‘P’ and ‘Q’ started a business with investment of Rs. 3,000 and Rs. 4,200, respectively. After 8 months, ‘P’ increased his investment by Rs. ‘8y’. After another 8 more months, ‘Q’ decreased his investment by Rs. ‘8y’ and ‘R’ joined the business with Rs. 4,800. At the end of 32 months, the ratio of profit share of ‘P’, ‘R’ and ‘Q’ was 137:100:167, respectively. Which of the following statement(s) is/are true regarding ‘y’?
I. Sum of squares of each digit of ‘y’ is 80.
II. 12y < √576 × √256
III. The tens digit of ‘y’ is a composite number.

Ratio of profit share of ‘P’, ‘Q’ and ‘R’, respectively: = [3,000 × 8 + (3,000 + 8y) × 24]:[4,200 × 16 + (4,200 − 8y) × 16]:[4,800 × 16] = (12,000 + 24y) : (16,800 − 16y) : 9,600 So, (12,000 + 24y):9,600 = 137:100 Or, 12,000 + 24y = 13,152 Or, 24y = 1,152 Or, ‘y’ = 48 Statement I:
Required sum = 4² + 8² = 16 + 64 = 80
So, statement I is true. Statement II:
12y = 12 × 48 = 576
√576 × √256 = 24 × 16 = 384
Here, 12y > √576 × √256
So, statement II is false. Statement III:
Tens digit of ‘y’ = 4, which is a composite number.
So, statement III is true. Therefore, statements I and III are true.