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1) What do you understand from Curl and Div of a vector field F? Explain them with examples. Is there any difference between Grad and Div of F? What are the applications of Grad, Div and Curl?2) Find the area of a parallelogram spanned by the vectors and where .3) State and Prove Cayley-Hamilton theorem. Using Caley-Hamilton theorem find the inverse of the matrix A = 4) Define Contra positive, Converse, and Inverse of a statement. What are the contra positive, the converse, and the inverse of the conditional statement "The home team wins whenever it is raining?". Give an example of statement is a Tautology. Justify your example. 5) Prove without contradiction, that if is an integer and is odd, then is odd. Can this be proved using contradiction? If so, prove using contradiction also. 6) Verify that the conditions of Rolle's Theorem are satisfied by the function and determine a value of c in for which Is Rolles' theorem related Mean value theorem? If yes, how is it related? What is the significance of Mean Value theorem?7) What do you understand by series being conditionally convergent and absolutely convergent? Determine whether the series and are absolutely convergent, conditionally convergent or divergent. 8) Determine the radius of convergence of each of the following power series
imagine a solar electric power plant that operates by using a gigantic mirror to concentrate light on the boiler for a
Consider the scond hand on a clock which is smoothly rotating. Does it have linear velocity at the tip? Does it have tangential acceleration? If you spin a bucket full of water in a circle, why doesnt the water just fall out?
The car has just achieved double its initial velocity when the driver spots a police officer behind him and applies the brakes. The car then slows down, coming to rest at the stoplight ahead.
Two capacitors C1 = 5.6 ?F, C2 = 13.7 ?F are charged individually to V1 = 16.4 V, V2 = 6.1 V. Calculate the final potential difference across the plates of the capacitors once they are connected
So a bike went 10m in 4 seconds and the acceleration was constant the whole way. What Was the instantaneous velocity at 5m? If it helps the whole race was 1000m
what are the currents that the block can carry
A tortoise and hare start from rest and have a race. As the race begins, both accelerate forward. What is the total time the hare is moving
assume that all gases are perfect and that data refer to 298.15 k unless otherwise stated. the enthalpy of vaporization
A circular loop of wire 0.2 m in diameter is placed with its normal at an angle of 30° to the direction of a uniform 0.14 T magnetic field. What emf appears in the loop
A box is being pushed to the right parallel to the x-axis by a force of 15.0 N. find the magnitude of the net force acting on the box
Two objects with masses of 4.00 kg and 5.00 kg are connected by a light string thatpasses over a frictionless pulley. Determine the tension in the string. Determine the acceleration of each object.
A person whose mass is m = 77.0 kg steps on a mechanical bathroom scale placed on an inclined plane that makes the angle α = 27.7° with the horizontal. What is the reading on the scale
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