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Clyde Thompson
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1 Q.1. Which one of the following is scalar quantity? Displacement Option Electric field Acceleration Work Correct Answer 4 w = F.ds; it does not have any direction, it s a scalar quantity. Q.. Which one of the following is not the vector quantity? Torque Option Displacement Velocity Speed Correct Answer 4 Speed has no direction, it s not a vector Q.3. A vector is not changed if It is rotated through an arbitrary angle Option It is multipled by an arbitrary scalar It is cross multiplied by a unit vector It is slid parallel to itself. Correct Answer 4 A vector does not change, if its magnitude and direction are same. Thus it does not change, if it is slid parallel to itself. Q.4. What is the numerical value of vector 3 i + 4 j+ 5k? 3 Option Correct Answer a= a x +a y + a z = Q.5. Option = 5 The x and y components of a force are N and 3N. The force is i 3j i+ 3j i 3j 3i+ j Correct Answer 1 x component, F = i x y component, F = 3j y F = Fi + F x y j F = i 3j

4 F 1 +F Correct Answer Resultant of vectors, R = F + F + F F cos Q.1. Vectors are acting at 9, cos =. R = F 1 +F 6 Option Correct Answer 3 R = A + B Given R = A + B and R = A = B R = A + B + ABcos But, A = B = R. 1 A = A (1+ cos ) cos =. = The angle between A and B is Q.13. Option The resultant of two forces, each P, acting at an angle is Psin Pcos Pcos P Correct Answer Resultant, R = P + P + P cos = P (1+cos ) = P (cos ) cos = cos 1 = Pcos Q.14. The resultant of two vectors of magnitudes A and A acting at an angle is 1 A. The correct value of is? 3 Option Correct Answer Vector 1 A 1 = A; vector A = A ; Resultant, R = 1 A R = A 1 + A +A1A cos 1A = 4A +A + 4 A 4A = 4 A cos cos

6 A + B + ABcos = A B A + B + ABcos = A + B ABcos 4ABcos = cos = = 9 Q.18. The simple sum of two co initial vectors is 16 units. Their vectors sum is 8 units. The resultant of the vectors is perpendicular to the smaller vector. The magnitudes of the two vectors are: units and 14 units Option 4 units and 1 units 6 units and 1 units 8 units and 8 units Correct Answer 3 Co initial vectors are those, which start with same point. Given A + B = 16 (i) Q.. Option 3 A + B = 8 Since, resultant is perpendicular to smaller vector bsin tan = = a+bcos a a+bcos = cos = b Now, A + B = A +B +AB cos A 64 = A +B + AB =B A B 64 = (A + B) (B A) B A = 4 (ii) By (i) and (ii) B = 1; A = 6 (F 1 + F ) =F 1 +F +F1 F = = 49 F + F = 7 1 Also, (F 1 F ) =F 1 +F F1 F = 5 4 = 1 F 1 F =1 F 1 = 4 and F = 3. If the sum of the two unit vectors is also a unit vector, then magnitude of their difference is 4 7 Correct Answer a = b, b = 1. r = a + b ; r = 1 If Angle between a and b is ' '. r = a +b + abcos

7 1 1 = 1+1+ cos cos =. To find : r ' = a b ; r =? 1 r' = a +b abcos = = 3 Q.3. Given that A + B + C =. Out of three vectors, two are equal in magnitude and the magnitude of third vector times that of either of the two having equal magnitude. Then the angles between vectors are given by: 45, 45, 9 Option 9, 135, 135 3, 6, 9 45, 6, 9 Correct Answer A + B + C =. Let A = B and C = A = B C = A + B. C = A +B +ABcos A = A + A cos = 9 Angle between A and B is 9 Angle of A + B with A or B is 45 C is opposite to A + B, i.e. at 18 to A + B and of 135 with A or B Q.4. The sum of the magnitudes of two forces acting at a point is 16N. The resultant of these forces is perpendicular to the smaller force has a magnitude of 8 N. If the smaller force is magnitude x, then the value of x is N Option 4 N 6 N 7 N Correct Answer 3 F 1 + F = 16 F 1 + F = 8 Co initial vectors are those, which start with same point. Given A + B = 16 (i) A + B = 8 Since, resultant is perpendicular to smaller vector

