Page 1 Section I Single Correct Option 1. Pressure ( ) p = Force Area \ [p] = - [MLT ] [L ] 2 2 = - - [ ] ML T 1 2 Option (d) is correct. 2. W I Rt = 2 \ [R] = = - - - [ML T ] [A T] [ML T A ] T 2 2 2 2 2 2 [ ] â€¦(i) V L dI dt = and W Vq = \ L W q dt dI = [L] = - [ML T ][T] [A T] 2 2 2 = - - [ML T A ] 2 2 2 Using Eq. (i) [R] = [ ] [ ] L T i.e., [ ] T L R = é ë ê ù û ú Option (c) is correct. 3. F av =6ph \ [ ] [ ] [ ] h = F av = - - [MLT ] LLT 2 [ ] 1 = - - [ML T ] 1 1 Option (d) is correct. 4. f=Li \ [f ] = [ ][ L i] = - - [ML T A ] A 2 2 2 [ ] = - - [ML T A ] 2 2 1 Option (a) is correct. 5. Linear impulse ( ) I F t = × D [I] = - [MLT ][T] 2 = - [MLT ] 1 Option (c) is correct. 6. F G m m d = 1 2 2 [G] = = - [ ][ ] [ ] [ML T ] [L ] [M ] 2 1 2 2 2 F d m m 2 = - - [M L T ] 3 1 2 Option (c) is correct. 7. F i i d = × m p 0 1 2 4 \ [ m 0 ] = - [ML T ][L] [A ] 2 2 = - - [ML T A ] 2 2 2 Option (c) is correct. 8. [ ] [L] [L T ] [T] k = = -1 Option (c) is correct. 9. [ ] a = = - - [ML T ] [ T ] [ML T ] 2 3 [ ] b = = - - [ML T ] [ T ] [ML T ] 2 2 4 Option (c) is correct. Units & Dimensions Vectors 2 Page 2 Section I Single Correct Option 1. Pressure ( ) p = Force Area \ [p] = - [MLT ] [L ] 2 2 = - - [ ] ML T 1 2 Option (d) is correct. 2. W I Rt = 2 \ [R] = = - - - [ML T ] [A T] [ML T A ] T 2 2 2 2 2 2 [ ] â€¦(i) V L dI dt = and W Vq = \ L W q dt dI = [L] = - [ML T ][T] [A T] 2 2 2 = - - [ML T A ] 2 2 2 Using Eq. (i) [R] = [ ] [ ] L T i.e., [ ] T L R = é ë ê ù û ú Option (c) is correct. 3. F av =6ph \ [ ] [ ] [ ] h = F av = - - [MLT ] LLT 2 [ ] 1 = - - [ML T ] 1 1 Option (d) is correct. 4. f=Li \ [f ] = [ ][ L i] = - - [ML T A ] A 2 2 2 [ ] = - - [ML T A ] 2 2 1 Option (a) is correct. 5. Linear impulse ( ) I F t = × D [I] = - [MLT ][T] 2 = - [MLT ] 1 Option (c) is correct. 6. F G m m d = 1 2 2 [G] = = - [ ][ ] [ ] [ML T ] [L ] [M ] 2 1 2 2 2 F d m m 2 = - - [M L T ] 3 1 2 Option (c) is correct. 7. F i i d = × m p 0 1 2 4 \ [ m 0 ] = - [ML T ][L] [A ] 2 2 = - - [ML T A ] 2 2 2 Option (c) is correct. 8. [ ] [L] [L T ] [T] k = = -1 Option (c) is correct. 9. [ ] a = = - - [ML T ] [ T ] [ML T ] 2 3 [ ] b = = - - [ML T ] [ T ] [ML T ] 2 2 4 Option (c) is correct. Units & Dimensions Vectors 2 10. E h = n \ [ ] [ML T ] [T ] [ML T ] 2 2 h = = - - - 2 1 1 Angular momentum ( ) J nh = 2p [ ] J h = = - [ ] [ML T ] 2 1 Option (b) is correct. 11. [Energy] = - [ML T ] 2 2 = - [M][LT ] 2 1 \ [Mass] = - [ ] Ev 2 Option (c) is correct. 12. 1 2 0 2 e = E Energy density = Energy Volume \ 1 2 0 2 2 3 e é ë ê ù û ú = - E [ML T ] L 2 [ ] = - - [ML T ] 1 2 Option (b) is correct. 13. [ ] a =[T ] 2 [ ] [ ] [ ][ ] b = - - T L ML T 2 1 2 \ a b é ë ê ù û ú = - [MT ] 2 Option (b) is correct. 14. Velocity gradient = dv dx [Velocity gradient] = - [L T ] [L] 1 = - [T ] 1 = - [ ] M L T 0 0 1 Option (a) is correct. 