Anna University Chennai 2008-1st Sem B.E Civil Engineering Anna university-Maths- - Question Paper

DMI INSTITUTION OF EDUCATION

SEMESTER:1/8

MA2111-MATHEMATICS-I

Time:3 hours
Marks:100


PART-A(10x2=20)
ans all ques.

1.For a provided matrix A of order 3, |A|=32 and 2 of its eigen values are eight and 2. obtain the sum of the eigen values.
2.Check whether the matrix B is orthogonal? Justify.
| cos? sin? 0|
B= |-sin? cos? 0|
| 0 0 1|
3.Write the formula of the tangent plane at (1,5,7) to the sphere
(x-2)2+(y-3)2+(z-4)2=14.
4.Find the formula of the right circular cone whose vertex is at the origin and axis is the line x/1=y/2=z/3 having semi vertical angle of 45°.
5.Find the envelope of the lines y=ms±v(a2m2+b2) where m is the parameter.
6. describe the circle of curvature at a point p(x1,y1) on the curve y=f(x).
7.Using Euler’s theorem, provided u(x,y) is a homogeneous function of degree n, prove that x2uxx+2xyuxy+y2uyy=n(n-1)u.
8.Using the definition of total derivatives, obtain the value of du/dt provided u=y2-4ax; x=at2,y=2at.
9.Write down the double integral, to obtain the area ranging from the circles r=2sin? and r=4sin?.
10.Change the order of integration in I=0?1 x2?2-x f(x,y)dxdy.

PART-B(5x16=80)
11(a)(i) Fine the characteristic formula of the matrix A provided
|2 -1 1|
A = |-1 two -1| . Hence obtain A-1 and A2.
|1 -1 2|
(ii) obtain the eigen values and eigen vectors of
|2 one 1|
A = |1 two 1|
|0 0 1|
OR
(b)Reduce the provided quadratic form Q to its canonical form using orthogonal transformation. Q=x2+3y2+3z2-2yz.

12(a)(i)Obtain the formula of the sphere having the circle x2+y2+z2=9, x+y+z=3 as a great circle.
(ii) obtain the formula of the right circular cylinder whose axis is the line x=2y=-z and radius 4.
OR
(b)(i)Find the formula of the right circular cone whose vertex is at the origin and axis is the line x/1=y/2=z/3 and which has semi vertical angle of 30°.
(ii) obtain the formula of the sphere defined on the line joining the points (2,-1,4) and (-2,2,-2) as diameter. obtain the area of the circle in which this sphere is cut by the plane 2x+y-z=3.

13(a)(i)Find the evolute of the hyperbola x2/a2-y2/b2=1 considering it as the envelope of its normals.
(ii) obtain the radius of curvature of the curve vx+vy=va at (a/4,a/4).
Or
(b)(i) obtain the formula of circle of curvature of the parabola y2=12x at the point (3,6).
(ii) obtain the envelope of the family of line x/a+y/b=1 subject to the condition that a+b=1.

14(a)(i)If u=xy, show that uxxy=uxyx.
(ii) If u=log(x2+y2) + tan-1(y/x) prove that uxx+uyy=0.
(iii) obtain the Jacobian ?(x,y,z)/?(r,?,z) of the transformation x=rsin?cosf, y=rsin?sinf and z=rcos?.
Or
(b)(i) obtain the maximum value of xmynzp subject to the condition x+y+z=a.
(ii) obtain the Taylors series expansion of exsiny at the point (-1,p/4) up to third degree terms.

15(a)(i)Find the area inside the circle r=asin? but lying outside the cardioid r=a(1-cos?).
(ii) Evaluate0 ?a 0 ?b 0 ?c (x2+y2+z2)dxdydz.
Or
(b)(i) change to spherical polar co-ordiantes and hence evaluate V???1/(x2+y2+z2)dxdydz, where V is the quantity of the sphere x2+y2+z2=a2.
(ii) change the order of integration and hence evaluate 1?3 y=0?6/x x2dydx.

Earning:   Approval pending.