8 bsin tan = = a+bcos a a+bcos = cos = b Now, A + B = A +B +AB cos A 64 = A +B + AB =B A B 64 = (A + B) (B A) B A = 4 (ii) By (i) and (ii) B = 1; A = 6 Q.5. Two vectors a and b are at an angle of 6 with each other. Their resultant makes an angle of 45 with a. If b = unit, then a is 3 Option Correct Answer = 6, a =? b = = 45 bsin tan = bsin = a + b cos a+bcos 3 1 = a + a = 3 1 Q.6. Two equal forces (F each) act a point inclined to each other at an angle of 1. The magnitude of their resultant is F Option F 4 F F Correct Answer R = F +F +F cos = F +F cos1 = R = F Q.7. If A and B are two vectors such that A + B = A B the angle between vectors A and B is :

9 Option Correct Answer 3 A + B = A B Q.8. A +B + ABcos = A +B ABcos 4 ABcos = cos = = 9 = Option = 3 = = Two vectors A and B are such that A + B = C and A + B = C. If is the angle between positive direction of A and B then the correct statement is Correct Answer 4 A + B = C and A + B = C...(1) C = A + B = A +B +AB cos () ABcos = cos = (By (1) and ()) =. Q.9. Given that P = 1, Q = 5 and R = 13 also P + Q = R, then the angle between P andq will be Option Zero 4 Correct Answer R = P + Q ; P = 1, Q = 5, R = 13 R =P + Q +PQ cos 13 = (1)(5)cos cos = = Q.3. 9 Option Between and only The angle between P + Q and P Q will be

10 None of these Correct Answer A = P + Q and B = P Q A + B = P A + B = A +B + A B = P + Q + PQ cos +P + Q PQ cos + AB cos { is angle between P and Q. is angle between A and B} P Q = AB cos cos can take all values between 1 to 1 is between & 18 Q.31. Two vectors of equal magnitude have a resultant equal to either of them, then the angle between them will be 3 Option Correct Answer F 1 = A ; F = A ; R = F 1 + F R = A Q R =F +F + F F co s ( is angle between F and F ) A = A +A cos 1 cos =. =1 Given that P + Q + R =. Two out of the three vectors are equal in magnitude. The magnitude of the third vector is times that of the other two. Which of the following can be the angles between these vectors? 9, 135, 135 Option 45, 45, 9 3, 6, 9 45, 9, 135 Correct Answer 1 A + B + C =. Let A = B and C = A = B C = A + B. C = A +B +ABcos A = A + A cos = 9 Angle between A and B is 9

11 Angle of A + B with A or B is 45 C is opposite to A + B, i.e. at 18 to A + B and of 135 with A or B Q.33. Given A = i + j 3k. When a vector B is added to A, we get a unit vector along X axis. Then, B is j + 3k Option i j i + 3k j 3k Correct Answer 1 A = i + j 3k Q.34. Let B = x i + y j + zk Given A + B = i 5 Option Correct Answer 1 A = 7 i + 6 j (1 + x) i +( + y) j + (z 3)k = 1 i Since, x, y and z components are independent to each other. Equating them on either side. 1 + x = 1 x = + y = y = z 3 = z = 3 B = j + 3k The magnitude of the X and Y components of A are 7 and 6. Also the magnitudes of X and Y components of A + B are 11 and 9 respectively. What is the magnitude of B? Let B = x i + y j + zk A + B = 11 i + 9 j (7 + x) i + (6 + y) j + zk = 11 i + 9 j Equating components. 7 + x = 11 x = y = 9 y = 3 z = z =

12 B = 4 i + 3 j Β = = 5units. Q.35. If the resultant of the vectors i + j 3k, i j + k and C is a unit vector along Option the y direction, then C is i k i + k i k i + k Correct Answer 1 A = i + j k ; B = i j + k Q.36. Let C = x i + y j + zk A + B + C = 1 i (+ x) i + (1+ y) j + (1 + z)k = 1 j Equating components. + x = x = 1 + y = 1 y = 1 + z = z = 1 C = i k What vector must be added to the sum of two vectors i j + 3k and 3 i j k so that the resultant is a unit vector along Z axis 5 i + k Option 5 i + 3 j 3 j + 5k 3 j + k Correct Answer A = i j + 3k ; B = 3 i j k Let a vector C = x i + y j + zk is added A + B + C = k (5+ x) i +(y 3) j +(1+ z)k = k Equating the components. 5 + x = x = 5 y 3 = y = z = 1 z = C = 5 i + 3 j