15. [Force] = - [MLT ] 2 \ [Mass] = - [F] [L T ] 2 = - [FL T ] 2 1 Option (a) is correct. 16. Coefficient of friction (m) = Limitting frictional force Normal force \ [m] =[M L T ] 0 0 0 Option (b) is correct. 17. q CV = and V iR = \ q iCR = i t iCR = Þ [CR] = = [ ] t [M L T A ] 0 0 0 Option (a) is correct. 18. F q q r = 1 4 0 1 2 2 pe \ Unit of e 0 = Newton-metre 2 /coulomb 2 . Option (b) is correct. 19. Angular momentum ( ) J nh = 2p I mr = S 2 \ h I J n mr = 2 2 p / S = mvr mr S 2 h I é ë ê ù û ú = é ë ê ù û ú = - - [LT ] L [T ] 1 1 = Frequency Option (a) is correct. 20. v at b t c = + + [ ] [ ] c T = b t c v + é ë ê ù û ú =[ ] or [ ] b = = - [LT ][T] [L] 1 [ ] [ ] at v = = - [LT ] 1 Þ [a] = - [LT ] 2 Option (a) is correct. 21. y A ct x = - é ë ê ù û ú sin ( ) 2p l = - é ë ê ù û ú A ct x sin 2 2 p l p l 2p l q x = (angle) \ [ ] [ ] [ ] x = = l L Further, y A = sinq \ [ ] [ ] [ ] A y = = L Option (a) is correct. 22. [ ] X = - - [M L T A ] 3 1 3 2 = - [TA ] [ML T ] 2 2 2 6 | Mechanics-1 Page 3 Section I Single Correct Option 1. Pressure ( ) p = Force Area \ [p] = - [MLT ] [L ] 2 2 = - - [ ] ML T 1 2 Option (d) is correct. 2. W I Rt = 2 \ [R] = = - - - [ML T ] [A T] [ML T A ] T 2 2 2 2 2 2 [ ] â€¦(i) V L dI dt = and W Vq = \ L W q dt dI = [L] = - [ML T ][T] [A T] 2 2 2 = - - [ML T A ] 2 2 2 Using Eq. (i) [R] = [ ] [ ] L T i.e., [ ] T L R = é ë ê ù û ú Option (c) is correct. 3. F av =6ph \ [ ] [ ] [ ] h = F av = - - [MLT ] LLT 2 [ ] 1 = - - [ML T ] 1 1 Option (d) is correct. 4. f=Li \ [f ] = [ ][ L i] = - - [ML T A ] A 2 2 2 [ ] = - - [ML T A ] 2 2 1 Option (a) is correct. 5. Linear impulse ( ) I F t = × D [I] = - [MLT ][T] 2 = - [MLT ] 1 Option (c) is correct. 6. F G m m d = 1 2 2 [G] = = - [ ][ ] [ ] [ML T ] [L ] [M ] 2 1 2 2 2 F d m m 2 = - - [M L T ] 3 1 2 Option (c) is correct. 7. F i i d = × m p 0 1 2 4 \ [ m 0 ] = - [ML T ][L] [A ] 2 2 = - - [ML T A ] 2 2 2 Option (c) is correct. 8. [ ] [L] [L T ] [T] k = = -1 Option (c) is correct. 9. [ ] a = = - - [ML T ] [ T ] [ML T ] 2 3 [ ] b = = - - [ML T ] [ T ] [ML T ] 2 2 4 Option (c) is correct. Units & Dimensions Vectors 2 10. E h = n \ [ ] [ML T ] [T ] [ML T ] 2 2 h = = - - - 2 1 1 Angular momentum ( ) J nh = 2p [ ] J h = = - [ ] [ML T ] 2 1 Option (b) is correct. 11. [Energy] = - [ML T ] 2 2 = - [M][LT ] 2 1 \ [Mass] = - [ ] Ev 2 Option (c) is correct. 12. 1 2 0 2 e = E Energy density = Energy Volume \ 1 2 0 2 2 3 e é ë ê ù û ú = - E [ML T ] L 2 [ ] = - - [ML T ] 1 2 Option (b) is correct. 13. [ ] a =[T ] 2 [ ] [ ] [ ][ ] b = - - T L ML T 2 1 2 \ a b é ë ê ù û ú = - [MT ] 2 Option (b) is correct. 14. Velocity gradient = dv dx [Velocity gradient] = - [L T ] [L] 1 = - [T ] 1 = - [ ] M L T 0 0 1 Option (a) is correct. 15. [Force] = - [MLT ] 2 \ [Mass] = - [F] [L T ] 2 = - [FL T ] 2 1 Option (a) is correct. 