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

 

Subject: ENGINEERING MATHEMATICS - I

Flowchart: Alternate Process: DECEMBER 2009
 


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.  The period of a simple pendulum is  maximum error in T due to the possible error up to 1% in  and 2.5% in g, equals   

 

                  (A) 1.63%                                            (B) 1.59%                                                                         

                  (C) 1.75%                                            (D) None of these                                                

 

             b. The value of integral is equal to 

 

                  (A)                                                   (B)

                  (C)                                                   (D)

 

             c.  The solution of the differential equation satisfying the initial conditions y(0) = 1, y(π/2) = 2 is

 

                  (A) y = 2cos(x) + sin(x)                        (B) y = cos(x) + 2 sin(x)

                  (C) y = cos(x) + sin(x)                          (D) y = 2cos(x) + 2 sin(x)

 

             d.  If  the matrix  then

 

                  (A) C=Acos(θ) – Bsin(θ)                     (B) C=Asin(θ) + Bcos(θ)

                  (C) C=Asin(θ) – Bcos(θ)                     (D) C=Acos(θ) + Bsin(θ)

 

             e.  The general solution of  is  

 

                  (A)                                 (B)

                  (C)                                   (D)

 

             f.   The function whose first difference , is

 

                  (A)                              (B)

                  (C)                              (D)

 

             g. The Bessel’s equation of order 0 is given as

 

                  (A)                          (B)

                  (C)                            (D)

            

             h.  In terms of Beta function  is  

 

(A)                                       (B)

                  (C)                                      (D) None of these

 

             i.   If λ is an eigen value of a non-singular matrix A then the eigen value of A-1 is

 

                  (A) 1/ λ                                                (B) λ

                  (C)                                                  (D) -1/ λ

 

             j.   The value of the integral is

 

                  (A)                                     (B)                                                                

                  (C)                                    (D)                         

 

 

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

 

 

  Q.2     a.   If  , show that .                         (8)

                  

             b.   If   where x-1+y-1+z-1=1, show that the stationary value of u is given by .                                                      (8)

 

                

  Q.3     a.   Evaluate the integral  by changing to polar coordinates, R is the region in the x-y plane bounded by the circles  and =9.        

                                                                                                                                            (8)

                                                                                                                                               

             b.   Evaluate the integral where T is region bounded by the cone  and the planes z=0  to z=h in the first octant.                                                             (8)

 

  Q.4     a.   Investigate the values of λ for which the equations are consistent,

                    

                   Hence find the ratios of x:y:z when λ has the smallest of these values.                   (8)

       

             b.   Find the eigen value and eigen vector of the matrix .            (8)

            

  Q.5     a.   Find the solution of the differential equation (y-x+1)dy – (y+x+2) dx = 0.            (6)

 

             b.   Solve the differential equation .     (6)

 

             c.   Show that the functions 1, sinx, cosx are linearly independent.                             (4)

 

  Q.6     a.   Prove that .                                                  (8)

 

             b.   Solve .                                               (8)

 

  Q.7     a.   Find the power series solution about the point  of the equation .                                                                     (11)

 

             b.   Express f(x)= in terms of Legendre Polynomial.              (5)

       

  Q.8     a.   Express  in terms of  and .                                                    (8)

 

             b.   Solve .                                                                   (8)

 

  Q.9     a.   Solve by Gauss-Seidel method, the following system of equations:                       (8)

                  

 

             b.   Using Runge-Kutta method  of fourth order, solve for y(0.1), y(0.2) given that  .                                                                (8)