20 Correct Answer 3 If angle between A and B is ' ', then angle between A and B is 18 A B = A + B + ABcos (18 ) = ( cos ) = (1 cos ) = sin ( cos = 1 sin ) = sin Q.5. Option Correct Answer 4 A particle s velocity changes from i + 3 j m / s to 3 i j m / s in s. acceleration in m/s is : i + 5 j i + 5 j Zero i 5 j Initial velocity, V i = i + 3 j and final velocity, V j = 3 i j V = V V = i 5 j j i i 5 j acceleration = V = ( time taken is sec) The Q.53. If P = 4 i j + 6k and Q = i j 3k, then the angle which P + Q makes with x axis is 1 3 cos 5 Option 1 4 cos cos cos 5 Correct Answer 3 P = 4 i j + 6k and Q = i j 3k P + Q = 5 i 4 j + 3k

21 Angle made by a vector with x axis is given by x component cos = magnitude of vector 5 cos = = cos 5 Q.54. Given P = 3 j + 4k and Q = j + 5k. The magnitude of the scalar product of these vector is Option Correct Answer 3 P = 3 j + 4k and Q = j + 5k P Q = Px i + Py j + Pz k Qx i + Qy j + Qz k = Px Q x + Py Q y + Pz Qz = 6 + = 6 Q.55. If P = i 3 j + k and Q = 3 i j, then P Q is Zero Option Correct Answer 3 P = i 3 j + k and Q = 3 i j P Q = = 1 YhVOa8 Q.56. If A B = AB, then the angle between A andb is Option Correct Answer 1 A B = AB cos ( = angle between A and B) Given A B = AB AB cos = AB cos = 1 = Q.57. A force of 1 i 3 j + 6k N acts on a body of mass 1 g and displaces it from

22 6 i + 5 j 3k m to 1 i j + 7k m. The work done is 1 J Option 11 J 361 J 1 J Correct Answer F = 1 i 3 j + 6k ; r = 6 i + 5 j 3k ; r = 1 i j + 7k Q.58. Displacement, i S = rj r i = 4 i 7 j + 1k Work done, W = F ds = F s = = 11 J j A force F = i + j N displace a particle through S = i + k m in 16 s. The power developed by F is.5 J s 1 Option 5 J s 1 5 J s 1 45 J s 1 Correct Answer 1 F = i + j, S = i + k, time Work done, w = F ds = F s = 4 dw w 4 Power = = = =.5 J/ s dt t 16 t = 16 sec Q.59. Option If A = B, then which of the following is not correct A = B AB = AB A = B AB BA Correct Answer If A = B Both vectors have same magnitude and direction. A = B and A = B AB = BA and A B = AB cos = AB But, AB = (1)(1)cos = 1 A B = AB

23 Q.6. If A and A 1 are two non collinear unit vectors and if A 1 + A = 3, then the value of A1 A A 1 + A 1 Option 1 3 Correct Answer A = A = 1 and A = A = A + A = 3 1 If angle between A 1 and A is. A A = A A cos A + A + A A = 3 1 A1 A = ( A 1 = A = 1) A A A + A = A A A + A A A 1 = 1 1 = Q.61. Option Consider a vector F = 4 i 3 j. Another vector that is perpendicular to F is 4 i + 3 j 6 j 7 j 3 i + 4 j Correct Answer 4 F = 4 i 3 j For perpendicular vectors, A and B, A B = If A F, Let A = x i + y j 4x 3y = 4x = 3y A = 3 i + 4 j Q.6. 3 The angle between the z axis and the vector i + j + k is

24 Option Correct Answer A = i + j + Q.63. k Let B = k, If angle between A and z axis is then A B = AB cos. = cos 1 cos = = 45 If A = i + 3 j + 4k and B = 4 i + 3 j + k, then angle between A and B is 1 5 sin 9 Option 1 9 sin cos cos 5 Correct Answer 3 A = i + 3 j + 4k and B = 4 i + 3 j + k Let angle between A and B is ' ' A B = AB cos = cos cos = 5 9 = cos = cos Q.64. What is the angle between i + j + k and i Option 6 3 None of these Correct Answer 4

