Why use arc length parameter in curvature/torsion definitions?

Arc length provides a natural parameter that measures intrinsic geometry of curves without depending on parameterization.

What is the core identity to prove for z in the mixed partials task (Q3d)?

Show that ∂^2 z/∂y∂x = (x^2 - y^2)/(x^2 + y^2) for z = x^2 arctan(y/x) − y^2 arctan(x/y).

Probability question about failures and passes (Q3e)

If 20% fail in a 6-candidate exam, find the probability that exactly 5 pass (i.e., 1 fails).

Matrix decomposition repeated (Q4a)

Express A = [[1+i, 2, 5-5i], [2i, 2+i, 4+2i], [-1+i, -4, 7]] as sum of Hermitian and skew-Hermitian matrices.

Mathematics Code-1141 2017 HCS mains question paper

Name: Mathematics Code-1141 2017 HCS mains question paper
Creator: Haryana Government
Keywords: Hermitian matrix, skew-Hermitian, unitary matrix, matrix decomposition, complex matrices, paraboloid volume, plane z=0, arc length parameter, curvature, torsion, partial derivatives, mixed partial derivative, z = x^2 arctan(y/x) - y^2 arctan(x/y), probability, six candidates

Haryana Government Jobs Administrative / Civil Services 2017

Year 2017
Conducted By Haryana Government
Questions 9
Maximum Marks 150
Duration 3 Hours
Languages Hindi & English

Exam Details

Detail	Information
Examination	Mathematics (Code-1141)
Year	2017
Conducting Body	Haryana Government
Paper	Mathematics
Subject	Mathematics
Duration	3 Hours
Maximum Marks	150
Number of Questions	9
Question Type	Descriptive / Subjective

This page consolidates the 2017 Haryana HCS Main Mathematics paper (Code-1141). It covers descriptive questions on linear algebra (Hermitian and unitary matrices), analytic geometry (volume cut by a plane from a paraboloid), and multivariable calculus (curl/torsion-like curvature via arc-length parameter). It includes both Hindi and English texts, with the Hindi sections preserved and English used for SEO clarity. Page 2 continues with repair OCR-proofs and supplementing questions on matrix decompositions and probability, ready for SEO-ready metadata and structured data schemas.

Major Topics Covered

Linear Algebra
Hermitian Matrix
Skew-Hermitian Matrix
Unitary Matrix
Complex Matrices
Matrix Decomposition
Analytic Geometry
Paraboloid Volume
Calculus
Partial Derivatives
Mixed Partial Derivatives
Arc Length Parameter
Curvature
Torsion
Probability
Binomial/Hypergeometric-like Scenarios

Why This Paper is Important

Useful for Mathematics (Code-1141) preparation
Helps understand the latest exam pattern
Useful for practice and self-assessment
Covers frequently asked General Studies topics
Helpful for analysing question trends

Related Resources

Instructions

MATHEMATICS (Code-1141) गणित (कोड-1141 ) Time: 3 Hours Maximum Marks : 150 अधिकतम अंक : 150 समय : 3 घंटे
Note: (i) Attempt five questions in all.
All questions carry equal marks.
Question number 1 is compulsory.
Answer any two questions from Part-I and two questions from Part-II.
The parts of the same question must be answered together and must not be interposed between answers to other questions.
In case of any discrepancy in the English and Hindi versions, English version will be
(i) Attempt five questions in all.
(ii) taken as final. पाँच प्रश्न हल कीजिए। सभी के अंक समान हैं। प्रश्न संख्या 1 अनिवार्य है। भाग-। से दो प्रश्नों नोट:
(i) तथा भाग-11 से वो प्रश्नों का उत्तर दीजिए। एक प्रश्नं के सभी अंशों का उत्तर एक साथ दीजिए। एक a in de प्रश्न के अंशों का उत्तर दूसरे प्रश्न के अंशों के मध्य न ले जायें। यदि अंग्रेजी एवं हिन्दी विवरण में कोई विसंगति हो, तो अंग्रेजी विवरण अंतिम माना जायेगा।
(ii) Answer any four of the following: The contract of the contract of the contract of

Questions (page 2)

Q1. (a) 1. Prove that matrix A = [[1, 1-i, 2], [1+i, 3, i], [2, -i, 0]] is Hermitian. सिद्ध कीजिए कि मैट्रिक्स A = [[1, 1-i, 2], [1+i, 3, i], [2, -i, 0]] हरमीशन मैट्रिक्स है।

(a) 1. Prove that matrix A = [[1, 1-i, 2], [1+i, 3, i], [2, -i, 0]] is Hermitian. सिद्ध कीजिए कि मैट्रिक्स A = [[1, 1-i, 2], [1+i, 3, i], [2, -i, 0]] हरमीशन मैट्रिक्स है।

(b) Find the volume cut from the paraboloid: x² + y²/4 + z = 1 by the plane z = 0.

