Name of the Student:

Engineering Mathematics 016 SUBJECT NAME : Engineering Mathematics - I SUBJECT CODE : MA111 MATERIAL NAME : Universit Questions REGULATION : R008 WEBSITE : wwwhariganeshcom UPDATED ON : Januar 016 TEXTBOOK FOR REFERENCE : Sri Hariganesh Publications (Author: C Ganesan) To bu the book visit wwwhariganeshcom/tetbook (Scan the above QR code for the direct download of this material) Name of the Student: Branch: Unit I (Matrices) Cale Hamilton Theorem 1 Find the characteristic equation of the matri satisfies the equation Hence evaluate Find the characteristic equation of the matri A given 1 A and 1 3 A = 1 4 and show that A 3 1 1 1 A (N/D 014) 1 1 A = 1 1 Hence find 1 1 4 A (Jan 009) Tetbook Page No: 154 1 1 1 3 Show that the matri 0 1 0 satisfies the characteristics equation and hence find 0 3 its inverse (Jan 011),(Jan 013) Tetbook Page No: 143 Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 1

Engineering Mathematics 016 1 3 7 4 Using Cale-Hamilton theorem, find the inverse of A = 4 3 (N/D 011) 1 1 Tetbook Page No: 145 5 Using Cale Hamilton theorem find 1 A 1 0 3 for the matri A = 1 1 (Jan 014) 1 1 1 1 0 3 6 Using Cale Hamilton theorem, find the inverse of the matri A = 8 1 7 3 0 8 Tetbook Page No: 154 (N/D 010) 7 Using Cale Hamilton theorem, find 1 A when 1 A = 1 1 (M/J 010) 1 1 8 Use Cale Hamilton theorem to find the value of the matri given b 1 1 8 7 6 5 4 3 A 5 A + 7 A 3 A + A 5 A + 8 A A + I, if the matri A = 0 1 0 1 1 Tetbook Page No: 150 (M/J 009) 1 1 1 9 Verif Cale Hamilton Theorem and hence find A for A = 1 1 1 1 Tetbook Page No: 154 (Jan 010) 10 Verif Cale Hamilton Theorem for the matri 1 3 A = 4 (A/M 011) 1 1 Tetbook Page No: 147 Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page

Engineering Mathematics 016 0 1 11 Verif Cale Hamilton Theorem for the matri 0 0 and hence find 1 0 4 A (Tetbook Page No: 154) (M/J 01) n 1 Find A using Cale Hamilton theorem, taking A 1 4 = 3 Hence find A3 Tetbook Page No: 15 (Jan 01) Eigen Values and Eigen Vectors of a given matri 1 A and 1 Find the eigenvalues and eigenvectors of Find the eigenvalues and eigenvectors of 1 1 A = 1 1 0 0 1 (Jan 009) 1 1 6 1 0 1 1 (Jan 013) Tetbook Page No: 117 3 Find all the eigenvalues and eigenvectors of the matri 1 1 4 3 1 (Jan 011) 1 1 Tetbook Page No: 114 11 4 7 4 Find the eigenvalues and eigenvectors of the matri A = 7 5 (N/D 011) 10 4 6 Tetbook Page No: 140 1 5 Find the eigenvalues and eigenvectors of the matri A = 1 3 1 1 Tetbook Page No: 11 (M/J 010),(N/D 010),(Jan 01) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 3

