B. Tech./ Semester-III / Engineering Mathematics-III / Lab .
B. Tech./ Semester-III / Engineering Mathematics-III / Lab ManualCourse outcomes:Engineering Mathematics-III1.2.3.4.5.6.Define eigen values, eigen vectors, Laplace transforms and Fourier series.Extend the knowledge of matrices to reduce quadratic form to canonical form.Apply Laplace transforms of commonly used functions and its properties.Examine Linearly dependence and independence of the vectors.Construct Fourier series for the function in the interval [α , α 2π ] and [α , α 2 c ] .Solve problems using applications of Laplace transforms, matrices and Half range sineand cosine series.TUTORIAL 11.2.Test for consistency the following equations and if possible solve them:(i)x 2 y z 1,x 2 y 2 z 9, 2 x y z 2.(ii)x1 3 x2 8 x3 10, 3 x1 x 2 4 x3 0, 2 x1 5 x2 6 x3 13 .(iii)6 x y z 4, 2 x 3 y z 0, x 7 y 2 z 7 .(iv)3 x 3 y 2 z 1,(v)x 3 y z 4, 2 x y z 7, 2 x 4 y 4 z 6, 3 x 4 y 11(vi)2 x1 x2 x3 4, x1 x2 3x3 3, 4 x1 x2 x3 2x 2 y 4, 10 y 3 z 2,For what value of λ the equations 3 x 2 y λ z 1,2x 3y z 52 x y z 2,x 2 y λ z 1 , willhave no unique solution? Will the equations have any solution for this value of λ .3.For what value of λ , the following system of equations possesses a non-trivial solution? Obtainthe solution for real values of λ :3x1 x2 λ x3 0, 4 x1 2 x2 3x3 0, 2λ x1 4 x2 λ x3 04.Investigate for what values of λ and µ the equations(1)x 2 y 3 z 4,x 3 y 4 z 5,x 3 y λ z µ have (i) no solution (ii) a uniquesolution (iii) an infinite no. of solutions.(2)2 x 3 y 5 z 9, 7 x 3 y 2 z 8, 2 x 3 y λ z µ have (i) no solution (ii) aunique solution (iii) an infinite no. of solutions.1
5.Solve the following equations:(i)x1 2 x2 3x3 0, 2 x1 5 x2 6 x3 0(ii)x1 x2 x3 0, x1 2 x2 x3 0, 2 x1 x2 3x3 0 .(iii)2 x1 x2 3 x3 0, 3 x1 2 x2 x3 0, x1 4 x2 5 x3 0 .(iv)x1 x2 x3 x4 0, x1 x2 2 x3 x4 0, 3x1 x2 x4 0(v)x1 2 x2 3x3 0, 2 x1 3x2 x3 0, 4 x1 5 x2 4 x3 0, x1 2 x2 2 x3 0TUTORIAL 21. Examine whether the following vectors are linearly independent or dependent. If linearlydependent find the relation between:(i) [1, 1,1], [2,1,1], [3,0,2] (ii) [3,1, 4], [2,2, 3], [0, 4,1](iv) [2,1,1], [1,3,1], [1,2, 1](iii) [1,1,1], [1,1,0], [1,0,0](v) [1, 2,1] , [ 2,1, 4 ] , [ 4, 5, 6 ] ,[1,8, 3](vi) [1, 2, 1, 0 ] , [1, 3,1, 2 ] , [ 4, 2,1, 0 ] ,[6,1, 0,1](vii) [1,3, 4, 6 ] , [ 0,1, 6, 0 ] , [ 2, 2, 2, 3] ,[1,1, 4, 4] , (viii) [ 2,1, 1,1] , [1, 2,1, 1] , [1, 2, 2,1]2. Find the characteristic equation, eigen values and eigen vectors of the following matrices: 2 1 1 (i) 1 2 1 1 1 2 2 1 1 (ii) 2 3 2 3 3 4 3 1 1 (iii) 1 5 1 1 1 3 3 10 5 (v) 2 3 4 3 5 7 7 2 0 (vi) 2 6 2 0 2 5 8 6 2 (ix) 6 7 4 2 4 3 2 2 2 1 (x) 1 1 1 3 1 6 2 2 (vii) 2 3 1 2 1 3 TUTORIAL 32 1 0 1 (iv) 1 2 1 2 2 3 1 2 2 (viii) 2 1 2 2 2 1