16. Coefficient of friction (m) = Limitting frictional force Normal force \ [m] =[M L T ] 0 0 0 Option (b) is correct. 17. q CV = and V iR = \ q iCR = i t iCR = Þ [CR] = = [ ] t [M L T A ] 0 0 0 Option (a) is correct. 18. F q q r = 1 4 0 1 2 2 pe \ Unit of e 0 = Newton-metre 2 /coulomb 2 . Option (b) is correct. 19. Angular momentum ( ) J nh = 2p I mr = S 2 \ h I J n mr = 2 2 p / S = mvr mr S 2 h I é ë ê ù û ú = é ë ê ù û ú = - - [LT ] L [T ] 1 1 = Frequency Option (a) is correct. 20. v at b t c = + + [ ] [ ] c T = b t c v + é ë ê ù û ú =[ ] or [ ] b = = - [LT ][T] [L] 1 [ ] [ ] at v = = - [LT ] 1 Þ [a] = - [LT ] 2 Option (a) is correct. 21. y A ct x = - é ë ê ù û ú sin ( ) 2p l = - é ë ê ù û ú A ct x sin 2 2 p l p l 2p l q x = (angle) \ [ ] [ ] [ ] x = = l L Further, y A = sinq \ [ ] [ ] [ ] A y = = L Option (a) is correct. 22. [ ] X = - - [M L T A ] 3 1 3 2 = - [TA ] [ML T ] 2 2 2 6 | Mechanics-1 = [ ][ ] [ ] t i 2 Work \ X is resistance. [ Q W i Rt = 2 ] 23. F i j k ® = - + 2 3 4 ^ ^ ^ r i j k ® = + + 3 2 3 ^ ^ ^ \ t ® ® ® = ´ r F = - ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ i j k ^ ^ ^ 3 2 3 2 3 4 or t ® = - - 17 6 13 i j k ^ ^ ^ 24. ( ) ( ) ( ) 0.5 0.8 2 2 2 1 + + = c or 0.25 0.64 + + = c 2 1 or c 2 1 = -0.89 c = 0.11 Option (b) is correct. 25. | | | | A B A B ® ® ® ® + = - ( ) ( ) ( ) ( ) A B A B A B A B ® ® ® ® ® ® ® ® + × + = - × - A B A B 2 2 2 2 2 2 + + × = + - × ® ® ® ® A B A B i.e., A B ® ® × =0 \ Angle between A ® and B ® = ° 90 26. ( ) ( ) A B A B ® ® ® ® + × - = 0 A A B A B B A B ® ® ® ® ® ® ® ® × + × - × - × = 0 A B 2 2 0 - = A B = ± | | | | A B ® ® = Option (d) is correct. 27. Work ( ) = × ® ® F s is a scalar quantity. Option (d) is correct. 28. Speed = ® | | v Option (d) is correct. 29. | | A ® = 3, | | B ® = 5 and angle between A ® and B ® is 60°. \ A B A B ® ® ® ® × = ° | || |cos 60 = æ è ç ö ø ÷ ( ) ( ) 3 5 1 2 = 7.5 Option (b) is correct. 30. A B C ® ® ® + = \ ( ) ( ) A B A B CC ® ® ® ® ®® + × + = × or AA AB BB CC ® ® ® ® ®® ®® × + × + × = × 2 or A B C 2 2 2 2 + × + = ® ® A B or A B ® ® × =0 or | || |cos A B ® ® = q 0 or cosq =0 or q p = 2 Option (d) is correct. 31. Magnetic field intensity. Option (d) is correct. 32. P Q R ® ® ® + = ( ) ( ) P Q P Q R R ® ® ® ® ® ® + × + = × P Q R 2 2 2 2 + + × = ® ® P Q 12 5 2 13 2 2 2 + + × = ® ® P Q P Q ® ® × = 0 \ Angle between P ® and Q ® = p 2 Option (b) is correct. 33. Option (b) is correct. 34. P Q R ® ® ® + + = 0 \ P Q R ® ® ® + = - or ( ) ( ) ( ) ( ) P Q P Q R R ® ® ® ® ® ® + × + = - × - or P P Q Q P Q R R ® ® ® ® ® ® ® ® × + × + × = × 2 or P Q R 2 2 2 2 + + × = ® ® P Q â€¦(i) Let Q P 2 2 = and R P = 2 Thus, Eq. (i) takes the form P P PQ P 2 2 2 2 2 + + = cos q or 2 0 PQcos q = Units & Dimensions Vectors 7 Page 4 Section I Single Correct Option 1. Pressure ( ) p = Force Area \ [p] = - [MLT ] [L ] 2 2 = - - [ ] ML T 1 2 Option (d) is correct. 