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}.
(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.