along Y-axis and release. Assume that it is allowed to move only along Y-axis. At position O,

ANSWER KEYELECTROSTATICS DPP-04

1. B 2. C 3. A 4. B 5. A 6. A

7. B 8. D 9. C 10. C 11. C 12. B

13. A 14. D 15. B 16. A 17. D 18. A

19. A,B,CD 20. A,D 21. C,D 22. A,C 23. A

24. C 25. A 26. (A- P, R, S) ( B – P, Q, S) (C –P, Q, S) (D – P, R, S)


PHYSICS



1. Two identical conducting spheres (of negligible radius), having charges of opposite sign,attract each other with a force of 0.108 N when separated by 0.5 meter. The spheres areconnected by a conducting wire, which is then removed (when charge stops flowing), andthereafter repel each other with a force of 0.036 N keeping the distance same. What werethe initial charges on the spheres?[Ans. ± 1.0 x 10-6 C, 3 x 10-6 C ]

2. Two identical balls of mass m = 0.9 g each are charged by the same charges, joined by a threadand suspended from the ceiling (Figure). What is the charge (in µC) should both balls have sothat the tension in both the threads is the same? The distance between the centers of balls R = 3m.

R

[Ans. 3]

3. Two spherical bobs of mass m each & identically charged with charge q each are suspendedon strings of length L each. They are suspended from a point O where there is a third chargeq. Find q, if the angle between the strings in equilibrium position is Can you find thevalue of charge at O, if it is not q but q1. If yes, find? If no, explain the physical significanceof it.

[Ans. q =2

tanmg2

sin4 0

]

4. Two small equally charged identical conducting balls are suspended from long threadssecured at one point. The charges and masses of the balls are such that they are in equilibrium.When the distance between them is a = 10 cm (the length of the threads L >> a). One of theballs is discharged. How will the balls behave after this? What will be the distance b betweenthe balls when equilibrium is restored?[Ans. The balls will first go down, touch each other and then move apart by a distance

b = a

( ) /4 1 3]


PHYSICS5. Two positive point charges each of magnitude 10 C are fixed at positions A & B at a separation

2 d = 6 m. A negatively charged particle of mass m = 90 gm & charge of magnitude 10 106 C is revolving in a circular path of radius 4 m in the plane perpendicular to the line ABand bisecting the line AB. Neglect the effect of gravity. Find the angular velocity of theparticle. If gravity is also considered will it still move in the circular path assuming AB tobe horizontal.

[ Ans : 400 rad/s ]

6. Two small identical balls having the same mass and charge are located in the same verticalline at heights h1 and h2 from ground are thrown in the same direction along the horizontalat the same velocity v. The first ball touches the ground at a distance from the initialvertical line. At what height H2 will the second ball be at this instant? Neglect the effect ofair friction on motion of the balls.

[Ans2

212 vghhH

]

7. An electrometer consists of a fixed vertical metal bar OB at the top of which is attached a thinrod OA which gets deflected from the bar under the action of an electric charge (fig.). The rodcan rotate in vertical plane about fixed horizontal axis passing through O. The reading is takenon a quadrant graduated in degrees . The length of the rod is and its mass is m . What will bethe charge when the rod of such an electrometer is deflected through an angle in equilibrium.Find the answer using the following two assumptions:(i) the charge on the electrometer is equally distributed between the bar & the rod(ii) the charges are concentrated at point A on the rod & at point B on the bar.

[ Ans : q = 4

2

sinmg4 0 sin 2 ]


PHYSICS8. A rigid insulated wire frame in the form of a right angled triangle ABC, is set in a vertical

plane as shown. Two bead of equal masses m each and carrying charges q1 & q2 are connectedby a cord of length & slide without friction on the wires . Considering the case when thebeads are stationary, determine.

(a) The angle .(b) The tension in the cord &(c) The normal reaction on the beads .(d) If the cord is now cut, what is the product of the charges for which the beads continue to remain stationary .

[Ans. (a) 60º (b) mg + 2

21qqk

(c) 3 mg , mg (d) q1 & q2 should have unlike charges

for the beads to remain stationary & q1q2 = k

mg 2 ]

9. The distance between two fixed positive charges 4e and e is . How should a third charge‘q’ be arranged for it to be in equilibrium? Under what condition will equilibrium of thecharge ‘q’ be stable (for displacement on the line joining 4e and e) or will it be unstable?

[Ans. 23 from charge 4 e ( If q is positive stable, If q is negative unstable)]

10. Two equally charged particles A and B, each having a charge Q are placed a distance dapart. Where should a third particle of charge q be placed on the perpendicular bisector ofAB so that it experiences maximum force? Also find the magnitude of the maximum force.

[Ans. d

2 2, 2

0d33Qq4πε

]

11. Four charges q1 = 1 c, q2 = 2 c, q3 = 3 c and q4 = 4 c are placed at (0, 0, 0), (1, 0, 0), (0,

1, 0), (0, 0, 1) respectively. Let F i be the net electric force acting on ith charge of the given

charges then F i = _____________.

[Ans. 0 ]

12. Calculate the magnitude of electrostatic force on a charge placed at a vertex of a triangularpyramid (4 vertices, 4 faces), if 4 equal point charges are placed at all four vertices ofpyramid of side ‘a’.

[Ans. 20

2

a4q6

]


PHYSICS



1. Two copper balls, each having weight 10 g are kept in air 10 cm apart. If one electron from every

106 atoms is transfered from one ball to the other, the coulomb force between them is (atomic

weight of copper is 63.5)

(A) 2.0 × 108 N (B) 2.0 × 106 N (C) 2.0 × 1010 N (D) 2.0 × 104 N2. The figure below shows the forces that three charged particles exert on each other. Which of the

four situations shown can be correct.

(I) (II) (III) (IV)

(A) all of the above (B) II, III & IV (C) II, III (D) none of the above3. Two charges 8 C and –6 C are placed with a distance of separation ‘d’ between them and exert

a force of magnitude F on each other. If a charge 8 C is added to each of these and they are

brought nearer by a distance 3d

, the magnitude of force between them will be-

(A) F31

(B) F49

(C) F23

(D) F32

4. Two identical spheres A and B having charge Q are placed at a distance r apart. The forceacting between them is F. An identical uncharged sphere C comes into contact with A. Afterthat it comes into contact with B and is then placed in middle of A and B. The net forceacting on the C is

(A) 83F

(B) F (C) 4

3F (D) 3F

5. Two identical positive charges are fixed on the y-axis, at equal distances from the origin O. Aparticle with a negative charge starts on the x-axis at a large distance from O, moves along the +x-axis, passes through O and moves far away from O. Its acceleration a is taken as positivealong its direction of motion. The particle’s acceleration a is plotted against its x-coordinate.Which of the following best represents the plot?

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


PHYSICS6. Two pith balls with mass m are suspended from insulating threads. When the pith balls are

given equal positive charge Q, they hang in equilibrium as shown.

Q Q

We now increase the charge on the left pith ball from Q to 2Q while leaving its mass essentially

unchanged. Which of the following diagrams best represents the new equilibrium configuration?

(A)

2Q Q

(B) 2Q

Q

(C)

2QQ

(D)

2Q Q

7. Two identical small balls each have a mass m and charge q. When placed in a hemisphericalbowl of radius R with frictionless, nonconductive walls, the beads move, and at equilibriumthe line joining the balls is horizontal and the distance between them is R.Neglect any

induced charge on the hemispherical bowl. Then the charge on each bead is: (K = 04

1 )

(A) 2/1

3KmgRq

(B)

2/1

3KmgRq

(C)

2/1

Kmg3Rq

(D)

2/1

Kmg3Rq

8. Two identical charged spheres suspended from a common point by two massless strings oflength are initially a distance d(d << 1) apart because of their mutual repulsion. The chargebegins to leak from both the spheres at a constant rate. As a result the charges approach eachother with a velocity v. Then as a function of distance x between them, [AIEEE-2011](A) v x–1/2 (B) v x–1 (C) v x1/2 (D) v x

9. In the figure shown, A is a fixed charged. B (of mass m) is given a velocity V perpendicularto line AB. At this moment the radius of curvature of the resultant path of B is

(A) 0 (B) (infinity)

(C) 2

220

qmvr4

(D) r


PHYSICS10. In gravity free space a ring of mass m and charge +q can move on a smooth circular wire

track of radius R. A point charge –q is fixed at the centre of the track and another charge +q isfixed on the track at the position shown. If the ring is released from the rest at the position A,then just after the release, the tangential acceleration and normal force on the ring will berespectively

(A) 132F,

m2F

(B) 2F,

m23F (C) 0,

m23F (D) 0, F/2

11. A simple pendulum of mass m and charge + q is suspended vertically by a massless threadof length . At the point of suspension, a point charge + q is also fixed. If the pendulum isdisplaced slightly from equilibrium position, its time period will be

(A) T = 2 2

2

mkqg

(B) T = 2 (C) T = 2 g

(D) will be greater than 2 g

12. An insulating long massless rod of length L, pivoted at its centre and balanced with a weight Wat a distance x from the left end, is shown in the figure. Charges q and 2q are attached at the leftand right ends of the rod. At a distance h directly below each of these charges is a positive chargeQ. The distance x in terms of q, Q, L and W is

(A) WhWLhqQL

20

20

(B)

Wh8WLhqQL4

2

20

(C) Wh8

WLh4qQL2

0

20

(D)

WhWLh4qQL

2

20

13. Two identical spheres of same mass and specific gravity (which is the ratio of density of asubstance and density of water) 2.4 have different charges of Q and – 3Q. They are suspendedfrom two strings of same length fixed to points at the same horizontal level, but distant fromeach other. When the entire set up is transferred inside a liquid of specific gravity 0.8, it isobserved that the inclination of each string in equilibrium remains unchanged. Then the dielec-tric constant of the liquid is(A) 2 (B) 3 (C) 1.5 (D) 4


PHYSICS14. Two point charges are kept separated by 4 cm of air and 6 cm of a dielectric of relative permittivity

4. The equivalent dielectric separation between them so far their colombian interaction isconserved is(A) 10 cm (B) 8 cm (C) 5 cm (D) 16 cm

15. Force between two identical charges placed at a distance of r in vacuum is F. Now a slab ofdielectric constant K = 4 is inserted between these two charges. The thickness of the slab is r/2.The force between the charges will now become -(A) F/4 (B) F/2 (C) 5

3 F (D) 9

4 F

Paragraph

Three charges are placed as shown in Fig. The magnitude of q1 is 2.00 mC, but its signand the value of the charge q2 are not known. Charge q3 is +4.00 C, and the net forceon q3 is entirely in the negative x-direction

y

x

q3

3cm4cm

q1 q2

F

5 cm

16. As per the condition given in the problem the sign of q1 and q2 will be

(A) +, + (B) +, (C) , + (D) , 17. The magnitude of q2 is

(A) C6427

(B) C3227

(C) C3213

(D) 6413

C

18. The magnitude of force acting on q3 is(A) 25.2 N (B) 32.2 N (C) 56.2 N (D) 13.5 N


1. A 2. C 3. C 4. C 5. B 6. D

7. A 8. A 9. C 10. B 11. C 12. C

13. C 14. B 15. D 16. C 17. B 18. C


PHYSICS

JRS TUTORIALSELECTROSTATICS (12-IIT)DPP-01 : Charge and its properties

1. Find the charge on an iron particle of mass 2.24 mg, if 0.02 % of electrons are removed from it.(A) 0.01996 (B) 0.01996 C (C) 0.02 C (D) 2.0 C

2. In 1 gm of a solid, there are 5 1021 atoms. If one electron is removed from everyone of 0.01%atoms of the solid, the charge gained by the solid is (given that electronic charge is 1.6 10–19 C)(A) + 0.08 C (B) + 0.8 C (C) – 0.08 C (D) – 0.8 C

3. The direction of electrostatic forces which are possible in the system of three charges A, Band C is :

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

4. Five Styrofoam balls are suspended from insulating threads. Several experiments are performedon the balls and the following observation are made

(i) Ball A repels C and attracts B

(ii) Ball D attracts B and has no effect on E

(iii) A negatively charged rod attracts both A and E. An electrically neutral Styrofoam ball getsattracted if placed nearby a charged body due to negative charge. What are the charges, if any on

each ball ?

