Indian Institute of Information Technology 2008 B.Tech MATHEMATICS - Question Paper

B.E MA 1101- MATHEMATICS- I ques. paper

B.E ./8.T ech.DEGREEE XAMINATION,M AY/JUNE 2007.

First Semester
Civil Engineering
MA 1101- MATHEMATICS- I


(Common to all branches)
(Regulation 2004)
Time : 3 hours Maximum : 100 marks

ans ALL ques..
PARTA-(fO x2=20 marks)


(t two 3')
1. obtain the rank of the matrix | four b six |
t l
(7 eight e)
2. obtain the sum and product of the eigenvalues of the square matrix using the
(s16)
propertieslS5 71.
t l
14 e 2)
3. Findthe acute angleindegrees betweenthelines + =+ = sand
4 - + =1.
! two -1 two one 1
4. obtain the formula of the sphere concentric with
x2 + y2 + z' -2x -2y -22 -1 = 0 and passingthrough the point (-2, I,S).
5. obtain the curvature at (3, -4) to the curve xt + y' = 25.
6. Findthe envelopeo f thefamilyof straightlines rcosd+ ysin0 =6 where d
is a parameter.
7 . obtain the stationary points of the function f(x,y) = x3 + y3 -I%cy .
8. If u,u,w are functions of independent variables tc,y,z urrd
0-\.u'''*)..=4,
find
d\x,y,z)
rL^ __^r^- -^ " 0(2u,2ur2w)
rne value or
aG,yF;

o
10.
, , d - \ t
Solve 4=y.
dx"
If x =e', express4 ttterms of the derivativeso fy w.r.t. z.
dx-



PARTB-(5x16=80marks)
11. (a) (i) Using Cayley-Hamilton theorem obtain A-L and A3 + AG ,
A=(2
-1.l
f5 -2)
if
(
(ii) obtain the eigen values and
(4 t rl
l1 411
[1 L4)
Or
(i) Examine the consistency of
3x - y +22 = 4, 5x -4y +32 = 4.
the system.
the eigen vectors of the matrix
the linear system x+2y+z=4,
If consistent, obtain the solution of
(
(b)
(ii ) If the quadratic form 3x2 +6y2 +322 -4xy +4yz -2zx is decreased to
the canonical form cJl + cry", + cry! by an orthogonal
transformation, obtain the constanLs c1,c2 and cr. Also state the
nature of the quadratic form. (6 + 2)
obtain the formula of the plane passing through the intersection of
the planesx +2y*32=1 and 2x-y*z=7 and perpendiculatro

12. (a) (i)
the plane rc +3y - z = 6.
Or
ft) (i) obtain the shortest distance ranging from the lines
. x-5 v-8
a n d - = ! - - =
z -t0
(ii) provethatthetineqs1 =! ..3= ":-5.nd tr3 =!:-7 -z-L0
245123
are coplanar. obtain the formula of the plane containing these lines.
(
-x=--3= - y-5 z-7
234
345
(ii) Prove that the plane 2x + 2y - z = eight touches the sphere
x'+y'+z'-4x+2y-Gz +5 =0. Finda isot hep ointofc ontact.( eight )
c 3272

13. (a) (i) Findtheradiusof curvatureat(1,1 )tothecurve x'lt +y213= 2. 6)
(ii) Prove that the envelope of the family of straight lines
! = mx -2am - amt is 27ay2 = 4(x -2a)3 . (
Or
(b) (i) obtain the locus of the centre of curvature of the paraboia y2 = 4ax .
(
(ii) Prove that the envelope of L + ! = t where the parameters a and b
ab
arec onnectebdy a+b =c is Ji * Ji =J; .
L4. (a) (i) Using Taylor's expansion, express f(x,y)-e* cos(by)in
.r and y upto 2nd degree terms.
(
powers of
(
(ii) Using Lagrange's multiplier method, determine the maximum
capacity ofa rectangular tank, open at the top, ifthe surface area is
(b) (i)
(ii)
15. (a) (i)
(ii )
(b) Solve
108 sq. m.
Or
Applying differentiation under integral sign,
7^* o - 1
l?d* = log(1+a).
"o logx
If u = yzlx,u = zJcy/ 'f,- w = JJcty lz,f rna {t4.
d ( u r u , w )
Solve (D2 - 4D + 4)y = e" xo + cos2x .
Solve (x2 D2 - 4xD + 6)y = x(l + x) .
prove that

-d+x y - e - ' el
dt
Or
',d! t * 4x = t, givent hat r(0) = two andy (0)= U4. (16)
c 3272

Earning:   Approval pending.