Visvesvaraya Technological University (VTU) 2006 B.E FIRST SEMESTER , ENGINEERING MATHEMATICS I - Question Paper

FIRST SEMESTER B.E. DEGREE EXAMINATION, JANUARY 2006
COMMON TO ALL BRANCHES
ENGINEERING MATHEMATICS I
Time: three hrs. Max. Marks:100
Note: ans any 5 ques.. Choosing are lowest 1 ques. from every part.
2. All ques. carry equal marks.
PART A
1. (a) A line makes angles, a, b, d with 4 diagonals of a cube. Show that Cos2 a
+cos two b + cos two ? + cos two d= 4/3 (6 marks)
(b) obtain the formula of the plane which passes through the point (3, 3,
1) and is
perpendicular to the line joining the points (3, 2, 1)
and (2, 1
5) (7 marks)
(c) obtain the formula of the plane which finds the line x1/
2 = y+1/1 = z3/
4 and is
perpendicular to the plane x+2x+z=12 (7 marks)
2. (a) obtain the points on the lies x3/
1 = y5/
2
= z7/
1 and x+1/7 = y+1/6=z+1/1 closest
to every other, and hence obtain the shortest distance ranging from the 2 lines.
(6 marks)
(b) obtain the formula of the right circular cone whose vertex is at the origin, semi
vertical angle 45°, and whose axis is the line x=2y=z (7 marks)
(c) obtain the formula of the right circular cylinder having radius three units, and the line
passing through the points(2, 3, 4) and (4, 4, 2) as its axis. (7 marks)
PART B
3. (a) obtain the n th derivative of sin three xcos two x (6 marks)
(b) Show that is y = logx/x then yn = 1
n n!/x n+1 {logx11/
2 – 1/ three ……
1/
n}
(7 marks)
(c) Show that the curves r = x two + y two + z two then show that
d two u / d x two + d two u / d y two + d two u / d z two = 0 (6 marks)
(b) if u = x two = 2y two and v=2x two – y two , where x = rcos?,
y= rsin?, show that d(u,v)/d(r,?) (7 marks)
(c) If T = 2p 1/g and g is a constant, obtain the fault in T due to an fault of 3% in the
value of 1, Also obtain the maximum fault n T due to possible errors up to 1% in one and
3% in g. (7 marks)PART – C
5. (a) find a reduction formula for ò cosec n xdx (6 marks)
(b) Evaluate two a ò x two 2 ax - x two dx (7 marks)
(c) Trace the curve y two (ax)
= x2 (ax), a > 0 (7 marks)
6. (a) obtain the surface are of the solid generated by revolving r two = a two cos 2? about the
initial line (6 marks)
(b) obtain the total perimeter of the curve x 2/3 + y 2/3 = a 2/3 (7 marks)
(c) obtain the quantity of the solid generated by revolving the curve xy two = 4a two (2ax)
about y –axis. (7 marks)
PART D
7. (a) Solve 1+x two +y two +x two y two +xy(dy/dx) = 0 (5 marks)
(b) ye x/y dx = (xc x/y +y two )dy (5 marks)
(c) {y (1+1/x) + cosy} dx+{x+logxxsiny}
dy = 0 (5 marks)
8. (a) provided y = ke 2x
+ 3x obtain member of its orthogonal trajectories passing through
the point (0, 3) (6 marks)
(b) Examine the convergence of the series.
(2 / one two / 1) (3 / two three / two ) (4 / three four / three ) ........... ....
(c) Using integral test show that the series å ¥
=1/ni np converges if p > one and diverges if0 < p £ 1

Earning:   Approval pending.