(a) 1. Prove that matrix A = [[1, 1-i, 2], [1+i, 3, i], [2, -i, 0]] is Hermitian. सिद्ध कीजिए कि मैट्रिक्स A = [[1, 1-i, 2], [1+i, 3, i], [2, -i, 0]] हरमीशन मैट्रिक्स है।
(b) Find the volume cut from the paraboloid: x² + y²/4 + z = 1 by the plane z = 0.

Q2. 2. Express the matrix A = [[1+i, 2, 5-5i], [2i, 2+i, 4+2i], [-1+i, -4, 7]] as the sum of a Hermitian matrix and a skew-Hermitian matrix.
(a) (Part of the same task) Matrix A can be written as H + K where H is Hermitian and K is skew-Hermitian.

Q3. (c) Why do we use the arc length parameters instead of general parameter 't' in defining the curvature and torsion? साधारण प्राचल 't' के स्थान पर चाप लम्बाई का प्रयोग क्यों किया जाता है, इसे वक्रता और टॉर्शन में परिभाषित कीजिए।

(c) Why do we use the arc length parameters instead of general parameter 't' in defining the curvature and torsion? साधारण प्राचल 't' के स्थान पर चाप लम्बाई का प्रयोग क्यों किया जाता है, इसे वक्रता और टॉर्शन में परिभाषित कीजिए।
(d) If z = x² arctan(y/x) - y² arctan(x/y), then prove that ∂^2 z/∂y∂x = (x² - y²)/(x² + y²).
(e) The overall percentage of failures in a certain examination is 20%. If six candidates appear in the examination, what is the probability that exactly five pass the examination?

(c) Why do we use the arc length parameters instead of general parameter 't' in defining the curvature and torsion? साधारण प्राचल 't' के स्थान पर चाप लम्बाई का प्रयोग क्यों किया जाता है, इसे वक्रता और टॉर्शन में परिभाषित कीजिए।
(d) If z = x² arctan(y/x) - y² arctan(x/y), then prove that ∂^2 z/∂y∂x = (x² - y²)/(x² + y²).

Q4. (a) Express the matrix A = [[1+i, 2, 5-5i], [2i, 2+i, 4+2i], [-1+i, -4, 7]] as the sum of a Hermitian matrix and a skew-Hermitian matrix.

(a) Express the matrix A = [[1+i, 2, 5-5i], [2i, 2+i, 4+2i], [-1+i, -4, 7]] as the sum of a Hermitian matrix and a skew-Hermitian matrix.

(b) Define a unitary matrix. If N = [[0, 1+2i], [-1+2i, 0]] is a matrix, show that (I - N)(I + N)^{-1} is a unitary matrix.

(a) Express the matrix A = [[1+i, 2, 5-5i], [2i, 2+i, 4+2i], [-1+i, -4, 7]] as the sum of a Hermitian matrix and a skew-Hermitian matrix.
(b) Define a unitary matrix. If N = [[0, 1+2i], [-1+2i, 0]] is a matrix, show that (I - N)(I + N)^{-1} is a unitary matrix.

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Frequently asked questions

What is the duration and maximum marks of this Mathematics paper?

Duration is 3 hours and maximum marks are 150.

Which question is compulsory in this paper?

Question 1 is compulsory.

From which parts should you answer the questions?

Answer any four of the following; the parts of the same question must be answered together.

What is the topic of Q1(a)?

Prove that the given matrix A is Hermitian.

What is the topic of Q1(b)?

Compute the volume cut from the paraboloid x^2 + y^2/4 + z = 1 by the plane z = 0.

What type of matrices are discussed in Q2?

Express a matrix as the sum of a Hermitian and a skew-Hermitian matrix; and define a unitary matrix.

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