2 2 1 1. For the matrix A 1 30 , prove that A 1 A2 5 A 9 I . 0 2 1 1 2 3 1 2 1A A 18I .2. For the matrix A 2 1 4 , prove that A 40 3 1 1 [] 2 1 1 1 2 13. For the matrix A 1 2 1 , prove that A A 6 A 9 I . 4 1 1 2 []4. Find the characteristic equation of the following matrices and obtain the inverse: 2 1 1 (i) 1 2 1 1 1 2 1 0 2 (ii) 0 2 1 2 0 3 13 1 (iii) 13 3 2 4 4 1 2 3 (iv) 3 1 1 0 1 2 5. Find the characteristic equation of the matrix given below and verify that it satisfies CayleyHamilton theorem: 1 0 0 (i) 1 0 1 0 1 0 2 1 1 (ii) 1 2 1 1 1 2 2 1 3 (iii) 1 1 2 1 2 1 1 1 3 (iv) 1 0 3 2 1 0 6. Find the characteristic equation of the matrix A and hence find A 1 and A 4 . 1 1 0 (i) A 0 0 1 2 1 2 13 2 0 1 1 (ii) A 0 2 0 (iii) A 13 3 1 0 2 2 4 4 7. Find the characteristic equation of the matrix A given below and hence, find the matrixrepresented by(i) 1 3 7 A 4 A 20 A 34 A 4 A 20 A 33 A I , where A 4 2 3 1 2 1 (ii) 3 10 5 A 6 A 9 A 4 A 12 A 2 A I , where A 2 3 4 357 766554433223
1 0 8. A , find eigen values of 4 A 1 3 A 2 I . (Ans: 9,15) 2 4 1 0 0 9. A 2 3 0 , find eigen values of A 2 . (Ans: 1,9,4) 14 2 3 0 210. A , find eigen values of A ' 3 A ' 4 I . (Ans: 4,2) 1 2 11. Two of the eigen values of a 3 x 3 matrix whose determinant is 6 are 1,3. Find the thirdeigen value. (Ans: 2)12. The sum of the eigen values of a 3 x 3 matrix is 6 and the product of the eigen values isalso 6. If one of the eigen value is one, find other two eigen values. (Ans: 2,3)13. Find the sum & the product of the eigen values of the matrix 8 6 2 2 9 5 8 4 a) A , b) A 6 7 4 , c) A 5 10 7 2 2 2 4 3 9 21 14 14.Using Cayley-Hamilton Theorem, find matrix represented by 1 4 A 7 9 A 2 I where A 1 1 15Using Cayley-Hamilton Theorem, find matrix represented by 1 2 2 A 4 5 A 3 7 A 6 I where A 2 2 16. Using Cayley-Hamilton Theorem, find matrix represented by 1 2 3 9876A 6 A 10 A 3 A A I where A 1 3 1 1 0 2 17. Using Cayley-Hamilton Theorem, find matrix represented by 2 1 1 A 5 A 7 A 3 A A 5 A 8 A 2 A I where A 0 1 0 1 1 2 18. Using Cayley-Hamilton Theorem, express matrix B as a quadratic polynomial in A, alsofind B, where 1 2 3 8765432B A 11A 4 A A A 11A 3 A 2 A I given A 2 4 5 3 5 6 87654324
