UPSC NDA I 2025 Mathematics Paper

Given,
The inequality is: k<(√2+1)3<k+2, where k is a natural number.
We need to compute (√2+1)3
(√‌2+1)3=2√‌2+6+3√‌2+1=5√‌2+7
Approximating √‌2≈1.414, we find:
5√‌2+7≈5×1.414+7=7.07+7=14.07
The inequality becomes:
k<14.07<k+2
This implies that k>12.07\ ), so the smallest integer value of k is:
k=13
∴ The value of k is 13 .