A B C D E A B C D E

(A) + + 0 + (B) + + + 0

(C) + + 0 0 (D) + 0 0


PHYSICS5. Five balls, numbered 1 to 5, are suspended using separate threads. Pairs (1, 2), (2, 4), (4, 1) show

electrostatic attraction, while pairs (2, 3) and (4, 5) show repulsion. Therefore ball 1 :(A) Must be positively charged (B) Must be negatively charged(C) May be neutral (D) Must be made of metal

6. Consider three identical metal spheres A, B & C. Sphere A carries charge + 6 q and sphere

B carries charge 3 q. Sphere C carries no charge. Spheres A & B are touched together and

then separated. Sphere C is then touched to sphere A and separated from it. Finally the

sphere C is touched to sphere B and separated from it. The final charge on the sphere C is:

(A) 3 q (B) 0.75 q (C) 1.25 q (D) 1.125 q7. A common concept on charge is given, Select the odd statement.

(A) Charge gained by an uncharged body by conduction from a charged body is equal to half of the total charge initially present(B) Magnitude of charge does not with its velocity(C) Charge cannot be coexisted without matter although matter can exist without charge(D) Between two substance repulsion is true test of presence of charge

8. Consider a neutral conducting sphere. A positive point charge is placed outside the sphere.The net charge on the sphere is then, [JEE-2007](A) negative and distributed uniformly over the surface of the sphere(B) negative and appears only at the point on the sphere closest to the point charge(C) negative and distributed non-uniformly over the entire surface of the sphere(D) zero

9. A positively charged insulator is brought near (but does not touch) two metallic sphere that arein contact. The metallic spheres are then separated. The sphere which was initially farthest fromthe insulator will have(A) no net charge (B) a negative charge(C) a positive charge (D) either a negative or a positive charge.

10. A circle of radius r has a linear charge density = 0 cos2 along its circumference. Totalcharge on the circle is

(A) 0(2 r) (B) 0( r) (C) 0r

2 (D) 0

r4

11. The charge density of a spherical charge distribution is given by

nr0

nrr)r(0

.

What is the total charge on the distribution

(A) B C 0

3

34

D 03

32


PHYSICS12. The volume charge density as a function of distance x from one face inside a unit cube is

varying as shown in figure. Find charge stored in cube.

0

41

43 )min(x

y

1y

x1x

(A) 0

4

(B) 0

2

(C) 034

(D) 0

13. When an isolated body is connected to earth, electrons from the earth flow into the body.This means the body is :(A) charged negatively (B) an insulator(C) uncharged (D) charged positively

14. When a negatively charged rod is brought near, but does not touch, the initially unchargedelectroscope shown below, the leaves spring apart (I). When the electroscope is then touchedwith a finger, the leaves collapse (II). When next the finger and finally the rod are removed, theleaves spring apart a second time (III). The charge on the leaves is

I II III

- - - -- - - - --

(A) positive in both I and III (B) negative in both I and III

(C) positive in I, negative in III (D) negative in I, positive in III


PHYSICSParagraphA leaf electroscope is a simple apparatus to detect any charge on a body. It consists of twometal leaves OA and OB, free to rotate about O. Initially both are very slightly separated.When a charged object is touched to the metal knob at the top of the conducting rod, chargeflows from knob to the leaves through the conducting rod. As the leaves are now chargedsimilarly, they start repelling each other and get separated, (deflected by certain angle).

The angle of deflection in static equilibrium is an indicator of the amount of charge on thecharged body.

15. When a + 20 C rod is touched to the knob, the deflection of leaves was 5°, and when anidentical rod of – 40 C is touched, the deflection was found to be 9°. If an identical rod of+30 C is touched, then the deflection may be :(A) 0 (B) 2° (C) 7° (D) 11°

16. If we perform these steps one by one.

(i) A positively charged rod is brought closer to initially (A)

++++++

uncharged knob

(ii) Then the positively charged rod is touched to the knob (B)

++++++

(iii) Now the +vely charged rod is removed, and a negatively charged. (C)

- - -- - -

rod of same magnitude is brought closer at same distance

In which case, the leaves will converge (come closer), as compared to the previous state ?(A) (i) (B) (i) and (iii)(C) only (iii) (D) In all cases, the leaves will diverge


PHYSICS

ANSWER KEY PHYSICS DPP-01

1. B 2. A 3. C 4. C 5. C 6. D

7. A 8. D 9. C 10. B 11. B 12. C

13. D 14. D 15. C 16. C



Trigonometry – I

Section -I Prove the following identities:

1. .tantansecsec 2424 2. .sec2sin1

cossin1

cos

3. .cossin31cossin 2266 4. .cotcoscos1cos1

ec

5. )cos1()sin1(2)cossin1( 2 AAAA 6. .cosseccottan 22 AecAAA 7. .cossec)cot1(cos)tan1(sin AecAAAAA

8. 1 1 1 1 .cos cot sin sin cos cotec ec

9. .cos21sec

tan1sec

tan

ec

10. .sin5sec2)1tan2()2(tancos 11. .secsec)tan(tan)tantan1( 2222

12. 22

22

22

tantancoscoscoscos 13. .cot

cossinsin1coscossin2

22

Section - II

Do as required in each of the following. 1a. Which trigonometric ratios are negative for the following angles

(i) 315 2. – 210 3. 6

5

1b. Find the values of the following trigonometric functions.

(i) sin 1830 (ii) sin 765 (iii) cos (– 1710) (iv)

311sin

(v) 3

31sin (vi)

415cot (vii)

313cot

2. Find the values of the other five trigonometric functions in the following

(i) 2

3,178cos

(ii) ,21cos lies is the third quadrant

(iii)

2

,5

12cot (iv)

22

3,5

13sec

3. If cos = – ,2

3and53

find the value of

cotcostansec

ec

4. If ,2

32

and21tan,

53sin

show that 8 tan – .27sec5


5. Show that

(i) cos 70 cos 10 + sin 70 sin10 = 21 (ii)

2318sin78cos18cos78sin

6. Find the value of the following: (i) sin 105 + cos105 (ii) sin 300 cosec 1050 – tan ( – 120) (iii). sin 690 cos 930 + tan (– 765) cosec (– 1170)

(iv) 6

tan36

5cos6

cot 22

ec (v) 4

cot6

5sin43

sec6

sin3

7. Prove that 4

2675sin

8. Find 15tan and hence show that .415cot15tan

9. Evaluate ,4

)1(tan

n when n is an integer.

10. If sin 5

1sin,101

(, and + are acute angles). Show that .4

11. If ,2

0,2

0,419cosand

53sin

BABA Find the value of the following

(i) sin (A + B) (ii) cos (A + B) (iii) sin(A – B) (iv) cos (A – B)

12. if ,2

0and2

where,23cos,

21sin

BABA find the following

(i) tan (A + B) (ii) tan (A – B)

13. If ,2

and2

where,1312cos,

53sin

yxyx show that

6556)sin( yx .

14. Evaluate: (i) 2 2cos sin4 4

x x

(ii) )15(sin)15(sin 22 AA (iii)

28sin

28sin 22 AA

15. Prove that

(i) 1)2cot(2

3cot)2cos(2

3cos

xxxx

(ii)

2cot

2cos)sin(

)cos()cos(

16. Show that

(i) 21)10sin()40cos()10cos()40sin(

(ii) xxnxnxnxn cos)2cos()1cos()2sin()1sin(


(iii) )sin(4

sin4

sin4

cos4

cos yxyxyx

(iv) 0cos)cos3(cossin)sin3(sin xxxxxx

17. Prove that 25tan

20sin20cos20sin20cos 18. Prove that cos9 + sin9 = .54sin2

19. Prove that )sin()sin(

tantantantan

BABA

BABA

20. Prove that (i) tan 8 – tan 6 – tan 2 = tan 8 tan 6 tan 2 (ii) tan 9 + tan 36 + tan 9 tan 36 = 1 (iii) tan 3x – tan 2x – tan x = tan 3x tan 2x tan x 21. If sin sin – cos cos = 1, show that tan + tan = 0.

22. If sin ( + ) = 1 and sin ( – ) = ,21 where 0 , ,

2 then find the values of

tan ( +2) and tan (2 + ).

SECTION – III 1. If AAAAppAA ifsinand,sec,tanfind,0,tansec is acute angle.

2. If A and B are acute angles and ,3tantanand2

sinsin

BA

BA find A and B.

3. If tan + cot = 2 find sin .

4. If sin x + cos x = a, then prove that .2where,)1(431cossin 22266 aaxx

5. Eliminate from the following equations: (i) sin,cos bkyahx (ii) byxayx cossin,sincos

6. If sin x + cos x = m and sec x + cosec x = n, prove that n (m2 – 1) = 2m.

7. If x = r sin . cos , y = r sin . sin , z = r cos , show that x2 + y2 + z2 = r2

8. Prove that xyyx

4sin

22

is possible for real values of x and y only if x = y and x 0.

9. Show that 2cossin 22 ec 10. Prove that sec + cos .23

11. If .0cossinandcossincossin 33 yxyx prove that x2 +y2 = 1.

12. Show that xy

yx2

sin22

2 is possible for real values of x and y only when x = y 0.

13. Show that the equation x

x 1sin is impossible if x is real

14. If .cos2sincos Prove that: sin2sincos


15. If ,1sinsin 24 AA prove that (i) 1tan

1tan

124

AA (ii) 1tantan 24 AA

16. If .cossincossinthatshow,tan 22

22

baba

baba

ba

17. If ,1tan 22 k show that .)2(costansec 2/323 kec 18. If ,tansincos 222 ABB prove that .tansincos1cos2 2222 BAAA 19. If .cossin nn

nT prove that .0132 46 TT 20. Can, 02sin7sin6 2 , hold for any real value of ?

21. Prove that if cos 2x – 4 cos x + 1 = 8 sin x – 2 sin 2x, then )2/tan(x

22. If the product of the sines of the angles of a triangle is p and the product of their

cosines is q, show that the tangents of the angles are the roots of the equation

.

23. If then prove that 222cossin cbaAbAa

24. If ,1sinsin

coscos

2

4

2

4

BA

BA prove that

(i) ;sinsin2sinsin 2244 BABA (ii) ,1sinsin

coscos

2

4

2

4

AB

AB

25. If mxx cossin and necxx cossec show that mmn 2)1( 2

ANSWERS Section: II

1a. (i) sin , tan, cot , and cosec (ii) cos , tan , cot , and sec

(iii) sin , cos , sec and cosec

1b. (i) 2 (ii) 2

1 (iii) 0 (iv) 23 (v)

23 (vi) 1 (vii)

31

2. (i) 1517cos,

817sec,

158cot,

815tan,

1715sin ec

(ii) 3

2cos,2sec,3

1cot,3tan,23sin ec

(iii) 1213cos,

1312sec,

125tan,

1312cos,

135sin ec

(iv) 1213cos,

125cot,

512tan,

135cos,

1312sin ec

qx3 px2 1 qx p 0

a cos A b sinA c,


6. (i) 2

1 (ii) 0 (iii) 143 (iv) 6 (v) 1

8. 1313

9. ( 1)n

11. (i) 205187 (ii)

20584

(iii) 205133

(iv) 205156

12. (i) 0 (ii) 3

14. (i) 0 (ii) A2sin21 (iii) Asin

21 22.