6 4 319. Find A , A , A if A 1 1 4 1 1 1 23420. Find A , A , A & A if A 3 1 2 3 1 2 36 2 3 2 1 1 TUTORIAL 41. Show that the matrix A is diagonalizable. Find the transforming matrix and thediagonal matrix.(i) 8 6 2 A 6 7 4 ( λ 0,3,15 ) 2 4 3 8 8 2 (ii) A 4 3 2 ( λ 1,3,2 ) 3 4 1 9 4 4 (iii) A 8 3 4 ( λ -1,-1,3) 16 8 7 1 6 4 2 ( λ 0,1,1)(iv) A 0 4 0 6 3 6 2 2 (v) A 2 3 1 ( λ 2,2,8) 2 1 3 2 2 3 1 6 ( λ 5,-3,-3)(vi) A 2 1 2 0 1 6 1 (vii) A 1 2 0 ( λ -1,3,4) 0 0 3 1 2 3 (viii) A 0 2 0 ( λ 1,2,2) 0 0 2 2 3 4 (ix) A 0 2 1 ( λ 2,2,1) 0 0 1 3 1 1 (x) A 1 5 1 ( λ 2,3,6) 1 1 3 5
8 8 2 2. Show that the matrix A 4 3 2 is diagonalisable. Find the transforming matrix 3 4 1 and the diagonal matrix. 2 3 4 3. Show that the matrix A 0 2 1 is not similar to a diagonal matrix. 0 0 1 1 6 4 4. Show that the matrix A 0 42 is similar to a diagonal matrix. Also find the 0 6 3 transforming matrix and the diagonal matrix.5. Reduce the following matrix to diagonal form: 3 1 1 (i) 1 3 1 1 1 3 9 1 9 (ii) 3 1 3 7 1 7 6 2 2 (iii) 2 3 1 2 1 3 1 2 2 (iv) 121 1 1 0 6. Show that the following matrices are similar to diagonal matrices. Find the diagonal form andthe diagonal matrix: 4 2 2 5 3 2 2 4 1 (i)(ii) 2 2 3 21 6 1 2 0 TUTORIAL 61. 1 4 then prove that 3 tan A A tan 3 . 2 1 If A 0 1 At, find e . 1 0 7 3 , find A n . 2 6 2.If A 3.If A 4. 1 0 0 If A 1 0 1 , find A50 . 0 1 0 5.If A 6.3 2If A , find A100 . 3 4 7. ππ 2 , find sin A .If A 0 3π 2 6 3 1 AA, find e , 5 . 1 3
3 2 1 28. Find A 50 , where i) A , ii) A 1 2 3 4 1 2 2 433 4 1. Find A100 , where i) A , ii) A , iii) A 0 2 1 7 8 1 1 0 0 1 1 3 1 2. Find e A & 4 A if A .2 1 3 π3. If A 0 4. Show thatπ 4 , find cos A . 2 cos O3 3 I 3 3π 14. Write down the matrix corresponding to each of the following quadratic forms(i)x 2 2 y 2 3z 2 2 xy 6 xz 10 yz (ii)x 2 2 y 2 3z 2 4 xy xz 2 yz22222 x1 3 x 2 4 x3 x 4 2 x1 x 2 3 x3 x1 4 x1 x 4 5 x 2 x 3 6 x 2 x 4 x3 x 4(iii)15. Reduce the quadratic form 2 x1 x2 2 x2 x3 2 x3 x1 into canonical form. Examine fordefiniteness.22216. Find the matrix of the quadratic form 6 x1 3 x 2 3 x3 4 x1 x 2 4 x3 x1 2 x 2 x3 and find thelinear transformation X QY which transforms the given form to sum of squares. Write alsorank, index, signature and nature of the quadratic form.22217. Reduce 8 x 7 y 3 z 12 xy 4 xz 8 yz into canonical form by orthogonal transformation.22218. Reduce the following quadratic form 6 x1 3 x 2 14 x3 4 x1 x 2 18 x3 x1 4 x 2 x3 todiagonal form through congruent transformations.19. Reduce the following quadratic form to canonical form and find its rank and signatures. Alsowrite linear transformation which brings about the normal reduction:22221x1 11x 2 2 x3 30 x1 x 2 12 x 3 x1 8 x 2 x3 .20. Reduce the