Engineering Mathematics 2018 : MA6151

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1 Engineering Mathematics 08 NAME OF THE SUBJECT : Mathematics I SUBJECT CODE : MA65 MATERIAL NAME : Universit Questions REGULATION : R 03 WEBSITE : wwwhariganeshcom UPDATED ON : November 07 TEXT BOOK 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) Unit I (Matrices) Cale Hamilton Theorem Show that the matri 0 0 satisfies the characteristics equation and hence find 0 3 its inverse (Jan 0),(Jan 03) Tet Book Page No: 43 Show that A 0 0 satisfies its own characteristic equation and hence find (N/D 07) A Using Cale-Hamilton theorem, find the inverse of A 4 3 (N/D 0) Tet Book Page No: 45 4 Verif Cale-Hamilton theorem for 3 A 4 Hence using it find A 3 (M/J 06) Sri Hariganesh Publications (Ph: , ) Page

2 Engineering Mathematics 08 5 Verif Cale Hamilton Theorem for the A 5 4 Hence find A (Jan 06) 6 Verif Cale Hamilton Theorem for the matri 3 A 4 (A/M 0) Tet Book Page No: 47 7 Find the characteristic equation of the matri A given A Hence find A and 4 A (Jan 009)(Jan 00)(M/J 00) (M/J 03)(N/D 04) Tet Book Page No: Using Cale Hamilton theorem, find the inverse of the matri A 8 7 Tet Book Page No: 54 (N/D 00) 9 Using Cale-Hamilton theorem find A and 4 A, if A 3 0 (Jan 04) 0 0 Use Cale Hamilton theorem to find the value of the matri given b A 5A 7A 3A A 5A 8A A I, if the matri A 0 0 Tet Book Page No: 50 (M/J 009) Verif Cale Hamilton theorem for the matri 0 3 A, hence find its A (M/J 04) Sri Hariganesh Publications (Ph: , ) Page

3 Engineering Mathematics 08 If 3 A 5, verif Cale-Hamilton theorem and hence find A 3 (M/J 05) 0 3 Verif Cale Hamilton Theorem for the matri 0 0 and hence find 0 A and 4 A (M/J 0) Tet Book Page No: 54 n 4 Find A using Cale Hamilton theorem, taking A 4 3 Hence find A3 Tet Book Page No: 5 (Jan 0) Eigenvalues and Eigenvectors of a given matri 3 Find the eigenvalues and the eigenvectors of the matri 5 (M/J 06) 3 Find all the eigenvalues and eigenvectors of the matri 4 3 (Jan 0) Tet Book Page No: 4 3 Find the eigenvalues and eigenvectors of 6 0 (Jan 03) Tet Book Page No: 7 4 Find the eigenvalues and eigenvectors of the matri A 3 Tet Book Page No: (M/J 00),(N/D 00),(Jan 0),(Jan 04) Sri Hariganesh Publications (Ph: , ) Page 3

4 Engineering Mathematics Find the eigenvalues and eigenvectors of 3 3 (M/J 05) Tet Book Page No: 35 6 Find the eigenvalues and eigenvectors of A (Jan 009) Find the eigenvalues and eigenvectors of the matri A Tet Book Page No: 40 (N/D 0),(N/D 06) 8 3 Find the eigenvalues and eigenvectors for the matri A 6 0 Tet Book Page No: 40 (M/J 009),(Jan 00),(M/J 04) 9 0 Find the eigenvalues and eigenvectors of the matri A Tet Book Page No: 39 (M/J 03),(Jan 06) Find the eigenvalues and eigenvectors of (N/D 04) Tet Book Page No: 40 Diagonalisation of a Matri Reduce the matri to diagonal form (A/M 07) Sri Hariganesh Publications (Ph: , ) Page 4

