Model Question Paper

MODEL QUESTION PAPER B.E. / B.Tech. Degree Examinations First Semester MA 131 - Mathematics I Time: 3 hours. Max.marks:100 Answer All Questions PART- A (10 x 2 Marks = 20 Marks) 1. If are the eigenvalues of a matrix what are the eigenvalues of and 2. If write in terms of and using Cayley-Hamilton theorem. 3. Find the radius of curvature at any point on the curve 4. Find the envelope of the family where is a parameter. 5. If evaluate 6. Find the direction cosines of the line drawn from the point (1,0,1) to (1,1,-1). 7. Find the equation of th sphere on the line joining (1,1,1) and (2,2,2) as diameter. 8. If prove that 9. Find the particular integral of 10. Solve:

PART-B (5 x 16 Marks = 80 Marks) Question No.11 has no choice; Questions 12 to 15 have one choice (EITHER-OR TYPE) each. 11. Reduce the quadratic form into a canonical form by means of an orthogonal transformation. Determine its nature. Find a set of non-zero values for for which the above quadratic form is zero. 12. (a) (i) Find the image of the point in the plane (6) (ii) Show that the lines and are coplanar. Find the coordinates of their point of intersection and the equation of the plane containing them. (10) OR (b) (i) Find the equation of the sphere passing through the points and having its centre on the line (8) (ii) Find the tangent planes to the sphere that are parallel to the plane (8) 13. (a) Find the evolute of the cycloid: OR (b) (i) Find the evolute of the parabola considering it as the envelope of its normals. (8) (ii) Find the equation of the circle of curvature of at (8) 14. (a) (i) Obtain terms up to the third degree in the Taylor series expansion of around the point (10) (ii) By differentiating under the integral sign, show that (6) OR (b) (i) If and find without actual substitution. (6) (ii) Show that the points on the surface nearest to the origin are at a distance from it. (10)

15. (a) (i) Solve given that (8) (ii) Solve by the method of variation of parameters: (8) OR (b) (i) Solve: (8) (ii) Solve by reducing the order given that is a solution. (8) MA 1X01 - ENGINEERING MATHEMATICS - I (Common to all Branches of Engineering and Technology) Regulation 2004 Time : 3 Hrs Maximum: 100 Marks

Answer all Questions PART – A (10 x 2 = 20 Marks) 1. Find the sum and product of the eigen values of the matrix ??????????----311151113 2. If x = r cos?, y = r sin?, find ),x(),(yr??? 3. Solve (D3+D2+4D+4)y = 0. 4. The differential equation for a circuit in which self-inductance L and capacitance C neutralize each other is L .022=+Cidtid Find the current i as a function of t. 5. Find, by double integration, the area of circle x2+y2 = a2. 6. Prove that curl grad f = o. 7. State the sufficient conditions for a function f(z) to be analytic. 8. State Cauchy’s integral theorem. 9. Find the Laplace transform of unit step function at t = a. 10. Find L-1 [13432+++sss]. 1 PART – B (5 x 16 = 80 marks) 11.(a).(i). Verify Cayley-Hamilton theorem for the matrix A = . ??????????----126216227 Hence find its inverse. (8) (ii). Find the radius of curvature at any point ‘t’ on the curve x = a (cost + t sint), y = a(sint-t cost) (8) (OR) (b).(i). Diagonalise the matrix by orthogonal transformation. (8). ??????????----342476268 (ii). A rectangular box open at the top is to have volume of 32 c.c. Find the dimensions of the box requiring least material for its construction, by Lagrange’s multiplier method. (8). 12(a). (i). Solve (3x+2)2 +22xdyd3(3x+2) =-yddy36x3x2+4x+1 (8) (ii). For the electric circuit gover ned by (LD2+RD+C1) q = E where D = dtd if L = 1 henry, R = 100 Ohms, C = 10-4 farad and E = 100 volts, q = dtdq= 0 when t = 0, find the charge q and the current i. (8) (OR) (b).(i). Solve 032x=++yxdtd, 3x+teydtdy222=+ (8) 2 (ii). The differential equation satisfied by a beam uniformly loaded (w kg/ metre) with one end fixed and the second end subjected to tensile force P is given by 22221wxPydxdEIy-= . Show that the elastic curve for the beam with conditions y = 0 = dxdy at x = 0 is given by y = 2Pnw (1-coshnx) + Pwx22where EIPn=2 (8) 13. a.(i). Change the order of integration in 220xxaaxxayddy-?? and hence evaluate the same. (8). (ii). Prove that F= (y2cosx + z3)i+(2ysinx-4)j+3xz2k is irrotational and find its scalar potential. (8) (OR) b.(i). By changing to polar co-ordinates, evaluate ??+aayydyd0222xxx (8) (ii). Verify Gauss divergence theorem for izFx4=-kyzjy+2, taken over the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1. (8) 14. (a).(i). If f(z) is an analytic function, prove that ??????????+??2222xy| f(z)|2 = 4|f '(z)|2. (8). (ii). Find the Laurent’s series expansion of the function f(z) = )2)(3)(1(162+----zzzzz in the region 3 < |z+2| < 5. (8). (OR) 3 (b).(i). Find the bilinear map which maps –1, 0, 1 of the z-plane onto –1,-i, 1 of the w-plane. Show that the upper half of the z-plane maps onto the interior of the unit circle | w | = 1. (8). (ii). Using contour integration, evaluate ?8++022222)x)(x(xbadx (8). 15.(a) (i). Find the Laplace transform of t sint sinh2t and tatcos1- (8) (ii). Using convolution theorem, find L-1 222)(1as+ (8) (OR) (b).(i).Find the Laplace transform of the function ???=+<<-<<=)()2(,2,20,)(tftftttttfppppp (8) (ii).Using Laplace transform technique, solve 2225sin0,00tdydyyetdtdtdyywhentdt-++==== ,

(8) B.E/B.Tech. DEGREE EXAMINATION, January 2006 First Semester (COMMON TO ALL BRANCHES OF ENGINEERING AND TECHNOLOGY) MA 1101-Mathematics-I Answer all the questions PART A(10*2=20) 123 1.What is the rank of A= -1 -2 -3 ? 246 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+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 or 12.(b).(i).Solve: x 2 y’’=2xy’-4y=x4 (ii).Solve:(dx/dt)+ 2x-3y=t (dy/dt)-3x+2y=e2t 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. Or 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 or 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. 12 15.(a).(i).If A= 3 4 find A-1 and A3 using Cayley-Hamilton theorem. (ii).Diagnoalize 6 -2 2 A=-2 3 -1

2 -1 3 by an orthogonal transformation. Or 100 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.