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. đź“š Paper 1: All PYQs & Syllabus(Subject/Topic … Read more
Looking for step-by-step solutions? This page contains topic-wise questions for practice. For complete solutions, please visit our dedicated solutions page: View Year-Wise Solutions Paper-II Topics Index Abstract Algebra Real Analysis Sequences & Series Metric Spaces Complex Analysis Partial Diff. Eqns Laplace Transforms Calculus of Variations Numerical Analysis & C Abstract Algebra Syllabus: Mappings, elementary properties … Read more
Looking for step-by-step solutions? This page contains topic-wise questions for practice. For complete solutions, please visit our dedicated solutions page: View Year-Wise Solutions Paper-I Topics Index Linear Algebra Calculus Analytic Geometry Ordinary Diff. Eqns Vector Analysis Statics Dynamics Linear Algebra Syllabus: Matrices, row and column reductions, echelon forms. Eigenvalues, eigenvectors and characteristic equation of a … Read more
HPAS 2025 Mathematics Optional Paper-1: Complete Solutions Welcome to the comprehensive solution guide for the Himachal Pradesh Administrative Service (HPAS) 2025 Mathematics Optional Paper-1. This resource provides detailed, step-by-step solutions designed specifically for civil service aspirants to master the core mathematical concepts and methodologies required for the exam. Whether you are revising key theorems, practicing … Read more
1. Nomenclature & Regional Identity 2. Ancient & Literary References 3. Social Structure & Dialects 4. The Bushahr & Trade Connection 5. Political Evolution & Transitions History of Rampur Bushahr: Final Exam Notes 1. Legendary Origins (The Three Devtas) 2. Raja Chatar Singh (110th Ruler) 3. Raja Kehri Singh (The “Chattarpati”) 4. Raja Mahender Singh … Read more