2. W I Rt = 2 \ [R] = = - - - [ML T ] [A T] [ML T A ] T 2 2 2 2 2 2 [ ] â€¦(i) V L dI dt = and W Vq = \ L W q dt dI = [L] = - [ML T ][T] [A T] 2 2 2 = - - [ML T A ] 2 2 2 Using Eq. (i) [R] = [ ] [ ] L T i.e., [ ] T L R = é ë ê ù û ú Option (c) is correct. 3. F av =6ph \ [ ] [ ] [ ] h = F av = - - [MLT ] LLT 2 [ ] 1 = - - [ML T ] 1 1 Option (d) is correct. 4. f=Li \ [f ] = [ ][ L i] = - - [ML T A ] A 2 2 2 [ ] = - - [ML T A ] 2 2 1 Option (a) is correct. 5. Linear impulse ( ) I F t = × D [I] = - [MLT ][T] 2 = - [MLT ] 1 Option (c) is correct. 6. F G m m d = 1 2 2 [G] = = - [ ][ ] [ ] [ML T ] [L ] [M ] 2 1 2 2 2 F d m m 2 = - - [M L T ] 3 1 2 Option (c) is correct. 7. F i i d = × m p 0 1 2 4 \ [ m 0 ] = - [ML T ][L] [A ] 2 2 = - - [ML T A ] 2 2 2 Option (c) is correct. 8. [ ] [L] [L T ] [T] k = = -1 Option (c) is correct. 9. [ ] a = = - - [ML T ] [ T ] [ML T ] 2 3 [ ] b = = - - [ML T ] [ T ] [ML T ] 2 2 4 Option (c) is correct. Units & Dimensions Vectors 2 10. E h = n \ [ ] [ML T ] [T ] [ML T ] 2 2 h = = - - - 2 1 1 Angular momentum ( ) J nh = 2p [ ] J h = = - [ ] [ML T ] 2 1 Option (b) is correct. 11. [Energy] = - [ML T ] 2 2 = - [M][LT ] 2 1 \ [Mass] = - [ ] Ev 2 Option (c) is correct. 12. 1 2 0 2 e = E Energy density = Energy Volume \ 1 2 0 2 2 3 e é ë ê ù û ú = - E [ML T ] L 2 [ ] = - - [ML T ] 1 2 Option (b) is correct. 13. [ ] a =[T ] 2 [ ] [ ] [ ][ ] b = - - T L ML T 2 1 2 \ a b é ë ê ù û ú = - [MT ] 2 Option (b) is correct. 14. Velocity gradient = dv dx [Velocity gradient] = - [L T ] [L] 1 = - [T ] 1 = - [ ] M L T 0 0 1 Option (a) is correct. 15. [Force] = - [MLT ] 2 \ [Mass] = - [F] [L T ] 2 = - [FL T ] 2 1 Option (a) is correct. 16. Coefficient of friction (m) = Limitting frictional force Normal force \ [m] =[M L T ] 0 0 0 Option (b) is correct. 17. q CV = and V iR = \ q iCR = i t iCR = Þ [CR] = = [ ] t [M L T A ] 0 0 0 Option (a) is correct. 18. F q q r = 1 4 0 1 2 2 pe \ Unit of e 0 = Newton-metre 2 /coulomb 2 . Option (b) is correct. 19. Angular momentum ( ) J nh = 2p I mr = S 2 \ h I J n mr = 2 2 p / S = mvr mr S 2 h I é ë ê ù û ú = é ë ê ù û ú = - - [LT ] L [T ] 1 1 = Frequency Option (a) is correct. 20. v at b t c = + + [ ] [ ] c T = b t c v + é ë ê ù û ú =[ ] or [ ] b = = - [LT ][T] [L] 1 [ ] [ ] at v = = - [LT ] 1 Þ [a] = - [LT ] 2 Option (a) is correct. 21. y A ct x = - é ë ê ù û ú sin ( ) 2p l = - é ë ê ù û ú A ct x sin 2 2 p l p l 2p l q x = (angle) \ [ ] [ ] [ ] x = = l L Further, y A = sinq \ [ ] [ ] [ ] A y = = L Option (a) is correct. 22. [ ] X = - - [M L T A ] 3 1 3 2 = - [TA ] [ML T ] 2 2 2 6 | Mechanics-1 = [ ][ ] [ ] t i 2 Work \ X is resistance. [ Q W i Rt = 2 ] 23. F i j k ® = - + 2 3 4 ^ ^ ^ r i j k ® = + + 3 2 3 ^ ^ ^ \ t ® ® ® = ´ r F = - ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ i j k ^ ^ ^ 3 2 3 2 3 4 or t ® = - - 17 6 13 i j k ^ ^ ^ 24. ( ) ( ) ( ) 0.5 0.8 2 2 2 1 + + = c or 0.25 0.64 + + = c 2 1 or c 2 1 = -0.89 c = 0.11 Option (b) is correct. 