following quadratic form to sum of squares and interpret your result:2223 x1 2 x 2 x3 4 x1 x 2 2 x 3 x1 6 x 2 x321. Reduce the following quadratic form to canonical form using congruenttransformation. Also write linear transformation. State rank, index , signature &nature of quadratic form222a) 3x1 4 x1 x2 2 x2 x3 2 x1 x3 6 x 2 x3 , find non-zero values of x1, x2 , x3 which willmake the quadratic form positive and negative respectively.7
222b) x1 2 x1 x2 2 x2 3x3 2 x3 x1 2 x 2 x3 , find non-zero values of x1, x2 , x3 which willmake the quadratic form positive and negative respectively.222c) 5 x1 6 x1 x 2 26 x2 10 x3 14 x3 x1 4 x2 x3 , find non-zero values of x1, x2 , x3 whichwill make the quadratic form zero.222d) x1 2 x1 x 2 2 x2 2 x3 x3 x1 2 x2 x3222e) 21x1 30 x1 x2 11x2 2 x3 12 x3 x1 8 x2 x3 , find non-zero values of x1, x2 , x3 whichwill make the quadratic form zero.222f) 2 x1 12 x1 x2 x2 3x3 4 x3 x1 8 x2 x3222g) 6 x1 4 x1 x2 3x 2 3x3 4 x3 x1 2 x2 x3222h) 10 x1 4 x1 x2 2 x2 5 x3 10 x3 x1 6 x2 x3i)2224 x1 8 x1 x2 3x2 x3 4 x3 x1 6 x2 x322x1 2 x1 x2 3 x2 4 x3 x1 12 x2 x3 , find non-zero values of x1, x2 , x3 which will makethe quadratic form positive.222k) 6 x1 4 x1 x2 3x 2 3x3 4 x3 x1 2 x2 x3j)2226 x1 4 x1 x2 3x2 14 x3 18 x3 x1 4 x2 x3m) 2 x1 x2 2 x3 x1 2 x 2 x3n) x1 x2 x3 x1 x2 x3o) 2 x1 x2 4 x3 x1 6 x 2 x3l)22. Reduce the following quadratic form to canonical form using orthogonaltransformation. Also write linear transformation. State rank, index , signature &nature of quadratic form.222a) 3x1 2 x1 x2 5 x2 3 x3 2 x3 x1 2 x2 x3222b) 7 x1 5 x 2 6 x3 4 x3 x1 4 x2 x3222c) 6 x1 4 x1 x2 3x2 3x3 4 x3 x1 2 x2 x3222d) x1 4 x1 x2 4 x2 9 x3 6 x3 x1 12 x2 x322222e) 7 x1 8 x1 x2 8 x2 8 x3 8 x3 x1 2 x2 x3f) 3x1 4 x1 x2 3x3 8 x3 x1 4 x2 x3222g) 10 x1 4 x1 x2 2 x2 5 x3 10 x3 x1 6 x2 x3h) 2 x1 x2 2 x3 x1 2 x2 x322222i)x1 2 x2 4 x1 x2 3 x3 4 x2 x3j)x1 3x2 3 x3 2 x2 x32222k) 8 x1 7 x2 12 x1 x 2 3 x3 8 x2 x3 4 x1 x38
(2)2l) 2 x1 x 2 x 2 x1m) 2(x1 x2 x3 x1 x2 x3 )22n) 17 x1 17 x 2 30 x 2 x1TUTORIAL 7Find the Laplace transforms of the following functions:1. t , 0 t Tf (t ) T 1,t T2. (t 1)2 , t 1f (t ) 0 t 1 0,3. t 2 , 0 t 1f (t ) t 1 1,4. t , 0 t af (t ) t a b,Find the Laplace transforms of the following functions:5.cos 3 2t6.sinh 2 2t7. (sin t cos t )9.sin (ϖt α )10.sin 5 t11.14.cosh 5 tcos tt15.2sin t 021.cosh 2t sin tet22.e t sinh t sin t24.cos t cos 2t cos 3t25.e 2 t sin 4 tt cos( 3t 4)sin 2 tt29.cosh 2t cos 2t 32. Evaluate: e Evaluate: et023.26.t 2 sin atsinh tt30. 2t3sin tdt035.19. sinh at sin at33. t t0sin tdttTUTORIAL 8Find the inverse Laplace transforms of the following functions:2( 2) 7 sin t13.sinh 5 t1, then find α .420.( 2 )sin t 2sinh tsin 2t cos t cosh 2t27. te 4 t sin 3t31.cos 2t cos 3tt t 3Evaluate: e t sin tdt92cos 5 t e 4 t sin 3 t28.12. tIf e 2 t sin (t α )cos(t α )dt 16.17.18.5e8.34. Evaluate: e 3t t sin tdt0
