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Mathematics Questions;

1. A 72 Ohm Coil of Wire Is Connected in Series with An Adjustable Resistor (rheostat) Whose Resistance Can Be Varied from 0 to 88 Ohms. if The Line Potential of Wire Is 115 Volts, What Is the Rheostat Resistance when Power Taken by The Coil Is 90 Watts?

Solution;

Power =VI

2. Solve for α in the oblique triangle ABC; AB = 30; AC = 15 and angle B = 20°

Oblique triangle ABC; AB = 30; AC = 15 and angle B = 20°

1. Type out the two equations substituting the numbers from the diagram.

2. Type out the Law of Sines set of relationships and type out the most appropriate version to use the Law of Cosines for this solution.

3. Solve for a using both methods (show step by step work)

4. A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix.

(a) Show that I − A is also idempotent.

(b) Show that if A is invertible, then A = I.

(c) Show that the only possible eigenvalues of A are 0 and 1. (Hint: Suppose x is an eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.)

5.Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwise indicated.

?C(lnx+y)dx−x2dy

where C is the rectangle with vertices (1, 1), (3, 1), (1, 4), and (3, 4)

6. Vivian and Noelle both leave the park at the same time, but in opposite directions. If Noelle travels 5 mph faster than Vivian and after 8 hours, they are 136 miles apart, how fast in mile per hour is each traveling?

7. An archer shoots an arrow at a 75 m distant target; the bull’s-eye of the target is at same height as the release height of the arrow. If d = 75 m and v = 37 m/s. At what angle must the arrow be released to hit the bull’s-eye if its initial speed is 37m/s?

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