31,3

Section – III

1. 2

222

11sin,

21sec,

21tan

ppA

ppA

ppA

2. 30,45 BA

3. 2

1 5. (i) 1

22

bky

ahx (ii) 2222 yxba

JRS Tutorials, Durgakund, Varanasi-221005, Ph No. (0542) 2311777, 2311922, 9794757100, 317347706

JRS TUTORIALS CHEMISTRY 20-21

Solid State XII – (IIT and PMT)–DPP-2

1. The formula for determination of density of unit cell is

1. 303

cmgMNNa −

××

2. 3

03 cmg

NaMN −

×× 3. 3

0

3

cmgNNMa −

×× 4. 3

30 cmg

NaNM −

××

2. The packing efficiency of two dimensional square unit cell shown

below is

1. 32.97 % 2. 68%

3. 74% 4. 78.5%

3. A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is

'a', the closest approach between two atoms in metallic crystal will be :-

1. 2a 2. a22 3. a2 4. 2

a

4. Body centered cubic lattice has a coordination number of 1. 4 2. 8 3. 12 4. 6

5. The element crystallizes in a body centered cubic lattice and the edge of the unit cell is 0.351 nm. The density is 0.533 g/cm3. What is the atomic weight?

1. 12.0 2. 6.94 3. 9.01 4. 10.8

6. An element crystallizes in a structure having fcc unit cell of an edge 200 pm. Calculate the density, if 100g of this element contains 12 × 1023 atoms

1. 41.66 g/cm3 2. 4. 166 g/cm3 3. 10.25 g/cm3 4. 1.025 g/cm3 7. The number of close neighbour in a body centred cubic lattice of identical sphere is 1. 8 2. 6 3. 4 4. 2 8. In the closest packed structure of a metallic lattice, the number of nearest neighbours

of a metallic atom is 1. Twelve 2. Four 3. Eight 4. Six 9. The number of atoms in 100 gm of an bcc crystal with density d = 10 g/cm3 and cell

edge equal to 100 pm, is equal to 1.

25104× 2. 25103× 3.

25102× 4. 25101×

10. The number of atoms in primitive cubic unit cell, body-centred cubic unit cell and face-

centred cubic unit cell are A, B and C, respectively. Select the correct values.

A B C A B C

1. 1, 2, 3 2. 1, 2, 4

3. 1, 1, 3 4. 2, 3, 4

L


2

11. The following crystallographic data were obtained for a protein. Volume of unit cube = 1.50 × 10–19 cm3, Density = 1.35 g cm–3 Z = 4 and protein fraction = 0.75. Thus, molar mass of protein is 1. 2.3 × 104 g mol–1 2. 3.048 × 104 g mol–1 3. 1.725 × 104 g mol–1 4. None of these 12. A element is having body centred unit cell arrangement. Length of body diagonal is

350 pm. Density of unit cell is 6 gm/cm3. Find how many atoms are present in 500 gm of element.

1. 20 NA 2. 25 NA 3. 34 NA 4. 40 NA 13. A metal crystallises in f.c.c. lattice with unit cell edge length of 4 . If 100 gm of this

metal contains 3 × 1023 atom, its density is 1. 1.44 gm/cm3 2. 14.4 gm/cm3 3. 10.4 gm/cm3 4. 20.8 gm/cm3

14. What are the number of atoms per unit cell and the number of nearest neighbours in a

sample cubic structure? 1. 1, 6 2. 4, 12 3. 2, 8 4. 2, 6 15. % of empty space in body centered cubic unit cell is nearly 1. 52.36 2. 68 3. 32 4. 26 16. Lithium has a bcc structure. Its density is 530 kg m–3 and its atomic mass is 6.94 gmol–1.

Calculate the edge length of a unit cell of Lithium metal. (NA = 6.02 × 1023) 1. 154 pm 2. 352 pm 3. 527 pm 4. 264 pm

17. A solid has a structure in which W atoms are located at the corners of a cubic lattice,

O atoms at the centres of edges and Na atom at the centre of the cube. The formula of the compound is

1. NaWO2 2. NaWO3 3. Na2WO3 4. NaWO4

18. A metal crystallizes with a face-centred cubic lattice. The edge of the unit cell is 408 pm. The diameter of the metal atom is

1. 288 pm 2. 408 pm 3. 144 pm 4. 204 pm

Answer DPP-2 Q.No. 1 2 3 4 5 6 7 8 9 10 Ans 2 4 4 2 2 1 1 1 3 2

Q.No. 11 12 13 14 15 16 17 18 Ans 2 3 4 1 3 2 2 1




1. Which of the following is not a crystalline solid ? 1. Common salt 2. Sugar 3. Iron 4. Rubber 2. A pseudo solid is : 1. Glass 2. pitch 3. KCl 4. Glass and pitch both 3. Solid CO2 is an example of, 1. Ionic crystal 2. Covalent crystal 3. Metallic crystal 4. Molecular crystal 4. A molecular crystalline solid, 1. is very hard 2. is volatile 3. has a high melting point 4. is a good conductor 5. Select the correct statement 1. Crystalline solids are anisotropic 2. Amorphous solids are isotropic 3. Both 1 and 2 4. None of the above 6. Amorphous materials are infact considered as 1. supercooled liquids 2. spercooled solids 3. covalent network 4. molecular crystals 7. The sharp melting point of crystalline solids compared to amorphous solids is due to 1. same arrangement of constituent particles in different directions 2. different arrangement of constituent particles in different directions

3. a regular arrangement of constituent particles observed over a long distance in the crystal lattice

4. a regular arrangement of constituent particles observed over a short distance in the crystal lattice

8. As it cools, olive oil solidifies and forms a solid over a wide range of temperature.

Which term best describes the solid? 1. Ionic 2. Covalent network 3. Metallic 4. amorphous solid 9. Which of the following can be regarded as molecular solids? 1. SiC 2. AlN 3. C(diamond) 4. Ne 10. A solid can be characterized by 1. definite mass, volume and shape 2. short intermolecular distances 3. strong intermolecular forces 4. All of the above 11. Which of the following are the correct axial distances and axial angles for rhombohedral

system ? 1. a = b = c, α =β = γ ≠ 90° 2. a = b ≠ c, α =β = γ = 90° 3. a ≠ b = c, α =β = γ = 90° 4. a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°


2 12. Select incorrect statement for polar molecular solids,

1. molecules have polar covalent bonds 2. molecules are held by relatively stronger dipole-dipole interactions 3. molecules are held by weak London forces 4. higher melting point as compared to non-polar molecular solids, is observed 13. a ≠ b ≠ c, α = γ = 90° β ≠ 90° represents 1. tetragonal system 2. orthorhombic system 3. monoclinic system 4. triclinic system 14. Bravais lattices are of, 1. 10 types 2. 8 types 3. 7 types 4. 14 types 15. In a simple cubic cell, each point on a corner is shared by, 1. 2 unit cells 2. 1 unit cell 3. 8 unit cells 4. 4 unit cells 16. In face centred cubic cell, an atom at the face centres is shared by, 1. 4 units cells 2. 2 unit cells 3. One unit cell 4. 6 unit cells 17. In a body centred cubic cell, an atom at the body centre is shared by, 1. 1 unit cell 2. 2 unit cell 3. 3 unit cells 4. 4 unit cells 18. Match Column A with B and select the correct option. Column A Column B

A. Ionic solid I. ZnS B. Metallic solid II. Au C. Covalent solid III. Diamond) D. Molecular solid IV. Ice 1. A - I, B - II, C - IV, D - III 2. A - I, B - II, C - III, D - IV 3. A - III, B - II, C - I, D - IV

4. A - II, B - IV, C - I, D - III

Answer DPP-1 Q.No. 1 2 3 4 5 6 7 8 9 10 Ans 4 4 4 2 3 1 3 4 4 4

Q.No. 11 12 13 14 15 16 17 18 Ans 1 3 3 4 3 2 1 2

JRS TUTORIALS, Durgakund, Varanasi – 221 005 (U.P.) Ph. No. (0542) 3290510, 2311922, 2311777

JRS TUTORIALS MATHEMATICS PROBLEMS SHEET

Functions - I 1. Find f∘g(x) in each of the following :

(a) f (x) = 1x , g(x) = x2 – 2x

(b) f (x) = x2 – 1, g(x) = 1x

(c) f (x) = ln x g(x) = x + 12 x

2. Find f (x) in each of the following

(a) f (g(x)) = 2x – 4, g(x) = x + 1 (b) f (g (x)) = x2 – 3x + 2, g(x) = x + 1 (c) g(f (x)) = x2 + 1, g (x) = x2 + 2x – 1 3. Find f (g (x)) and g (f (x)) if ( ) = 2 – 3, – 2 1 and ( ) = + 1, – 1 2

4. If f (x) =

0,0,

2 xxxx

Find f (f (x).

5. Let f (x) = x1

1 find ∘ ∘ ( ). What is domain?

6. Let f (x) =

20,3202,1

xxxx

find f∘f (x).

7. If 24

4

x

x

xf , then show that 11 xfxf

8. Let f be a function defined for all x > 1 such that 1214 8 ( 1) logxf f x xx

, then value of

4( (10) (13) (17))f f f is

9. f (x + y, x – y) = xy, then the arithmetic mean of f (x, y) and f (y, x) is 0.

10. If 11)(

xxxf then show that f (f (ax)) in terms of f (x) is equal to

)1)((1)(

xfaxf

.

11. Determine if the following functions are periodic if yes find their period

(i) 2cos32 xxf (ii) xxxxf tancos3sin 2

(iii) 3

sin4

sin xxxf (iv) .

72sin

53cos xxxf

(vii) xxf sin (viii) xxxf sin


12. Determine whether the following functions are even or odd or neither even nor odd :

(i)

11

x

x

aaxxf (ii) 1log 2 xxxf

(iii) xxxf cossin (iv) xxxf 12

13. If

1)1ln(10cos

xxxxxx

xf

then extend the definition of 0,xforxf such that f (x) becomes (i) An even function (ii) An odd function 14. If f (x + y) = f (x) + f (y) – 1 for all ,1)1(and, fyx R then the number of solutions of

Nnnnf ,)( is one. 15. Let be a function with domain [–3, 5] and let g(x) = 3x + 4, find the domain of (og) (x). 16. The function f (x) = sin 5x + cos 3 x is non-periodic. Why?

17. Let f be a real valued function with domain R satisfying 21)(0 xf and for some fixed a,

,))(()(21)( 2 R xxfxfaxf then the period of the function f (x) is 2a.

18. Let f (x) = max R xxx ,2,1,1 . Then 1

111

,1,2,1

xx

x

x

xxf .

19. If ,)})({()(and))(()(,11)( xfffxhxffxgx

xf

then the value of

xhxgxf is – 1.

20. Determine a function f satisfying the functional relation

2 1 211 1

xf x f

x x x

21. Determine f(x) such that (| |) + | ( )| = 3 + 4. 22. If 3f(x) + af(1/x) = x and f(2) = 0 find a. 23. If f(x) is a polynomial satisfying f(x) + f(1/x) = f(x)f(1/x) determine f(x). 24. If f(x + y) = f(x) + f(y) for all x, y then f is odd, show. 25. If f(x - 1) + f(x + 1) = 4 then f(x) is periodic.


ANSWERS

1. (A) |x-1| (B) x (C) ln 1x2

2. (A) 2x 6 (B) x2 5x + 6 (C) 1 ± 3x2

3. ( ) = 2 − 1, −1 ≤ ≤ 0, ( ) = 2 − 2, = 1

4. ( ( ) = , < 0 ( ( ) = , ≥ 0

5. ∘ ∘ ( ) = , ≠ 0,1

6. ( ) =

⎩⎪⎨

⎪⎧ + 2, −2 ≤ < −1

2 − 1, −1 ≤ < 02 − 2, ≤ <

4 − 9, 3/2 ≤ ≤ 2

11. (A) T = 2 (B) T = 2 (C) T = 24 (D) T = 70

(E) not periodic (F) not periodic

12. (A) even (ii) odd (C) neither (D) even

15. −3, 20. ( ) = , ≠ 0,1 21. ( ) =, ≥ 0

, − ≤ < 0

22. 12 23. ( ) = 1 ±

Page No. - 1

JRS TUTORIALS, Durgakund, Varanasi – 221 005 (U.P.) Ph. No. (0542) 2311922, 2311777

Mob. : 9794757100, 7317347706 e-mail : [email protected] Web site : www.jrstutorials.ac.in

JRS TUTORIALS JEE PRACTICE SHEET No. – 1

Mathematics – Target & XII Time : 1 Hr.

