Engineering Mathematics–I — Major (End Semester) 2024 question paper
MMMUT Chemical Engineering previous year question paper for Engineering Mathematics–I (BSM 110), semester 1, Major (End Semester) 2024. All 24 questions are listed below, each with a written answer on Nexsus.
Paper details
Subject: Engineering Mathematics–I (BSM 110)
Branch: Chemical Engineering
Semester: 1
Exam: Major (End Semester) 2024
Questions: 24
Questions asked in Engineering Mathematics–I Major (End Semester) 2024
Q1(a). If u=sinx+cosx, then show that u r =r![1+(−1) r sin2x] −1/2 where u r ∈R. Differentiate 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 3 +y 3 −3x−12y+20. [2 marks]
Q1(d). If u=f(x 2 −y 2 ,y 2 −z 2 ,z 2 −x 2 ), then prove that x 1 ∂x ∂u + y 1 ∂y ∂u + z 1 ∂z ∂u =0. [2 marks]
Q1(e). The eigen values of a matrix [ a 1 4 b ] are −2 and 3, then find the value of a and b. [2 marks]
Q1(f). Check the consistency of the system of equations: x+2y−z=6, 3x−y−2z=3, and 4x+3y+z=9. [2 marks]
Q1(g). Find inverse of the following matrix using Gauss Jordan Method: A= 2 3 3 1 2 4 1 3 9 (g) Find inverse of the following matrix using Gauss Jordan Method: [2 marks]
Q2(a). Show that Γ( n 1 )Γ( n 2 )Γ( n 3 )⋯Γ( n n−1 )= n 1/2 (2π) 2 n−1 . [5 marks]
Q2(b). Show that Γ(m)Γ(m+ 2 1 )= 2 2m−1 π Γ(2m). [5 marks]
Q3(a). Evaluate ∬ R (x−y) 2 e x+y dxdy, where R is the square with vertices at (1,0), (2,1), (1,2) and (0,1). [5 marks]
Q3(b). Change the order of integration: ∫ 0 a ∫ x 2a−x xydydx [5 marks]
Q3(c). (i) Using double integration, find the area enclosed by curves y=2x 2 andy 2 =4x. (ii) Find the volume of the solid region contained in the first octant of the ellipsoid a 2 x 2 + b 2 y 2 + c 2 z 2 =1. [5 marks]
Q4(a). Evaluate ∬ S F ⋅ n ^ dS where F =4y i ^ +18z j ^ −x k ^ and S is the surface of the plane 3x+2y+6z=6, contained in the first octant. [5 marks]
Q4(b). State Stoke’s theorem. Verify Stoke’s theorem for the vector field F =5x(2x−y) i ^ −yz 2 j ^ −y 2 zk 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 ∮ C (x−2y)dx+xdy taken around the circle x 2 +y 2 =4. [5 marks]
Q5(a). (i) Show that r n is solenoidal for n=−3 and irrotational for all values of n. (ii) Find the directional derivative of ϕ=xyz 2 +xz at (1,1,1) in the direction of the normal to the surface 3xy 2 +y+z=1. [5 marks]
Q5(b). Verify Gauss divergence theorem for F =(x 2 −yz) i ^ +(y 2 −zx) j ^ +(z 2 −xy) k ^ taken over the parallelepiped bounded by 0≤x≤a,0≤y≤b,0≤z≤c. [5 marks]
Q5(c). (i) Show that ∇⋅( r 3 r )=0, where r is the magnitude of position vector r =x i ^ +y j ^ +z k ^ . (ii) Find the work done in moving a particle by the force field F =3x 2 y i ^ +(2x 3 −y) j ^ −z k ^ from t=0 to t=1 along the curve x=2t 2 ,y=t,z=4t 3 . [5 marks]