25. | | | | A B A B ® ® ® ® + = - ( ) ( ) ( ) ( ) A B A B A B A B ® ® ® ® ® ® ® ® + × + = - × - A B A B 2 2 2 2 2 2 + + × = + - × ® ® ® ® A B A B i.e., A B ® ® × =0 \ Angle between A ® and B ® = ° 90 26. ( ) ( ) A B A B ® ® ® ® + × - = 0 A A B A B B A B ® ® ® ® ® ® ® ® × + × - × - × = 0 A B 2 2 0 - = A B = ± | | | | A B ® ® = Option (d) is correct. 27. Work ( ) = × ® ® F s is a scalar quantity. Option (d) is correct. 28. Speed = ® | | v Option (d) is correct. 29. | | A ® = 3, | | B ® = 5 and angle between A ® and B ® is 60°. \ A B A B ® ® ® ® × = ° | || |cos 60 = æ è ç ö ø ÷ ( ) ( ) 3 5 1 2 = 7.5 Option (b) is correct. 30. A B C ® ® ® + = \ ( ) ( ) A B A B CC ® ® ® ® ®® + × + = × or AA AB BB CC ® ® ® ® ®® ®® × + × + × = × 2 or A B C 2 2 2 2 + × + = ® ® A B or A B ® ® × =0 or | || |cos A B ® ® = q 0 or cosq =0 or q p = 2 Option (d) is correct. 31. Magnetic field intensity. Option (d) is correct. 32. P Q R ® ® ® + = ( ) ( ) P Q P Q R R ® ® ® ® ® ® + × + = × P Q R 2 2 2 2 + + × = ® ® P Q 12 5 2 13 2 2 2 + + × = ® ® P Q P Q ® ® × = 0 \ Angle between P ® and Q ® = p 2 Option (b) is correct. 33. Option (b) is correct. 34. P Q R ® ® ® + + = 0 \ P Q R ® ® ® + = - or ( ) ( ) ( ) ( ) P Q P Q R R ® ® ® ® ® ® + × + = - × - or P P Q Q P Q R R ® ® ® ® ® ® ® ® × + × + × = × 2 or P Q R 2 2 2 2 + + × = ® ® P Q â€¦(i) Let Q P 2 2 = and R P = 2 Thus, Eq. (i) takes the form P P PQ P 2 2 2 2 2 + + = cos q or 2 0 PQcos q = Units & Dimensions Vectors 7 or cosq =0 or q = ° 90 \ Angle between P ® and Q ® is 90° P Q R ® ® ® + + = 0 \ P R Q ® ® ® + = - or ( ) ( ) ( ) ( ) P R P R Q Q ® ® ® ® ® ® + × + = - × - or P R PR Q 2 2 2 2 + + f = cos or 2 2 2 2 PR Q P R cosf = - - or 2 2 PR R cosf = - or 2P R cosf = - or 2 2 P P cosf = - or cosf = - 1 2 \ f = ° 135 \ Angle between P ® and R ® is 135°. Option (a) is correct. 35. Angle ( ) f between P Q ® ® + and P Q ® ® - tan sin cos f = + Q P Q q q Angle f¢ between P Q ® ® - and P ® tan sin ( ) cos ( ) f¢ = + + + Q P Q p q p q = - - Q P Q sin cos q q tan[ ( )] tan tan tan tan f + -f¢ = f - f¢ + f f¢ 1 = + - - - - + × - Q P Q Q P Q Q P Q Q sin cos ( sin ) cos sin ( cos ) ( s q q q q q q 1 in ) ( cos ) q q P Q - = + 2 2 2 2 PQ P Q sin cos q q This implies that angle between P Q ® ® + and P Q ® ® - will vary from 0 to p. Option (b) is correct. 36. R P Q PQ 2 2 2 2 = + + cos q for R P Q = = P P P PP 2 2 2 2 = + + cos q or cosq = - 1 2 or q = ° 120 Option (b) is correct. 37. W = × ® ® F s = + × + ( ) ( ) ^ ^ ^ ^ 3 4 3 4 i j i j =25 J Option (b) is correct. 38. P Q i j k i j k ® ® × = + + × - - ( ) ( ) ^ ^ ^ ^ ^ ^ a a a 3 2 = - - a a 2 2 3 For P Q ® ® ^ , P Q ® ® × =0 i.e., a a 2 2 3 0 - - = or ( )( ) a a - + = 3 1 0 Þ a =3 Other value is - ive. Option (d) is correct. 