Rajasthan Technical University 2009-2nd Sem B.Tech(Main) Engineering Mathematics-II - Question Paper

Unit – I
1. Find the formula of the sphere passing through the 4 points (0,0,0), (-1,2,3), (1,-2,3) and (1,2,-3) and determine its radius.
OR
obtain the centre and radius of the circle x2+y2+z2=9; x+y=3. obtain the formula of the sphere having this circle as a great circle.

Unit – II
a. Find the inverse of the matrix
0` one 2
1 2` 3 using elementary row transformation.
3 one 1
b. Examine the consistency of the provided formula 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
two -1 1
A = -2 two -1
one -1 2
b. Find the eigen values and corresponding eigen vectors for the matrix.
-1 two -2
A = one two 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 ??= 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) x2 j`+ (1-a) rz2k`. compute 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` + z2k` over the cylindrical region bounded by x2 + y2 =9, z = 0 and z = 2.

Unit – IV
1.a. A particle begin from origin in the direction of initial line with velocity f and moves with constant angular velocity ? 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 formula of its path is ?2r =f (1– e-?).
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. 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 reduced 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/ under root u2+v2 after a time
v/g[tan-1 u/v+ tanh-1 V/v]

Unit – V
1.a. Solve the provided differential formula in series
x2 d2y/d2y+ x dy + (x2-4) y = 0

a. Solve the provided partial differential formula.
x2 (y-z) del z/del x + y2 (z-x) del z/ del y = z2 (x-y)

OR
a. obtain the complete integral of (x+y) (p+q)2 +(x-y)(p-q)2 = 1. where
p= delz upon del x and q = del z upon del y
b. Use Charpit's method to solve(x2-y2) pq-xy(p2-q2) =1. where
p= delz upon del x and q = del z upon del y

Earning:   Approval pending.