1.6.11.2 11 3 s ss 4s 2(s 3)(s 1)32s 3s2 92.7.2ss 443.4 s 1516 s 2 258.ss s2 14.2s 3s 2s 25.s 2s 4s 71s 2s 310.2s πs (s π )29.4423Find the inverse Laplace transforms by using convolution theorem:(i)1s ( s a)(iv)s22( s a )(s 2 b 2 )(vii)12( s 3)(s 2 s 2)(x)(xiii)(ii)1(s 3)(s 3)21s 2 (s 1)1s ( s a) 22s2( s 2 a 2 )(s 2 b 2 )(iii)1( s a)(s b) 2(v)s2(viii)(s2 a2(ix))2s2s2 4 s2 1(xi)()((xii))1(xiv)(s2 4 s 132s2(s2 a2)2s 2 2s 3s 2 2s 2 s 2 2s 5()()(s 3 )2(xv))( s 2) 2( s 2 4s 8) 2(vi)(s2 6s 5)2TUTORIAL 9tfor 0 t T and f (t ) f (t T ) .T0 t a 1,f (t ) and f (t ) is periodic function 1, a t 2a1.Find Laplace transform of f (t ) K2.Find the Laplace transform ofwith period 2a.3.4.5.Find Laplace transform of a sin pt , f (t ) 0, 0 t ππpp t 2πpand f (t ) f t 2π .p E,0 t p 2Find Laplace transform of f (t ) , f (t p ) f (t ) .p E , t p 2Find Laplace transform of sin t H t π H t 3π .22[ ( )] [ ( )]((1 3t 4t)6.Find Laplace transform of 1 2t 3t 2 4t 3 H (t 2) .7.Find Laplace transform of8.Find inverse Laplace transform of the following:2) 2t 3 H (t 3) .10
e ase 4 3 s55(i)(s b ) 2(e s 1 ss39.10.20.(s 4) 2(ii))(iii)se(v)(vi)Find Laplace transform of : s π e ss2 π 2(vii)2(iv)s e ass2 b2(ii)t[H (t 4)] t 2δ (t 4)(iii) π sin 2tδ t t 2δ (t 2) 2 (iv)t 4 [H (t 2)] t 2δ (t 2)Find: (i) s L 1 s 1 L 1 (e as sin a)(ii) e (1 2t 3tEvaluate:(i)2) 4t 3 H (t 2) dt0 e (1 t t )H (t 3) dt 2t(ii)2012.Find Laplace Transform of tH (t 2)13.Find Laplace Transform of t 2 H (t 3)14.15.Find Laplace Transform of t 4 H (t 3)Find Laplace Transform of sin tH (t π )16.Find Laplace Transform of g (t ) (t 2 ) , t 2 and g (t ) 0,0 t 2 π 3π Find Laplace Transform of sin t.H t H t 2 2 218.Find Laplace Transform of (1 2t 3t 2 4t 3 ).H (t 2)19.Find Laplace Transform of (1 3t 4t 2 2t 3 ).H (t 3)20.Find Laplace Transform of f (t ) (t 3) , t 3 and f (t ) 0,0 t 321.using Laplace Transform4 e t(1 2t 3t 2 4t 3 ).H (t 2)dt t(1 2t t 2 t 3 ).H (t 1)dt0 22.using Laplace Transform e023.24.(s 4 )3sin 2tδ (t 2) t17.e 3 s(i) 11.e π ss 2 2s 2 1 2t t 2 , t 2Find Laplace Transform of f (t ) 0,0 t 2Find:11