39. If a vector makes angles a, b and g with the co-ordinate axes, then cos cos cos 2 2 2 1 a b g + + = Now, 3 7 9 49 2 æ è ç ö ø ÷ = , 6 7 36 49 2 æ è ç ö ø ÷ = , 2 7 4 49 2 æ è ç ö ø ÷ = and 9 49 36 49 4 49 1 + + = \ Option (a) is correct. 40. A i j ® = - 4 3 ^ and B i j ® = + 8 8 ^ $ \ A B C i j ® ® ® + = = + 12 5 ^ ^ 8 | Mechanics-1 P 90° Q R 135° 135° ® ® ® â€“ Q Q Q P â€“ Q P + P q f f' ® ® ® ® ® ® ® Page 5 Section I Single Correct Option 1. Pressure ( ) p = Force Area \ [p] = - [MLT ] [L ] 2 2 = - - [ ] ML T 1 2 Option (d) is correct. 2. W I Rt = 2 \ [R] = = - - - [ML T ] [A T] [ML T A ] T 2 2 2 2 2 2 [ ] â€¦(i) V L dI dt = and W Vq = \ L W q dt dI = [L] = - [ML T ][T] [A T] 2 2 2 = - - [ML T A ] 2 2 2 Using Eq. (i) [R] = [ ] [ ] L T i.e., [ ] T L R = é ë ê ù û ú Option (c) is correct. 3. F av =6ph \ [ ] [ ] [ ] h = F av = - - [MLT ] LLT 2 [ ] 1 = - - [ML T ] 1 1 Option (d) is correct. 4. f=Li \ [f ] = [ ][ L i] = - - [ML T A ] A 2 2 2 [ ] = - - [ML T A ] 2 2 1 Option (a) is correct. 5. Linear impulse ( ) I F t = × D [I] = - [MLT ][T] 2 = - [MLT ] 1 Option (c) is correct. 6. F G m m d = 1 2 2 [G] = = - [ ][ ] [ ] [ML T ] [L ] [M ] 2 1 2 2 2 F d m m 2 = - - [M L T ] 3 1 2 Option (c) is correct. 7. F i i d = × m p 0 1 2 4 \ [ m 0 ] = - [ML T ][L] [A ] 2 2 = - - [ML T A ] 2 2 2 Option (c) is correct. 8. [ ] [L] [L T ] [T] k = = -1 Option (c) is correct. 9. [ ] a = = - - [ML T ] [ T ] [ML T ] 2 3 [ ] b = = - - [ML T ] [ T ] [ML T ] 2 2 4 Option (c) is correct. Units & Dimensions Vectors 2 10. E h = n \ [ ] [ML T ] [T ] [ML T ] 2 2 h = = - - - 2 1 1 Angular momentum ( ) J nh = 2p [ ] J h = = - [ ] [ML T ] 2 1 Option (b) is correct. 11. [Energy] = - [ML T ] 2 2 = - [M][LT ] 2 1 \ [Mass] = - [ ] Ev 2 Option (c) is correct. 12. 1 2 0 2 e = E Energy density = Energy Volume \ 1 2 0 2 2 3 e é ë ê ù û ú = - E [ML T ] L 2 [ ] = - - [ML T ] 1 2 Option (b) is correct. 13. [ ] a =[T ] 2 [ ] [ ] [ ][ ] b = - - T L ML T 2 1 2 \ a b é ë ê ù û ú = - [MT ] 2 Option (b) is correct. 14. Velocity gradient = dv dx [Velocity gradient] = - [L T ] [L] 1 = - [T ] 1 = - [ ] M L T 0 0 1 Option (a) is correct. 15. [Force] = - [MLT ] 2 \ [Mass] = - [F] [L T ] 2 = - [FL T ] 2 1 Option (a) is correct. 16. Coefficient of friction (m) = Limitting frictional force Normal force \ [m] =[M L T ] 0 0 0 Option (b) is correct. 17. q CV = and V iR = \ q iCR = i t iCR = Þ [CR] = = [ ] t [M L T A ] 0 0 0 Option (a) is correct. 18. F q q r = 1 4 0 1 2 2 pe \ Unit of e 0 = Newton-metre 2 /coulomb 2 . Option (b) is correct. 19. Angular momentum ( ) J nh = 2p I mr = S 2 \ h I J n mr = 2 2 p / S = mvr mr S 2 h I é ë ê ù û ú = é ë ê ù û ú = - - [LT ] L [T ] 1 1 = Frequency Option (a) is correct. 20. v at b t c = + + [ ] [ ] c T = b t c v + é ë ê ù û ú =[ ] or [ ] b = = - [LT ][T] [L] 1 [ ] [ ] at v = = - [LT ] 1 Þ [a] = - [LT ] 2 Option (a) is correct. 