Tutorial Problems for MA3151 Matrices and Calculus (2023-2024) PDF

ANNA UNIVERSITY, CHENNAI DEPARTMENT OF MATHEMATICS First Year 2023 - 2024 Tutorial Problems for MA3151- Matrices and Calculus (Regulation 2023) Unit 1: Matrices 1. If the eigenvalues of the matrix A of order 3x3 are 2, 3...

ANNA UNIVERSITY, CHENNAI DEPARTMENT OF MATHEMATICS First Year 2023 - 2024 Tutorial Problems for MA3151- Matrices and Calculus (Regulation 2023) Unit 1: Matrices 1. If the eigenvalues of the matrix A of order 3x3 are 2, 3 and 1, then find the eigenvalues of adjoint of A. 2. If  is the eigenvalue of the matrix A , then prove that 2 is the eigenvalue of A 2.  5 4 3. Find the eigenvalues of 3A+2I , where A=  .  0 2 2 1 2 4. Find the sum and product of the eigenvalues of the matrix A  1 3 1  2 1  6.  7 3 5. Use Cayley Hamilton theorem to find the inverse of A   .  2 6 1 2    6. Use Cayley Hamilton theorem to find A 4  4 A 3  5 A 2  A  2 I , when A    4 3 7. What is the nature of the quadratic form x 2  y 2  z 2 in four variables? 8. Find the nature of the quadratic form 5 x 2  5 y 2  14 z 2  2 xy  16 yz  8 xz without reducing to canonical form. 2 2 1  9. Find the eigenvalues and the eigenvectors of the matrix 1 3 1 . 1 2 2  0 1 1 10. Find the eigenvalues and eigenvectors of the matrix A   1 0 1 .    1 1 0 1001 1 3  11. Find eigenvalues and eigenvectors for the matrix A   1 1005 1 .   3 1 1001 1 2  2 12. Using Cayley –Hamilton theorem find A 1  and A , if A   1 3 4 0 .  0  2 1  1  1 2 3 13. Verify Cayley –Hamilton theorem for A  2  1 4 . Hence using it find A 1.  3 1  1 1 0 0 14. If A  1 0 1 , then show that A n  A n2  A 2  I , for n  3 using Cayley – Hamilton 0 1 0 theorem. 5 3 15. Find the eigenvalues of A and hence find A n (n is a positive integer), given that A   1 3  16. The eigenvectors of a 3x3 real symmetric matrix A corresponding to eigenvalues -1,1, and 4 0  2 1 are     and   respectively. Find the matrix A by using suitable transformation. 1   1 1 1 1   1       2 2  7  17. Diagonalize the matrix A  2 1 2  by similar transformation. 0 1  3 2 0 2 18. Diagonalize the matrix  0 2 0  by means of orthogonal transformation.  2 0 2 19. Reduce the quadratic form 6 x 2  3 y 2  3 z 2  4 xy  2 yz  4 xz into a canonical form by an orthogonal reduction. Hence find its rank and nature. 2 2 2 20. Reduce the quadratic form x1  2 x 2  x3  2 x1 x 2  2 x 2 x3 to canonical form through an orthogonal transformation and hence show that it is positive semi-definite and find its rank, index, signature. ***** Unit 2: Functions of Several Variables 1  y   2u  2u 1. If u  log( x  y )  tan   , then prove that 2 2   0. x  x2  y2 dy 2. If x y  y x  1 , then find. dx 2      9 3. If u  log( x  y  z  3 xyz ) , then prove that  3 3 3    u  x  y z ( x  y  z) 2 dz 4. If z  u 2  v 2 , where u  at 2 , v  2at , then find. dt 2   (u , v) 5. If x  u (1  v) and y  uv , find.  ( x, y )  ( x, y , z ) 6. If x  r sin  cos  , y  r sin  sin  , z  r cos  , then find.  (r ,  ,  )  y2   2u  2u 2  u 2 7. If u  tan 1   , then prove that x 2  2 xy  y   sin 2 u sin 2u.  x  x 2  x y  y 2  x y  8. If u  sin 1   , then prove that x u  y u  tan u and  x y x y 2    2u  2u  2u sin u cos 2u tan u x 2 2  2 xy  y2 2  . x xy y 4 cos 3 u 9. If f ( x, y )   (u , v) , where u  x 2  y 2 , v  2 xy , show that 2 f 2 f 2   2   2   4 ( x 2  y )    u2  v2 .  x2  y2   z z z 10. If z  f ( x, y ) and x  eu cos v and y  eu sin v , then prove that x y  e2u and v u y 2 2  z   z   2u  z  2  z   2       e      .  x   y   u   v   11. At a given instant the sides of a rectangle are 4 ft. and 3ft. respectively and they are increasing at the rate of 1.5ft./sec. and 0.5ft./sec. respectively, find the rate at which the area is increasing at that instant. 12. Obtain the Taylor’s series of f ( x, y )  x y in powers of x  1 and y  1 up to the third terms. 13. Obtain the Taylor’s series of log(1  x  y ) in powers of x and y up to the third terms. 