Model Question Paper-1 with effect from 2018-19

(CBCS Scheme)

USN 18MAT21

Second Semester B.E. Degree Examination

Advanced Calculus and Numerical Methods
(Common to all Branches)
Time: 3 Hrs Max.Marks: 100

Note: Answer any FIVE full questions, choosing at least ONE question from each module.

Module-1

1. (a) Find the angle between the surfaces x 2  y 2  z 2  4 and z  x 2  y 2  13 at 2,1,2 (06 Marks)
  
(b) If F  xy 3 z 2  , find divF and curlF at the point 1,1,1. (07 Marks)
   
(c) Find the value of a, b, c such that F  (axy  bz 3 )i  (3x 2  cz ) j  (3xz 2  y)k is a

conservative force field. Hence find the scalar potential  such that F  . (07 Marks)
OR

2. (a) Use Green’s theorem to find the area between the parabolas x 2  4 y and y 2  4 x. (06 Marks)

(b) Using Gauss divergence theorem, evaluate  F  nˆdS over the entire surface of the region above
s
   
xy-plane bounded by the cone z  x  y and the plane z  4 , where F  4 xzi  xyz 2 j  3zk . (07 Marks)
2 2 2

   
(c) Find the work done by the force F  3x 2 i  2 xz  y  j  zk , when it moves a particle from the
point t  0 to t  2 along the curve x  t , y  t 2 4 , z  3t 3 8. (07 Marks)

Module-2

3. (a) Solve: D 3  D 2  4D  4y  3e  x  4 x  6, where D 

d
. (06 Marks)
dx
d2y 1
(b) Solve: y , using the method of variation of parameters. (07 Marks)
dx 2
1  sin x
 
(c) Solve: x 2 D 2  3xD  4 y  1  x  , where D  .
2 d
dx
(07 Marks)

OR
4. (a) Solve: D 3  8y  x 4  2 x  1, where D 
d
. (06 Marks)
dx
d2y
(b) Solve: 3x  2  33x  2  36 y  8 x 2  4 x  1
2 dy
2
(07 Marks)
dx dx
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 (c) The differential equation of the displacement x(t) of a spring fixed at the upper end and
d 2 x dx
a weight at its lower end is given by 10 2   200 x  0. The weight is pulled down 0.25 cm,
dt dt
below the equilibrium position and then released. Find the expression for the displacement of the
weight from its equilibrium position at any time t during its first upward motion. (07 Marks)

Module-3

5. (a) Form the partial differential equation by eliminating the arbitrary constants from
x  a 2   y  b2  z 2  c 2 (06 Marks)
2z
(b) Solve  z, given that when y  0, z  e x and z  e  x (07 Marks)
y 2
(c) Derive one-dimensional wave equation in the standard form. (07 Marks)

OR

6. (a) Form the partial differential equation by eliminating the arbitrary function from
 
f x 2  y 2 , z  xy  0 (06 Marks)
   
(b) Solve: x 2  yz p  y 2  zx q  z 2  xy (07 Marks)
(c) Solve one dimensional heat equation, using the method of separation of variables. (07 Marks)

Module-4


n!
7. (a) Test for the convergence or divergence of the series :  (06 Marks)
n 1 n n 2

(b) Solve Bessel’s differential equation leading to J n x . (07 Marks)

(c) Express f x   x 4  3x 3  x 2  5x  2 in terms of Legendre polynomials. (07 Marks)

OR

n2
8. (a) Test for the convergence or divergence of the series :  n (06 Marks)
n 1 3
1

(b) If  and  are two distinct roots of J n x   0, prove that  xJ x J x dx  0 if    .
0
n n (07 Marks)

(c) Use Rodrigues’s formula to show that P4 cos    35 cos 4  20 cos 2  9
1
(07 Marks)
8

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 Module-5

9. (a) Find a real root of the equation x sin x  cos x  0, near x   correct to four decimal places,
using Newton- Raphson method. (06 Marks)

(b) Use an appropriate interpolation formula to compute using the following data: (07 Marks)
x 1.7 1.8 1.9 2.0 2.1 2.2
f(x) 5.474 6.050 6.686 7.389 8.166 9.025
 2
(c) Use Weddle’s rule to evaluate  cos xdx , by dividing  
 2
2,  2 into six equal parts. (07 Marks)

OR

10. (a) Find a real root of x log10 x  1.2  0 , correct to three decimal places lying in the interval (2,3),
using Regula-Falsi method. (06 Marks)

(b) Using Lagrange’s interpolation formula to fit a polynomial for the following data: (07 Marks)

2 10 17
1 3 4
3
dx
 1  x 
th
(c) Using Simpson’s rule, evaluate 2
taking 4 equidistant ordinates. (07 Marks)
0

*****

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