Correct option is A
Solution:
All three numbers are prime
All are distinct
All are greater than 13
So we’ll consider primes like: 17, 19, 23, 29, 31, ...
Option A: Mean of the three numbers may be 19
Let’s test this:
Try primes: 17, 19, 21 (21 not prime)
Try: 13, 19, 25 (13 and 25 not valid – 13 is not >13, 25 not prime)
Try: 17, 19, 21 (21 not prime)
Try: 17, 19, 21 again invalid.
Try: 17, 19, 21 → no combination gives mean = 19
Let’s use formula: Mean = (a + b + c) / 3 = 19 → a + b + c = 57
Try finding 3 distinct primes >13 whose sum is 57:
Try: 17 + 19 + 21 = 57 21 not prime
Try: 17 + 19 + 23 = 59
Try: 17 + 19 + 29 = 65
Try: 17 + 19 + 13 = 49 13 not allowed
Try: 19 + 17 + 21 = 57 again 21 not prime
We keep failing to find 3 valid primes >13 that add to 57.
So, no such set exists → This statement is NOT true.
Option B: Median of the three numbers may be 19
Try: 17, 19, 23 → all valid primes >13
Median = 19 → Possible
️ This is true
Option C: Standard deviation of the three numbers may be greater than 1
Try widely spaced primes: 17, 19, 31
Clearly spaced enough to give standard deviation >1
️ True
Option D: Standard deviation is always greater than 0.5
Try closely spaced primes: 17, 19, 23
Compute SD (roughly):
Mean = 19.67
Deviations: −2.67, −0.67, +3.33 → squares = ~7.11, 0.45, 11.09
Sum = ~18.65 → Variance = 18.65/3 = ~6.2 → SD = √6.2 ≈ 2.49
️ Definitely > 0.5
Try: 17, 19, 23 → all sets will give SD > 0.5
️ Always true
Final Answer:
S. Ans. (a) Mean of the three numbers may be 19