Engineering Mathematics 016 6 Find the eigenvalues and the eigenvectors of 8 6 A = 6 7 4 (Jan 014) 4 3 3 7 Find the eigenvalues and eigenvectors for the matri A = 1 6 1 0 Tetbook Page No: 140 (M/J 009),(Jan 1010),(N/D 014) Diagonalisation of a Matri 1 The eigenvectors of a 3X3 real smmetric matri A corresponding to the eigenvalues T T, 3,6 are ( 1, 0, 1 ),( 1,1,1) and ( ) 1,, 1 T respectivel Find the matri A Tetbook Page No: 185 (A/M 011) If the eigenvalues of 8 6 A = 6 7 4 are 0, 3, 15, find the eigenvectors of A and 4 3 diagonalize the matri A (Jan 013) Tetbook Page No: 181 Quadratic form to Canonical form 1 Through an orthogonal transformation, reduce the quadratic form + 5 + + + + 6 to a canonical form (N/D 014) 1 3 1 3 3 1 Reduce the given quadratic form Q to its canonical form using orthogonal transformation Tetbook Page No: 1113 3 Reduce the quadratic form Q 3 3z z = + + (Jan 009) 5 3 4 + + z + to the Canonical form b orthogonal reduction and state its nature (M/J 010),(Jan 01) Tetbook Page No: 195 4 Reduce the quadratic form 1 1 3 3 + to a canonical form b an orthogonal reduction Also find its nature (Tetbook Page No: 1100) (A/M 011) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 4

Engineering Mathematics 016 5 Reduce the quadratic form 1 3 3 1 Tetbook Page No: 1114 6 Reduce the quadratic form + + into canonical form (Jan 013) + + + 4 to canonical form 1 3 1 1 3 3 b an orthogonal transformation Also find the rank, inde, signature and nature of the quadratic form (Tetbook Page No: 1113) (N/D 010) 7 Reduce the quadratic form 3 3 5 6 6 z z z to its canonical form using orthogonal transformation Also find its rank, inde and signature (Jan 014) 8 Find a change of variables that reduces the quadratic form 1 3 3 3 + 5 + 3 1 3 1 + to a sum of squares and epress the quadratic form in terms of new variables (Tetbook Page No: 1113) (Jan 011) 9 Reduce the quadratic form 8 + 7 + 3 1 8 + 4 into canonical 1 3 1 3 3 1 form b means of an orthogonal transformation (N/D 011) Tetbook Page No: 1113 10 Reduce the quadratic form + + + to the Canonical form 1 3 1 3 through an orthogonal transformation and hence show that is positive semi definite Also given a non zero set of values (,, ) which makes this quadratic form zero 1 3 Tetbook Page No: 1113 (M/J 009) 11 Reduce the quadratic form 10 + + 5 + 6 10 4 to a Canonical 1 3 3 3 1 1 form through an orthogonal transformation and hence find rank, inde, signature, nature and also give n0n zero set of values for 1 3,, (if the eist), that will make the quadratic form zero (Tetbook Page No: 1106) (Jan 010) 1 Reduce the quadratic form z z z + + to canonical form through an orthogonal transformation Write down the transformation (M/J 01) Tetbook Page No: 1114 Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 5

Engineering Mathematics 016 Unit II (Three Dimensional Analtical Geometr) Spheres 1 Find the equation of the sphere passing through the points ( 0,0,0 ),( 0,1, 1 ), ( 1,,0) and ( ) 1,,3 (N/D 011) Obtain the equation of the sphere having the circle + + z = 9, z 3 + + = as a great circle (Jan 009) 3 Obtain the equation of the sphere having the circle + + z + 10 4z 8= 0, + + z = 3 as the greatest circle (Jan 01),(M/J 01) 4 Find the equation to the sphere passing through the circle + + z = 1 and cuts orthogonall the sphere z z + + z = 9, + + + 4 16 + 17 = 0 (M/J 010) 5 Find the equation of the sphere passing through the circle z z + + + 3 + 1= 0, 5 z 7 0 sphere z z + + = and cuts orthogonall the + + 3 + 5 7 6= 0 (N/D 010) 6 Find the equation of the sphere having its centre on the plane 4 5 z = 3 and passing through the circle z z + + 3 + 4 + 8= 0 ; z 8 + = (N/D 010) 7 Find the equation of the sphere described on the line joining the points (, 1,4) and (,, ) as diameter Find the area of the circle in which this sphere is cut b the plane + z = 3 (Jan 009) 8 Find the equation of the sphere of radius 3 and whose centre lies on the line 1 1 z 1 = = at a distance from ( ) 1,1,0 (A/M 011) 9 Find the equations of the spheres which pass through the circle + + z = 5 and + + 3 z = 3 and touch the plane 4 + 3 = 15 (M/J 009) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 6

