Verify which of the following statements are true. (i) (k + 1) (k + 2) - (k + 3) is always 2. Class 8

Verify which of the following statements are true. (i) (k + 1) (k + 2) - (k + 3) is always 2. Class 8

Question 1.

Verify which of the following statements are true. Class 8

(i) (k + 1) (k + 2) - (k + 3) is always 2.

(ii) (2q + 1) (2q - 3) is a multiple of 4.

(iii) Squares of even numbers are multiples of 4, and squares of odd numbers are 1 more than multiples of 8.

(iv) (6n + 2)² - (4n + 3)² is 5 less than a square number.

Solution:

(i) (k + 1)(k + 2) - (k + 3)

= k² + 2k + k + 2 - k - 3

= k² + 2k - 1 ≠ 2

So, statement is not true.

(ii) (2q + 1) (2q - 3)

= 4q² - 6q + 2q - 3

= 4q² - 4q - 3 = 4 (q² - q) - 3 is not a multiple of 4.

So, statement is not true.

(iii) (2)² = 4, (4)² = 16 = 4 × 4, (6)²

= 36 = 4 × 9, ...

So, this statement is true.

3² = 9 = 8 + 1, 5² = 25 = 3 × 8 + 1,

7² = 49 = 3 × 16 + 1, and so on.

So, this statement is true.

(iv) (6n + 2)² - (4n + 3)²

= (6n + 2 + 4n + 3) {(6n + 2) - (4n + 3)}

= (10n + 5) (6n + 2 - 4n - 3)

= (10n + 5) (2n - 1)

= 2n² - 10n + 10n - 5 = 20n² - 5

20n² is not a square number.

So, this statement is not true.