ENGG. MATHS-III-TRANSFORMS AND PDE-PRACTICE QUESTIONS FOR Z-TRANSFORMS

                                          ”Z”-TRANSFORMS

Marks:100                                                                                        Time:3hours

                                            PART—A (16marks)                                          

                   ATTEMPT ANY 8 OF THE FOLLOWING:(8*2=16):                      

01.Define a two sided Z-transform. State the applications of it.

02.Find the Z-transform of aⁿ

                                              n!

03.Evaluate the Z-transforms of rⁿcosnф and rⁿsinnф.

04.Define the following functions and derive their Z-transforms:

(a)Unit impulse function

(b)Unit function

05.state and prove the scaling in Z-domain property.

06.Find the Z-transforms of the following functions:

(a)cos³t

(b)sin³nП/6

07.Find the inverse Z-transform of F(z) given by

               F(z) = log(1 divided by (1 – az^-1))

08.State and prove the time shifting property of Z-transforms.

09.Derive an expression to find the Z-transform of the derivative of a function

from the Z-transform of the actual function.

10.Find the Z-transform of (n+1)(n+2).

11.Form the difference from the following by eliminating the arbitary constants:

(a) U_n=a2ⁿ⁺¹

(b) y_n=a+b3ⁿ

12.Evaluate the Z-transform of { t ^k}.

                                            PART—B:(84 marks):                                           

    ANSWER ANY 12 FROM THE FOLLOWING QUESTIONS:(12*7=84):   

13.Find the inverse Z-transform of the following using the residue theorem:

                                                            z

                                      (z – 1)(z² + 1) (7marks)

14.Use the power series method to evaluate inverse Z-transform of the

following:

(a) 10z               (b)         z                           (4+3)marks

(z - 1)(z - 2)           z² + 7z + 10

15.Solve the following equation using Z-transforms only:

y(n+2) + 6y(n+1) + 9y(n) = 2ⁿ provided y₀=y₁=0. (7marks)

16.By the method of partial fractions find the inverse Z-transform of the

following:

         z(z2- z + 2)                                                               (7marks)

       (z + 1)( z - 1)²

17.State and prove the following theorems: ((2+3+3)marks)

(a)Initial value theorem

(b)Final value theorem

(c)Frequency shifting theorem

18.(a)State and prove the convolution theorem for Z-transforms.

     (b)Using the above concept alone, find the Z^-1 of the following:

                             8Z²                (3+4marks)

                      (2z – 1)(4z + 1)

19.Evaluate the Z-transforms of the following: (2+2+3marks)

(a) 3ⁿ cosh 2n

(b)(k-1)a^(k-1)

(c)    2n+3

    (n+1)(n+2)

20.Find the inverse Z-transform of the following using the residue theorem:

                             z(3z² - 6z + 4)

                              (z + 2)(z – 1)²             (7marks)

21.Solve the following difference equation using Z transformation method:

x(n+2) - 4x(n+1)+ 3x(n) = 2ⁿ.n² ,x(0)=0 and x(1)=0. (7marks)

22.Find the inverse Z-transform of the following. Ensure that the final answer

is a pre - defined function:

                           4z² – 2z

                      z³ – 5z² + 8z – 4             (7marks)

23.Find the Z-transform of f(n)*g(n) if (4+3marks)

(a) f(n)=U(n) and g(n)= δ(n) + (1/2)ⁿ U(n).

(b)f(n)=2ⁿ u(n) and g(n)=3ⁿ u(n).

24.(a)Find the inverse Z-transform of

                      z – 4

               (z – 1)(z - 2)²                   (4+1+2marks)

(b)Find the initial and final value of { f(n) } if

F(z)  =                   0.4 z²

                  (z-1)(z² – 0.736z +0.136)

25.Find the Z-transform of the following: (2+2+1+2marks)

(a)aⁿ and naⁿ

(b)cosnθ and ncosnθ

(c)e^-at sinbt

(d)(t+T) e^-(t+T)

26.Solve the following difference equation by (4+3marks)

(a)Residue

(b)partial fractions:

y(n) – y(n-1) = u(n) + u(n-1)

27.(a)Discuss the differences between L and Z-transforms. (2+3+2marks)

     (b)Find the inverse Z-transform of

                                       z2 – 3z

                                    (z – 5)(z + 2)

(c)Find the initial value of the following function:

         F(z)   =       ze^aT (ze^aT – cosbT)

                         z²e2aT – 2ze^aTcosbT + 1

NOTE:

1.Apply the conditions in the correct order.

2.Write down all the corresponding formulas.

ALL THE BEST AND BEST OF EFFORTS!!!

set by,

RANGARAJAN T.R.