1. The value of such that

8

9ln

)1x()x(

dx2

1

is

A. – 3 B. 2

1 C. –

2

1 D. 3

2. If

then,2

dxxsecxtan

xtan2/

0

=

A. 1 B. – 1 C. 2

1 D. –

2

1

3. The value of

n2

0

Nn,4dx)]x3cos1(x2[sin and [t] denotes the greatest integer

function then the value of n = ……..

A. 2 B. 3 C. 4 D. 5

4. Let x

0

dt)t(g)x(f , where g is not zero function as well as even. If f(x + 5) = g(x)

then x

0

dt)t(f

A.

5

5x

dt)t(g5 B. 5x

5

dt)t(g C. 5x

5

dt)t(g2 D.

5

5x

dt)t(g

5. The integral

1

x3x2

dxxlnx

e

e

x is equal to

then,

e

e

1 3

2= …………..

A. 3 B. 5 C. 7 D. 9

6. If xcos)xx1(ln2

xcos)xx1ln(2)x(f

2

2

and g(x) = ln x, (x > 0) when the value of the integral

4/

4/

dx))x(f(g is

A. ln 3 B. ln 1 C. ln 2 D. ln 3

Page No. - 2



7. Let

b

a

24 dx)xx(I . If I is minimum in the interval (–2, 2) then | a | + | b | = ………

(where N)

A. 2 B. 4 C. 6 D. 8

8. The value of

2/

2/

x

2

dxe1

xcosx is equal to

A. 24

2

B. 24

2

C. 2/2 e D. 2/2 e

9. The value of

1

3

23 dx)}2xcos()2x()10x12x6x({ is equal to ………..

A. 0 B. 3 C. 4 D. 1

10. Let

........,3,2,1,0n,dxxsin)e1(

nxsinI

xn

I1 + I2 + I3 + ………… + In = 99 and maximum of n = N then the value of

.....12

2N

11. The value of

1

0

2

44

n

mdx

x1

)x1(x

(where HCF (m, n) = 1) then (m – 3n) = ……….

12. If L

dt)atcosat(sine

dt)atcosat(sine

0

50100t

8

0

50100t

Where a is an even natural number then the value of ...........)L2.()1e(

)1e(8

13. Let 2448 tan3xtan3xtan7xtan7)x(f

for all L)x(fxand2

,2

x

4/

0

then

the value of L4

1= …………..

Page No. - 3



14. The value of integral

2/1

0

4/162 ])x1()1x[(

dx)31( is …………..

15. The value of

dx)x1(dx

dx)x1(5050

10150

1

0

1

0

10050

then (L – 5050) = ………..

16. Let f : [1, ) [2, ) be differential function.

If 1xx)x(fx3dt)t(f6 3

x

1

If | f(x) | = K has maximum number of solution for K (0, a) then the value of a

1=……

17. If f(x) = x + sin x and y = g(x) is the inverse of y = f(x).

If

2

3dx)x(g

2

then + = ……………

18. If ......I

Ithendx))x1(x(gIanddx))x1(x(gxI,

e1

e)x(f

1

2

2)a(f

2)a(f

2

2)a(f

2)a(f

1x

x

19. Let f(x) be continuous and differentiable in the interval (a, b) and ,1)x(flimax

.3)x(flim 4/1

bx

If )x(f

1)x(f)x(f 3 then value of [24(b – a)] = ……………

Where [t] is greatest integer function.

20. Let 2

2187

v

uif,dx)x1(xvanddx)x3(xu

n

n

1

0

nn

n

2/3

0

nn

n then n = ……….

Page No. - 4



JRS TUTORIALS JEE PRACTICE SHEET No. – 1

Answer

Question Answer Question Answer

1 B 11 1

2 A 12 2

3 C 13 3

4 D 14 2

5 B 15 1

6 B 16 9

7 B 17 4

8 A 18 2

9 C 19 3

10 9 20 3


JRS TUTORIALS MATHEMATICS FOUNDATION COURSE

PRACTICE SHEET: TRIGONOMETRY II

1. The value of cos 135 sin 135 is

1. 0 2. < (1) 3. > 1 4. None

2. cot 1. cot 2. cot 3…….cot 89 =

1. 2. 0 3. 1 4. 2

1

3. Which of the following is correct?

1. cos 1 > cos 1 2. cos 1 < cos 1

3. cos 1 = cos1 4. cos 1 = 1cos180

4.

75tan1

75tan12

2

1. 2

1 2.

2

3 3.

3

2 4.

3

2

5. sin 75 =

1. 22

13 2.

22

13 3. 32 4. 32

6. If

sinthen4

3cot

1. 5

4notbut

5

4 2.

5

4or

5

4

3. 5

4notbut

5

4 4. None

7. cot 15 tan 15 =

1. 32 2. 32 3. 32 4. None

8.

11sin11cos

11sin11cos

1. tan 45 2. tan 34 3. tan 60 4. cot 11

9.

14sin

14

3sin

14

5sin

1. 8

1 2.

4

1 3.

4

1 4.

8

1

10. The minimum value of ,cos24sin7 for all real values of is

1. 1 2. 24 3. 25 4. 31


Page No. 2

11. is45cos45sin 1. cos2 2. sin2 3. non zero constant 4. 0

12.

4tan .

4tan =

1. 1 2. 0 3. 1 4. infinite

13. If cos =

24

costhen2

3,

5

4

1. 5

1 2.

10

1 3.

5

1 4.

10

1

14. If and3

1cot lies in the 3rd quadrant, then cosec

1. 10

1 2. 10 3.

3

10 4.

3

10

15. If

costhen2

,5

3sinand

20,

13

12sin

1. 65

56 2.

65

16 3.

65

56 4.

65

16

16. cos 40 cos 20 + sin 10 =

1. 1 2. 0 3. 2

1 4. 2

17. 20sin20cos3

1. 2 2.

40sin

20sin2 3. 40sin2 4. None

18. If cos (B A) = 5

3 and tan A = 2 cot B then

1. 5

1coscos

BA 3.

5

1sinsin

BA

3. 5

1cos

BA 4.

5

4sinsin

BA

19. If tan = tan 60 then the value of in the 3rd quadrant is 1. 240 2. 210 3. 200 4. 220 20. Which of the following is correct? 1. sin + sin 2 + sin 3 = 3 is true for some real value of 2. If = 155 then sin + cos is negative 3. cos 11 cos 2 is positive

4. cos (1044) = 4

15


Page No. 3

21. The maximum value of 8 + 4 cos x + 3 sin x is 1. 12 2. 13 3. 8 4. 5

22. If sec 2 = 7

9 then sin =

1. 3

1notbut

3

1 2.

3

1notbut

3

1

4. 3

1or

3

1 4. None

23. sin 420 cos 390 cos (300) sin 330 =

1. 0 2. 1 3. 1 4. 2

13

24. tan 760 cot 760 + tan 225 cos 450 = 1. 0 2. 1 3. 2 4. 2

25. The signs of 5

12sin

5

12cosand

7

13sin

are respectively

1. +, + 2. , 3. +, 4. , +

26.

6

17sin

3

11cos

3

13sin

6

19cos

1. 1 2. 2

1 3.

2

1 4. 1

27. tan 6

11sin

4

17

1. 2

1 2. 1 3.

2

1 4.

2

3

28. cos 5

13

1. 5

2cos

2.

5

2sin

3.

10sin

4.

10

cos4

29. If 5 then to....3sin2sin(sin 222 18 terms) is equal to

1. 7 2. 8 3. 9 4. 2

19

30. Which of the following is a rational number? 1. sin 15 2. cos 15

3. sin 15 cos 15 4. sin 15 cos 75

31. 22 coscossinsin

1. 2cos4 2. 2sin4 3. 2

cos4 2 4. 2

sin4 2


Page No. 4

32. If tanthen1cossin

1. 1 2. 2

1 3. 0 4. none

33. If n tanthen2sinsin m

1. m

m

1

1 2.

nm

nm

3. nm

nm

4. None

34. If 22thensincosandcossin banbamba

1. m + n 2. mn 3. m2 + n2 4. mn

35. If 2

cotthen1cos2cos2cos

1. 2

cot3

2. 2

cot

3. 2

cot3

1 4.

2cot

3

1

36. If 0 < A, B < lyrespectiveareBandAthen2

1sinand

2

1cos,

2 BABA

1. 45, 45 2. 60, 45 3. 45, 15 4. 45, 60 37. The minute-hand of a clock is of length 10 cm. How far the tip of the hand move in

10 minutes?

1. cm3

10 2. cm

27

11160 3. cm

27

113050 4. None

38. If the angles of a triangle are in A.P. and the greatest angle is 2

then the smallest angle is

1. 3

2.

6

3.

4

4.

39. If sincosthensin2sincos

1. cos2 2. sin2 3. sincos2 4. None

40. If

sin

cos1,

cos

sin1

yx then

1. xy + 1 = x y 2. xy + 1 = 2y 3. xy +1 = x + y 4. xy +1 = y x

ANSWERS 1. 2 2. 3 3. 1 4. 4 5. 2 6. 2 7. 1 8. 2 9. 4 10. 3 11. 4 12. 3 13. 1 14. 3 15. 1 16. 2 17. 3 18. 3 19. 1 20. 2 21. 2 22. 3 23. 3 24. 2 25. 2 26. 4 27. 4 28. 3 29. 4 30. 3 31. 1 32. 3 33. 4 34. 3 35. 4 36. 3 37. 1 38. 2 39. 1 40. 4


PHYSICS



1. An infinite number of charges, each of charge 1 C, are placed on the x-axis with co-ordinates

x = 1, 2, 4, 8, .... . If 1 C is kept at the origin, then what is the net force acting on 1 C charge

(A) 9000 N (B) 12000 N (C) 24000 N (D) 36000 N2. Given are four arrangements of three fixed electric charges. In each arrangement, a point labeled

P is also identified — test charge, +q, is placed at point P. All of the charges are the samemagnitude, Q, but they can be either positive or negative as indicated. The charges and point Pall lie on a straight line. The distances between adjacent items, either between two charges orbetween a charge and point P, are all the same.

I. II.

III. IV.

Correct order of choices in a decreasing order of magnitude of force on P is(A) II > I > III > IV (B) I > II > III > IV (C) II > I > IV > III (D) III > IV > I > II

3. Four positive point charges are arranged as shown in the accompanying diagram. The forcebetween charges 1 and 3 is 6.0 N; the force between charges 2 and 3 is 5.0 N; and the forcebetween charges 3 and 4 is 3.0 N. The magnitude of the total force on charge 3 is most nearly

(A) 6.3 N

(B) 8.0 N

2 3 4

1

(C) 10 N

(D) 11 N4. A point charge +Q is placed at the centroid of an equilateral triangle. When a second charge +Q

is placed at a vertex of the triangle, the magnitude of the electrostatic force on the central charge

is 4N. What is the magnitude of the net force on the central charge when a third charge +Q is

placed at another vertex of the triangle?

(A) zero (B) 4 N (C) 4 3N (D) 8 N5. Three identical spheres each having a charge q and radius R, are kept in such a way that

each touches the other two. The magnitude of the net electric force on any sphere is

(A) 2

0

314 4

qR

(B) 2

0

212 4

qR

(C) 2

0

214 4

qR

(D) 2

0

312 4

qR


PHYSICS6. A charge Q is placed at each of the opposite corners of a square. A charge q is placed at

each of the other two corners. If the net electrical force on Q is zero, then Q/q equals

(A) – 22 (B) – 1 (C) 22 (D) –21 [AIEEE-2009]

7. Given figure shows an arrangement of six charged particles. The net electrostatic force F acting

on charge +q at the origin due to other charges is

(A) 20

2

a4q6

(B) zero (C) 20

2

a2q7

(D)

3

23

a4q

20

2

8. The magnitude of electric force on 2 C charge placed at the centre O of two equilateraltriangles each of side 10 cm, as shown in figure is P. If charge A, B, C, D, E & F are 2 C,2 C, 2 C, -2 C, - 2C, - 2 C respectively, then P is:(A) 21.6 N

(B) 64.8 N OA

B

C

D

E

F

(C) 0

(D) 43.2 N9. Through the exact centre of a hydrogen molecule, an -particle passes rapidly, moving on

a line perpendicular to the internuclear axis. The distance between the two hydrogen nucleiis b. The maximum force experienced by the -particle is

(A) 2

20

43 3

eb (B)

2

20

83

eb (C)

2

20

83 3

eb (D)

2

20

43

eb

10. Which of the following graphs best represents the force acting on a charged particle kept at

distance x from the centre of a square and on the axis of the square whose corners haveequal charges.

