2008 Anna University Chennai B.E Civil Engineering Anna universityMathsFirst sem Question paper
DMI INSTITUTION OF EDUCATION
SEMESTER:1/8
MA2111MATHEMATICSI
Time:3 hours Marks:100
PARTA(10x2=20) Answer all questions
1.For a given matrix A of order 3, A=32 and two of its eigen values are 8 and 2. Find 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 equation of the tangent plane at (1,5,7) to the sphere (x2)2+(y3)2+(z4)2=14. 4.Find the equation 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. Define the circle of curvature at a point p(x1,y1) on the curve y=f(x). 7.Using Euler’s theorem, given u(x,y) is a homogeneous function of degree n, prove that x2uxx+2xyuxy+y2uyy=n(n1)u. 8.Using the definition of total derivatives, find the value of du/dt given u=y24ax; x=at2,y=2at. 9.Write down the double integral, to find the area between the circles r=2sin? and r=4sin?. 10.Change the order of integration in I=0?1 x2?2x f(x,y)dxdy.
PARTB(5x16=80) 11(a)(i) Fine the characteristic equation of the matrix A given 2 1 1 A = 1 2 1 . Hence find A1 and A2. 1 1 2 (ii) Find the eigen values and eigen vectors of 2 1 1 A = 1 2 1 0 0 1 OR (b)Reduce the given quadratic form Q to its canonical form using orthogonal transformation. Q=x2+3y2+3z22yz.
12(a)(i)Obtain the equation of the sphere having the circle x2+y2+z2=9, x+y+z=3 as a great circle. (ii) Find the equation of the right circular cylinder whose axis is the line x=2y=z and radius 4. OR (b)(i)Find the equation 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) Find the equation of the sphere described on the line joining the points (2,1,4) and (2,2,2) as diameter. Find the area of the circle in which this sphere is cut by the plane 2x+yz=3.
13(a)(i)Find the evolute of the hyperbola x2/a2y2/b2=1 considering it as the envelope of its normals. (ii) Find the radius of curvature of the curve vx+vy=va at (a/4,a/4). Or (b)(i) Find the equation of circle of curvature of the parabola y2=12x at the point (3,6). (ii) Find 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) + tan1(y/x) prove that uxx+uyy=0. (iii) Find the Jacobian ?(x,y,z)/?(r,?,z) of the transformation x=rsin?cosf, y=rsin?sinf and z=rcos?. Or (b)(i) Find the maximum value of xmynzp subject to the condition x+y+z=a. (ii) Find the Taylors series expansion of exsiny at the point (1,p/4) up to 3rd degree terms.
15(a)(i)Find the area inside the circle r=asin? but lying outside the cardioid r=a(1cos?). (ii) Evaluate0 ?a 0 ?b 0 ?c (x2+y2+z2)dxdydz. Or (b)(i) Change to spherical polar coordiantes and hence evaluate V???1/(x2+y2+z2)dxdydz, where V is the volume of the sphere x2+y2+z2=a2. (ii) Change the order of integration and hence evaluate 1?3 y=0?6/x x2dydx.
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