5 Engineering Mathematics 08 The eigenvectors of a 3X3 real smmetric matri A corresponding to the eigenvalues,3,3 are, 0, T T,,, and,, T respectivel Find the matri A Tet Book Page No: 85 (A/M 0) 3 The eigenvectors of a 3X3 real smmetric matri A corresponding to the eigenvalues,3,6 are,0, T T,,0, and orthogonal transformation 0,,0 T respectivel Find the matri A b an If the eigenvalues of A are 0, 3, 5, find the eigenvectors of A and 4 3 diagonalize the matri A (Jan 03) Tet Book Page No: 8 Quadratic form to Canonical form Reduce the quadratic form 5 3z 4 to the Canonical form b orthogonal reduction and state its nature (M/J 00),(Jan 0) Tet Book Page No: 95 Reduce the quadratic form 3 3 to a canonical form b an orthogonal reduction Also find its nature (A/M 0) Tet Book Page No: 00 3 Reduce the given quadratic form Q to its canonical form using orthogonal transformation Q 3 3z z (Jan 009) Tet Book Page No: 3 4 Reduce the quadratic form 3 3 into canonical form(jan 03) Tet Book Page No: 4 5 Reduce the quadratic form 5 6 to the canonical form through orthogonal transformation and find its nature (M/J 04) 6 Reduce the quadratic form 5 z z 6z into canonical form and hence find its rank (M/J 05) Sri Hariganesh Publications (Ph: , ) Page 5

6 Engineering Mathematics 08 7 Reduce the quadratic form 4 to canonical form b an orthogonal transformation Also find the rank, inde, signature and nature of the quadratic form (N/D 00) Tet Book Page No: 3 8 Find a change of variables that reduces the quadratic form to a sum of squares and epress the quadratic form in terms of new variables (Jan 0) Tet Book Page No: 3 9 Reduce the quadratic form 3 5 3z z z into canonical form through orthogonal transformation (M/J 03),(N/D 04) 0 Reduce the quadratic form into canonical form b means of an orthogonal transformation (N/D 0) Tet Book Page No: 3 Reduce the quadratic form 6 3 3z 4 z 4z into a canonical form b an orthogonal reduction Hence find its rank and nature (Jan 04),(Jan 06),(M/J 06),(N/D 06),(N/D 07) Reduce the quadratic form to a Canonical form through an orthogonal transformation and hence find rank, inde, signature, nature and also give n0n zero set of values for,, 3 (if the eist), that will make the quadratic form zero (Jan 00) Tet Book Page No: 06 3 Reduce the quadratic form to the Canonical form 3 3 through an orthogonal transformation and hence show that is positive semi definite Also given a non zero set of values,, 3 which makes this quadratic form zero Tet Book Page No: 3 (M/J 009) 4 Reduce the quadratic form z z z to canonical form through an orthogonal transformation Write down the transformation (M/J 0) Tet Book Page No: 4 Sri Hariganesh Publications (Ph: , ) Page 6

7 Engineering Mathematics 08 General Problems Prove that the eigenvalues of a real smmetric matri are real (M/J 04) If is an eigenvalue of a matri, then prove that is the eigenvalue of A Unit II (Sequences and Series) Comparison Test (N/D 04) 3 4 Test the convergence of the sum (A/M 07) Test the convergence of the series (M/J 04) Tet Book Page No: 3 3 Show b direct summation of n terms that the series is 3 34 convergent (N/D 04) 4 Using comparison test, eamine the convergence or divergence of 3 5 (M/J 05),(Jan 06),(M/J 06) Tet Book Page No: 0 Integral Test Find the nature of the series b Cauch s integral test (M/J 04) p n n log n Tet Book Page No: 4 n Test the convergence of the series ne (Jan 04) n0 Tet Book Page No: 45 3 Test the convergence of the series sin n n n (M/J 06) D Alembert s Ratio Test Sri Hariganesh Publications (Ph: , ) Page 7