Engineering Mathematics 016 10 Find the equation of the sphere which passes through the circle z + + 4 = 0, 3 z 0 + + = and touch the plane 4 + 3 = 5 11 Show that the plane + z + 1 = 0 touches the sphere z z + + 4 + = 3and find also the point of contact 1 Find the two tangent planes to the sphere (Jan 011)(AUT) (M/J 009), (N/D 011),(Jan 013) z z + + 4 + 6 + 5= 0, which are parallel to the plane + 4 + 8 z = 0 Find their point of contact 13 Obtain the equation of the tangent planes to the sphere + + z + 4+ 6z 7= 0, which intersect in the line (Jan 010),(M/J 01),(Jan 013) 6 3 3 = 0 = 3z + (Jan 01) 14 Find the equation of the tangent lines to the circle 3 3 3z 3 4z 0 + + =, 3 4 5 z 6 0 + + = at the point (1,, 3) (Jan 011) 15 Find the centre, radius and area of the circle given b z z + + + 4 19 = 0, z 7 0 + + + = 16 Find the centre and radius of the sphere whose equation is (Jan 010),(M/J 010),(Jan 014) 6z = 0 Show that the intersection of this sphere and the plane z 0 0 + + = is a circle whose centre is the point ( ) z 4 + +,4,5 and find the radius of the circle (N/D 014) Cone 1 Find the equation of the right circular cone whose verte is at the origin and ais is the z line = = and which has semi vertical angle of 30 (Jan 009),(N/D 010) 1 3 Find the equation of the right circular cone whose verte is (,1,0 ), semi vertical angle is 30 and the ais is the line 1 z = = (Jan 013) 3 1 Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 7

Engineering Mathematics 016 3 Find the equation of the right circular cone generated b revolving the line = 0, z = 0 about the ais = 0, z = (M/J 009) 4 Find the equation of the right circular cone generated when the straight line which is the intersection of the planes + 3z = 6 and = 0 revolves about the z ais with constant angle (Jan 011),(M/J 01) 5 Find the equation of the cone whose verte is ( 1,,3 ) and whose guiding curve is the circle + + z = 4, + + z = 1 (M/J 009), (N/D 011) 6 Find the equation of the cone with verte at ( 1,1,1 ) and passing through curve of intersection of + + z = 1and z 1 + + = (A/M 011) 7 Find the equation of the cone whose verte is the origin and guiding curve is z + + = 1, + + z = 1 (N/D 014) 4 9 9 8 Find the equation of the cone formed b rotating the line + 3 = 5, z = 0 about the ais (Jan 010) 9 Find the equation of the cone whose verte is the point ( 1,1,0 ) and whose base is the curve = 0, + z = 4 (M/J 010),(Jan 014) 10 Find the equation of the cone formed b rotating the line + 3 = 6, z = 0 about the ais (Jan 01) Clinder 1 Find the equation of the right circular clinder whose ais is the line = = z and radius 4 (Jan 009) 1 3 z 5 Find the equation of the right circular clinder of radius 3 and ais = = 1 (Jan 010),(M/J 010),(A/M 011),(M/J 01) 1 z 3 3 Find the equation of the right circular clinder whose ais is = = and 1 radius (N/D 010),(Jan 01) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 8

