Engineering Mathematics-1 — Major (End Semester) 2025 question paper
MMMUT Internet of Things previous year question paper for Engineering Mathematics-1 (BSM-110), semester 1, Major (End Semester) 2025. All 23 questions are listed below, each with a written answer on Nexsus.
Paper details
Subject: Engineering Mathematics-1 (BSM-110)
Branch: Internet of Things
Semester: 1
Exam: Major (End Semester) 2025
Questions: 23
Questions asked in Engineering Mathematics-1 Major (End Semester) 2025
Q1(a). If u = sin nx + cos nx, then show that u_r = n^r [1 + (-1)^r \sin 2nx]^{1/2} , where u_r is the r^th differential coefficient of u w.r.t. x. [2 marks]
Q1(b). Find the n^th derivative of e^ax sin(bx + c). [2 marks]
Q1(c). Find the extreme values of the function f(x,y) = x^2 + y^2 - 3x - 32y + 20. [2 marks]
Q1(d). If u = f(x^2 - y^2, y^2 - z^2, z^2 - x^2), then prove that frac{1}{x}\{\partial u}{\partial x} + \frac{1}{y}\frac{\partial u}{\partial y} + \frac{1}{z}\frac{\partial u}{\partial z} = 0. [2 marks]
Q1(f). The eigen values of a matrix $\begin{bmatrix} 8 & 4 \\ 1 & b \end{bmatrix}$ are -2 and 3, then find the value of a and b. [2 marks]
Q1(g). Find inverse of the following matrix using Gauss Jordan Method. [2 marks]
Q2(a). Show that [5 marks]
Q2(b). Show that [5 marks]
Q2(c). Evaluate [5 marks]
Q3(a). Evaluate R (x - y)^2 e^{x+y} dx dy, where R is the square with vertices at (1,0), (2,1), (1,3) and (0,1). [5 marks]
Q3(b). Change the order of integration and hence evaluate it. [5 marks]
Q3(c). (i) Using double integration, find the area enclosed by curves y=2x2 and y2=4x. (ii) Find the volume of the solid region contained in the first octant of the ellipsoid a2x2+b2y2+c2z2=1. [5 marks]
Q4(a). Evaluate \iint_{S} F \cdot \hat{n} dS.Here F = 4xi\hat{i} + 10xj\hat{j} - zk\hat{k} and S is the surface of the plane 3x + 2y + 6z = 6, contained in the first octant.(Note: The coefficients in the image appear printed as 4yi and 10xj, but are standardly written in vector notation as \hat{i}, \hat{j}, \hat{k} components. [5 marks]
Q4(b). State Stoke's theorem. Verify Stoke's theorem for the vector field F = (2x - y)\hat{i} - yz^2\hat{j} - y^2zk\hat{k}, over the upper half surface of the sphere x^2 + y^2 + z^2 = 1, bounded by its projection on the xy-plane. [5 marks]
Q4(c). Verify Green's theorem in a plane for the integral [5 marks]
Q5(a). (i) Show that x^n F is solenoidal for n = -3 and irrotational for all values of n. (ii) Find the directional derivative of xy^2 + xz at (1,1,1) in the direction of the normal to the surface 3xy^2 + y + z at (0,1,1). [5 marks]
Q5(b). Verify Gauss divergence theorem for F = (x^2 - yz)\hat{i} + (y^2 - zx)\hat{j} + (z^2 - xy)\hat{k} taken over the rectangular parallelepiped bounded by 0<= x<=a, 0 < y < b<0<z< c. [5 marks]
Q5(c). (i) Show that div r = 0, where r is the magnitude of position vector \vec{r} = x\hat{i} + y\hat{j} + z\hat{k}.(ii) Find the work done in moving a particle by the force field $F = 3x^2\hat{i} + (2xz - y)\hat{j} - z\hat{k}, from x = 0 to x = 1 along the curve x = 2t^2, y = t, \z = 4t^3. [5 marks]