8 Engineering Mathematics 08 Using D Alembert s ratio test, eamine the convergence or divergence of Tet Book Page No: 8 Eamine the convergence and the divergence of the following series 3 3 (M/J 05) Tet Book Page No: 7 n n 3 n 0 (Jan 04) Test the convergence of the series b D Alembert s ratio test (N/D 04) p p p Test the convergence of the series b D Alembert s ratio test! 3! 4! Tet Book Page No: 59 (M/J 04) 3 5 Test the convergence of the series 3 Tet Book Page No: 6 (Jan 06),(M/J 06) n n 6 Test the convergence of the series, n 0 (N/D 04) n Eamine convergence of the series n n Tet Book Page No: 36 8 Test the series n n n (Jan 06) n (A/M 07),(ND 07) Alternating Series for Leibnitz Test 3 4 Test the convergence of the series, Tet Book Page No: 95 (Jan 04),(N/D 07) Test the convergence and absolute convergence of the series (A/M 07) Find the interval of the convergence of the series: 4 Discuss the convergence and the divergence of the following series 3 4 (M/J 06) (Jan 04) Tet Book Page No: 87 Sri Hariganesh Publications (Ph: , ) Page 8

9 Engineering Mathematics 08 5 Test for convergence or divergence of Tet Book Page No: 97 (M/J 05) Absolute and Conditional Convergence cos n Determine convergence of an alternating series and test for absolute and n conditional convergence (N/D 04) cos n Test for convergence of the series n (A/M 07) n 3 3 Test for absolute convergence of (M/J 05)!! 3! Tet Book Page No: 08 n 4 Test whether the series is conditionall convergent or absolutel n n convergent (N/D 07) General Problems Prove that the harmonic series is divergent (M/J 04) Unit III (Applications of Differential Calculus) n Radius of Curvature and Circle of curvature a a Find the radius of curvature of the curve a at, 4 4 (Jan 009) Tet Book Page No: 33 a a Find the equation of the circle of curvature at, 4 4 on a (M/J 00),(N/D 00),(A/M 0),(N/D 0),(Jan 0),(M/J 0),(Jan 04), (N/D 04),(Jan 06),(M/J 06) Tet Book Page No: 33 3 Find the equation of circle of curvature of the parabola at the point 3,6 Tet Book Page No: 334 (Jan 009),(N/D 06),(N/D 07) Sri Hariganesh Publications (Ph: , ) Page 9

10 Engineering Mathematics 08 4 Find the equation of circle of curvature of the rectangular hperbola at the point 3,4 (Jan 00),(A/M 07) Tet Book Page No: Find the equation of the circle of curvature of 6 Find the center of curvature of the curve at,3 (M/J 04) at the point, 3 Tet Book Page No: 39 (M/J 03) 7 Find the center of curvature of at (3,3) (M/J 05) Tet Book Page No: 346 0,c on the curve ccosh c Tet Book Page No: 35 (M/J 009) 8 Find the radius of curvature at the point 9 Find the radius of curvature at an point of the catenar ccosh c Tet Book Page No: 35 (Jan 06) 3a 3a 0 Find the radius of curvature at the point, on the curve 3 3 3a Tet Book Page No: 37 (N/D 0) Find the radius of curvature of the curve at 3,3 (M/J 03) Tet Book Page No: 33 Find the radius of curvature at the point acos 3, asin 3 on the curve /3 /3 /3 a (M/J 009),(M/J 05) Tet Book Page No: 37 3 Find the radius of curvature at a,0on Tet Book Page No: a (Jan 00),(N/D 04) Sri Hariganesh Publications (Ph: , ) Page 0