(i) e 4 3 s L 1 52 (s 4 ) (iv) e L 4 (s 2 ) (ii) (s 1)e s L 1 2 s s 1 (v) e πs L 2 s 9 5 s 1(vii) se 3s L 1 2 s 1 (x) e 4 s L 2s 7 (xiii)(viii) 1(vi) 1 s L 1 e s 3 s (xi) se 2 s L 2 s 2s 2 (xii) e πs L 1 2 2 s s 1 (xv) π L sin 3t.δ t 3 (xiv)(L[sin 3t.δ (t 4)](xviii)L t 2 .δ (t 4 ) t.H (t 4)(xx) π L sin t.δ t t 2 .δ (t 2 ) 2 (xxi)][)]L t 4 .δ (t 2) t 2 .H (t 2)(xix) se 3se s 1 L 1 L (xxii)2 s2 a2 ( s 1)( s 2 ) Application Of Laplace Transforms To Solve Differential Equations :Solve:dy 2 y e 3t , y 1 atdt) π L cos t.δ t t 3 4t 3 δ ( t 5 ) 2 (xvii)(xvi)3( 1TUTORIAL 101. se as L 1 22 s b (ix) e bs L 1 2 s (s a ) 1[ se as L 2 s 3s 2 1 1 s 2 L e s 2 s 1 8e 3 s L 1 2 s 4 (iii)t 0.12() 2
2. Solve: L3. Solve:dI RI Ee at , y 1 wheredtdy 3 y 2 e t , ifdt(y 1 atI ( 0) 0 .t 0.)4. Solve: D 2 3D 2 y 4e 2t with y (0) 3 andy ' (0) 5 .d2ydydy 2 3 y sin t , when t 0, y 0 and 0.2dtdtdt22t'6. Solve ( D 3D 2) y 4e with y (0) 3 and y (0) 35. Solve:2'7. Solve ( D D 2) y 20 sin 2t with y (0) 1 and y (0) 22(2 t2)'8. Solve ( D 3D 2) y 2 t t 1 with y (0) 2 and y (0) 0'9. Solve ( D 2 D 5) y e sin t with y (0) 0 and y (0) 132'"10. Solve ( D 2 D 5D) y 0 with y (0) 0 , y (0) 0 and y (0) 111. Solve12. Solve13. Solve14. Solved2ydy 9 y δ (t ) given that y 0 , 0 at t 02dtdtd2ydy 16 y δ (t ) given that y 0 , 0 at t 02dtdtd2ydydy 3 2 y tδ (t 1) given that y 0 , 0 at t 02dtdtdtd2ydy 4 y f (t ) given that y 0 , 1 at t 02dtdtwhen 0 t 1 1f (t ) when t 1 0Where (i)(ii) f (t ) H (t 2)dxdy y sin t , x cos t ;where x 0 , y 2 at t 0dtdt tt2 t t16. Solve Dx 2 y x 2te e 6t , D x Dy te 2e 3 ; where x 0 ,15. Solve17. y 0 & Dx 1 at t 0TUTORIAL 11Find the Fourier series for the following functions:1.f ( x) e x in (0, 2π ) .2.π2 1 1 1 1 π x .in(0,2π).Hencededucethatf ( x) 12 12 2 2 3 2 4 2 2 3.f ( x) x 3 in (0, 2π ) also in ( π , π ) .213
4.f ( x) e x in ( π , π ) .5.f ( x ) 1 cos x in (0, 2π ) .6. π x 0 0,1 12 cos 2nxf ( x) prove that f ( x ) sin x 2.π 2π n 1 4n 1 sin x, 0 x πHence show that111π 2 . .1.3 3.5 5.747. π ,f ( x) x,8. x0 x π 2 ,f ( x) x π , π x 2π 29.0 x π a ,f ( x) a, π x 2π10. 0, π x 0f ( x) 0 x π π ,11.f ( x) x 2 in (0, a ) .12. 2,f ( x) x,13.f ( x) x x 2 in 1 x 1 .14.f ( x) π 2 x 2 in π x π .15.f ( x) 1 x 2 in 1 x 1 .16.π x 2 ,f ( x) π x, 2 π x 00 x π 2 x 00 x 2 π x 0, deduce that0 x πUsing parseval’s identity prove thatπ496 π28 1 11 2 2 . .21 3 51 11 4 4 .41 3 514