Engineering Mathematics 016 4 Find the equation of the right circular clinder of radius whose ais is the line 1 1 z 3 = = (N/D 014) 1 5 Find the equation of the right circular clinder of radius 5 whose ais is the line 1 z 3 = = (N/D 011) 1 6 Find the equation of the right circular clinder which passes through the circle z z + + = 9, + = 3 (Jan 011) 7 Find the equation of the right circular clinder which passes through the circle z z + + = 9, + + = 3 (Jan 014) z 8 Find the equation of the clinder whose generators are parallel to = = and 3 whose guiding curve is the ellipse 3 3 Unit III (Differential Calculus) + = (Jan 013) Radius of Curvature and Circle of curvature a a 1 Find the radius of curvature of the curve + = a at, 4 4 (Jan 009) Tetbook Page No: 331 a a Find the circle of curvature at, 4 4 on + = a Tetbook Page No: 331 (M/J 010),(N/D 010),(A/M 011), (N/D 011),(Jan 01),(M/J 01) 3 Find the equation of circle of curvature of the parabola = 1 at the point ( 3,6 ) Tetbook Page No: 334 (Jan 009) 4 Find the equation of circle of curvature of the rectangular hperbola = 1 at the point ( 3,4 ) (Tetbook Page No: 336) (Jan 010) 5 Find the equation of the circle of curvature at ( c, c) on = c (N/D 014) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 9

Engineering Mathematics 016 6 Show that the radius of curvature at an point of the catenar c cosh = c is Also find ρ at ( 0,c ) (Tetbook Page No: 315) (N/D 014) 0,c on the curve c cosh = c Tetbook Page No: 315 (M/J 009) 7 Find the radius of curvature at the point ( ) 3a 3a 8 Find the radius of curvature at the point, on the curve 3 3 + = 3a Tetbook Page No: 37 (N/D 011) 3 3 9 Find the radius of curvature at the point ( a cos, a sin ) θ θ on the curve /3 /3 /3 + = a (Tetbook Page No: 317) (M/J 009) 10 Find the radius of curvature at ( a,0) on Tetbook Page No: 36 11 Prove that the radius of curvature of the curve 3 3 a = (Jan 010) c 3 3 = a at the point ( a, 0) is 3 a Tetbook Page No: 36 (N/D 010),(Jan 014) 1 Find the radius of curvature at an point of the ccloid a( θ sinθ ) ( 1 cos ) = +, = a θ (M/J 010),(M/J 01),(Jan 013) Tetbook Page No: 31 13 Find the radius of curvature of the curve = 3a cosθ a cos 3 θ, = 3a sinθ a sin 3θ (Tetbook Page No: 319) (A/M 011) / 3 a 14 If =, prove that ρ a + = + a, where ρ is the radius of curvature (Tetbook Page No: 34) (Jan 01),(Jan 014) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 10

Engineering Mathematics 016 Evolute 1 Show that the evolute of the parabola 4a = is the curve 3 7a = 4( a) Tetbook Page No: 348 (Jan 010),(M/J 010) Find the equation of the evolute of the parabola = 4a Tetbook Page No: 348 (Jan 011)(AUT),(Jan 01),(M/J 01),(N/D 014) 3 Find the radius of curvature and centre of curvature of the parabola 4a = at the point t Also find the equation of the evolute (Jan 013) Tetbook Page No: 348 4 Find the evolute of the hperbola a = (N/D 010),(N/D 011) b 1 Tetbook Page No: 356 5 Obtain the equation of the evolute of the curve a( cosθ θ sinθ ) a( sinθ θ cosθ ) = +, = (Tetbook Page No: 369) (M/J 009) 6 Show that the evolute of the ccloid = a( θ sinθ ), a( 1 cosθ ) = is another ccloid (Tetbook Page No: 361) (A/M 011) Envelope 1 Find the envelope of the famil of straight lines cos α + sin α = c sin α cos α, α being the parameter (Tetbook Page No: 384) (A/M 011) Find the envelope of the straight line + = 1, where aand bare parameters that are a b connected b the relation a + b = c (Jan 009),(M/J 009) Tetbook Page No: 387 3 Find the envelope of + = 1, where aand bare connected b the relation a b a + b = c, c being constant (N/D 010),(Jan 013) Tetbook Page No: 389 Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 11

