

InterviewSolution
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State which of the following statements are true and which are false. Justify your answer.(i) 35 ∈ {x | x has exactly four positive factors}.(ii) 128 ∈ {y | the sum of all the positive factors of y is 2y}(iii) 3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0}(iv) 496 ∉ {y | the sum of all the positive factors of y is 2y}. |
Answer» (i) True According to the question, 35 ∈ {x | x has exactly four positive factors} The possible positive factors of 35 = 1, 5, 7, 35 35 belongs to given set Since, 35 has exactly four positive factors ⇒ The given statement 35 ∈ {x | x has exactly four positive factors} is true. (ii) False According to the question, 128 ∈ {y | the sum of all the positive factors of y is 2y} The possible positive factors of 128 are 1, 2, 4, 8, 16, 32, 64, 128 The sum of them = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255 2y = 2 × 128 = 256 Since, the sum of all the positive factors of y is not equal to 2y 128 does not belong to given set ⇒ The given statement 128 ∈ {y | the sum of all the positive factors of y is 2y} is false. (iii) True According to the question, 3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0} x4 – 5x3 + 2x2 – 112x + 6 = 0 On putting x = 3 in LHS: (3)4 – 5(3)3 + 2(3)2 – 112(3) + 6 = 81 – 135 + 18 – 336 + 6 = –366 ≠ 0 So, 3 does not belong to given set ⇒ The given statement 3 ∉ {x | x4 – 5x3 + 2x2 – 112x + 6 = 0} is true. (iv) False According to the question, 496 ∉ {y | the sum of all the positive factors of y is 2y} The possible positive factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248, 496 The sum of them = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 = 996 2y = 2 × 496 = 992 Since, the sum of all the positive factors of y is equal to 2y 496 belongs to given set ⇒ The given statement 496 ∉ {y | the sum of all the positive factors of y is 2y} is false. |
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