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Material Type: Exam; Class: College Algebra; Subject: Mathematical Sciences; University: University of Wisconsin - Milwaukee; Term: Spring 2008;
Typology: Exams
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(a) Using the variables x, y and z , what system does this augmented matrix represent? (b) Either solve the system of equations or explain why it has no solution.
PQ the line segment with endpoints P and Q.
(a) Give the coordinate-free definition of a circle. (b) Find the standard equation of the circle for which PQ is a diameter. (c) Give the coordinates of the points where this circle intersects the line 4 ( x+ 5 )+ 3 (y+ 1 )= 0 (d) Find the equation of the line tangent to this circle and passing through the point P.
(e) Give the coordinate-free definition of a hyperbola. (f) Find the center, vertices, foci, and asymptotes of the hyperbola x 2 − y^2 − 10 x+ 10 y− 1 = 0 and sketch its graph.
(b) Simplify
998
4
your answer. (c) Use the binomial theorem to simplify (^) ( 1 + 2 i )^5. Simplify all powers of i and collect like terms according to i. (d) A pizza parlor offers a choice of 15 different toppings. How many five-topping pizzas are possible? Express your answer as a product of integers.
(c) If x must be real number, what is the domain of the function 4 3
x x
x f x.
Find the partial fraction decomposition of f (x).
(d) Graph the function
2
3
−
x f x. Include the horizontal and vertical
asymptotes, if any, x- and y-intercepts, if any, and clearly label 3 points on the graph with their coordinates. (e) Give the domain, range, and the rule of the inverse of the preceding function. (f) A bottle of beer is initially at 35 degrees C. It is placed into a water bath kept at a constant temperature of 10 degrees C. According to Newton’s law of cooling, the temperature of the bottle, B( t), will follow the rule B (t )= 10 + Aektwhere t is time measured in minutes. After 30 minutes in the water bath the temperature of the bottle is 19 degrees C. What was the temperature of the bottle after only 15 minutes in the water bath? Express your answer without exponents or logarithms.
(a) Use the rational zero theorem to list all the possible rational zeros of P( x). (b) Explain why P ( x)< 0 if x < 0. (c) Find all three solutions of P ( x)= 0. Which of these solutions, if any, lie in the interval (0, 11/2)? (d) An open box is to be made from a rectangular piece of cardboard that measures 21 by 11 inches, by cutting out squares of the same size from each corner and bending up the sides. Note that the height of the box will be x. If the volume of the box to be 225 cubic inches, is x = 3 is the only possible choice? Explain.