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


PHYSICS11. Three charges +Q1, +Q2 and q are placed on a straight line such that q is somewhere in between

+Q1 and +Q2. If this system of charges is in equilibrium, what should be the magnitude and sign

of charge q ?

(A) ve,)QQ(

QQ2

21

21 (B) ve,

2QQ 21

(C) ve,)QQ(

QQ2

21

21 (D) ve,

2QQ 21

12. Three charges + 4q, -q and +4q are kept on a straight line at position (0, 0, 0), (a, 0, 0) and(2a, 0, 0) respectively. Considering that they are free to move along the x-axis only(A) all the charges are in stable equilibrium(B) all the charges are in unstable equilibrium(C) only the middle charge is in stable equilibrium(D) only middle charge is in unstable equilibrium

13. A mass particle (mass = m and charge = q) is placed bewteen two point charges of charge

q separtion between these two charge is 2L. The frequency of oscillation of mass particle,

if it is displaced for a small distance along the line joining the charges–

(A) 30Lm

12q

(B) 30Lm

42q

(C) 30Lm4

12q

(D)2q

30mL161

14. Two positive point charges are fixed at some distance apart. A third negative charge ' q ' is placedat the centre of the line joining the two charges. Then the number of lines along which ' q ' canperform SHM for small displacements is(A) 1 (B) 2 (C) 3 (D)

15. Four point positive charges are held at the vertices of a square in a horizontal plane. Theirmasses are 1 kg, 2 kg, 3 kg & 4 kg. Another point positive charge of mass 10 kg is kept onthe axis of the square. The weight of this fifth charge is balanced by the electrostatic forcedue to those four charges. If the four charges on the vertices are released such that they canfreely move in any direction (vertical, horizontal etc) then the acceleration of the centre ofmass of the four charges immediately after the release is: (Use g = 10 m/s2)

(A) 10 m/s2 (B) 20 m/s2 (C) zero (D) 10 m/s2 16. Under the influence of the Coulomb field of charge +Q, a charge –q is moving around it in an

elliptical orbit.Find out the correct statement(s). [JEE -2009](A) The angular momentum of the charge –q is constant.(B) The linear momentum of the charge –q is constant.(C) The angular velocity of the charge – q is constant.(D) The linear speed of the charge –q is constant.


PHYSICS17. A ring of radius R is made out of a thin metallic wire of area of cross section A. The ring has a

uniform charge Q distributed on it. A charge q0 is placed at the centre of the ring. If Y is theyoung’s modulus for the material of the ring and R is the change in the radius of the ring, then

(A) R = RAY4Qq

0

0

(B) RAY4QqR

0

0

(C) RAY8

QqR 20

0

(D) RAY8

QqR0

20

18. Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap

film of side ‘ ’. The surface tension of the soap film is . The system of charges and planar

film are in equilibrium, and = k1/N2q

, where ’k’ is a constant. Then N is

(A) 3 (B) 2 (C) 4 (D) 6 [JEE-2011]

MULTIPLE CORRECT ANSWERS19. Three charged particles are in equilibrium under their electrostatic forces only

(A) The particles must be collinear.(B) All the charges cannot have the same magnitude.(C) All the charges cannot have the same sign.(D) The equilibrium is unstable.

20. Two equal negative charges –q each are fixed at the points (0, a) and (0, -a) on the y-axis .Apositive charge Q is released from rest at the point (2a, 0) on the x-axis. The charge Q will:(A) At origin velocity of particle is maximum.(B) Execute simple harmonic motion about the origin(C) Move to infinity(D) Execute oscillatory but not simple harmonic motion.

21. A negative point charge placed at the point A is

(A) in stable equilibrium along x-axis

(B) in unstable equilibrium along y-axis

(C) in stable equilibrium along y-axis

(D) in unstable equilibrium along x-axis22. Two identical charges +Q are kept fixed some distance apart. A small particle P with charge q is

placed midway between them. If P is given a small displacement , it will undergo simpleharmonic motion if(A) q is positive and is along the line joining the charges.(B) q is positive and is perpendicular to the line joining the charges.(C) q is negative and is perpendicular to the line joining the charges.(D) q is negative and is along the line joining the charges.


PHYSICSParagraph

Three charged particles each of +Q are fixed at the corners of anequilateral triangle of side ‘a’. A fourth particle of charge –q andmass m is placed at a point on the line passing through centroid of triangle and perpendicular to the plane of triangle at a distance xfrom the centre of triangle

23. Force on the fourth particle is

(A) 2/3220 )ax3(

Qx394

1 (B) 2/322

0 )ax3(Qx33

41

(C) 2/3220 )ax2(

Qx224

1 (D) 2/322

0 )ax2(Qx24

41

24. Value of x for which the force is maximum is

(A) 3a

(B) 2a

(C) 6a

(D) 5a

MATCH THE COLUMN25. Four charge Q1,Q2,Q3, and Q4,of same magnitude are fixed along the x axis at x = –2a –a, +a

and +2a, respectively. A positive charge q is placed on the positive y axis at a distance b > 0.Four options of the signs of these charges are given in List-I . The direction of the forces on thecharge q is given in List- II Match List-1 with List-II and select the correct answer using thecode given below the lists [JEE (ADV)-2014]

List-I List-IIP. Q1,Q2,Q3, Q4, all positive 1. +xQ. Q1,Q2 positive Q3,Q4 negative 2. –xR. Q1,Q4 positive Q2, Q3 negative 3. +yS. Q1,Q3 positive Q2, Q4 negative 4. –yCode :(A) P-3, Q-1, R-4,S-2 (B) P-4, Q-2, R-3, S-1(C) P-3, Q-1, R-2,S-4 (D) P-4, Q-2, R-1, S-3


PHYSICS26. In the following Fig. charges, each +q, are fixed at L and M. O is the mid point of distance LM.

X- and Y –axes are as shown. Consider the situations given in column I and match them with

the information in column II

Column I Column –II(A) Let us place a charge +q at O, displace it (P) force on the charge is zero slightly along X-axis and release. Assume that it is allowed to move only along X-axis. At position O,(B) Place a charge –q at O, Displace it slightly (Q) potential energy of the system along X-axis and release. Assume that it is maximum is allowed to move only along X-axis. At position O,(C) Place a charge +q at O. Displace it slightly (R) potential energy of the system is along Y-axis and release. Assume that it is minimum allowed to move only along Y-axis. At position O,(D) Place a charge –q at O. Displace it slightly (S) the charge is in equilibrium along Y-axis and release. Assume that it is allowed to move only along Y-axis. At position O,


1. B 2. C 3. A 4. B 5. A 6. A

7. B 8. D 9. C 10. C 11. C 12. B

13. A 14. D 15. B 16. A 17. D 18. A

19. A,B,CD 20. A,D 21. C,D 22. A,C 23. A

24. C 25. A 26. (A- P, R, S) ( B – P, Q, S) (C –P, Q, S) (D – P, R, S)



MOLE CONCEPT –DPP-1

Subjective 1. Find out number of moles in 0.0036 gram of water?

2. Find out number of atoms in following a. 5 moles of oxygen b. 5 moles of water c. 5 moles of glucose d. 5 moles of CuSO4.5H2O

3. Find out number of atoms in following A. 25 moles of nitrogen B. 0.24 moles of methane C. 1.2 moles of glucose D. 200 milimoles of water Hint: 1 mole = 1000 millimoles

4. Convert following into moles A. 3.011 × 1023 molecules of water B. 1.2044 × 1025 molecules of water C. 3.011 × 1020 molecules of oxygen D. 2.4088 × 1022 molecules of nitrogen

5. Find out number of electrons in following A. 6.022 × 103 molecules of water B. 10 moles of water C. 0.0024 moles of carbon dioxide D. 400 milimoles of oxygen

6. What is mass of five sodium atoms in amu

7. What should be mass of one molecule of water in gram?

8. Find out mass of one Mg atom in gram

9. One atom of an element X weighs 6.644 × 10–23 g. Calculate the number of moles of X in 40 kg of it

10 4.6 × 1022 atoms of an element weigh 13.8 gm. The gram atomic mass of the element

is: (NA = 6 × 1023)

11. Find out number of moles of CH4 in 5.6 L of CH4 at STP.

12. Find out volume of 10 moles methane at STP.

Objective (Only one option is correct) 1. The mass of 3.2 ×105 atoms of an element is 8.0 ×10–18 g. the atomic mass of the

element is about (NA = 6×1023) 1. 2.5 ×10–22 2. 15 3. 8.0×10–18 4. 30 2. Number of electrons in 18 g H2O 1. 6.022 × 1022 2. 6.022 × 1023 3. 6.022 × 1024 4. 6.022 × 1025 3 The number of neutrons present in 9 mg of O18 is

1. 10 2. 5NA 3. 0.005 NA 4. 0.0005 NA


2 4. Which of the following have largest number of atoms?

1. 4 gm oxygen (at mass O = 16) 2. 16 gm sulphur (at mass S = 32) 3. 35.5 gm chlorine (at mass Cl = 35.5) 4. 14 gm lithium (at mass Li = 7) 5. Number of oxygen molecules having weight equal to weight of 20 molecules of SO3. 1. 100 2. 50 3. 15 4. 8 6. The weight of 3.2 × 105 atoms of an element is 8.0 × 10–18 gm. The atomic weight of

the element should be about 1. 2.5 × 10–22 2. 15 3. 1.5 4. 150

7. The number of molecules of water in 333 g of Al2(SO4)3. 18H2O is

1. 18.0 × 6.02 × 1023 2. 9.0 × 6.02 × 1023

3. 18.0 4. 36.0 8. The mass of 2 gram atoms of calcium (atomic mass = 40) 1. 2 g 2. 0.05 g 3. 0.5 g 4. 80 g 9. The number of molecules present in 88 g of CO2 (Relative molecular mass of

CO2 = 44) 1. 1.24 × 1023 2. 3.01 × 1023 3. 6.023 × 1024 4. 1.2046 ×1024

10. One a.m.u. is equivalent to 1. 1.66 × 10–24 kg 2. 1.66 × 10–25 kg 3. 1.66 × 10–26 kg 4. 1.66 × 10–27 kg 11. The number of atoms present in 0.05g of water is 1. 1.67 × 1023 2. 1.67 × 1022 3. 5.02 × 1021 4. 1.67 × 1021 12. How many moles of Magnesium phosphate Mg3(PO4)2 will contain 0.25 mole of

oxygen atoms? 1. 0.02 2. 3.125 × 10–2 3. 1.25 × 10–2 4. 2.5 × 10–2

13. Which has the maximum no. of atom? 1. 6 g C 2. 1 g H2 3, 12 g Mg 4. 30 g Ca

Answers – Mole concept –DPP-1 Subjective 1. [0.0002] 2. [(a ) 6.022 × 1024 (b) 9.033 × 1024 (c) 7.2264 × 1025 (d) 6.323 × 1025 3. [A. 25× 2 × NA B. 0.24 × 5 × NA C. 1.2× 24× NA D. 200× 10–3× 3× NA ] 4. [A. 0.5 B. 20 C. 5 × 10–4 D. 0.04] 5. [A. 6.022 × 104 B. 10 ×10 × 6.022 × 1023 C. 3.17 × 1022 D. 3.85 × 1024 ] 6. [115 amu] 7. [29.88 × 10-24 gram] 8. [39.84 × 10-24 gram] 9. [1000] 10. [180]11. [0.25] 12. [224]

