banogyaniofficial@gmail.comHere is a structured, highly scannable version of your historical notes, optimized for quick revision and easy reference. Chronological Overview of History The Paleolithic Era During the Paleolithic Era, early humans were nomadic hunter-gatherers. They primarily stayed in the Shiwalik Foothills to avoid the harsh, glaciated climate of the higher altitudes. Key Paleolithic Sites: Prehistoric … Read more
BAGHAT STATE: COMPREHENSIVE NOTES 1. Geographical Profile & Etymology 2. Foundation & Early Expansion 3. 18th Century & The Gurkha Period 4. The Era of Escheat & Annexation (1839–1862) 5. Later Rulers & Land Concessions Here are the comprehensive and final notes on Baghal (Arki) State, synthesized from all the provided historical texts. BAGHAL (ARKI) … Read more
Naming of Sirmaur Establishment of the Princely State Rulers of Sirmaur mughal period sikhs and gorkhas british rule
The present-day district of Shimla is a modern amalgamation that includes the territories of numerous erstwhile princely states within its borders. History of Balsan State (Shimla Hill States) 1. Geographical Overview & Location 2. Origin and Early History 3. Gurkha Invasion & British Alliance 4. The Revolt of 1857 5. Administrative Status & State Relations … Read more
HPAS 2015 Maths Optional Paper-1 Question 1(a) Find characteristic equation and roots of the following matrix: Solution: HPAS 2015 Maths Optional Paper-1 Question 1(b) Prove that the set forms a basis of vector space . Solution: HPAS 2015 Maths Optional Paper-1 Question 1(c) If and , , then find the value of with the help … Read more
HPAS 2014 Maths Optional Paper-1 Question 1(a) Find rank of the matrix : Solution: HPAS 2014 Maths Optional Paper-1 Question 1(b) Show that the set : , is a subspace of the vector space . Solution: HPAS 2014 Maths Optional Paper-1 Question 1(c) Examine for continuity of the following function at : Is it differentiable? … Read more
HPAS 2015 Maths Optional Paper-2 Question 1(a) Show that the set: is a finite group of order 5 with respect to addition modulo 5. Solution: HPAS 2015 Maths Optional Paper-2 Question 1(b) Prove that the sequence , where: is bounded monotonically increasing and convergent. Solution: HPAS 2015 Maths Optional Paper-2 Question 1(c) Prove that in … Read more
HPAS 2014 Maths Optional Paper-2 Question 1(a) If is a permutation on five symbols, then find and order of A. Solution: HPAS 2014 Maths Optional Paper-2 Question 1(b) Test the convergence of the integral : Solution: HPAS 2014 Maths Optional Paper-2 Question 1(c) Prove that every Cauchy sequence is bounded. Solution: HPAS 2014 Maths Optional … Read more
HPAS 2025 Mathematics Optional Paper-2: Complete Solutions Comprehensive solutions designed specifically for HPAS aspirants. Table of Contents Question 1(a): Centre of non-singular matrices group Question 1(b): Complete integral of PDE Question 1(c): Limit of inductive sequence Question 1(d): Uniform continuity of Question 1(e): Gauss elimination method Question 2(a): Group of order Question 2(b): Compact subset … Read more
Welcome to the complete repository for HPAS Mathematics Optional Previous Year Questions (PYQs) and Solutions. Here you will find detailed, step-by-step answers to help streamline your preparation. You can choose to navigate the syllabus comprehensively via the Subject-Wise compilations, or practice full-length exams using the Year-Wise indexes below. 📚 Paper 1: All PYQs & Syllabus(Subject/Topic … Read more
