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2007 Anna University Chennai B.Tech Information Technology Mathematics Question paper

Course: B.Tech Information Technology   University/board: Anna University Chennai

Model Question Paper
First Semester
MA 1101-Mathematics-I

Answer all the questions
PART A(10*2=20)

1 2 3
1.What is the rank of A= -1 -2 -3 ?
2 4 6
2.If A is an orthogonal matrix prove that |A|=±1.
3.Prove, by direction ratios, the points (1,2,3); (4,0,4); (-2,4,2) are collinear.
4.Write down the equation of the sphere whose diameter is the line joining (1,1,1) and
5.What is the curvature of x 2+y2-4x-6y+10=0 at any point on it?
6.Find the envelope of the family of straight lines y=mx±v(m2-1), where m is the parameter.
7.If u=exyz2 find du.
8.If x=r cos?, y=r sin? find ?(r, ?)/?(x,y).
9.Solve r(d2u/dr2)+r(du/dr)=0
10.Find the particular integral of y’’+2y’’+5y=e-x cos 2x

PART B(5*16=80 marks)

11.(i).Find the evolute of the rectangular hyperbola xy=c2
(ii).Find the radius of curvature at ‘t’ on x=et cost, et sint

12.(a).(i).Find the particular integral of y’’+7y’-8y=e2x by the method of variation of parameters.
(ii).Solve: y’’+2y’+y=x cosx

12.(b).(i).Solve: x 2 y’’=2xy’-4y=x4
(ii).Solve:(dx/dt)+ 2x-3y=t

13.(a).(i).Expand f(x,y)=sin(xy) in powers of (x-1) and (y-p/2) upto second degree terms
(ii).If T=x3-xy+y3 , x=?cosf,y= ?sinf find ?T/??, ?T/?f
(iii).If y=f(x+at)+g(x-at), show that ?2y/?t2 =a2(?2y/?x) where a is constant.

13.(b).(i).Evaluate ?e-ax(sinx/x)dx, with limits (0,8) where a=0 and hence show that

?e-x(sinx/x)dx = p/4 with limits (0, 8)
(ii).Find the shortest distance from the origin to the curve x2+8xy+7y2 = 225.

14.(a).(i).Show that the lines
(x+3)/2=(y+5)/3=(z-7)/-3 and (x+1)/4=(y+1)/5=(z+1)/-1 are co-planar and find the equation of the plane containing them.
(ii).Find the equation of the plane through the point (-1,3,2) and perpendicular to the planes
x+2y+2z=5 and 3x+3y+2z=8

14.(b).(i).Find the centre, radius and area of the circle in which the sphere
x2+y2+z2+2x-2y-4z-19=0 is cut by the plane x+2y+2z+7=0.
(ii).Find the two tangent planes to the sphere x2+y2+z2-4x+2y-6z+5=0, which are parallel to the plane 2x+2y=z. Find their points of contact.
1 2
15.(a).(i).If A= 3 4 find A-1 and A3 using Cayley-Hamilton theorem.
(ii).Diagonalize 6 -2 2
A=-2 3 -1
2 -1 3
by an orthogonal transformation.

1 0 0
15.(b).(i).If A= 1 0 1 then show that An=An-2+A2-I for n=3 using
0 1 0,
Cayley-Hamilton theorem.
(ii).Reduce the quadratic form q=2x1x2+2x2x3+2x3x1 to canonical form using orthogonal transformation.

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