14. Find the percentage of error in the area of an ellipse if one per cent error is made in measuring the major and minor axes. 15. Find the maximum and minimum values of the function f ( x, y )  x 3  3 xy 2  15 x 2  15 y 2  72 x 16. Find the dimension of the rectangular box open at the top is maximum capacity whose surface is 432 square cm. 17. A rectangular box open at the top is to have volume of 256 cubic feet. Find the dimensions of the box requiring least material for its construction. 18. The sum of three numbers is constant. Prove that their product is maximum when they are equal. 19. Find the points on the surface z 2  xy  1 nearest to the origin. 20. The temperature T at any point ( x, y, z ) in space is T  400xyz 2. Find the highest temperature on the surface of the unit sphere x 2  y 2  z 2  1. ***** 3 Unit 3: Integral Calculus  dx 1. Find the values of p such that the improper integral x 0 p convergent or divergent.   1 2. Evaluate the improper integral (i)   (1  2 x)e x dx (ii) dx  1  x 2 0 2 1 1 dx dx du 3. Evaluate the improper integral (i) 2 x 2 , (ii) 5 10  2 x ,(iii) 3 u 2  2u  dx 4. Evaluate the improper integral  x 1. 0  5 dx sin 3 xdx 5. Does the improper integrals (i) 0 x 3 1 (ii)  0 x5 converges or diverges? Justify your answer.  dx  6. If  a  b cos x  0 a2  b2 for a  b , then using the concept of differentiation under integral   dx cos xdx sign find  and  a  b cos x . 0 a  b cos x  2 2 0 2  a  x 2   7. Using the concept of differentiation under integral sign, if F (a )   e x dx , then prove 0 dF (a ) that  2 F (a ) and hence evaluate F (a ). da 8. Using the concept of differentiation under integral sign, show that a2 d x 1  da 0 tan 1  dx  2a tan 1 (a )  log( a 2  1). a 2 1 x m 9. Using the concept of differentiation under integral sign, evaluate (log x) n dx. 0 10. Using the concept of differentiation under integral sign, show that  tan 1 ax  0 x(1  x 2 ) dx  2 log(1  a) , for a  0.    1 x2 1 1   xe x e    x8 2  x4 11. Prove that (i) dx  dx  , (ii) dx  dx  ( Hint: 0 0 16 2 0 1 x 4 0 1 x 4 4  x 2 dx  put x 2  sin t ), (iii)  1  x  0 4 2  8 2 (Hint: put x 2  tan t ). 4   1 x m1 x m1  x n1 12. Prove that  m, n    dx. Hence deduce that  m, n    dx. 0 (1  x) m n 0 (1  x) m n  ( m ) ( n ) 1 13. Prove that  m, n  . Hence deduce that    .  ( m  n) 2   ( n) 1  1  1  14.  n, n   (or)  n, n   2 n 1   n,  and (n) n    2 n 1 2n .Hence 1 2  2  2 2 2 2 n1 (n  ) 2   3 1 deduce that      2. 4 4 1 x (1  x n ) p dx in terms of Gamma function. (Hint: put x n  t ) m 15. Evaluate 0  x m1 16. Evaluate 0 (1  x n ) p dx in terms of Gamma function and deduce that   x m 1  1  1 0 1  x n dx . Hence show that  dx .(Hint put t)  m  1  x 4 2 2 1  x n n sin   0  n    x n1    Assume the result (n)(1  n)   dx .   0 1 x sin n   b  ( x  a) m 1 17. Evaluate (b  x) n1 dx in terms of Beta function. a x2 y2 18. Find the value of  x y dxdy over the positive quadrant of the ellipse 2  2  1 ,in m 1 n 1 a b terms of Gamma functions. dxdydz 19. Evaluate  1  x2  y2  z 2 taken over the region of space in the positive octant bounded by the sphere x 2  y 2  z 2  1 , by using Beta and Gamma functions. d erf (ax)  2a e a x 2 2 20. Prove that (i) erf ( x)  erf ( x)  0 and ercf ( x)  ercf ( x)  2 , (ii) dx  d ercf (ax)   2a e a x. 2 2 and dx  ***** 5 Unit 4: Multiple Integral 1 x 1. Evaluate   xy( x  y)dxdy. 0 x 2. Evaluate   xydxdy over the region bounded by y 2  x and the lines y  0 and y  x  2. a 2a x 3. Change the order of integration and hence evaluate   xydydx. 0 x2 a 4 a 2 ax 4. Change the order of integration and hence evaluate   xydydx. 0 x2 4a 1 2 x 2 x 5. Change the order of integration and hence evaluate   0 x x2  y2 dydx. 6. Find the area bounded by the parabolas y 2  4  x and y 2  x by double integration. 2 a 2 ax  x 2   (x  y 2 )dydx. 2 7. By changing into polar coordinates evaluate 0 0 a a x2 2 dydx 8. By changing into polar coordinates evaluate   0 2 a x y 2 2 2. ax  x xy 9. By changing into polar coordinates evaluate  R x y2 2 dxdy over the region R in the first quadrant bounded by the circles x  y  a and x 2  y 2  4a 2. 