17.f ( x) x cos x in ( π , π ) .18. k ,f ( x) k,19.f ( x) x x 2 in 1 x 1 .20. π x 00 x πf ( x ) π x, π x 0 π x,0 x π3x 2 6 xπ 2π 2π2 1 1 1 .in (0, 2π ) . Hence deduce that126 12 2 2 3221.f ( x) 22. 2 x 1 0, 1 x, 1 x 0 f ( x) 0 x 1 1 x, 0,1 x 2TUTORIAL 121.Find half range cosine series for f ( x) x in 0 x 2 . Using parseval’s identity, deduceπ41 11 4 4 4 .that (i)96 1 3 5(ii)π490 1 111 4 4 4 .41 2342.Find half range sine series for f ( x) x sin x in (0, π ) .3. kx, Expand f ( x) 0,0 x l2l x l2Deduce the sum of the seriesinto half range cosine series.1 11 2 2 .21 35 1,0 x a 2f ( x) .a 1, x a 24.Find half range cosine series for5. kx,0 x l 2 .Obtain a half range cosine series for f ( x ) k (l x ), l 2 x lDeduce the sum of the series1 1 1 .12 32 5215
6.Find a half range cosine series to represent f ( x) sin x in 0 x π .7.Obtain a half range sine series for f ( x) x(π x) in ( 0, π ) .8. 1 x, 4Obtain a half range sine series for f ( x) x 34 , 9. Find half range cosine series for f ( x ) a 1 10.Obtain a half range sine series for f ( x) x(2 x) in 0 x 2 .11.Show that a constant c can be expanded in a infinite series4c sin 3 x sin 5 x . in the range 0 x π . sin x π 35 12.Find half range cosine series for the function f ( x ) ( x 1) in the interval 0 x 1 .0 x (1 )( )2.1 x 1( ) ( 2)x 0 x l.l 2 1 1 1 Hence show that π 2 8 2 2 2 . . 1 3 5 13.Obtain a half range sine series for f (t ) t t 2 , 0 t 1 .14.Find the half range sine series for f ( x) x cos x in ( 0, π ) .15.Obtain the half range sine series for e x in 0 x 1 .16. kx, 0 x l 2 into half range cosine series. Deduce the sum of theExpand f ( x ) l 0, 2 x l1 1 1series 2 2 2 .1 3 517.Show that the set of functions πx 3πx 5πx sin , sin , sin ,. is orthogonal over (0, L ) . 2L 2L 2L 18.Show that the set of functions1, sinπxL, cosπxL, sin2πx2πx, cos,.LL16
form an orthogonal set in ( L, L ) and construct an orthonormal set.()19.2Prove that f1 ( x ) 1, f 2 ( x ) x, f 3 ( x) 3 x 120.Show that the set of functions 1, sin21. πx 3π x 5π x Show that the set of functions sin , sin , sin ,. is orthogonal over 2L 2L 2L are orthogonal over ( 1,1) .2πxπx2π x2π x, cos,sin, cos,. form an orthogonalLLLLset in ( L, L ) and construct an orthonormal set.( 0, L ) .22.Show that the set of functionscos x cos 2 x cos 3 x,,,. form an orthonormal set in theπππinterval ( π , π ) .23.Show that the set of functionssin x sin 2 x sin 3 x,,,. form an orthonormal set in theπππinterval ( π , π ) .24.Prove that the set of functions sin x,sin 2 x,sin 3x,. is orthogonal over [ 0, 2π ] . Henceconstruct orthonormal set of functions.25.Show that the set of functions cos x, cos 2 x, cos3x,. is orthogonal over [ π , π ] . Henceconstruct orthonormal set of functions.26.Show that the set of functions sin x,sin 2 x,sin 3x,. is orthogonal over [ 0, π ] .27.Show that the set of functions cos x, cos 3x, cos 5 x,. is orthogonal over 0, π(17)2 .