Objective

Q. No. 1 2 3 4 5 6 7 8 9 10 Answer 2 3 3 4 2 2 2 4 4 4 Q. No. 11 12 13 Answer 3 2 2



Trigonometry – III 1. Express each of the following products into sums or difference of sines and cosines. (i) 2 cos 3 sin 2 (ii) 2 sin 5 cos 3 (iii) cos 9 cos 4 (iv) sin 75 cos 15

2. Prove that )140(sin21115cos25sin

3. Prove that : (i) 16380sin60sin40sin20sin

(ii) 8180cos40cos20cos (iii) 380tan60tan40tan20tan

4. Prove that : (i) 16370cos50cos30cos10cos

(ii) tan 20 tan 30 cos40 cos80 = 1

5. Prove that : (i)

3cos3

cos3

coscos4

(ii) 2

5sin5sin2

9cos3cos2

cos2cos xxxxxx

6. show that: 0)(cos)(sin)(cos)(sin)(cos)(sin DCBADBACDACB



9. If ,3

1sin,2


cot2

tan

10. Express each of the following as product of sines and cosines (i) cos 9 + cos 3 (ii) sin 2 + cos 4 (iii) cos 12 – cos 4 (iv) sin 9 + sin 5 11. Prove that (i) 20cos265cos65sin (ii) 17cos77cos47sin 12. Prove that

(i) xxxxx cot

5sin7sin5cos7cos

(ii)

xx

xxxx

10cos2sin

3sin17sin5cos9cos

(iii) xxxxx 2tan

3coscos3sinsin

(iv) )3sin5(sincot)3sin5(sin4cot xxxxxx

(v) xxx

xx sin2cossin

3sinsin22


13. Prove that

(i) cos10 sin10 tan 35cos10 sin10

(ii) 020cos40cos80cos

(iii) 80sin70sin50sin40sin20sin10sin

(iv) 0140cos100cos20cos

14. Prove that

(i) 05

7cos5

6cos5

2cos5

cos

(ii) 1cos sin

12 12 2

(iii) 9

sin39

4cos185sin

(iv) xxx sin24

3cos4

3cos

(v) xxx cos24

cos4

cos

15. Prove that

(i) cos + cos + cos + cos( + + ) = 2

cos2

cos2

cos4

16. (i) 2

tancoscossinsin yx

yxyx

(ii)

2tan

coscossinsin yx

yxyx

(iii)

2cot

2tan

coscossinsin yxyx

yxyx

17. Prove that

(i) sin 3x + sin 2x – sin x = 4 sin x cos 2x cos

23x

(ii) xxxxxxx 4sin2coscos47sin5sin3sinsin

(iii) 3tan

5cos3coscos5sin3sinsin (iv) cos 4 cos3 cos 2 cot 3

sin 4 sin 3 sin 2

(v) sin 2sin 3 sin 5 sin 3sin 3 2sin 5 sin 7 sin 5

(vi) 3tan2sin2cos

5cos3cos2cos7cos5cos23cos

(vii)

tancos5cos

sin3sin25sin


18. If cosec A + sec A = cosec B + sec B, Prove that tan A tan B = 2

cot BA

19. Prove that

(i) (cos – cos )2 + (sin – sin )2 = 4 sin2

2

(ii) sin + sin + sin – sin ( + + ) = 4 sin

2sin

2sin

2

20. If ,0)cos()cos(

)cos()cos(

DCDC

BABA prove that 1tantantantan DCBA

21. If ,4

BA show that .2)1(cot)1(cot BA

22. If 0.cot 7cot that show,8

23. If x tantan and ,cotcot y prove that xy

yx )cot(

24. If ,tan2tan show that .3)sin()sin(

25. If ,cossinandsincos nBAmBA prove that. .2)sin(2 22 nmBA

26. If ,33

2tan3

tantan

xxx then prove that 1tan31

tantan32

3

x

xx

27. If a right angle be divided in to three parts , and , prove that cot cot cot cot cot cot 28. If sin sin – cos cos = 1, show that tan + tan = 0.



tan ( +2) and tan (2 + ).

30. If m tan ( – 30) = n tan ( + 120). Show that )(2

2cosnm

nm

31. If sin 2A = sin 2B, prove that 11

)tan()tan(

BABA

32. If cos ( + ) sin ( + ) = cos ( – ) sin ( – ), prove that cot cot cot = cot 33. If y sin = x sin (2 + ) show that (x + y) cot ( + ) = (y – x) cot

34. If + and ,tantan

yx

prove that .sin)sin(

yxyx

35. If and are the solutions of the equation a cos + b sin = C. then show that

(i) 22

22

)cos(baba

(ii) 22

222 )(2)cos(ba

bac


36. Find the maximum and minimum values of the following expressions: (i) a cos – b sin (ii) 7 cos + 24 sin 37. Show that

(i) AAAAA

2cotcot3cot

1tan3tan

1

(ii) AAAAA

4cotcot3cot

1tan3tan

1

38. Prove that

(i) xxxx 8sin4sin6cos2cos 22 (ii)

22

22

sincossinsin)tan()tan(

39 Prove that (i) 0)sin()sin()sin()sin()sin()sin( ACACCBCBBABA (ii) .0})12tan{(})12{(tan nn

40. If ,12

1tan,1

tan

mm

m prove that .4

ANSWERS

1. (i) sin 5 – sin (ii) sin 8 – sin 2 (iii) )5cos13(cos21

10. (i) 2 cos 6 cos 3 (ii)

3

4cos

4cos2 (iii) – 2sin 8 sin 4

(iv) 2 sin 7 cos 2 36. (i) maximum = ,22 ba minimum = – 22 ba (ii) maximum = 25, minimum = – 25

Page No. - 1

JRS TUTORIALS, Durgakund, Varanasi – 221 005 (U.P.) Ph. No. (0542) 2311922, 2311777 Mob. : 9794757100, 7317347706 e-mail : [email protected] Web site : www.jrstutorials.ac.in

JRS TUTORIALS Mathematics

Vectors (Collinearity and Coplanarity) DPP – 1

1. Prove that points A(1, 3, 2), B(–2, 0, 1) and C(4, 6, 3) are collinear. 2. If position vectors of points A, B, C are baba 23,, respectively (where a and b are

non-collinear vectors). Then prove that A, B, C are collinear points. 3. Points P, Q, R, S have position vectors cb32,c4b33a5,c4a2 and

ca2 respectively. Prove that line segment PQ is parallel to line segment RS. (where candb,a are linearly independent vectors).

4. If points (1, x, 3), (3, 4, 7) and (y, –2, –5) are collinear. Prove that 1 yx . 5. a and b are non-collinear vectors. Find ‘x’ for which (x–2) ba)1x2(,ba are

collinear vectors. 6. Let candb,a are three vectors of which every pair is non-collinear. If ba is collinear

with cbandc is collinear with a . Prove that 0cba . 7. Prove that vectors k4j3i7,kj3i2,kji are non-coplanar. 8. If vectors k5jai3andk3j2i,kji2 are coplanar. Prove that | a | = 4. 9. If candb,a are non-coplanar, prove that vectors cbacba 432,32 and c2b are coplanar. 10. If candb,a are non-coplanar vectors. Prove that four points ,c3b2a,cb3a2 c2b4a3 and c6b6a are coplanar.

Page No. - 2

JRS TUTORIALS, Durgakund, Varanasi – 221 005 (U.P.) Ph. No. (0542) 2311922, 2311777 Mob. : 9794757100, 7317347706 e-mail : [email protected] Web site : www.jrstutorials.ac.in

11. Let b)1y3x2(a)2x2y(qandb)1yx2(a)y4x(p where banda are non-collinear and q2p3 . Find x and y.

12. Find ‘a’ if points j52iaandj8i40,j3i60 are collinear. 13. kjic,k4j3i4b,kjia where candb,a are linearly dependent

and 1|| c then prove that = 1, = 1. 14. k5j2i3randk3ji2c,k2j3ib,kji2a 1 . If crbqapr1

then prove that p = q + r. 15. Let candb,a are three non-zero vectors, no two of which are collinear. If b7a3 is

collinear with c2b3andc is collinear with a . Prove that 0c14b21a9 . 16. )kxj5i4(Dand)k2ji(C),k4j3i2(B),kj2i3(A are coplanar points.

Find x. 17. Let candb,a are three non-coplanar vectors, such that acbrcbar 21 , , cbarandbacr 4323 . If 321 rzryrxr then prove that x + z = 3. 18. If vectors kji2andkji,k2ji are coplanar. Then find .

19. Let a, b and c are distinct non-negative numbers. If vectors andki,kcjaia kbjcic lie in a plane then prove that c is geometric mean of a and b. 20. The vectors jiandjiji 865,32 have their initial points at (1, 1). Find the

value of so that the vectors terminate on one straight line.

ANSWERS

5. 31 11. x = 2 and y = –1 12. a = – 40

16. 17146 18. = –2, 1 3 20. 9




Subjective 1. A solid contains A+ and B¯ ions. The structure of solid is fcc for B¯ ions and A+ ions are

present in one fourth of tetrahedral voids as well as in one fourth of octahedral voids. What is the simplest formula of solid ?

2. A cubic solid is made by atoms A forming close pack arrangement, B occupying one.

Fourth of tetrahedral void and C occupying half of the octahedral voids. What is the formula of compound

3. Spinel is a important class of oxides consisting of two types of metal ions with the oxide

ions arranged in CCP pattern. The normal spinel has one-eight of the tetrahedral holes occupied by one type of metal ion and one half of the octahedral hole occupied by another type of metal ion. Such a spinel is formed by Zn2+, Al3+ and O2–, with Zn2+ in the tetrahedral holes. Give the formulae of spinel

4. A closed packed structure of uniform spheres has the edge length of 534 pm. Calculate

the radius of sphere, if it exist in (a) simple cubic lattice (b) BCC lattice (c) FCC lattice

5. An element crystallizes into a structure which may be described by a cubic type of unit cell having one atom on each corner of the cube and two atoms on one of its body diagonals. If the volume of this unit cell is 24×10–24 cm3 and density of element is 7.2 g cm–3, calculate the number of atoms present in 200 g of element.

6. A cubic unit cell contains manganese ions at the corners and fluoride ions at the center of each edge. (a) What is the empirical formula of the compound? (b) What is the co-ordination number of the Mn ion? 7. An element (atomic mass = 100) having BCC structure has unit cell edge length 400

pm. The density of this element will be (NA = 6 × 1023) 8. A solid has three types of atoms X, Y and Z. ‘X’ forms a FCC lattice with ‘Y’ atoms

occupying all the tetrahedral voids and ‘Z’ atoms occupying half the octahedral voids. The simplest formula of solid is

9. Lithium metal crystallizes in a body-centred cubic crystal. If the length of the side of the

unit cell of lithium is 351 pm, the atomic radius of lithium will be 10. A crystalline solid of a pure substance has a face-centred cubic structure with a cell edge

of 400 pm. If the density of the substance in the crystal is 8 g cm–3, Calculate the number of atoms present in 256 g of the crystal :


Page-2

Only one option is correct 1. Copper crystallizes in a face-centred cubic lattice with a unit cell length of 361 pm.