Engineering Mathematics 016 4 Find the envelope of the straight line + = 1 where the parameters aand bare a b connected b the relation a n + b n = c n, cbeing a constant (N/D 011) Tetbook Page No: 391 5 Find the envelope of + = 1, where the parameters l and m are connected b the l m relation l + m = 1 (a and b are constants) (Jan 01) a b Tetbook Page No: 3 6 Find the envelope of the straight line + = 1, where a and b are connected b the a b relation ab = c, c is a constant (Jan 010),(M/J 010) Tetbook Page No: 3100 7 Find the envelope of the ellipses + = 1, where aand bare connected b the a b relation a + b = c (Tetbook Page No: 393) (Jan 014) 8 Find the envelope of the sstem of ellipses + = 1, where aand bare connected b a b the relation ab = 4 (Tetbook Page No: 395) (M/J 01) 9 Find the envelope of the circles drawn upon the radius vectors of the ellipse a + = as diameter (Tetbook Page No: 3) (Jan 013) b 1 Evolute as the envelope of normals 1 Find the evolute of the hperbola = 1 considering it as the envelope of its a b normals (Tetbook Page No: 3107) (Jan 009) Considering the evolute as the envelope of the normals, find the evolute of the asteroid /3 /3 /3 + = a (N/D 014) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 1

Engineering Mathematics 016 Unit IV (Functions of several variables) Euler s Theorem 1 If u u u =, show that = (Jan 009) = + + prove that u u 0 If u log ( ) tan 1 ( / ) Tetbook Page No: 46 + = (Jan 009),(N/D 010) 3 If 4 If u = cos u = sin 1 + + 1 + + u u 1, prove that + = cot u (N/D 011), prove that u sin cos u u + + = u u 3 4cos u (A/M 011) 1 5 If u sin + u u =, prove that (1) + = tan u and () + u u u + + = tan u (Jan 011) 6 If 1 + u = sin, find + u u + (N/D 014) Total derivatives z u u u 1 If u = f,, z, prove that + + z = 0 (M/J 009) z If z = f (, ), where = u v, = uv, prove that z z 4 ( u v ) z z + = + + (Jan 010),(Jan 01) u v 3 If = u cosα v sin α, = u sin α + v cosα and V = f (, ), show that V V V V + = + (Tetbook Page No: 49) (Jan 011) u v Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 13

Engineering Mathematics 016 4 If u = e, show that (Jan 013) u u 1 u u + = u + Tetbook Page No: 43 u u 5 If F is a function of and and if = e sin v, = e cos v, prove that F F u F F e + = + u v (Jan 013) Tetbook Page No: 433 6 If u = f (, ) where = r cos θ, = r sinθ, prove that u u u 1 u + = + r r (M/J 010),(Jan 014) θ Tetbook Page No: 44 7 If u z = + + and e t, e t cos3 t, z e t sin3t = = =, Find d u d t (N/D 011) Talor s epansion 1 Find the Talor series epansion of sin e at the point ( 1, π / 4) up to 3 rd degree terms (Tetbook Page No: 458) (Jan 009),(M/J 009) π in the neighborhood of the point 1, 4 upto third degree terms (Tetbook Page No: 468) (N/D 010) Find the Talor s series epansion of e cos 3 Epande log(1 + ) in power of and upto terms of third degree using Talor s theorem (Tetbook Page No: 461) (N/D 011),(N/D 014) 4 Find the Talor s series epansion of + + 3 in powers of ( ) ( 1) upto 3 rd degree terms (Jan 010),(M/J 010),(Jan 01) Tetbook Page No: 454 5 Use Talor s formula to epand the function defined b + f(, ) and 3 3 = + + in powers of ( 1) and ( ) (Tetbook Page No: 468) (A/M 011) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 14

