THEORY EXAMINATION 2024-25 ENGINEERING MATHEMATICS-I

B.Tech Engineering 53 downloads

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📄 Question Paper Overview

The examination is for the 2024-25 academic session, with a total duration of 3 hours and a Maximum Marks (M.Marks) of 70.

Subject: Engineering Mathematics-I

Subject Code: BAS103

Level: B.Tech (SEM I) Theory Examination

Time: 3 HRS

Max Marks: 70

Identification Code: QP25DP2_290

Date/Time of Print: 03-Mar-2025 9:02:08 AM

🔢 Structure and Content

The paper is divided into three sections (A, B, and C), covering core areas of Engineering Mathematics, including Linear Algebra, Differential Calculus, Integral Calculus, and Vector Calculus.

Section A: Short Answer Questions (Total Marks: 14)

Consists of seven compulsory questions (Q. No. 1, parts a-g), each carrying 2 marks.

Questions require brief answers, direct computations, or statements of definitions/theorems (K1, K2, K3, K4 levels).

Topics covered:

Finding eigen values of a specific $2\times 2$ matrix.

Verifying Euler's theorem for homogeneous functions ($x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}$).

Difference between total and partial derivatives.

Applications of Jacobians.

Statement of Liouville’s Theorem (in the context of calculus/integrals).

Evaluating a simple double integral.

Proving $\text{curl } \vec{r} = 0$.

Section B: Medium Answer Questions (Total Marks: 21)

Requires attempting any three out of five given questions (Q. No. 2, parts a-e).

Each question carries 7 marks.

These questions primarily involve standard derivations, finding series, or applying theorems (K2, K3, K4, K5 levels).

Topics covered:

Finding non-singular matrices P and Q to reduce a matrix A to its Normal Form (Linear Algebra).

Finding the $n^{th}$ derivative of $\tan^{-1}(x/a)$ (Differential Calculus).

Finding the volume of the largest rectangular parallelepiped inscribed in an ellipsoid (Maxima/Minima, Application of Multivariable Calculus).

Applying Dirichlet’s theorem to evaluate a triple integral over an ellipsoid.

Showing that the vector $\vec{F} = f(r)\vec{r}$ is irrotational (Vector Calculus).

Section C: Long Answer/Numerical Questions (Total Marks: 35)

Consists of five main questions (Q. No. 3 to Q. No. 7).

In each of the five questions, candidates must attempt any one part (a or b), with each part carrying 7 marks.

The questions are analytical, descriptive, or involve extensive calculations/proofs (K1 to K5 levels).

Q. No.	Topic Area	Part (a)	Part (b)
3	Linear Algebra	Find eigen values and eigen vectors of a given $3\times 3$ matrix $\text{A}$.	Discuss the existence and nature of the solution for a given system of linear equations for all values of $\text{K}$.
4	Differential Calculus	Trace the curve $y^2(a+x) = x^2(3a-x)$.	Prove the relation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f''(r) + \frac{1}{r} f'(r)$, where $u = f(r)$ and $r^2 = x^2 + y^2$.
5	Multivariable Calculus	Find the Jacobian $\frac{\partial(x, y, z)}{\partial(u, v, w)}$ given $u=xyz$, $v=x^2+y^2+z^2$, $w=x+y+z$.	Find the maxima and minima of the function $\sin x + \sin y + \sin(x+y)$.
6	Integral Calculus	Find the area inside the circle $r=2a \cos\theta$ and outside the circle $r=a$ (Polar Coordinates).	Change the order of integration and then evaluate the given double integral.
7	Vector Calculus	Show that $\text{div}(\text{grad } r^n) = n(n+1)r^{n-2}$.	Verify Stokes' theorem for the vector field $\vec{F} = (x^2+y^2)\hat{i} - 2xy\hat{j}$ over a rectangle bounded by $x=0, x=a, y=0, y=b$.