Punjab Engineering College 2008 B.E MATHEMATICS _ I - Question Paper

B.E./B.TechD.E GREEE XAMINATIONA, PRILA4AY2 008.
First Semester
(Regulation 2004)
Civil Engineering


MA 1101- MATHEMATICS _ I
(Common to all branches of B.E./B.Tech.)


Time : 3 hours Maximum : 100 marks
ans ALL ouestions
.
PARTA- (10x2=20 marks)

1. provide 2 uses of Cayley Hamilton theorem.
2. If the sum of 2 eigen values and trace of a three x three matrix A are equal,
obtain lAl.
3. elaborate the direction cosines of ihe lines equallf inclined to the axes?
4. obtain the formula of the plane passing through the intersection of the planes
x + y +z =6 and 2x+3y +42 +5 = 0 andthe point ( t , t , t ) .
5. obtain the raoius of cun'ature of the curve )' = e' at the point where it crosses
the y - axis.
6. Show that the family of straight lines 2y - 4x + ) = 0 has no envelope where
.,i is the parameter.
7. rf x =rcos| / = rsind proveth at$ = + dx dr
6 o I o s I lf D three o
obtain the stationeryp ointso f f(x,y) --x y +9 * 9.
)cy
dx
:iolve
dt
10. Write Euler's Homogeneousl inear differential formula. How will you convert
it to a linear differential formula with constant coefficients?


PARTB-(5xL6=80marks)
11. (a) (i) Test whether equations 2x-31'+72=5,3x+y-Bz=13
2x + 19v - 47 z = 32 are consistent or not. (
(ii) decrease the quadratic form xl +2t"2 - -rrt - 2xrx, =2xqx3 to the
canonical form through an orthogonal transformation. (
Or
tr two si
+2 four s l,,sing CaYle-Y Hamilton
[e b 6.]
8.
9.
,'l ^, - U -'-d- -t1+ -T=U.
theorem. (
-1
0 0-
(ii) Diagonalize the matrix A= 0 three -1 using an orthogonal
o -1 3J
transformation. (
12. (a) (i) obtain the formula of the piane passing through
*
^l =
y +-r - Lr and perpendicular t'o x +2y + z =12. (6)
2-1 4
(ii) Prove that the li.
x+1 v-3
nes
==;=zt2
and
v-7 z+7
-32
intersect. obtain the co-ordinates of the point of intersection and
formula of the plane containing them. (10)
Or

(b) (i) obtain the formula of the tangent plane to the sphere
3(x2 + y2 + t2) -2x -3y - 4y -22 = 0 at the point 0,2, 3). Also obtain
the formula of the normal to the sphere at (1,2, 3'). (6)
(ii) obtain the formula of the sphere which
3x +2y - z +2= 0 at the point (L, -2, 1) and
thes phere*' + y' + z2- 4x +6y*4 = 0.
touches the plane
also cuts orthogonally
(10)
13. (a) (i) obtain the radius of curvature of the parabola x = at2 , ! = 2at at t .
(6)
ft) (i)
, ,
^ , J _ 1
az b"
obtain the evolute of ellipse
normals.
(10)
treatingi t as envelopeo fits
(10)
of the parabola x2 = 4ay is
(6)
(b) (i)
- x - 20y + 2lin Taylors series
(
r =rcos 0, y = rsind evaluate
(
+y+z=24.using
(
(
(ii) Showth att hec ircleo fc urvaturoef Ji * Ji = Ji at %,%)*
= a' / two .
Or
(.-u.x(')+ ' )'
(ii) Show that the formula of the evolute
4(y -2a)3 --27ax2 .
L4. (a) (i) Expand f(r,r1 = 4x2 + xy + 61'2
about (-1, 1).
(ii) If u -- 4x2 + 6xy u = 2y2 + xJ'
0(u,u)
0(r,01'
Or
obtain the minimum value of xy2z2 subject to r
Lagrange Multiplier.
(rr) .bjva. luate f e ' t ' ' ' ' s i n . T . f s l n . r , * /
J--;-.r.
d > U and deduce J-------{lx
='/2.
u * " o x '

15 ( a )
dy
Reducteh ee qua"t^i ,#.(*)' =f*'"
lzrtx
the substitution 1, =e and hence solve'
a Linear formula uslng
(16)
Or
(b) Solveb y methodo f variationo f parame tersy " - af l' *4/rY=x2+I'
/ x -
(16)

Earning:   Approval pending.