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148 If the discriminant in a quadratic equation is a negative real number, it means that (a) the equation has two different real roots (b) the equation has one real root with multiplicity 2 (c) the equation has one imaginary root with multiplicity 2 (d) the equation has two different imaginary or complex roots (e) the equation has no roots at all, real or complex 149 A mapping is a bijection if and only if it is (a) reflexive (b) a function (c) one-to-one and onto (d) a subset of its co-domain (e) transitive 150 Consider the following pair of equations as a two-by-two system: y = a1x + b1 and y = a2x + b2 where a1, a2, b1, and b2 are real numbers, and neither a1 nor a2 are equal to 0 What are the smallest and largest numbers of elements that the solution set of such a system can have Assume that the two equations are not identical, and are not some constant multiple of each other (a) None, and one (b) None, and two (c) None, and infinitely many (d) One, and infinitely many (e) Two, and infinitely many
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APPENDIX
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Worked-Out Solutions to Exercises: s 1 to 9
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These worked-out solutions do not necessarily represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see here, by all means go ahead! But always check your work to be sure your alternative answer is correct
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1
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1 There are several ways to figure this out, but they re all a little messy You can count the days out loud and write hash marks on a sheet of paper, or you can use a calendar and write numerals on it day by day, starting with 0 on June 24 and working up If you want to avoid all that counting, you can figure out the number of days of interest in June, July, August, September, and October, and then add them up Remember that in any year, June has 30 days, July has 31 days, August has 31 days, and September has 30 days In June, you have 30 24 = 6 days of interest; in July you have all 31 days; in August you have all 31 days; in September you have all 30 days; in October you have 2 days of interest The total is therefore 6 + 31 + 31 + 30 + 2 = 100 days 2 Remember that 1 = I, 5 = V, 10 = X, 50 = L, and 100 = C The answers are as follows, broken down into sums for clarification (a) 200 = 100 + 100 = C + C = CC (b) 201 = 100 + 100 + 1 = C + C + I = CCI (c) 209 = 100 + 100 + 9 = C + C + IX = CCIX (d) 210 = 100 + 100 + 10 = C + C + X = CCX 3 Remember that M = 1,000, C = 100, X = 10, V = 5, and I = 1 The answers are as follows, broken down into sums for clarification (a) MMXX = 1,000 + 1,000 + 10 + 10 = 2,020 (b) MMXIX = 1,000 + 1,000 + 10 + 9 = 2,019
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588 Worked-Out Solutions to Exercises: s 1 to 9
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(c) MMIX = 1,000 + 1,000 + 9 = 2,009 (d) MMVI = 1,000 + 1,000 + 6 = 2,006 4 The number three hundred two trillion, seventy billion, one hundred forty-nine million, six thousand, one hundred ten looks like this as a decimal numeral: 302,070,149,006,110 Note the placement of the commas and the ciphers Also note that we use the American billion, which is equivalent to a thousand million 5 This is a slightly whimsical problem You can add as many ciphers as you want to the left of the digit 3 in the answer to the previous problem, and it does not change the value of the number it represents A mathematician might write it as ,000,000,,302,070,149,006,110 You can keep attaching ciphers forever in the left-hand direction! 6 You can make any number in the decimal system ten times as large by adding one cipher to its right and then repositioning the commas Based on the numeral shown for the answer to Prob 4 above, you would get 3,020,701,490,061,100 If you want to make a number a hundred times as large, add two ciphers to its right and then reposition the commas Starting with the numeral shown for the answer to Prob 4, that gives you 30,207,014,900,611,000 If you want to make a number a thousand times as large, add three ciphers to its right and then reposition the commas Starting with the numeral shown for the answer to Prob 4, that gives you 302,070,149,006,110,000 7 The number in the final answer to Prob 6 would be written out in words as three hundred two quadrillion, seventy trillion, one hundred forty-nine billion, six million, one hundred ten thousand That s based on the United States terminology where a billion is a thousand million, and a trillion is a million million 8 To solve this problem, tally the values of each digit in the decimal system and then add them up, as follows One times one gives you 1 One times two gives you 2 Zero times four gives you 0
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