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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
Two cars start from rest at a red stop light. When the light turns green, both cars accelerate forward. What is the acceleration of the yellow car
A converging lens (f = 13.4 centimeter) is placed 33.5 centimeter to the left of a diverging lens (f = -5.37 centimeter). Evaluate what is the overall magnification
Light of wavelength 5.70 x102 nm passes through a slit of width 0.220 mm. Determine the width of the first-order bright fringe
What must be the distance in meters between point charge q1 = 28.3 µC and point charge q2 = -58.1 µC for the electrostatic force between them to have a magnitude of 5.58 N
A cylinder (radius = 0.11 m, center-of-mass rotational inertia = 0.015555 kg·m2, and mass = 1.34 kg) starts from rest and rolls without slipping down a plane by an angle of inclination of28.5°. Discover the time it takes it to travel 1.57m along the ..
A solid block is suspended from a spring scale. Whenthe block is in air the scale reads 34.7 N, when immersed inseawater the scale reads 30.1 N, What is the density of the block
A father pulls his children in a wagon for 25 minutes covering 1.25 miles on a level sidewalk. What is the average horsepower required
After the NEAR spacecraft passed Mathilde, on some occasion's rocket propellant was expelled to adjust the spacecraft's momentum in order to follow a path that would approach the asteroid Eros, final destination for the mission.
Two forces, F1and F2 , act on an object. The magnitude of F1 is 12.45 N, and it is at an angle of 32o above the x-axis in the second quadrant. What are the magnitude and direction of the resultant force
Differentiate this equation, q=qpcosωt to find the current in the LC circuit, and use q=CV to find the voltage. From these obtain the magnetic energy in the inductor and the electric energy in the capacitor, and sum to show that the total energy rema..
An insect that is 5.90 mm long is placed 11.5 cm from a converging lens with a focal length of 12.0 cm. The position of the image is 276 cm from the lens and the size of the image is 141.6 mm. What is the angular magnification if the lens is close to..
A pair of eyeglasses has an index of refraction nglass = 1.52. It is covered with an antireflection layer that has a thickness of 120 nm, what is its index of refraction
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