Engineering Mathematics 016 6 Epand + 3 in powers of ( 1) and ( + ) upto 3 rd degree terms Tetbook Page No: 468 (M/J 01) Maima and Minima 1 Find the etreme values of the function 3 3 f (, ) = + 3 1 + 0 Tetbook Page No: 470 (Jan 010),(A/M 011),(Jan 01) Find the maimum and minimum values of Tetbook Page No: 47 3 Discuss the maima and minima of the function + + (M/J 01) f (, ) 4 4 4 = + + Tetbook Page No: 476 (N/D 010) 4 Test for an etrema of the function Tetbook Page No: 4111 5 Discuss the maima and minima of Tetbook Page No: 411 6 Eamine the function f (, ) 3 ( 1 ) Tetbook Page No: 480 4 4 f(, ) = + 1 (Jan 011) 1 1 f (, ) = + + + + (Jan 014) = for etreme values (M/J 009) 7 Test for the maima and minima of the function f (, ) 3 ( 6 ) Tetbook Page No: 411 8 Find the maimum value of = (Jan 013) m n p z subject to the condition + + z = a Tetbook Page No: 4103 (Jan 009) 9 A rectangular bo open at the top, is to have a volume of 3 cc Find the dimensions of the bo, that requires the least material for its construction Tetbook Page No: 494 (M/J 010),(N/D 011)(AUT),(M/J 01),(N/D 014) 10 Find the volume of the greatest rectangular parallelepiped inscribed in the ellipsoid 11 If z whose equation is + + = 1 (M/J 009) a b c Tetbook Page No: 4100 + + z = r, show that the maimum value of z + z + is r and the r minimum value is (Jan 013) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 15

Engineering Mathematics 016 Jacobians 1 Find the Jacobian (,, z ) ( r, θ, φ) and of the transformation r sinθ cos φ, = = r sin θ sin φ z = r cosθ (Tetbook Page No: 444) (Jan 009),(A/M 011) If + + z = u, + z = uv, z = uvw prove that (,, z) = ( u, v, w) u v Tetbook Page No: 446 (Jan 010),(Jan 01) 3 Find the Jacobian of 1,, 3 with respect to 1,, 3 if =, =, 3 3 1 1 1 1 3 = (Tetbook Page No: 447) (N/D 010),(Jan 014) 3 Unit V (Multiple Integrals) Change of order of integration 1 Evaluate e dd b changing the order of integration (N/D 010),(A/M 011) 0 Tetbook Page No: 537 Change the order of integration in a 0 a dd and then evaluate it (M/J 009) a Tetbook Page No: 551 3 Change the order of integration 1 0 dd and hence evaluate Tetbook Page No: 560 (Jan 010),(M/J 01) 4 Change the order of integration in the interval a a 0 / a dd and hence evaluate it Tetbook Page No: 547 (M/J 010),(Jan 013) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 16

Engineering Mathematics 016 1 5 Change the order of integration and hence find the value of dd Tetbook Page No: 554 (N/D 011) 6 Change the order of integration and hence evaluate 0 3 6/ dd (Jan 009) 1 = 0 a a + a 7 Change the order of integration dd and hence evaluate it 0 a a Tetbook Page No: 544 (Jan 011) 8 Change the order of integration in a ( ) b a a 0 0 dd and then evaluate it Tetbook Page No: 541 (Jan 01) 9 Change the order of integration in a a dd and hence evaluate it (Jan 014) 0 + Tetbook Page No: 535 Change into polar coordinates 1 Epress a a dd in polar coordinates and then evaluate it (M/J 009) 3/ 0 ( + ) Tetbook Page No: 5100 Evaluate of + 0 0 ( ) e dd b converting to polar coordinates Hence deduce the value e d (Jan 010),(N/D 010),(N/D 014) 0 Tetbook Page No: 510 3 Transform the integral ( + ) 0 0 dd into polar coordinates and hence evaluate it (Tetbook Page No: 510) (A/M 011) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 17