2 2 2   e e 2 ( x  y2 )  x2 10. By changing into polar coordinates evaluate dxdy. Hence evaluate dx. 0 0 0 11. Find the area of the cardioid r  a (1  cos  ) by double integration. 12. Find the area that lies inside the cardioids r  a (1  cos  ) and outside the circle r  a by double integration. log 2 x x  log y   e x y z 13. Evaluate dzdydx. 0 0 0 dzdydx 14. Evaluate  ( x  y  z  1) V 3 , where V is the region bounded by x  0 , y  0 , z  0 , x  y  z  1. dzdydx 15. Evaluate  ( V 1 x2  y2  z2 , where V is the region bounded by the coordinate planes and the sphere x 2  y 2  z 2  1 and contained in the positive octant. 16. Find the volume bounded by the cylinder x 2  y 2  4 and the planes y  z  4 , z  0. 6  x y z 17. Find the volume of the tetrahedron bounded by the coordinate planes and    1. a b c a a2  x2 a2  x2  y2 18. Evaluate   0 0  xyzdxdydz , by transforming to spherical polar coordinates. 0 19. Evaluate  a 2  x 2  y 2  z 2 dxdydz , taken over the volume of the sphere x 2  y 2  z 2  a 2 , by transforming to spherical polar coordinates.    (x  y 2  z 2 )dxdydz , taken over the volume of the space bounded by 2 20. Evaluate x 2  y 2  a 2 and 0  z  1 by transforming to cylindrical polar coordinates. ***** Unit 5: Vector Calculus 1. Find a unit normal vector to the surface x 2  y 2  z  4 at (1,2,5). 2. Find the directional derivative of f ( x, y, z )  xy 2  yz 3 at the point (2,-1,1) in the direction    i  2 j  2k. Also find the maximum directional derivative of f ( x, y, z ) at the point (2,-1,1). 3. The temperature of points in space is given by T ( x, y, z )  x 2  y 2  z. A mosquito located at (1,1,2) desires to fly in such a direction that it will get warm as soon as possible. In what direction should it move?      4. Prove that r n  nr n  2 r , where r  xi  yj  zk.     5. Show that F  3 yz 2 i  (2 z 3  4 x) j  2 x 2 y 3 k ) is solenoidal.     6. Show that F  yz 2 i  ( xz 2  1) j  (2 xyz  2)k is irrotational.     7. Find ‘a, b, c’ if F  ( x  2 y  az )i  (bx  3 y  z ) j  ( 4 x  cy  2 z ) k is irrotational.  8. Find the value of n such that the vector r n r is both solenoidal and irrotational,     where r  xi  yj  zk.     9. Show that  2 r n  n(n  1)r n  2 , where r  xi  yj  zk and r  x 2  y 2  z 2.    10. Find the work done in moving a particle in the force field F  3 xyi  y 2 j , in the XY-plane along the curves i) y  2x 2 from the point (0, 0) to (2,8) and ii) y  x from (1,1) to (2,2).     11. Show that F  (2 xy  z 3 )i  x 2 j  3 xz 2 k is a conservative force field. Find the scalar potential  and the work done by F in moving an object in this field from (1,-2,1) to (3,1,4).     12. Verify Gauss divergence theorem for F  ( x 2  yz )i  ( y 2  xz ) j  ( z 2  xy ) k taken over the rectangular parallelopiped bounded by the planes x  0 , y  0 , z  0 , x  a , y  b , and z c. 7  13. Verify Gauss Divergence theorem for F  y i  x j  z 2 k over the cylindrical region bounded by x 2  y 2  9 , z  0 and z  2.           2 2 14. Using divergence theorem, evaluate F.d s for F 4 x i 2 y j z k , taken over the surface S region bounded by cylinder x  y  4 and z  0 , z  3. 2 2    15. Verify Stokes theorem for F  ( x 2  y 2 )i  2 xyj in the rectangular region of z  0 plane bounded by the lines x  0 , y  0 , x  a and y  b.     16. Verify Stoke’s theorem for the vector field F  (2 x  y )i  yz 2 j  y 2 zk over the upper half surface of x 2  y 2  z 2  1 , bounded by its projection on the XY-plane. 17. Apply Stoke’s theorem evaluate  ( x  y)dx  (2 x  z )dy  ( y  z )dz  , where C is the boundary c of the triangle with vertices (2,0,0),(0,3,0) and (0,0,6).   18. Verify Greens theorem for  ( xy  y 2 )dx  ( x 2 )dy , where c is the boundary of the region c bounded by y  x and y  x 2.  (3x   8 y 2 )dx  (4 y  6 xy )dy , where C is the boundary of the 2 19. Verify Green’s theorem for c square bounded by lines x  0 , x  a , y  0 and y  a.  (2 x   y 2 )dx  ( x 2  y 2 )dy , where C is the boundary of the 2 20. Apply Green’s theorem evaluate c area enclosed by the X-axis and the upper half of the circle x 2  y 2  a 2. ***** 8