Practice • Improve • Repeat
Ask any ISS aspirant to name a geography topic they are confident about and El Nino comes up almost immediately. Ask them to write 200 words on it and something odd happens. The first hundred words are perfect ocean science, the next fifty drift into general climate change, and the answer ends without a single … Read more
A student in our community wrote last month with a familiar complaint. She had studied the Indus Valley Civilization three times, could list Mohenjo daro, Harappa, Dholavira and Lothal without pausing, and still froze when a mock test asked what the Rakhigarhi excavations had added to our understanding of Harappan society. She knew the sites. … Read more
Welcome to Part 16, the final part of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. So far the problem was always the same: given the arguments xx and their entries yy, predict a missing yy. That is direct interpolation. Today the examiner flips the script: given a value of yy, find … Read more
Welcome back to Part 15 of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. In Part 14 we met Newton’s Divided Difference formula. Powerful, but it still needs a table. Today we go table free with the formula the syllabus names as “Lagrange’s formula for unequal intervals”: Lagrange’s Interpolation Formula, the examiner’s … Read more
Welcome back to Part 14 of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. In Part 13 we defined divided differences and discovered their symmetry. But a single difference is not a prediction. To actually build a polynomial and estimate a missing value from scattered data, we need the most versatile weapon … Read more
Welcome to Part 13 and the opening of Module 4, updated for the UPSC ISS 2026 to 2027 cycle. In Modules 1 to 3 we lived in a perfect world where every formula, from Newton Gregory to Stirling and Bessel, demanded equally spaced arguments. Real data is rarely that polite. When the intervals turn unequal, … Read more
Welcome to Part 12, the final part of Module 3, updated for the UPSC ISS 2026 to 2027 cycle. In Part 11 we met the Gauss Forward and Backward formulas. They work, but their zig zag paths are clumsy for hand calculation. Mathematicians fixed this by averaging them, and the averages became the two most … Read more
Welcome back to Part 11 of Module 3, updated for the UPSC ISS 2026 to 2027 cycle. In Part 10 we met the central operators δ\delta and μ\mu. Today we put them to work with the two foundational pillars of central interpolation: Gauss’s Forward and Backward formulas. Both formulas measure the step value u=x−x0hu = … Read more
Welcome to Part 10 and the opening of Module 3, updated for the UPSC ISS 2026 to 2027 cycle. In Module 2 we predicted values near the beginning and the end of a table. But when the missing value sits exactly in the middle, both Newton Gregory formulas converge slowly. The accurate tools for the … Read more
Welcome to Part 9, the final part of Module 2, updated for the UPSC ISS 2026 to 2027 cycle. The forward and backward formulas need a full difference table. But when a question asks for one missing value from a small dataset, there is a far faster route. The hack: assume the nn known values … Read more