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

Monday, 28 January 2013 10:45Web

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 formul

**a.**How will you convert

it to a linear differential formula with constant coefficients?

PARTB-(5xL6=80marks)

**1**

**1.****(**

**a)****(**Test whether equations 2x-31'+72=5,3x+y-Bz=13

**i)**2x + 19v - 47 z = 32 are consistent or not. (

**(**decrease the quadratic form xl +2t"2 - -rrt - 2xrx, =2xqx3 to the

**i****i)**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-

**(**Diagonalize the matrix A= 0 three -1 using an orthogonal

**i****i)**o -1 3J

transformation. (

**1**

**2.****(**

**a)****(**obtain the formula of the piane passing through

**i)***

^l =

y +-r - Lr and perpendicular t'o x +2y + z =

**1**(6)

**2.**2-1 4

**(**Prove that the li.

**i****i)**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)****(**obtain the formula of the tangent plane to the sphere

**i)**3(x2 + y2 + t

**2)**-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)

**(**obtain the formula of the sphere which

**i****i)**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)

**1**

**3.****(**

**a)****(**obtain the radius of curvature of the parabola x = at2 , ! = 2at at t .

**i)**(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)

**(**(i)

**b)**- x - 20y + 2lin Taylors series

(

r =rcos 0, y = rsind evaluate

(

+y+z=24.using

(

(

**(**Showth att hec ircleo fc urvaturoef Ji * Ji = Ji at %,%)*

**i****i)**= a' / two .

Or

(.-u.x(')+ ' )'

**(**Show that the formula of the evolute

**i****i)**4(y -2a)3 --27ax2 .

L

**4.**

**(**

**a)****(**Expand f(r,r1 = 4x2 + xy + 61'2

**i)**about (-1, 1).

**(**If u -- 4x2 + 6xy u = 2y2 + xJ'

**i****i)**0(u,u)

0(r,01'

Or

obtain the minimum value of xy2z2 subject to r

Lagrange Multiplier.

(rr) .bjv

**a.**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

**(**Solveb y methodo f variationo f parame tersy " - af l' *4/rY=x2+I'

**b)**/ x -

(16)

Earning: Approval pending. |