AMIETE – ET/CS/IT (NEW SCHEME)   –   Code: AE51/AC51/AT51

Subject: ENGINEERING MATHEMATICS - I

Time: 3 Hours                                                                                                     Max. Marks: 100

NOTE: There are 9 Questions in all.

· Question 1 is compulsory and carries 20 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Out of the remaining EIGHT Questions answer any FIVE Questions. Each question carries 16 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1       Choose the correct or the best alternative in the following:                                  (210)

             a.  If z = f(x+ct)+g(x-ct), then 

(A) (B)

                  (C) (D)

             b. One of the stationary values of the function f(x,y) = x⁴+y⁴-2x²+4xy-2y² is 

(A) (B) (2,-2)

(C) (D) (-2,2)

             c.  Value of the integral is

(A) 1                                                    (B) 0

(C) (D) None

             d.  Rank of the matrix A =  is

(A) 0                                                    (B) 1

(C) 3                                                    (D) 2

             e.  Eigen values of the matrix A = are

(A) 2, 3, 5                                            (B) –2, 0, 5

(C) 2, 2, 5                                            (D) 2, 5, -3

             f.   Root of the equation xe^x = cos x in (0,1) using Regula-Falsi method after two iteration is

(A) 0.5362                                           (B) 0.4467

(C) 0.1932                                           (D) None

             g. Solution of  is

(A) y=c₁e^x+c₂e^2x (B) y=c₁+(c₂+c₃x)e^-x

(C) y=(c₁+c₂x+c₃x²)e^-x (D) y=c₁+c₂e^-x

             h.  General solution of linear differential equation of First order  (where P and Q are constants or functions of y) is

(A) (B)

(C) (D)

             i.   The value of  is

(A) (B)

(C) (D)

             j.   Value of

(A) (B)

                  (C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

  Q.2     a.   If u is a homogeneous function in x, y of degree ‘n’. Prove the following results.                 

                   (i)

                   (ii)  (8)

b.   A rectangular box open at the top is to have volume of 32 cubic feet. Find the dimension of the box requiring least material for its construction.          (8)

Q.3     a.   By changing the order of integration evaluate

(8)

b.   Using triple integration find the volume of the sphere x²+y²+z²=a² (8)

Q.4     a.   Find all the eigen values and the eigen vector corresponding to the dominant eigen value of the matrix

                            A= (8)

             b.   Investigate for consistency of the following equations and if possible find the solutions 

                   4x-2y+6z=8

                   x+y-3z=-1

                   15x-3y+9z=21                                                                                                  (8)

  Q.5     a.   Solve the following equations by Gauss-Siedal Method

      2x+15y+6z=72

                   54x+y+z=110

                   –x+6y+27z=85                                                                                                 (8)

             b.   Solve x sin x+cos x=0, near x=  using Newton-Raphson method. Carry out three iterations.                                                                   (8)

Q.6     a.   Solve the differential equation (8)

             b.   A body of mass m, falling from rest is subject to the  force of gravity and an  air resistance proportional to the  square of the velocity (v²). If it falls through a distance x and possesses a velocity v at that instant, prove that where mg=ka². Mention one important observation.         (8)

Q.7     a.   Solve by the method of variation of parameter.                   (6)

             b.   Solve the simultaneous equations

                   being given x=y=0, when t=0                                                                           (10)

  Q.8     a.   Solve in series the equation .                                                      (10)

             b.   Prove that (6)

Q.9     a.   If are two distinct roots of J_n(x)=0. Prove that

(8)

             b.   Prove that (8)