Question

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 1

(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