11 Engineering Mathematics 08 4 Prove that the radius of curvature of the curve 3 3 a at the point ( a,0) is 3 a Tet Book Page No: 36 (N/D 00),(N/D 06),(N/D 07) 5 Find the radius of curvature at an point of the ccloid a sin, a cos (M/J 00),(M/J 0),(Jan 03),(Jan 04),(A/M 07) Tet Book Page No: 3 6 Find the radius of curvature of the curve acos t t sint ; asin t t cost at ' t ' (M/J 03) Tet Book Page No: 33 7 Find the radius of curvature of the curve 3a cos a cos 3, 3a sin a sin 3 (A/M 0) Tet Book Page No: 39 t t 8 Find the radius of curvature at an point on e cos t, e sint Tet Book Page No: 346 (M/J 04),(M/J 06) /3 a 9 If a a, where is the radius of curvature (Jan 0) Tet Book Page No: 34 Evolute Show that the evolute of the parabola 3 4a is the curve 7a 4( a) Tet Book Page No: 348 (Jan 00),(M/J 00) Find the equation of the evolute of the parabola 4a Tet Book Page No: 348 (Jan 0),(Jan 0),(M/J 0),(Jan 04),(Jan 06),(M/J 06) 3 Find the evolute of the parabola 4a (M/J 03),(N/D 07) Tet Book Page No: 350 Sri Hariganesh Publications (Ph: , ) Page

12 Engineering Mathematics 08 4 Find the evolute of the hperbola a (N/D 00),(N/D 0) b Tet Book Page No: Find the equation of the evolute of the curve acos t t sin t, a sint t cost (M/J 009),(N/D 06) Tet Book Page No: Show that the evolute of the ccloid a sin, a cos is another ccloid (A/M 0) Tet Book Page No: 36 7 Find the evolute of the ccloid a sin, a cos Tet Book Page No: 36 8 Obtain the evoluteof a sin, a cos Tet Book Page No: 376 (N/D 04) (M/J 05) 9 Find the evoluteof a (M/J 04) Envelope Find the envelope of m a m b, where m is the parameter Tet Book Page No: 379 (Jan 06) Find the envelope of the famil of straight lines m am am 3, where m is the parameter (Jan 04),(M/J 06) Tet Book Page No: Find the envelope of the famil of straight lines cos sin csin cos, being the parameter (A/M 0) Tet Book Page No: Find the envelope of the famil of straight lines given b cos sin asec, where is the parameter (N/D 07) Sri Hariganesh Publications (Ph: , ) Page

13 Engineering Mathematics 08 5 Find the envelope of the straight line, where a and b are parameters that a b are connected b the relation a b c (Jan 009),(M/J 009) Tet Book Page No: Find the envelope of, where a and b are connected b the relation a b a b c, c being constant (N/D 00),(Jan 03),(M/J 05),(A/M 07) Tet Book Page No: Find the envelope of the famil of straight lines, where a and b are a b connected b a b 64 (N/D 04) Tet Book Page No: Find the envelope of the straight line where the parameters a and b are a b connected b the relation a n b n c n, c being a constant (N/D 0),(M/J 04) Tet Book Page No: 39 9 Find the envelope of the straight line, where a and b are connected b the a b relation ab c, c is a constant (Jan 00),(M/J 00) Tet Book Page No: Find the envelope of the ellipse relation a where a and b are connected b the b a b c, c being a constant (Jan 04),(N/D 06) Tet Book Page No: 393 Find the envelope of the sstem of ellipses, where a and b are connected a b b the relation ab 4 (M/J 0) Tet Book Page No: 395 Sri Hariganesh Publications (Ph: , ) Page 3

14 Engineering Mathematics 08 Evolute as the envelope of normals Find the evolute of the hperbola considering it as the envelope of its a b normals (Jan 009) Tet Book Page No: 307 Find the evolute of the ellipse, considering it as the envelope of its a b normal (A/M 07) Unit IV (Differential Calculus of Several Variables) Partial Derivatives If u, show that u u (Jan 009) prove that u u 0 (Jan 009),(N/D 00) If u log tan / Tet Book Page No: 46 u, find sin (M/J 05) 3 If u log tan tan tan z Euler s theorem and Total derivatives If u cos, then prove that u u cot u (A/M 07) If u z z where 3 If w f z, z,, t e and t z e t find d dt w w w, then show that 0 z (M/J 03) Tet Book Page No: 4 (Jan 04),(Jan 06),(M/J 06) Sri Hariganesh Publications (Ph: , ) Page 4

