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) Given: 35 ∈ {x | x has exactly four positive factors}

To check: 35 belongs to given set or not

The possible positive factors of 35 are 1, 5, 7, 35

Since, 35 has exactly four positive factors

⇒ The given statement is true.

(ii) Given: 128 ∈ {y | the sum of all the positive factors of y is 2y}

To check: 128 belongs to given set or not

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

⇒ The given statement is false.

(iii) Given: 3 ∉ {x | x⁴ – 5x³ + 2x² – 112x + 6 = 0}

To check: 128 belongs to given set or not

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

So, 3 does not belong to given set

⇒ The given statement is true.

(iv) Given: 496 ∉ {y | the sum of all the positive factors of y is 2y}

To check: 128 belongs to given set or not

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 is false.