What is the radius of copper atom in pm? 1. 157 2. 181 3. 108 4. 128 2. If the radius of the anion in an ionic solid is 200 pm, what would be the radius of the

cation that fits exactly into a cubic hole 1. 146.4 pm 2. 82.8 pm 3. 45 pm 4. 60.8pm

3. The decreasing order of size of the void is 1. cubic > octahedral > tetrahedral > Trigonal 2. trigonal > tetrahedral > octahedral > cubic 3. Trigonal > octahedral > tetrahderal > cubic 4. cubic > tetrahedral > octahedral > trigonal 4. For the structure given below the site marked as S is a 1. Tetrahedral void 2. Cubic void 3. Octahedral void 4. None of these 5. If a stands for the edge length of the cubic system: simple cubic, body centred cubic and

face centred cubic, then the ratio of radii of the spheres in these systems will be respectively

1. a22:a

23:a

21 2. 1a : a3 : a2

3. a22

1:a43:a

21 4. aaa

21:3:

21

6. In which of the following substances the carbon atom is arranged in a regular tetrahedral structure. 1. Diamond 2. Benzene 3. Graphite 4. Carbon black 7. For a solid with the following structure, the co-ordination number

of the point B is 1. 3 2. 4 3. 5 4. 6 8. In a hexagonal close packed (hcp) structure of spheres, the fraction of the volume occupied by the sphere is A. In a cubic close packed structure, the fraction is B. The relation for A and B is 1. A = B 2. A < B 3. A > B 4. A is equal to the fraction in a simple cubic lattice. 9. In a close packed array of N spheres, the number of tetrahedral holes are 1. 2/N 2. 4N 3. 2N 4. N

S

A B


Page-310. If the radius ratio is in the range of 0.225–0.414, then the coordination number

will be 1. 2 2. 4 3. 6 4. 8

11. In a solid C atom are forming CCP lattice with A atoms

occupying all tetrahedral voids and B atoms occupying all octahedral voids. Which of the following arrangement is obtained along the plane shown in diagram?

1. 2. 3. 4.

19. A metallic crystal crystallizes into a lattice containing a sequence of layers AB AB

AB....Any packing of spheres leaves out voids in the lattice. What percentage of volume of this lattice is empty space, 1. 74% 2. 26% 3. 50% 4. 40%

13. The fraction of octahedral voids filled by Al3+ ions in Al2O3 )43.0/( 23 =−+ OAl

rr is 1. 0.43 2. 0.287 3. 0.667 4. 1

14. Which of the following expressions is correct for an NaCl unit cell with lattice parameter a

1. 2a

rr ClNa =+ −+ 2. −+ + ClNa rr = 4a

3. 4a

r ClNa r =+ −+ 4. a43

rr ClNa =+ −+

15. In a unit cell atoms (A) are present at all corners atom(B) are present at 50% faces and all edge centre, Atoms(C) are present at face centres left from (B) and 1 at each body diagonal at distance of 1/4 th of body diagonal from corner. Formula of the given solid is

1. A3B8C7 2. AB4C6 3. A6B4C8 4. A2B9C11

16. Which of the following statements is correct for 3CsBr 1. It is a covalent compound

2. It contains +3Cs and −Br ions 3. It contains +Cs and −

3Br ions. 4. It contains +Cs , −Br and lattice 2Br molecule

17. A crystal is made of particles X and Y. X forms fcc packing and Y occupies all the

octahedral voids. If all the particles along one body diagonal are removed then the formula of the crystal would be

1. X4Y3 2. X5Y4 3. X4Y5 4. X3Y4

CCP unit cell

C C

C C

B

B

B B A

B A

C C C

C C C

B B B

C C C

C C C

B B B A A A A

C C C

C C C

B B B


Page-4

18. In a face centred cubic arrangement of A and B atoms when A atoms are at the corner of the unit cell and B atoms of the face centres. One of the A atom is missing from one corner in unit cell. The simplest formula of compound is

1. 37BA 2. 3AB 3. 247BA 4. 38/7 BA

19. Ferrous oxide has a cubic structure and each edge of the unit cell is 5.0 Å. Assuming density of the oxide as 4.0 g 3cm−− , then the number of +2Fe and −2O ions present in each unit cell will be 1. Four +2Fe and four −2O 2. Two +2Fe and four −2O 3. Four +2Fe and two −2O 4. Three +2Fe and three −2O

20. For an octahedral arrangement the lowest radius ratio limit is 1. 0.155 2. 0.732 3. 0.414 4. 0.225

Answer DPP-3

Subjective 1. [A3 B4] 2. [A4B2C2] 3. [ZnAl2O4] 4. [ 267pm , 231.2pm ,188.8pm

5. [3.472 × 1024 atoms] 6. [ (a) MnF3 (b) 6 ] 7. [5.2 g/ml]

8. [X2Y4Z] 9. [151.8 pm] 10. [2 × 1024]

Only one option is correct

Q.No. 1 2 3 4 5 6 7 8 9 10 Ans 4 1 1 3 3 1 4 1 3 2

Q.No. 11 12 13 14 15 16 17 18 19 20 Ans 3 3 3 1 4 3 2 3 1 3




Subjective 1. Cesium chloride forms a body centered cubic lattice. Cesium and chloride ions are in

contact along the body diagonal of the unit cell. The length of the side of the unit cell is 412 pm and Cl– ion has a radius of 181 pm. Calculate the radius of Cs+ ion.

2. If the radius of Mg2+ ion, Cs+ ion, O2– ion, S2– ion and Cl– ion are 0.65 Å , 1.69 Å, 1.40

Å, 1.84 Å, and 1.81 Å respectively. Calculate the co-ordination numbers of the cations in the crystals of MgS, MgO and CsCl.

3. KCl crystallizes in the same type of lattice as does NaCl. Given that 5.0=−

+

Cl

Na

rr

and

7.0=+

+

K

Na

rr

Calculate:

(a) The ratio of the sides of unit cell for KCl to that for NaCl and (b) The ratio of densities of NaCl to that for KCl. 4. A unit cell of sodium chloride has four formula units. The edge of length of the unit

cell is 0.564 nm. What is the density of sodium chloride. 5. In a cubic crystal of CsCl (density = 3.97 gm/cm3) the eight corners are occupied by Cl–

ions with Cs+ ions at the centre. Calculate the distance between the neighbouring Cs+ and Cl– ions.

6. An ionic compound AB has ZnS type structure. if the radius A+ is 22.5 pm, then

calculate the ideal radius of B- would be 7. KF has NaCl structure. What is the distance between K+ and F– in KF if density of KF is

2.48 gm/cm3. 8. NaH crystallizes in the same structure as that of NaCl. The edge length of the cubic unit

cell of NaH is 4.88 Å. (a) Calculate the ionic radius of H–, provided the ionic radius of Na+ is 0.95 Å. (b) Calculate the density of NaH.

9. AgCl has the same structure as that of NaCl. The edge length of unit cell of AgCl is

found to be 555 pm and the density of AgCl is 5.561 g cm–3. Find the percentage of sites that are unoccupied.

10 A crystal of lead(II) sulphide has NaCl structure. In this crystal the shortest distance

between Pb+2 ion and S2– ion is 297 pm. What is the length of the edge of the unit cell in lead sulphide? Also calculate the unit cell volume.


Page-2

Only one option is correct 1. A binary solid (AB) has a rock salt structure. If the edge length is 400 pm, and radius of

cation is 80 pm the radius of anion is 1. 100 pm 2. 120 pm 3. 250 pm 4. 325 pm 2. In which case, can we see 12 as coordination number : 1. ZnS 2. CaF2 3. NaCl 4. Ag 3. In CsCl structure, the coordination number of +Cs is 1. Equal to that of −Cl , that is 6 2. Equal to that of −Cl , that is 8 3. Not equal to that of −Cl , that is 6 4. Not equal to that of −Cl , that is 8 4. Which of the following statement (s) is incorrect

1. The coordination number of each type of ion in CsCl crystal is 8 2. A metal that crystallizes in bcc structure has a coordination number of 12 3. A unit cell of an ionic crystal shares some of its ions with other unit cells. 4. The adge length of the unit cell in NaCl is 552 pm ( +Na

r = 95 pm; −Clr = 181 pm)

5. If the radius of Cs+ = 1.69 Å and Br- = 1.95 Å, then which of the following is correct

statement? 1. Edge length of until cell is 8.2 Å 2. Coordination number of Cs+ is 6 3. CsBr has BCC type structure 4. Br– ions touch each other along the edge 6. RbCl has NaCl type lattice and its unit cell length is 0.30 Å greater than that for KCl.

If +Kr = 1.33 Å , the ionic radius of Rb+ is

1. 1.48 Å 2. 1.63 Å 3. 1.03 Å 4. 1.75Å 7. 8 : 8 co-ordination is noticed in – 1. MgO 2. Al2O3 3. CsCl 4. All 8. A certain sample of cuprous sulphide is found to have composition Cu1.8S, because of

incorporation of Cu2+ ions in the lattice. What is the mole % of Cu2+ in total copper content in this crystal?

1. 99.8% 2. 11.11% 3. 88.88% 4. 94% 9. A solid +A −B has a body centred cubic structure. The distance of closest approach between the two ions is 0.767 Å. The edge length of the unit cell is

1. 23 pm 2. 142 = 2 pm 3. 2 pm 4. 81.63 pm.

10. For an ionic crystal of the general formula AX and co-ordination number 6, the radius ratio value will be, 1. greater than 0.73 2. between 0.73 and 0.41

3. between 0.41 and 0.22 4. less than 0.22 11. NH4Cl crystallizes in a body-centered cubic type lattice with a unit cell edge length of

387 pm. The distance between the oppositively charged ions in the lattice is 1. 335.1 pm 2. 83.77 pm 3. 274.46 pm 4. 137.23 pm 12. If the distance between Na+ and Cl– lons in NaCl crystal is 'a' pm, what is the length of

the cell edge? 1. 2a pm 2. a/2 pm 3. 4a pm 4. a/4 pm


Page-3

13. Which statement is correct in antifluorite structure of Na2O 1. O2– ions have CCP arrangement 2. It has 4 :8 coordination 3. Na+ ions occupy all the tetrahedral sites 4. All are correct 14. The number of nearest neighbours and next nearest neighbours of a Na+ ion in a crystal

of NaCl are respectively : 1. 6Na+, 12Cl– 2. 6Cl–, 12Na+ 3. 12Cl–, 6Na+ 4. 6Cl–, 6Na+ 15. In solid CsCl each Cl– is closely packed with how many Cs+ 1. 8 2. 6 3. 10 4. 2 16. For an ionic crystal of the general formula AX and coordination number 6, the value of radius ratio will be 1. Greater than 0.73 2. In between 0.73 and 0.41 3. In between 0.41 and 0.22 4. Less than 0.22. 17. Which of the following statements is correct in the zinc-blende type structure of an ionic

compound? 1. Coordination number of each cation and anion is two 2. Coordinate number of each cation and anion is four 3. Coordination number of each cation and anion is six 4. Coordination number of each cation and anion is eight 18. Select the incorrect statement 1. CsCl changes to NaCl structure on heating 2. NaCl changes to CsCl structure on applying pressure 3. Coordination number increases on applying pressure

4. Coordination number increases on heating

19. CsBr has bcc type structure with edge length 4.3 pm. The shortest inter ionic distance in between Cs+ and Br– is

1. 3.72 pm 2. 1.86 pm 3. 7.44 pm 4. 4.3 pm

20. In a solid ‘AB’ having the NaCl structure, ‘A’ atoms occupy the corners of the cubic unit cell. If all the face centered atoms along one of the axes are removed, then the resultant stoichiometry of the solid is :

1. AB2 2. A2B 3. A4B3 4. A3B4

Answer –DPP-4

Subjective

1. [175.8 pm] 2. [4, 6, 8] 3. [(a) 1.143, (b) 1.172] 4. [2.16 gm/cm3]

5. [3.57 Å] 6. [100pm] 7. [2.685 Å] 8. [(a) 1.49 Å, (b) 1.37 g/cm3] 9. [0.24%]

10. [a =5.94 ×10–8 cm, V=2.096×10–22 cm–3]

Only one option is correct

Q.No. 1 2 3 4 5 6 7 8 9 10 Ans 2 4 2 2 3 1 3 2 4 2

Q.No. 11 12 13 14 15 16 17 18 19 20 Ans 1 1 4 2 1 2 2 4 1 4

PHYSICS



DPP-05 : ELECTRIC FIELD1. Two charges 10 C and 25 C are placed at (1,0) and (4,0), electric field at (0,3) =

(A) ˆ ˆ(63 27 )i j

(B) 2 ˆ ˆ10 (63 27 )i j

(C) ˆ ˆ(27 63 )i j

(D) 2 ˆ ˆ10 (27 6 )i j2. Four charges are placed on the circumference of a circle of radius R, 90° apart as shown in the

fig. The electric field strength at the centre of the circle is

(A) 20 R

Q524

1 , making angle tan–12 with the – ve x-axis.