15 Engineering Mathematics 08 4 If z f (, ), where u v, uv, prove that z z 4 u v z z (Jan 00),(Jan 0) u v Tet Book Page No: 46 5 If ucos vsin, usin vcos and V f (, ), show that V V V V (Jan 0) u v Tet Book Page No: 49 6 If u e, show that u u u u u (Jan 03) Tet Book Page No: 43 u u 7 If F is a function of and and if e sin v, e cos v, prove that F F u F F e (Jan 03) u v Tet Book Page No: If u f (, ) where r cos, r sin, prove that u u u u r r (M/J 00) Tet Book Page No: 44 9 If u ( ) f, then find u u u (M/J 04) Talor s epansion Find the Talor s series epansion of 3 in powers of ( ) and ( ) upto 3 rd degree terms (Jan 00),(M/J 00),(Jan 0) Tet Book Page No: 454 Sri Hariganesh Publications (Ph: , ) Page 5

16 Engineering Mathematics 08 Use Talor s formula to epand the function defined b f (, ) 3 3 in powers of ( ) and ( ) (A/M 0),(M/J 05),(A/M 07) Tet Book Page No: Epand 3 in powers of ( ) and ( ) upto 3 rd degree terms Tet Book Page No: 468 (M/J 0) 4 Find the Talor series epansion of sin e at the point, / 4 up to 3 rd degree terms (Jan 009),(M/J 009) Tet Book Page No: Epand e sin in powers of and as far as the terms of the 3 rd degree using Talor s epansion (M/J 03),(Jan 06),(N/D 06) Tet Book Page No: Find the Talor s series epansion of e cos in the neighborhood of the point, 4 upto third degree terms (N/D 00) Tet Book Page No: Epand e cos at 0, upto the third term using Talor s series (M/J 04) Tet Book Page No: Epand e log( ) in power of and upto terms of third degree using Talor s theorem (N/D 0),(Jan 04),(M/J 06) Tet Book Page No: 46 9 Epand sin at, upto second degree terms using Talor s series Tet Book Page No: 463 (N/D 04),(N/D 07) Maima and Minima Find the etreme values of the function 3 3 f (, ) 3 0 Tet Book Page No: 470 (Jan 00),(A/M 0),(Jan 0),(N/D 04) Sri Hariganesh Publications (Ph: , ) Page 6

17 Engineering Mathematics 08 Test for maima and minima of the function 3 3 f (, ) 3 0 Tet Book Page No: 4 (M/J 03) 3 Eamine 3 f (, ) for etreme values Tet Book Page No: 473 (Jan 06) 4 Find the maimum and minimum values of Tet Book Page No: 47 (M/J 0) 5 Discuss the maima and minima of the function f (, ) Tet Book Page No: 476 (N/D 00) 6 Test for an etrema of the function Tet Book Page No: f (, ) (Jan 0) f, (N/D 06) 7 Eamine the etrema of 8 Eamine the function f, 3 for etreme values Tet Book Page No: 480 (M/J 009),(N/ D 07) 9 Test for the maima and minima of the function f, 3 6 Tet Book Page No: 4 0 Discuss the maima and minima of f, 3 Tet Book Page No: 483 Find the maimum value of (Jan 03) (Jan 04) m n p z subject to the condition z a Tet Book Page No: 403 (Jan 009) Find the minimum values of z a 3 z subject to the condition 3 (A/M 07) 3 Find the etreme value of z subject to the condition z 3a Tet Book Page No: 4 (M/J 04) 4 The temperature T at an point,, z in a space is temperature on the surface of the unit sphere T 400z Find the highest z (N/D 07) Sri Hariganesh Publications (Ph: , ) Page 7