25 A = i + j + k and B = i, angle between A and B = ' ' A B = AB cos Q = cos = cos For what value of a, A = i + a j + k will be perpendicular to B = 4 i j k 4 Option Zero Correct Answer 3 A = i + a j + k and B = 4 i j k For perpendicular vectors, = 9 cos = A B = 8 a 1 = a = 7 7 a = = 3.5 Q.66. The vector sum of two forces is perpendicular to their vector differences. In that case, the forces Are not equal to each other in magnitude Option Cannot be predicted Are equal to each other Are equal to each other in magnitude Correct Answer 4 R = A + B and P = A B Given that R is perpendicular to P R P = A + B A B = A B = A = B A = B Q.67. Option Projection of P onq is PQ PQ PQ

26 PQ Correct Answer 1 Projection of P onq = P cos = angle between P and Q P Q = PQ cos PQ Pcos = = P Q Q Q.68. The component of vector A = ax i + ay j + az k along the direction of i j is (a x a y + a z ) Option (a x + a y ) a a x (a x a y + a z ) Correct Answer 3 A = a i + a j + a k, B = i j y x y z Component of A along B = Acos = A B i j = a i + a j + a k x y z x = a a y Q.69. Given is the angle between A and B. Then A B is equal to sin Option cos tan cot Correct Answer 1 A B = AB sin A B = AB sin A B = (1) (1)sin = sin A = B = 1 Q.7. If PQ =, then PQ is

27 P Q Option Zero 1 PQ Correct Answer 1 PQ = angle between P andq = 9 P Q = P Q sin(9 ) = P Q Q.71. Option Correct Answer 3 c = a Q.7. Given c = a b. The angle which a makes with c is b By definition of cross product, c is a vector perpendicular to both a and b The angle a makes with c is 9 The magnitudes of the two vectors a and b are a and b respectively. The vector product a and b cannot be Equal to zero Option Less than ab Equal to ab Greater than ab Correct Answer 4 a b = absin sin 1 a b ab a b cannot be greater than ab. Q.73. Option Given r = 4 j and p = i + 3 j + k. The angular momentum is 4 i 8k 8 i 4k 8 j

28 9k Correct Answer 1 r = 4 j and p = i + 3 j + k Angular momentum, L = r p i j k = = i(4) j() + k( 8) = 4 i 8k Q.74. Option Given A = 4 i + 6 j and B = i + 3 j. Which of the following is correct? A B = A B = 4 A B 1 = A and B are anti parallel Correct Answer 1 A = 4 i + 6 j and B = i + 3 j i j k (a) A B = 4 6 = i () j() + k(1 1) = 3 (b) A B = (8 + 18) = 6 A (c) = = = B (d) A = i + 3 j = B A and B are parallel. Q.75. If A B = and A B = 1, then A and B are Perpendicular unit vectors Option Parallel unit vectors Parallel Perpendicular Correct Answer 4

30 Correct Answer 3 Since, A B = A is perpendicular to B And, A C = A is perpendicular to C Now, By definition of vector product, B C is a vector perpendicular to both B and C A is parallel to B C. Q.79. If the magnitudes of scalar and vectors products of two vectors are 6 and 6 3 respectively, then the angle between two vectors is 15 Option Correct Answer 3 A B = 6 and A B = 6 3 ABcos = 6 and ABsin = tan = = 3 6 = 6 Q.8. Given that A and B are greater than 1. The magnitude of A B Equal to AB Option Less than AB More than AB Equal to A/B Correct Answer 3 a b = absin sin 1 a b ab cannot be a b cannot be greater than a b. A B AB Q.81. If A = i + 3 j + 6k and B = 3 i 6 j + k then vector perpendicular to both A and B has magnitude k times that of 6 i + j 3k. That k is equal to 1 Option Correct Answer 3

31 A = i + 3 j + 6k and B = 3 i 6 j + k A vector perpendicular to both B and A is A B i j k A B = 3 6 = i(6 + 36) j(4 18) + k( 1 9) 3 6 Q.8. = 4 i + 14 j 1k = k 6 i + j 3k k = 7 A proton of velocity 3 i + j 5 1 ms 1 enters a magnetic field i + 3k T. If the 7 specific charge is C kg 1, the acceleration of the proton in ms is 6 i 9 j + 4 k Option 6 i + 9 j + 4k i 9 j 4k i + 9 j 4 k Correct Answer 3 V = 3 i + j 5 1 B = i + 3k Q.83. Option i j k V B = 3 1 = i(6) j(9) + k( 4) Force, F = q V B = i 9 j 4k 1 = 6 i 9 j 4k Angle between A and B is What is the value of A B A? A Bcos A Bsin cos A Bsin zero Correct Answer 4