28.Solve: y ' ' y t , y (0) y ' (0) 1 .14.dxSolve: dt 5 x 2 y t ,15.Solve: y ' ' ' 2 y ' ' y ' 2 y 0, y (0) y ' (0) 0 and y ' ' (0) 6.16.Solve: y ' ' 5 y ' 4 y 3 δ (t 2)17.Solve:18.d2y 4 y f (t ),Solve: dt 2dy 2 x y 0, x y 0 at t 0.dtd2ydy 3 2 y tδ (t 1),2dtdtat t 0, y (0) 2 & y ' (0) 2. .y (0) 0, y ' (0) 0. .y (0) 0, y ' (0) 1 &18 1, 0 t 1f (t ) 0, t 1 .
B. Tech./ Semester-III / Engineering Mathematics-III / Lab Manual Course outcomes: Engineering Mathematics-III 1. Define eigen values, eigen vectors, Laplace transforms and Fourier series. 2. Extend the knowledge of matrices to reduce quadratic form to canonical form. 3. Apply Laplace tra
2-semester project Semester 1 Semester 2 Semester 3 1-semester project Semester 2 Semester 3 BA to MA Student Semester 1 Semester 2 Select a faculty member to se rve as Project A dvisor Co m plete Project F orm #1, with A dviso r‘s signature, and file it with the Program Director Deve lop a wri tten Project Pr oposa l, and
Manajemen Akuntansi Keuangan Lanjutan 1 Taufan Adi K Muda Setia Hamid Aplikasi Perpajakan Uum Helmina C SEMESTER 5 A1 SEMESTER 5 A2 SEMESTER 5 A3 SEMESTER 5 A4 SEMESTER 5 A5 SEMESTER 5 A6 SEMESTER 5 A7 SEMESTER 5 A8 07.30-10.00 102 203 103 10.10-12.40 104 13.00-15.30 104 307 306 15.30-18.00 306 308 LTD 305 306
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MASTER OF SCIENCE (M.Sc.) ZOOLOGY First Semester – Fourth Semester (2-year programme) I Semester Examination November 2007 II Semester Examination April 2008 III Semester Examination November 2008 IV Semester Examination April 2009 Syllabus applicable for the students seeking a
Syllabus Name of Academic Unit: Computer Science and Engineering Level: B. Tech. Programme: B.Tech. i Title of the course CS 201 Data Structures and Algorithms ii Credit Structure (L -T P C) (3-0-0-6) iii Type of Course Core course iv Semester in which normally to be offered Autumn v Whether Full or Half Semester Course Full vi Pre-requisite(s), if any (For the
Mr Nagaraj Kamath, M.Tech Mr Ramakrishna Nayak, M.Tech, MBA Mr Vinod T Kamath, M.Tech, MBA Mr Ramnath Shenoy, M.Tech, MBA Assistant Professors Ms Soujanya S Shenoy, BE, MBA Mr Sandeep Nayak Pangal, MBA, M.Tech Ms Bhagya R S, BE, M.Tech Mr Devicharan R, BE, M.Tech Mr Nithesh Kumar
University of Calicut Syllabus - B.Tech – Civil Engineering - 2014 8 th Semester Hours/ Week Marks Duration of End Code Subject End Semester Credits Internal Semester examination L T P/D Environmental CE14 801 Engineering II 31 0 50 100 4 Quantity Surveyin
Mary plans to take Colin to see the secret garden. Mary’s visits make Colin feel a lot better. Martha’s brother, Dickon, visits Colin one day with Mary and brings lots of tame animals with him. Colin is delighted. Mary and Dickon take Colin secretly into the garden. Colin realises it is his mother’s garden, and says he will come every day. Colin spends a lot of time in the garden with .
