State which of the following statements are true and which are false. Justify your answer.

(i) False

Since, 37 has exactly two positive factors, 1 and 37, 37 belongs to the set.

(ii) True

Since, the sum of positive factors of 28

= 1 + 2 + 4 + 7 + 14 + 28

= 56 = 2(28)

(iii) False

7,747 is not a multiple of 37.

(i) The factors of 35 are 1, 5, 7 and 35. So, 35 is an element of the set. Hence, statement is true.

(ii) The factors of 128 hre 1,2,4, 8, 16, 32, 64 and 128.

Sum of factors = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255 * 2 x 128 Hence, statement is false.

(iii) We have, x⁴ – 5x³ + 2x² – 112x + 6 = 0 Forx = 3, we have

(3)⁴ – 5(3)³ + 2(3)² – 112(3) + 6 = 0

=> 81 – 135 + 18-336 + 6 = 0

=> -346 = 0, which is not true.

So 3 is not an element of the set

Hence, statement is true

iv) 496 = 2⁴ x 31

So, the factors of 496 are 1, 2, 4, 8, 16, 31, 62,124, 248 and 496.

Sum of factors = 1 +2 + 4 + 8+ 16 + 31 + 62 + 124 + 248 + 496 = 992 = 2(496)

So, 496 is the element of the set Hence, statement is false

(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.