NDA Mathematics Question (2025) — Solution

Step-by-step solution

Let x = ( 2 +1)^3. Let x = I + f, where I is the integer part and 0 Let f' = ( 2 -1)^3. Since 0 Now, evaluate x - f': x - f' = ( 2 +1)^3 - ( 2 -1)^3 x - f' = (2 2 + 6 + 3 2 + 1) - (2 2 - 6 + 3 2 - 1) x - f' = (7 + 5 2 ) - (5 2 - 7) = 14 Since x = I + f, we get I + f - f' = 14, which implies f - f' = 14 - I. Since 14 - I is an integer and -1 Thus, f = f' and I = 14. This gives x = 14 + f, meaning 14 We are given the inequality k Checking the given options: For k = 11, the interval is (11, 13), which does not contain x. For k = 13, the interval is (13, 15), which contains x since 14 For k = 15, the interval is (15, 17), which does not contain x. For k = 17, the interval is (17, 19), which does not contain x. Thus, the correct value of k from the options is 13. Answer: 13