Visvesvaraya Technological University (VTU) 2005 B.E Engineering Mathematics-1 - Question Paper
First Semester B.E Degree Examination, Feb/March 2005
Common to all Branches
Engineering Mathematics-1
Full ques. Paper in attachment
Page No... 1 MATH
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First Semester B.E Degree Examination, February/March 2005
Common to all Branches Engineering Mathematics -1
Time: 3 hrs.] [Max.Marks : 100
Note: 1. Answer FIVE full questions, choosing at least ONE question from EACH PART. 2. All questions carry equal marks.
1. (a) Show that the lines whose direction cosines satisfy the relations I + m + 4ra = 0
and mn + nl + Im = 0 are parallel. (Q Marks]
(b) Derive the equation of the plane in the intercept form. Also find the equation of the plane having y * Intercept 10, z - intercept 4 and perpendicular to the plane 7x + y + 13Z -17 = 0. (4+3=7 Marks)
(c) Find the image of the point (i, -1, 2) in the plane 2x 4- 2y + z = 11. [7 Marks)
2. (a) Show that the lines 2--i = and xJr2y + Zz-S 0 = 2z + 3y + 4z -11
intersect. Also find their point of Intersection. (6 Marks)
(b) Find the coordinates of the point of intersection of the line of S,D with the lines
and = = r and hence find the shortest
distance. (7 Marks)
(c) Find the equation of the right circular cone with vertex {2. -3, -4). semivertical angle 30 and whose axis is equally inclined to the coordinate axes. (7 Marks)
3. (a) if y = (x2 - l)n show that yn satisfies the equation:
(I - X2) - + n(n + l)y = 0. (6 Marks)
(b) Establish the pedal equation of the polar curve :
rn = dn sin n 8 t b7i cos n 9 in the form p2(a-n + b2n) = r2n+2. (7 Marks) (c) If u = log (a:3 + y3 + z3 - 3xyz) show that
m + m= wriand henceshow that
{+$j}+) U= {X + ylZ)2 (7MarkS)
4. (a) State and prove Euler's theorem for a homogeneous function u(x,y) of degree
n and hence show that
X2 Uxx + 2xy Uxy + y2 Uyy = n(n - l)tt (7 Marks)
if x = au cov v and y = au sin v show that J J* 1. (7 Marks)
The current measured by a tangent golvanometer is given by the relation c = k tan 8 where 9 is the angle of deflection. Show that the relative error in c due to a given error in 9 is minimum when 9 = 45. (6 Marks)
(b)
(c)
5. (a) (b)
(c|
6. (a)
(b)
TcT
7. (a)
(b)
8. (a)
ZL
4
Obtain the reduction formula for fn= J Secnx where n is a posiiive integer
o
and hence find Jc. [6 Marks)
Show that when n is a positive integer
2 a
2 a
and hence find J x3 y 2ax x2dx. (7 Marks)
o
2 2 2
Trace the astroid : + yZ = o.3. (7Marks)
Find the length of an arch of the cycloid
x = a(8 - sin 9), y = o{l - cos 9). (6Marks)
Find the surface area of the solid generated by revolving the cycloid x = a{8 - sin6)-, y = a( 1 cos 8) about the base, (7Marks)
Find the volume of solid generated by the revolution of the cardiod r ~ a(l + cos 9) about the initial line. (7 Marks)
Solve
i) = x tan (y ~ x) -\- 1
ii) { - 4y - 9) dx + {Ax + y ~ 2) dy = 0
iii) \xy sin(xy) -h cos{xy)]y dx + (zy sin(xy) cos(a:y)] xdy = 0
(5x3=15 Marks)
Find the orthogonal trajectories of the family of curves cos 8 = a,
a' being the parameter. (5 Marks)
Examine the nature of the following series.
i) -W + -fc&x + 21+:+3 , + .... (6 Marks)
1J 1 + 2 l* + 2-+3z
2 3
ii) i+fD + d) x2 + (I) x2 +......;>0 (7 Marks)
iii)
21- 3 - 5* 6
** *
2 Z2 i2 52 02 7- 8-
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Attachment: |
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