(b) Sofia’s expression, which is Mark’s expression. 39. This question assesses the student’s understanding of the properties of real number system and its subsets, specifically rational and irrational numbers. (a) An irrational number is a real number that cannot be expressed as the ratio of two integers. (b) We know 2 is rational and is irrational. Assume , where is a rational number. Then . Since and 2 are both n n rational, and the rational numbers are closed under subtraction, 2 is also rational. That means is rational. But n – that contradicts what was given, that is irrational. Thus, must be irrational. 41. This question assesses the student’s understanding of the properties of real number system and its subsets, specifically rational and irrational numbers. (a) . Both addends are ratios of integers, as is the sum. (b) . Radicals are rational if they are perfect squares. Only one factor is rational; the product is irrational. (c) . Both factors are irrational, but the product is rational. 42. This question assesses the student’s understanding of the properties of real number system and its subsets, specifically rational and irrational numbers. (a) (b) (c) Let be a rational number and be an irrational number. Assume + is a rational number. Then . a b a b, n Since and are both rational, and the rational numbers are closed under subtraction, is also rational. That n a n a – means is rational. But that contradicts what was given, that is irrational. Thus, + must be irrational. b b a b 60. This question assesses the student’s ability to use solve a quadratic equation. (a) (b) (c) 68. This question assesses the student’s understanding that polynomials are similar to systems of numbers.
(b) Sofia’s expression, which is Mark’s expression. 39. This question assesses the student’s understanding of the properties of real number system and its subsets, specifically rational and irrational numbers. (a) An irrational number is a real number that cannot be expressed as the ratio of two integers. (b) We know 2 is rational and is irrational. Assume , where n is a rational number. Then . Since n and 2 are both rational, and the rational numbers are closed under subtraction, n – 2 is also rational. That means is rational. But that contradicts what was given, that is irrational. Thus, must be irrational. 41. This question assesses the student’s understanding of the properties of real number system and its subsets, specifically rational and irrational numbers. (a) . Both addends are ratios of integers, as is the sum. (b) . Radicals are rational if they are perfect squares. Only one factor is rational; the product is irrational. (c) . Both factors are irrational, but the product is rational. 42. This question assesses the student’s understanding of the properties of real number system and its subsets, specifically rational and irrational numbers. (a) (b) (c) Let a be a rational number and b be an irrational number. Assume a + b, n is a rational number. Then . Since n and a are both rational, and the rational numbers are closed under subtraction, n – a is also rational. That means b is rational. But that contradicts what was given, that b is irrational. Thus, a + b must be irrational. 60. This question assesses the student’s ability to use solve a quadratic equation. (a) (b) (c) 68. This question assesses the student’s understanding that polynomials are similar to systems of numbers.