18 Engineering Mathematics 08 5 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 Tet Book Page No: 494 (M/J 00),(N/D 0),(M/J 0),(M/J 06),(A/M 07) 6 A rectangular bo open at the top, is to have a capacit of 08 cu ms Find the dimensions of the bo requiring the least material for its construction (Jan 04) Tet Book Page No: 4 7 Find the dimensions of the rectangular bo, open at the top, of maimum capacit whose surface area is 43 square meter (M/J 03) Tet Book Page No: Find the volume of the greatest rectangular parallelepiped inscribed in the ellipsoid z (M/J 009),(M/J 05) a b c Tet Book Page No: Find the length of the shortest line from the point 0,0, 9 to the surface z (N/D 04) 0 Find the shortest and longest distances from the point,, to the sphere z 4 (N/D 06) Jacobians Find the Jacobian and (,, z) ( r,, ) of the transformation rsincos, rsinsin z rcos (Jan 009),(A/M 0),(Jan 06),(M/J 06) Tet Book Page No: 444 If z u, z uv, z uvw prove that (,, z) ( u, v, w) uv Tet Book Page No: 446 (Jan 00),(Jan 0) 3 Find the Jacobian of u z, v z z, Tet Book Page No: 449 w z (M/J 05) Sri Hariganesh Publications (Ph: , ) Page 8

19 Engineering Mathematics 08 4 Find the Jacobian of,, with respect to 3,, if 3,, (N/D 00) 3 Tet Book Page No: If z z u, v, w, find z ( u, v, w) (Jan 04),(M/J 04) (,, z) Tet Book Page No: 447 Unit V (Multiple Integrals) Double integration Evaluate a a a dd (N/D 06) 0 0 Change of order of integration Evaluate e dd b changing the order of integration (N/D 00),(A/M 0) 0 Tet Book Page No: 537 Change the order of integration / e dd and hence evaluate it(n/d 04) 0 0 Tet Book Page No: 56 3 Change the order of integration in 4 dd and evaluate it (N/D 06) 0 0 4a a 4 Change the order of integration and hence evaluate it dd (A/M 07) 0 4a Sri Hariganesh Publications (Ph: , ) Page 9

20 Engineering Mathematics 08 a 0 a 5 Change the order of integration in dd and then evaluate it (M/J 009) Tet Book Page No: 55 6 Change the order of integration 0 a dd and hence evaluate Tet Book Page No: 560 (Jan 00),(M/J 0),(Jan 04),(Jan 06),(M/J 06),(N/D 07) 7 Change the order of integration in the interval a a 0 / a dd and hence evaluate it Tet Book Page No: 547 (M/J 00),(Jan 03),(M/J 04) 8 Change the order of integration and hence find the value of dd (N/D 0) Tet Book Page No: 554 a a 9 Change the order of integration in dd and hence evaluate it (M/J 03) Tet Book Page No: B changing the order of integration, evaluate dd (M/J 05) 0 Tet Book Page No: 535 Change the order of integration a a a dd and hence evaluate it (Jan 0) 0 a a Tet Book Page No: 544 Change the order of integration in a b a a 0 0 dd and then evaluate it(jan 0) Tet Book Page No: 54 Sri Hariganesh Publications (Ph: , ) Page 0

21 Engineering Mathematics 08 Change into polar coordinates a a Epress dd in polar coordinates and then evaluate it (M/J 009) 3/ 0 Tet Book Page No: 500 Evaluate of 0 0 e dd b converting to polar coordinates Hence deduce the value e d (Jan 00),(N/D 00),(Jan 04),(Jan 06),(M/J 06),(N/D 06) 0 Tet Book Page No: 50 3 Transform the integral 0 0 dd into polar coordinates and hence evaluate it (A/M 0),(N/D 04) Tet Book Page No: 50 4 B Transforming into polar coordinates, evaluate dd over annular region between the circles Tet Book Page No: 58 6 and 4 (M/J 00) 5 B Transforming into polar coordinates, evaluate dd over annular region between the circles Tet Book Page No: 53 6 Transform the double integral a and b, ( b a) (Jan 03) a a 0 dd a a into polar co-ordinates and then evaluate it (Jan 0) Tet Book Page No: 506 Sri Hariganesh Publications (Ph: , ) Page

