2007 Anna University Chennai B.Tech Information Technology Mathematics Question paper
Model Question Paper First Semester (COMMON TO ALL BRANCHES OF ENGINEERING AND TECHNOLOGY) MA 1101MathematicsI
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 (1,1,1). 5.What is the curvature of x 2+y24x6y+10=0 at any point on it? 6.Find the envelope of the family of straight lines y=mx±v(m21), 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=ex 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
or 12.(b).(i).Solve: x 2 y’’=2xy’4y=x4 (ii).Solve:(dx/dt)+ 2x3y=t (dy/dt)3x+2y=e2t
13.(a).(i).Expand f(x,y)=sin(xy) in powers of (x1) and (yp/2) upto second degree terms (ii).If T=x3xy+y3 , x=?cosf,y= ?sinf find ?T/??, ?T/?f (iii).If y=f(x+at)+g(xat), show that ?2y/?t2 =a2(?2y/?x) where a is constant.
Or 13.(b).(i).Evaluate ?eax(sinx/x)dx, with limits (0,8) where a=0 and hence show that
?ex(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=(z7)/3 and (x+1)/4=(y+1)/5=(z+1)/1 are coplanar 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
or 14.(b).(i).Find the centre, radius and area of the circle in which the sphere x2+y2+z2+2x2y4z19=0 is cut by the plane x+2y+2z+7=0. (ii).Find the two tangent planes to the sphere x2+y2+z24x+2y6z+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 A1 and A3 using CayleyHamilton theorem. (ii).Diagonalize 6 2 2 A=2 3 1 2 1 3 by an orthogonal transformation.
Or 1 0 0 15.(b).(i).If A= 1 0 1 then show that An=An2+A2I for n=3 using 0 1 0, CayleyHamilton theorem. (ii).Reduce the quadratic form q=2x1x2+2x2x3+2x3x1 to canonical form using orthogonal transformation.
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