21. y A ct x = - é ë ê ù û ú sin ( ) 2p l = - é ë ê ù û ú A ct x sin 2 2 p l p l 2p l q x = (angle) \ [ ] [ ] [ ] x = = l L Further, y A = sinq \ [ ] [ ] [ ] A y = = L Option (a) is correct. 22. [ ] X = - - [M L T A ] 3 1 3 2 = - [TA ] [ML T ] 2 2 2 6 | Mechanics-1 = [ ][ ] [ ] t i 2 Work \ X is resistance. [ Q W i Rt = 2 ] 23. F i j k ® = - + 2 3 4 ^ ^ ^ r i j k ® = + + 3 2 3 ^ ^ ^ \ t ® ® ® = ´ r F = - ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ i j k ^ ^ ^ 3 2 3 2 3 4 or t ® = - - 17 6 13 i j k ^ ^ ^ 24. ( ) ( ) ( ) 0.5 0.8 2 2 2 1 + + = c or 0.25 0.64 + + = c 2 1 or c 2 1 = -0.89 c = 0.11 Option (b) is correct. 25. | | | | A B A B ® ® ® ® + = - ( ) ( ) ( ) ( ) A B A B A B A B ® ® ® ® ® ® ® ® + × + = - × - A B A B 2 2 2 2 2 2 + + × = + - × ® ® ® ® A B A B i.e., A B ® ® × =0 \ Angle between A ® and B ® = ° 90 26. ( ) ( ) A B A B ® ® ® ® + × - = 0 A A B A B B A B ® ® ® ® ® ® ® ® × + × - × - × = 0 A B 2 2 0 - = A B = ± | | | | A B ® ® = Option (d) is correct. 27. Work ( ) = × ® ® F s is a scalar quantity. Option (d) is correct. 28. Speed = ® | | v Option (d) is correct. 29. | | A ® = 3, | | B ® = 5 and angle between A ® and B ® is 60°. \ A B A B ® ® ® ® × = ° | || |cos 60 = æ è ç ö ø ÷ ( ) ( ) 3 5 1 2 = 7.5 Option (b) is correct. 30. A B C ® ® ® + = \ ( ) ( ) A B A B CC ® ® ® ® ®® + × + = × or AA AB BB CC ® ® ® ® ®® ®® × + × + × = × 2 or A B C 2 2 2 2 + × + = ® ® A B or A B ® ® × =0 or | || |cos A B ® ® = q 0 or cosq =0 or q p = 2 Option (d) is correct. 31. Magnetic field intensity. Option (d) is correct. 32. P Q R ® ® ® + = ( ) ( ) P Q P Q R R ® ® ® ® ® ® + × + = × P Q R 2 2 2 2 + + × = ® ® P Q 12 5 2 13 2 2 2 + + × = ® ® P Q P Q ® ® × = 0 \ Angle between P ® and Q ® = p 2 Option (b) is correct. 33. Option (b) is correct. 34. P Q R ® ® ® + + = 0 \ P Q R ® ® ® + = - or ( ) ( ) ( ) ( ) P Q P Q R R ® ® ® ® ® ® + × + = - × - or P P Q Q P Q R R ® ® ® ® ® ® ® ® × + × + × = × 2 or P Q R 2 2 2 2 + + × = ® ® P Q â€¦(i) Let Q P 2 2 = and R P = 2 Thus, Eq. (i) takes the form P P PQ P 2 2 2 2 2 + + = cos q or 2 0 PQcos q = Units & Dimensions Vectors 7 or cosq =0 or q = ° 90 \ Angle between P ® and Q ® is 90° P Q R ® ® ® + + = 0 \ P R Q ® ® ® + = - or ( ) ( ) ( ) ( ) P R P R Q Q ® ® ® ® ® ® + × + = - × - or P R PR Q 2 2 2 2 + + f = cos or 2 2 2 2 PR Q P R cosf = - - or 2 2 PR R cosf = - or 2P R cosf = - or 2 2 P P cosf = - or cosf = - 1 2 \ f = ° 135 \ Angle between P ® and R ® is 135°. Option (a) is correct. 35. Angle ( ) f between P Q ® ® + and P Q ® ® - tan sin cos f = + Q P Q q q Angle f¢ between P Q ® ® - and P ® tan sin ( ) cos ( ) f¢ = + + + Q P Q p q p q = - - Q P Q sin cos q q tan[ ( )] tan tan tan tan f + -f¢ = f - f¢ + f f¢ 1 = + - - - - + × - Q P Q Q P Q Q P Q Q sin cos ( sin ) cos sin ( cos ) ( s q q q q q q 1 in ) ( cos ) q q P Q - = + 2 2 2 2 PQ P Q sin cos q q This implies that angle between P Q ® ® + and P Q ® ® - will vary from 0 to p. Option (b) is correct. 