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Name of the University: Sathyabama University
Name Of The Exam : Engineering Mathematics
Document type : Sample Question Paper

Website : sathyabamauniversity.ac.in

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Mathemativcs III : https://www.pdfquestion.in/uploads/5127-ASSIGNMENT.docx
Mathematics IV : https://www.pdfquestion.in/uploads/5127-ENGINEERING_MATHEMATICS_IV_QUESTION_BANK.pdf

Sathyabama Engineering Mathematics Question Paper

ASSIGNMENT PROBLEMS

UNIT – I

LAPLACE TRANSFORM :

Related : Sathyabama University BDS Entrance Exam Question Bank : www.pdfquestion.in/5451.html

1.Find the Laplace transform of (a) t e-2t sin3t (b) sinh 3t cos2 t
2.Find the Laplace transform of (a) (b)
3.Find the Laplace transform of (a) e –t (b)
4.(a) Find the Laplace transform
5.Find the value of the integral using Laplace Transform (a) (b)
6.Find (a) L-1 {(5s + 3)/(s 2 + 2s+ 5)} (b) L-1 {(3s+2)/(3s2 + 4s + 3)2}
7.Find (a) L-1 { s / (s 2 +1)( s 2 +4)} (b) L -1 { s/(s+2)3}
8.Find (a) L-1{ e -2s/ s(s+1)} (b) L -1 {(s + 1)/ .(s2 + s + 1) }
9.Using Convolution find L -1 { s2 / (s2 +a2)( s2+b2) }
10.Using Convolution find (a) L -1 {s/(s2+1)2 }

UNIT – II

APPLICATIONS OF LAPLACE TRANSFORMS
Solve the following differential equations
11.y”+4y=sin2t, given y (0)= y’(0)= 0.
12.y” – 2 y’ + 2y = 0 y = y’ = 1 at x = 0
13.y” -2y’ + x = e –t x( 0) = 2 x’(0) =1
14.y”–y’-2y = 20 sin 2t given y(0) = 0 y’(0) = 2
15.y” + 9 y = 18 t given y(0) = 0 = y(p/2)
16.y” – 3y’ + 2y = e –t given y(0) = 1 & y’(0) = 0
17.y’’ + 2y’ -5y = e-t sin t given y(0) = 0 and y’(0) = 1
18. given that x(0) = 0, y(0) = 0, x’(0) = 0.
19. given that x(0) = 1, y(0) = 0.
20. given that x(0) = 8, y(0) = 3.

UNIT – III

Complex Variables
Assignment – I
1.Test the analyticity of the functions (i) f(z) = (cos y + i siny) (ii) f(z) =
2.Prove that if w = u +iv is an analytic function then the curves of the family u(x,y)
3.(i) If u(x,y) = (x cosy – y sin y) find f(z) so that f(z) is analytic
(ii) Find f(z) whose imaginary part is v = x2 – y2 + 2xy – 3x -2y
4. (i) If u + v = (x – y) (x2+4xy +y2) and f(z) = u + iv find f(z) in terms of z
(ii) If u – v = (cos y – siny) find f(z) in terms of z
5. If f(z) is regular function of z prove that 2 = 4 (z) 2

Assignment – II
1.Find the image of the circle |z| = 2 by the transformation w = z + 3 +2i
2.Find the image of the circle |z-1| = 1 in the complex plane under the mapping w =
3.Find the bilinear transformation which maps the points z1 = -1 z2 = 0 z3 = 1 into the points w1
4.Determine the bilinear transformation which maps z1 = 0 z2 = 1 z3 = 8 into w1 = i w2 =
5.Find the bilinear transformation which transforms (0, -i, -1) into the points (i, 1, 0)

UNIT IV

ASSIGNMENT I
1. Using Cauchy’s integral formula, evaluate
where C is the circle |z + 1 –i| = 2.
2. Using Cauchy’s integral formula evaluate
where C is the circle |z| = 2.
3. Evaluate using Cauchy integral formula
where C is the circle |z| = 3.
4. Find Laurent’s expansion of
5. Expand in Laurent’s series if
(i) |z| < 2 (ii) |z| > 3 (iii) 2 < |2| < 3
6. Find all possible Laurent’s expansions of about z=0

Department Of Mathematics :
Engineering Mathematics-IV (SMTX1010) :
Question Bank :
UNIT III :
PART A :
1. The steady state temperature distribution is considered in a square plate with sides x = 0, y = 0, x = a and y = a. The edge y = 0 is kept at a constant temperature T and the other three edges are insulated. The same state is continued subsequently. Express the problem mathematically.

2. State any two laws which are assumed to derive one dimensional heat equation
3. State two-dimensional Laplace equation.
4. What is meant by Steady state condition in heat flow?

5. An insulated rod of length 1cm has its end A and B maintained at 0oC and 100oC respectively and the rod is in steady state condition. Find the temperature at any point in terms of its distance x from one end.
6. A bar 20 cm long with insulated sides has its ends kept at 40°C and 100°C until steady state conditions prevail. Find the initial temperature distribution in the bar.

7. A tightly stretched string with fixed end points x = 0 and x = l is initially in a position given by y(x,0 ) = f(x). If the string is released from rest, write down the most general solution of the vibrating motion of the string.

8. A bar 10 cm long with insulated sides has its ends kept at 30°C and 60°C until steady state conditions prevail. Write the initial and boundary conditions of one dimensional heat equation.
9. Write down various solutions of the one dimensional wave equation.