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65 When we say that a real number u is the natural logarithm of some other real number v, and that e is Euler s constant (an irrational number equal to about 271828), we are in effect saying that (a) v equals e to the uth power (b) u equals e to the vth power (c) v equals u to the eth power (d) u equals v to the eth power (e) v to the uth power equals e 66 A three-by-three linear system is consistent and not redundant if and only if (a) it has a single, unique solution (b) it has two distinct solutions (c) it has three distinct solutions (d) it has infinitely many solutions (e) it has no solutions 67 Under what circumstances can we divide both sides of an equation by the same variable and get another valid equation (a) Never (b) Only if the variable can never become equal to 0 (c) Only if the variable can never become negative (d) Only if the variable can never become irrational (e) Always 68 The relation graphed in Fig FE-6 is not a function of x if we think of it as a mapping from values of x to values of y We can see this because (a) the curve is not a straight line (b) the domain is not the entire set of real numbers (c) the relation is not one-to-one (d) there are values of x that map into more than one value of y (e) there no values of y that map into more than one value of x 69 How can we restrict the range of the relation graphed in Fig FE-6 so that it becomes a function of x if we think of it as a mapping from values of x to values of y (a) We can restrict the range to the set of non-negative reals (b) We can restrict the range to the set of negative reals (c) We can restrict the range to the set of reals larger than 1 (d) We can restrict the range to the set of reals smaller than 1 (e) Any of the above 70 The relation graphed in Fig FE-6 is a function of y if we think of it as a mapping from values of y to values of x We can see this because (a) there are values of x that map into more than one value of y (b) there no values of y that map into more than one value of x
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Final Exam y 6 4 (0,0) 2 y = +2 x1/2 x 6 4 2 2 4 6 2 4 6
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Figure FE-6 Illustration for Final Exam Questions 68,
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69, and 70
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(c) the curve is symmetrical (d) the domain is the entire set of real numbers (e) the relation is a surjection 71 Suppose that the coefficients and constant in a polynomial equation are placed in a synthetic division array We try a negative real-number test root and get a nonzero remainder The numbers in the last row alternate between positive and negative This tells us that our test root is (a) larger than all the real roots of the equation (b) equal to the largest real root of the equation (c) somewhere between the smallest and the largest real roots of the equation (d) equal to the smallest real root of the equation (e) smaller than all the real roots of the equation 72 State the solution set X for the quadratic equation x 2 + 100 = 0 (a) X = {j10, j10} (b) X = {10, 10} (c) X = {(10 + j10), (10 j10)}
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564 Final Exam
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(d) X = {j10} (e) X = 73 Which of the following equations is true for all positive real numbers u and v (a) ln u + ln v = ln (u + v) (b) ln u + ln v = ln (uv) (c) ln u + ln v = ln (uv) (d) ln u ln v = ln (uv) (e) None of the above 74 Imagine that we come across two different functions, both of which are of degree larger than 2, and where all the coefficients and constants are real numbers We want to solve these functions as a two-by-two system What is the maximum number of real solutions that such a system can have (a) Three (b) Four (c) Five (d) Infinitely many (e) We must have more information before we can answer this 75 When we say that the common logarithm of a real number w is equal to z, we are in effect saying that (a) z equals 10 to the wth power (b) w equals 10 to the zth power (c) z equals w to the 10th power (d) w equals z to the 10th power (e) w to the z power equals 10 76 The octal numeral 700 represents the same quantity as the base-10 numeral (a) 448 (b) 589 (c) 600 (d) 816 (e) 1,023 77 In Fig FE-7, the straight line is the graph of a function where t is the independent variable and x is the dependent variable What is the slope of this line (Be careful! Note that the horizontal axis represents t, and the vertical axis represents x !) (a) 1/2 (b) 1 (c) 1/2 (d) 1 (e) None of the above
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