32 . B A is a vector perpendicular to both A and B angle of A with B A is 9 A B A = A B A cos9 = Q.84. If A B = B A, then the angle between A and B is : Option 3 4 Correct Answer 1 A B = AB sin is between A and B B A = AB sin A B = B A AB sin = AB sin sin = = Q.85. The area of a parallelogram farmed from the vector A = i j + 3k and B = 3 i j + k as adjacent side is 8 3 units Option 64 units 3 units 4 6 units Correct Answer 4 A = i j + 3k and B = 3 i j + k Area of parallelogram with sides as A and B, i j k Area = A B = = i( + 6) j(1 9) + k( + 6) = 4 i + 8 j + 4k Area = = 96 = 4 6 units

33 Q.86. Option A vector F 1 is along the positive Y axis. If its vector product with another vector F is zero, then F could be 4 j j + k j k 4 i Correct Answer 1 Let F = k j Q Let F = x i + y j + zk i j k F F = k = i (kz) j() + k( kx) = 1 x y z z = and x = F is along y axis F can be 4 j If the vector A = i + 4 j and B = 5 i p j are parallel to each other, the magnitude of B is 5 5 Option Correct Answer 1 A = i + 4 j and B = 5 i p j Since, A is parallel to B angle between them is A B = AB sin() = i j k 4 = i() j() + k( p ) = 5 p p = p = 1 B = 5 i + 1 j B = = 15 = 5 5 Q.88. The vectors i + 3 j k, 5 i + a j + k and i + j + 3k are coplanar when a is

34 9 Option Correct Answer 4 A = i + 3 j k, B = 5 i + a j + k, C = i + j + 3k Q.89. For A, B and C to be coplanar, a vector perpendicular to A and C is perpendicular to B. A C is perpendicular to B. A C is perpendicular to both A and C A C B = ( dot product of vectors, is zero) i j k A C = 3 = i(9 + 4) j(6 + 3) + k(4 + 3) 1 3 = 13 i 9 j + 7k A C B = 13(5) 9a + 7 = 9a = 7 a = 8 The area of the parallelogram represented by the vectors, A = 4 i + 3 j and B = i + 4 j as adjacent side is 14 units Option 7.5 units 1 units 5 units Correct Answer 3 A = 4 i + 3 j and B = i + 4 j Q.9. 3 Option 6 Area of parallelogram with A and B as adjacent side. i j k Area = A B = = i() j() + k(16 6) = 1 units. 1 If A and B denote the sides of a parallelogram and its area is AB (A and B are the magnitude of A and B respectively), the angle between A andb is

35 45 1 Correct Answer 1 Area of parallelogram with A and B as adjacent side. Q.91. AB Area = A B = ABsin = (given) 1 sin = = 3 Given, C = A B and D = B A. What is the angle between C and D? 3 Option Correct Answer 4 C = A B and D = B A D = A B A B = B A C and D are anti parallel. angle between C andd is 18. Q.94. Option If vectors A and B are given by A = 5 i + 6 j + 3k and B = 6 i j 6k. Which is/are of the following correct? A and B are mutually perpendicular Product of A B is the same B A The magnitude of A and B are equal The magnitude of A B is zero A = 5 i + 6 j + 3k and B = 6 i j 6k i j k A B = = i ( ) j( 3 18) + k( 1 36) 6 6 i j k = 3 i + 48 j 46k B A = 6 6 = i ( ) j(18 + 3) + k(36 + 1) A B = = = 3 i 48 j + 46k

36 A and B are perpendicular Q.95. Option Which of the following statements is/are correct The magnitude of the vector 3 i + 4 j is 5 A force 3 i + 4 j N acting on a particle cause a displacement 6 j. The work done by the force is 3 N If A and B represent two adjacent sides of a parallelogram, then A B give the area of that parallelogram. A force has magnitude N. Its component in a direction making an angle 6 with the force is 1 3 N. a) Magnitude of 3 i + 4 j = = 5 b) Work done = F ds = F s = 3 i + 4 j 6 j = 4 N c) conceptual d) Component of vector in direction making angle with the vector = Acos 1 = cos6 = = 1 N Q.96. Option Two vectors A and B are inclined to each other at an angle. Which of the following? is the unit vector perpendicular to both A and B A B A B A B sin A B AB sin A B AB cos A vector perpendicular to both A and B is C = A B Unit vector perpendicular to A B C A B A B C = = = AB sin C A B A A B B A B Also, C = = AB sin sin

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