(B) 20 R

Q524

1 , making angle tan–12 with the + ve y-axis.

(C) 20 R

Q244

1 , making angle tan–1

21

with the – ve x-axis.

(D) 20 R

Q244

1 , making angle tan–1

21

with the + ve y-axis.

3. A point charge q is placed at origin. Let AE

, BE

and CE

be the electric field at three pointsA (1, 2, 3), B (1, 1, – 1) and C (2, 2, 2) due to charge q. Then [i] AE

BE

[ii] | BE

| = 4 | CE

|select the correct alternative(A) only [i] is correct (B) only [ii] is correct(C) both [i] and [ii] are correct (D) both [i] and [ii] are wrong

4. A positive charge +Q located at the origin produces an electric field E0 at point P (x = +1, y = 0).A negative charge –2Q is placed at such a point as to produce a net field of zero at point P. Thesecond charge will be placed on the

(A) x-axis where x > 1

(B) x-axis where 0 < x < 1

y

x+Q (1,0)

P-x(C) x-axis where x < 0

(D) y-axis where y > 05. Charges q, 2q, 4q, 8q, ..... are placed along x-axis at r, 2r, 4r, 8r, ..... from origin respectively.

The net electric field at origin is

(A) Infinite (B) 20

q4 r (C) 2

0

q2 r (D) 2

0

q8 r

PHYSICS


6. Four point charges q, q, Q and 2Q are placed in order at the corners A, B, C and D of a square.

If the field at the midpoint of CD is zero then the value of Qq

is

(A) 1 (B) 25

(C) 5

22(D)

255

7. Two point charges q and 2q are placed at (a, 0) and (0, a). A point charge q1 is placed at a pointP on the quarter circle of radius a as shown in the diagram so that the electric field at the originbecomes zero:

(A) the point P is

3a2,

3a

(B) q1 = – 5q s

(C) the point P is

5a2,

5a

(D) both (B) and (C)

8. An equilateral triangle wire frame of side L having 3 point charges at its vertices is kept in x-yplane as shown. Component of electric field due to the configuration in z direction at (0, 0, L)is [origin is centroid of triangle]

(A) 2L8kq39

(B) zero

(C) 2L8kq9

(D) None

9. Six charges q,q,q, – q, –q and –q are to be arranged on the vertices of a regular hexagon PQRSTUsuch that the electric field at centre is four times the field produced when only charge ‘q’ isplaced at vertex R. The sequence of the charges from P to U is(A) q, –q, q, q, –q, –q(B) q, q, q, –q, –q, –q(C) –q, q, q, –q, –q, q(D) –q, q, q, q, –q, –q

10. Two point like charges Q1 and Q2 of whose strength are equal in absolute value are placed at acertain distance from each other. Assuming the field strength to be positive in the positive directionof x-axis, the signs of the charges Q1 and Q2 for the graphs (field strength versus distance)shown in Fig.

(A) (Q1 positive; Q2 negative); (both positive); (Q1 negative; Q2 positive); (both negative)(B) (Q1 negative; Q2 positive); (Q1 positive; Q2 negative); (both positive); (both negative)(C) (Q1 positive; Q2 negative; (both negative); (Q1 negative; Q2 positive); (both positive)(D) (both positive; (Q1 positive, Q2 negative); (Q1 negative, Q2 positive); (both negative)

PHYSICS


11. Four equal positive charges are fixed at the vertices of a square of side L. Z-axis is perpendicularto the plane of the square. The point z = 0 is the point where the diagonals of the square intersecteach other. The plot of electric field due to the four charges, as one moves on the z-axis.

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

12. A wire is bent in the form of a regular hexagon of side a and a total charge Q is distributeduniformly over it. One side of the hexagon is removed. The electric field due to the remainingsides at the centre of the hexagon is

(A) 20

Q12 3 a (B) 2

0

Q16 3 a (C) 2

0

Q8 3 a (D) 2

0

Q12 3 a

13. The direction () of E at point P due to uniformly charged finite rod will be

(A) at angle 300 from x-axis(B) 450 from x - axis(C) 600 from x-axis(D) none of these

14. The charge per unit length of the four quadrant of the ring is 2 , – 2 , and – respectively..The electric field at the centre is

(A) – iR2 0

(B) j

R2 0

(C) iR4

2

0

(D) – 0

i4 R

15. In x-y plane a circular wire AB, of radius R is uniformly charged with linear charge density(0) which is constant, is shown in the figure. Centre of the arc conside with the origin. Theelectric field intensity at the origin is :

(A) ji3R2

K 0 (B) ji

R2K3 0

(C) j3iR2

K 0 (D) ki3

R2K 0

16. The ratio of the magnitude of electric field at O due to inner (r1 to r2) and outer (r3 to r4) partof the disc :

(A) )r/r(n)r/r(n

43

12

(B) )r/r(n)r/r(n

43

21

(C) )r/r(n)r/r(n

34

21

(D) )r/r(n)r/r(n

31

42

PHYSICS


17. The electric field at the centre of a hemispherical surface having uniform surface chargedensity is

(A) 0

σε (B)

0

σ2ε (C)

0

σ4ε (D)

0

σ8ε

18. The maximum electric field at a point on the axis a uniformly charged ring is E0. At how manypoints on the axis will the magnitude of electric field be E0/2(A) 1 (B) 2 (C) 3 (D) 4

19. The column I gives the two point charge system separated by 2a and the column II gives thevariation of magnitude of electric field intensity along x-axis.Match the situation in Column Iwith the results in Column II

Column – Column –

(A) + +(0, 0) a

q q

(a, 0)(-a, 0)xx' (p) Increases as x increases

in the interval 0 x < a

(B) + –(0, 0) a

q -q

(a, 0)(-a, 0)xx' (q) Decreases as x increases

in the interval 0 x < a

(C)

++

(0, 0)

q

q

(0,+a)

(0,–a)

x

y

(r) Zero at x = 0

(D)

–

+

(0, 0)

–q

q

(0,+a)

(0,–a)

x

y

(s) Decreases as x increasesin the interval a < x <

ANSWER KEY1. B 2. A 3. C 4. C 5. C 6. D7. C 8. B 9. D 10. A 11. D 12. A13. A 14. A 15. A 16. B 17. C 18. D19. (A) (p, r, s), (B) (p, s), (C) (r, s), (D) (q, s)



Trigonometry – III 1. Express each of the following products into sums or difference of sines and cosines. (i) 2 cos 3 sin 2 (ii) 2 sin 5 cos 3 (iii) cos 9 cos 4 (iv) sin 75 cos 15

2. Prove that )140(sin21115cos25sin

3. Prove that : (i) 16380sin60sin40sin20sin

(ii) 8180cos40cos20cos (iii) 380tan60tan40tan20tan

4. Prove that : (i) 16370cos50cos30cos10cos

(ii) tan 20 tan 30 cos40 cos80 = 1

5. Prove that : (i)

3cos3

cos3

coscos4

(ii) 2

5sin5sin2

9cos3cos2

cos2cos xxxxxx

6. show that: 0)(cos)(sin)(cos)(sin)(cos)(sin DCBADBACDACB



9. If ,3

1sin,2


cot2

tan

10. Express each of the following as product of sines and cosines (i) cos 9 + cos 3 (ii) sin 2 + cos 4 (iii) cos 12 – cos 4 (iv) sin 9 + sin 5 11. Prove that (i) 20cos265cos65sin (ii) 17cos77cos47sin 12. Prove that

(i) xxxxx cot

5sin7sin5cos7cos

(ii)

xx

xxxx

10cos2sin

3sin17sin5cos9cos

(iii) xxxxx 2tan

3coscos3sinsin

(iv) )3sin5(sincot)3sin5(sin4cot xxxxxx

(v) xxx

xx sin2cossin

3sinsin22


13. Prove that

(i) cos10 sin10 tan 35cos10 sin10

(ii) 020cos40cos80cos

(iii) 80sin70sin50sin40sin20sin10sin

(iv) 0140cos100cos20cos

14. Prove that

(i) 05

7cos5

6cos5

2cos5

cos

(ii) 1cos sin

12 12 2

(iii) 9

sin39

4cos185sin

(iv) xxx sin24

3cos4

3cos

(v) xxx cos24

cos4

cos

15. Prove that

(i) cos + cos + cos + cos( + + ) = 2

cos2

cos2

cos4

16. (i) 2

tancoscossinsin yx

yxyx

(ii)

2tan

coscossinsin yx

yxyx

(iii)

2cot

2tan

coscossinsin yxyx

yxyx

17. Prove that

(i) sin 3x + sin 2x – sin x = 4 sin x cos 2x cos

23x

(ii) xxxxxxx 4sin2coscos47sin5sin3sinsin

(iii) 3tan

5cos3coscos5sin3sinsin (iv) cos 4 cos3 cos 2 cot 3

sin 4 sin 3 sin 2

(v) sin 2sin 3 sin 5 sin 3sin 3 2sin 5 sin 7 sin 5

(vi) 3tan2sin2cos

5cos3cos2cos7cos5cos23cos

(vii)

tancos5cos

sin3sin25sin


18. If cosec A + sec A = cosec B + sec B, Prove that tan A tan B = 2

cot BA

19. Prove that

(i) (cos – cos )2 + (sin – sin )2 = 4 sin2

2

(ii) sin + sin + sin – sin ( + + ) = 4 sin

2sin

2sin

2

20. If ,0)cos()cos(

)cos()cos(

DCDC

BABA prove that 1tantantantan DCBA

21. If ,4

BA show that .2)1(cot)1(cot BA

22. If 0.cot 7cot that show,8

23. If x tantan and ,cotcot y prove that xy

yx )cot(

24. If ,tan2tan show that .3)sin()sin(

25. If ,cossinandsincos nBAmBA prove that. .2)sin(2 22 nmBA

26. If ,33

2tan3

tantan

xxx then prove that 1tan31

tantan32

3

x

xx

27. If a right angle be divided in to three parts , and , prove that cot cot cot cot cot cot 28. If sin sin – cos cos = 1, show that tan + tan = 0.



tan ( +2) and tan (2 + ).

30. If m tan ( – 30) = n tan ( + 120). Show that )(2

2cosnm

nm

31. If sin 2A = sin 2B, prove that 11

)tan()tan(

BABA

32. If cos ( + ) sin ( + ) = cos ( – ) sin ( – ), prove that cot cot cot = cot 33. If y sin = x sin (2 + ) show that (x + y) cot ( + ) = (y – x) cot

34. If + and ,tantan

yx

prove that .sin)sin(

yxyx

35. If and are the solutions of the equation a cos + b sin = C. then show that

(i) 22

22

)cos(baba

(ii) 22

222 )(2)cos(ba

bac


36. Find the maximum and minimum values of the following expressions: (i) a cos – b sin (ii) 7 cos + 24 sin 37. Show that

(i) AAAAA

2cotcot3cot

1tan3tan

1

(ii) AAAAA

4cotcot3cot

1tan3tan

1

38. Prove that

(i) xxxx 8sin4sin6cos2cos 22 (ii)

22

22

sincossinsin)tan()tan(

39 Prove that (i) 0)sin()sin()sin()sin()sin()sin( ACACCBCBBABA (ii) .0})12tan{(})12{(tan nn

40. If ,12

1tan,1

tan

mm

m prove that .4

ANSWERS

1. (i) sin 5 – sin (ii) sin 8 – sin 2 (iii) )5cos13(cos21

10. (i) 2 cos 6 cos 3 (ii)

3

4cos

4cos2 (iii) – 2sin 8 sin 4

(iv) 2 sin 7 cos 2 36. (i) maximum = ,22 ba minimum = – 22 ba (ii) maximum = 25, minimum = – 25

mathematics problem sheet trigonometry – iv - jrs tutorials

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