22 Engineering Mathematics 08 7 Transform the integral into polar coordinates and hence evaluate a a dd (Jan 0) 0 0 Tet Book Page No: 504 Area as a double integral Find the area bounded b the parabolas 4 and b double integration Tet Book Page No: 568 (N/D 00) Find, b double integration, the area enclosed b the curves 4a and 4a Tet Book Page No: 566 (Jan 00),(A/M 0),(M/J 03) 3 Find, b double integration, the area between the two parabolas 3 5and 5 9 (M/J 0) Tet Book Page No: Find the area common to 4and 4using double integration(n/d 0) Tet Book Page No: Using double integral find the area of the ellipse a (M/J 03),(N/D 06) b Tet Book Page No: Evaluate dd over the positive quadrant of the circle a Tet Book Page No: 59 (Jan 04), (Jan 06),(M/J 06) over the region between the line 7 Evaluate ( ) dd and the parabola (Jan 0),(A/M 07) Tet Book Page No: 57 8 Find the smaller of the areas bounded b the ellipse and the straight line 36 (Jan 0) Tet Book Page No: 594 Sri Hariganesh Publications (Ph: , ) Page

23 Engineering Mathematics 08 9 Find the surface area of the section of the clinder a made b the plane z a (M/J 04) 0 Find the area inside the circle r asin but ling outside the cardioids r a cos (Jan 009) Tet Book Page No: 590 Find the area which is inside the circle r 3acos and outside the cardioids r a cos (Jan 03) Tet Book Page No: 588 Find the area of the cardioid r a cos Tet Book Page No: 580 (M/J 04),(N/D 04),(M/J 05) 3 3 Evaluate r drd over the area bounded between the circles r cos and r 4cos (N/D 07) Triple integral a b c z dddz (A/M 07) Evaluate log Evaluate z e dddz (M/J 009) Tet Book Page No: Evaluate log log z e dzdd (M/J 03) Tet Book Page No: 54 a a a 4 Evaluate dzdd (N/D 0)(AUT) a z Tet Book Page No: 533 Sri Hariganesh Publications (Ph: , ) Page 3

24 Engineering Mathematics 08 5 Evaluate dzdd (Jan 0),(Jan 03),(M/J 05) Tet Book Page No: 533 z 6 Using triple integration, find the volume of the sphere z a Tet Book Page No: 546 (N/D 00) 7 Find the volume of Tet Book Page No: 546 z r using triple integral (M/J 05) z 8 Find the volume of the ellipsoid (Jan 00),(A/M 0) a b c Tet Book Page No: 548 z 9 Find the volume of the tetrahedron bounded b the plane and the a b c coordinate plane 0, 0, z 0 (M/J 00),(N/D 04) Tet Book Page No: Find, b using triple integrals, the volume of the tetrahedron bounded b the planes 0, 0, z 0 and z a (N/D 07) Find the value of z dddz through the positive spherical octant for which z a (A/M 07) Evaluate z dddz taken over the tetrahedron bounded b the planes z 0, 0, z 0 and (Jan 0) a b c Tet Book Page No: 56 3 Evaluate dzdd wherev is the region bounded b 0, 0, z 3 z 0, z (N/D 0),(Jan 04),(Jan 06),(M/J 06) Tet Book Page No: 560 Sri Hariganesh Publications (Ph: , ) Page 4

25 Engineering Mathematics 08 4 Evaluate dddz, where V is the volume of the sphere z V z a b changing to spherical polar coordinates (N/D 07) 5 Find the volume of the region bounded b the paraboloid z and the plane z 4 (M/J 04) Tet Book Page No: 555 Tetbook for Reference: ENGINEERING MATHEMATICS - I Publication: Sri Hariganesh Publications Author: C Ganesan Mobile: , To bu the book visit wwwhariganeshcom/tetbook ----All the Best---- Sri Hariganesh Publications (Ph: , ) Page 5