Engineering Mathematics-I — Minor 2024 question paper
MMMUT Computer Science and Engineering previous year question paper for Engineering Mathematics-I (BSM-110), semester 1, Minor 2024. All 15 questions are listed below, each with a written answer on Nexsus.
Paper details
Subject: Engineering Mathematics-I (BSM-110)
Branch: Computer Science and Engineering
Semester: 1
Exam: Minor 2024
Questions: 15
Questions asked in Engineering Mathematics-I Minor 2024
Q1(a). If u, v, w are the roots of the cubic equation (lambda - x)^3 + (lambda - y)^3 + (lambda - z)^3 = 0 in lambda, then find ∂(u, v, w)/∂(x, y, z). [4 marks]
Q1(b). If x^2/(a^2+u) + y^2/(b^2+u) + z^2/(c^2+u) = 1, then prove that (∂u/∂x)^2 + (∂u/∂y)^2 + (∂u/∂z)^2 = 2(x^2(∂u/∂x)^2 + y^2(∂u/∂y)^2 + z^2(∂u/∂z)^2)/(x^2+y^2+z^2). [4 marks]
Q1(c). Find the inverse of the matrix A = [[0, 2, 1, 3], [1, 1, -1, -2], [1, 2, 0, 1], [-1, 1, 2, 6]]. [4 marks]
Q1(d). Let A = [[6, -2, 2], [-2, 3, -1], [2, -1, 3]]. Find matrix P such that P^-1AP is a diagonal matrix. [4 marks]
Q2(a). The Temperature T at any point (x,y,z) in space is T(x,y,z) = kxyz^2, k is positive constant. Find the highest temperature on the surface of sphere x^2 + y^2 + z^2 = a^2. [3 marks]
Q2(b). If u be the homogeneous function of degree n in x and y, then show that ∂^2u/∂x^2 + 2xy ∂^2u/∂x∂y + y^2 ∂^2u/∂y^2 = n(n-1)u. [3 marks]
Q2(c). Expand f(x,y) = tan^-1(xy) in powers of (x - 1) and (y - 1) up to the second-degree term. Hence compute f(0.9, 1.1) approximately. [3 marks]
Q2(d). If log y = tan^-1 x, show that (1 + x^2)y_{n+2} + 2(n+1)x y_{n+1} + n(n+1)y_n = 0. [3 marks]
Q3(a). Find the rank of the matrix A by changing it into normal form. A = [[2, 3, -1, -1], [1, -1, -2, -4], [3, 1, 3, -2], [6, 3, 0, -7]]. [3 marks]
Q3(b). Find the value of a and b for which the system of equations 3x - 2y + z = b, 5x - 8y + 9z = 3, 2x + y + az = -1 has (i) unique solution (ii) no solution (iii) infinitely many solutions. [3 marks]
Q3(c). State the Cayley-Hamilton theorem. Verify this theorem for the matrix A = [[1, 2], [-1, 2]]. Hence express A^6 - 4A^5 + 8A^4 - 12A^3 + A - 6I as a linear polynomial in A. [3 marks]
Q3(d). Find the value of k such that the system of equations x + ky + 3z = 0, 4x + 3y + kz = 0, 2x + y + 2z = 0 has nontrivial solution. [3 marks]