Rajasthan Technical University 2009-2nd Sem B.Tech Civil Engineering (Main) - - exam paper

 

 

 

 

 

Unit I

1.    Find the equation of the sphere passing through the four points (0,0,0), (-1,2,3), (1,-2,3) and (1,2,-3) and determine its radius.

OR

Find the centre and radius of the circle x2+y2+z2=9; x+y=3. Find the equation of the sphere having this circle as a great circle.

Unit II

1.a. Find the inverse of the matrix

0` 1 2

1 2` 3 using elementary row transformation.

3 1 1

b. Examine the consistency of the given equation and solve them

x+y+z =6; 2x+y+3z = 13; 5x+2y+z = 12; 2x-3y-2z = - 10

OR

a.     State Cayley-Hamilton theorem and verify it form the matrix

2 -1 1

A = -2 2 -1

1 -1 2

b.    Find the eigen values and corresponding eigen vectors for the matrix. -1 2 -2

A = 1 2 1

-1 -1 0

Unit III

1.a. Find by vector method, the tangential and normal accelerations of a point moving in a plane curve.

b. Find the values of the constants a,b,c so that the directional derivatives of f = axy² + byz + cz² x³ at, (1,2,-1) has a maximum of magnitude 64 in the direction parallel to the z-axis.

OR

a.     Find the constant a so that V` is a conservative vector field, where V` = (axy z³) i` + (a-2) x<sup>2</sup> j`+ (1-a) rz²k`. Calculate its potential and work done in moving a particle from (1,2,-3) to (1,-4,2) in this field.

b.    State Gauss divergence theorem and verify it for the function F` = yi` + xj` + z<sup>2</sup>k` over the cylindrical region bounded by x2 + y2 =9, z = 0 and z = 2.

Unit IV

w

1.a. A particle start from origin in the direction of initial line with velocity f and moves with constant angular velocity w about the origin and with constant negative radial acceleration -f . Show that the rate of growth of radial velocity is never positive but tends to limit zero and rove that the equation of its path is w²r =f (1 e^-q).

b. A point moves along a circle. Prove that its angular velocity about any point on the circle is half of that about the centre of the circle.

OR

a.    

2u, 1. u, 1 u 3v 2 v 6 v

v

A particle is projected upwards in medium whose resistance is ^gv where v is its velocity. If v the terminal velocity is large compared to u, the velocity of projection, prove that the vertical height reached, the time ascent and the time of dissent are decreased respectively by the fraction of their values when there is no resistance.

b.    A particle is projected upwards with a velocity u in a medium whose resistance varies as the square of the velocity. Prove that it

will return to the point of projection with velocity v = UV after a time

U2+V2

g v V

v tan -1 u + tan h-1 v

 

Unit V

1.a. Solve the given differential equation in series

dx2 dx

_x2 d²y _{+ x} dy _{+ (x2-4) y = 0}

 

a.     Solve the given partial differential equation.

x y

_x2 _(y-z) z _{+ y2 (z-x)} z _{= z2 (x-y)}

OR

a. Find the complete integral of (x+y) (p+q)² +(x-y)(p-q)² = 1. where

x y

p ₌ z and q ₌ z

b.    Use Charpits method to solve(x²-y²) pq-xy(p²-q²) =1. where

x y

_p= z _{and q =} z