Engineering Mathematics 016 4 B Transforming into polar coordinates, evaluate dd + over annular region between the circles Tetbook Page No: 5118 + = 16 and + = 4 (M/J 010) 5 B Transforming into polar coordinates, evaluate region between the circles Tetbook Page No: 5113 6 Transform the double integral + dd over annular + = a and + = b, ( b> a) (Jan 013) a a 0 dd a a into polar co-ordinates and then evaluate it (Tetbook Page No: 5106) (Jan 01) 7 Transform the integral into polar coordinates and hence evaluate a a + dd (Tetbook Page No: 5104) (Jan 01) 0 0 Area as a double integral 1 Find the area bounded b the parabolas = 4 and = b double integration Tetbook Page No: 568 (N/D 010) Find, b double integration, the area enclosed b the curves 4a = and 4a = Tetbook Page No: 566 (Jan 010),(A/M 011) 3 Find, b double integration, the area between the two parabolas 5 9 3 5 = and = (Tetbook Page No: 594) (M/J 01) 4 Find the area common to 4 Tetbook Page No: 594 5 Evaluate ( ) dd = and 4 = using double integration (N/D 011) over the region between the line = and the parabola = (Tetbook Page No: 517) (Jan 011) Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 18

Engineering Mathematics 016 6 Evaluate ( + ) dd over the area bounded b the ellipse + = 1 a b Tetbook Page No: 55 (N/D 014) 7 Find the smaller of the areas bounded b the ellipse 4 + 9 = 36 and the straight line + 3 = 6 (Tetbook Page No: 594) (Jan 01) 8 Find the area inside the circle r = asinθ but ling outside the cardioids ( 1 cos ) r = a θ (Tetbook Page No: 590) (Jan 009) 9 Find the area which is inside the circle r = 3a cosθ and outside the cardioids ( 1 cos ) r = a + θ (Tetbook Page No: 588) (Jan 013) 3 3 10 Evaluate ( 3 + ) + ( + 3 ) C ( 0,0) to (, ) Triple integral d d where C is the parabola 4a = from a a (M/J 009) a b c 1 Evaluate ( + + ) 0 0 0 Tetbook Page No: 5144 log Evaluate z d d d z (Jan 009) + + + z e dddz (M/J 009) 0 0 0 Tetbook Page No: 5141 3 Evaluate a a a 1 dzdd (N/D 011) 0 0 0 Tetbook Page No: 5133 a z 4 Evaluate 1 1 1 0 0 0 1 dddz z (Jan 01),(Jan 013) Tetbook Page No: 5133 Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 19

Engineering Mathematics 016 1 1 1 5 Evaluate z dddz (N/D 014) 0 0 0 6 Using triple integration, find the volume of the sphere + + z = a Tetbook Page No: 5146 (N/D 010) z 7 Find the volume of the ellipsoid + + = 1 (Jan 010),(A/M 011) a b c Tetbook Page No: 5148 z 8 Find the volume of the tetrahedran bounded b the plane + + = 1 and the a b c coordinate plane = 0, = 0, z = 0 (M/J 010) Tetbook Page No: 5150 9 Evaluate z dddz taken over the tetrahedron bounded b the planes z = 0, = 0, z = 0 and + + = 1 (Jan 011) a b c Tetbook Page No: 5161 1 10 Change to spherical polar co ordinates and hence evaluate dddz, + + z where V is the volume of the sphere + + z = a (Jan 009) 11 Find the value of z dddz through the positive spherical octant for which + + z a (M/J 010) 1 Evaluate dzdd ( + + z + 1) wherev is the region bounded b = 0, = 0, 3 z = 0, + + z = 1 (N/D 011) Tetbook Page No: 5160 V Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 0

Engineering Mathematics 016 Tet Book for Reference: ENGINEERING MATHEMATICS - I Publication: Sri Hariganesh Publications Author: C Ganesan Mobile: 9841168917, 8939331876 To bu the book visit wwwhariganeshcom/tetbook ----All the Best---- Sri Hariganesh Publications (Ph: 9841168917, 8939331876) Page 1