36. R P Q PQ 2 2 2 2 = + + cos q for R P Q = = P P P PP 2 2 2 2 = + + cos q or cosq = - 1 2 or q = ° 120 Option (b) is correct. 37. W = × ® ® F s = + × + ( ) ( ) ^ ^ ^ ^ 3 4 3 4 i j i j =25 J Option (b) is correct. 38. P Q i j k i j k ® ® × = + + × - - ( ) ( ) ^ ^ ^ ^ ^ ^ a a a 3 2 = - - a a 2 2 3 For P Q ® ® ^ , P Q ® ® × =0 i.e., a a 2 2 3 0 - - = or ( )( ) a a - + = 3 1 0 Þ a =3 Other value is - ive. Option (d) is correct. 39. If a vector makes angles a, b and g with the co-ordinate axes, then cos cos cos 2 2 2 1 a b g + + = Now, 3 7 9 49 2 æ è ç ö ø ÷ = , 6 7 36 49 2 æ è ç ö ø ÷ = , 2 7 4 49 2 æ è ç ö ø ÷ = and 9 49 36 49 4 49 1 + + = \ Option (a) is correct. 40. A i j ® = - 4 3 ^ and B i j ® = + 8 8 ^ $ \ A B C i j ® ® ® + = = + 12 5 ^ ^ 8 | Mechanics-1 P 90° Q R 135° 135° ® ® ® â€“ Q Q Q P â€“ Q P + P q f f' ® ® ® ® ® ® ® C C i j ^ ^ ^ | | = = + + ® C 12 5 12 5 2 2 = + 12 13 5 13 i j ^ ^ Option (b) is correct. 41. A i j k ® = + - 2 3 2 ^ ^ $ , B i j k ® = + + 5 ^ ^ $ n , C i j k ® = - + + ^ ^ $ 2 3 \ Vectors A B C ® ® ® , ,and will be coplanar if their scalar triple product is zero i.e., ( ) A C B ® ® ® ´ × = 0 $ $ $ ( ) ^ ^ ^ i j k i j k 2 3 2 1 2 3 5 0 - - ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ × + + = n or ( ) ( ) ^ ^ ^ ^ ^ ^ 13 4 7 5 0 i j k i j k - + × + + = n or 65 4 7 0 - + = n or n = 18 Option (a) is correct. 42. Option (a) is correct. 43. ( ) ( ) a b a b ® ® ® ® + ´ - = ´ + ´ - ´ - ´ ® ® ® ® ® ® ® ® a a b a a b b b = - ´ - ´ - ® ® ® ® 0 0 a b a b =- ´ ® ® 2( ) a b = ´ ® ® 2( ) b a Option (a) is correct. 44. A i j k ® = + + 3 4 5 ^ ^ ^ B i j k ® = + - 3 4 5 ^ ^ ^ cos ( ) | || | q= × ® ® ® ® A B A B = + - + + 9 16 25 3 4 5 2 2 2 =0 Þ q = ° 90 Option (c) is correct. 45. A B + =7 A B - =3 \ B=2 N Option (c) is correct. 46. Angle between A i j ® = + 2 3 ^ ^ and B i j ® = + ^ ^ q= × ® ® ® ® A B |A||B| = + + × 2 3 2 3 2 2 2 = × 5 13 2 = 5 26 Component of A ® along i j ^ ^ + C i j ® = + 5 26 2 3 ( ) ^ ^ | | C ® = 5 2 Option (a) is correct. 47. R P P P P 2 2 2 3 2 2 3 2 = + + ´ ´ ( ) ( ) cos q or R P P 2 2 2 13 12 = + cos q â€¦(i) Further ( ) ( ) ( ) cos 2 6 2 2 6 2 2 2 2 R P P P P = + + ´ ´ q or 4 40 24 2 2 2 R P P = + cos q â€¦(ii) Dividing Eq. (ii) by Eq. (i), 10 6 13 12 2 2 2 2 P P P P + = + cos cos q q or 6cosq = - P or q = ° 120 Option (b) is correct. 48. tan sin cos q a a = + Q P Q As q = ° 90 , tana = ¥ \ P Q + = cosa 0 i.e., cosa = - P Q = - P P 2 Units & Dimensions Vectors 9 q = 90° R P Q = 2PRead More

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### Chapter 3 - Motion in One Dimension (Part - 1) - Physics, Solution by D C Pandey, NEET

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