M 2 QuestionBank

RAJALAKSHMI ENGINEERING COLLEGE MA 2161

UNIT I - ORDINARY DIFFERENTIAL EQUATIONS

PART A

1. Solve (D2 +D 2)y = 0.2. Solve (D2 + 6D + 9)y = 0.

3. Solve (D4 + 4)x = 0 where D = ddt

4. Find Particular Integral: (D + 2)(D 1)2y = e2x.5. Find Particular Integral: (D3 + 1)y = cos(2x 1).6. Find Particular Integral: (D2 +D)y = x2 + 2x+ 4.

7. Reduce to linear differential equation x2 d2y

dx2 x dydx + y = log x.

8. Reduce to linear differential equation x d2y

dx2 2yx = x+ 1x2 .

9. Reduce to linear differential equation (1 + x)2 d2y

dx2+ (1 + x)x dydx + y =

2 sinx.

10. Reduce to linear differential equation (2x+3)2 d2y

dx2(2x+3) dydx12yy =

6x.

PART B

1. Solve d2y

dx2 4y = x sinhx.

2. Solve (D2 1)y = x sin 3x+ cosx.3. Solve (D2 4D + 3)y = sin 3x cos 2x.4. Solve d

2ydx2

+ a2y = sec ax.

5. Solve x2 d2y

dx2+ 4x dydx + 2y = e

x.

6. Solve x3 d3y

dx3+ 2x2 d

2ydx2

+ 2y = 10(x+ 1x).

7. Solve (1 + x)2 d2y

dx2+ (1 + x) dydx + y = sin[2 log(1 + x)].

8. Solve (2 + 3x)2 d2y

dx2+ 3(2 + 3x) dydx 36y = 3x2 + 4x+ 1.

9. Solve dxdt + y = sin t;dydt + x = cos t Given that x = 2 and y = 0 when

t = 0.

10. Solve Dx+Dy + 3x = sin t;Dx+ y x = cos t

QUESTION BANK 1 MATHEMATICS II

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UNIT II - VECTOR CALCULUS

PART A

1. Find .(rr )2. Prove that (rn) = nrn2r3. Find the unit normal vector to the surface x2 + xy + z2 = 4 at the

point (1,1, 2).4. What is the greatest rate of increase of = xy2z at (1, 0, 3)?

5. The temperature at a point (x, y, z) in space is given by T (x, y, z) =x2 + y2 z. A mosquito located at (4, 4, 2) describes to fly in sucha direction that it gets cooled faster. Find the direction in which itshould fly.

6. If = yzi + xzj + xyk , then find .7. Write down in orthogonal curvilinear co-ordinates.8. For what value of k is the vector rkr solenoidal?9. Determine f(r) so that the vector f(r)r is solenoidal.10. Find a such that F = (3x2y+z)i +(4x+ayz)j +(xy+2z)k

PART B

1. If r is the position vector of the point P (x, y, z), prove (rn) =nrn2r where r = |r |.

2. Find the directional derivative of = xy2z3 at the point (1, 1, 1) alongthe normal to the surface x2 + xy + z2 = 3 at the point (1, 1, 1).

3. Find the directional derivative of = 3x2+2y+z2 at the point (1, 1, 1)in the direction 2i + 2j k .

4. Find a and b such that the surface ax3 by2z = (a + 3)x2 and4x2y z3 = 11 may cut orthogonally at (2,1,3).

5. Prove that divcurlF = 0.6. Show that F = (6xy+z3)i +(3x2z)j +(3xz2y)k is irrotational

vector and find the scalar potential function F = .




7. Apply Greens Theorem in the plane to evaluateC(3x

2 8y2)dx +(4y 6xy)dy where C is the boundary of the region defined by x =0, y = 0, x+ y = 1.

8. Apply Greens Theorem in the plane to evaluateC(3x

2 8y2)dx +(4y 6xy)dy where C is the boundary of the region defined by y =x, y = x2.

9. Verify Gausss divergence theorem for F = 4xzi y2j +yzk takenover the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.

10. Verify Stokes theorem for F = (x2 y2)i 2xyj taken over thecube bounded by x = a, x = 1, y = 0, y = b.




UNIT III - ANALYTIC FUNCTIONS

PART A

1. What is the necessary conditions for the existence of the derivative off(z)?

2. If w = log z, then determine where w is non analytic.

3. Define singular point of the function with an example.

4. Show that f(z) = xy + iy is not analytic.

5. Define holomorphic function.

6. Show that v = ex sin y is harmonic function.

7. Show that v = ex cos y is harmonic function.

8. Find the invariant points of the transformation w = z1z+1 .

9. Under the transformation w = 1z , find the image of |z 2i| = 2.10. Show that the transformation w = 1z transforms all circles and straight

lines to circles and straight lines in the w-plane.

11. State two important properties of Mobius transformation.

PART B

1. If f(z) is a regular function of z prove that(

2

x2+

2

y2

)|f(z)|2 =

4|f (z)|2.

2. If f(z) is an analytic function of z, prove that(

2

x2+

2

y2

)log |f(z)| =

0.

3. Show that the function f(z) =|xy| is not analytic at the origin even

though C-R equations are satisfied.

4. Determine the analytic function whose real part is sin 2x(cosh 2ycos 2x) .

5. Determine the analytic function whose real part is e2x(x cos 2yy sin 2y).6. Determine the analytic function whose imaginary part is ex sin y.




7. Determine the analytic function whose imaginary part is ex(x sin y y cos y).

8. find the analytic function z = u+ iv, if u v = xyx2+4xy+y2

9. Find the bilinear transformation that maps the points 1 + i,i, 2 iof the z-plane into the points 0, 1, i of the w-plane

10. Find the bilinear transformation that maps the points i,1, 1 of thez-plane into the points 0, 1, of the w-plane




UNIT IV - COMPLEX INTEGRATION

PART A

1. State Cauchys Theorem.

2. EvaluateC(za)1dz where C is a simple closed curve and the point

z = a is (i) outside C, (ii) inside C.

3. Prove thatC(z a)ndz = 0, [n 6= 1] where C is the circle |z a| = r.

4. Find the Taylors series expansion of 1z2 in |z| < 1.5. Find the Taylors series expansion of sin z in about z = pi4 .

6. Find the poles of f(z) = (z1)2

z(z2)2 .

7. Find the zeros of f(z) = (z2+1)2

(1z2) .

8. What are the critical points of the transformation w = z + 1z .

9. State Cauchys integral formula.

10. State Cauchys Residue formula.

PART B

1. Find the Laurents series of f(z) 1z(1z) valid in the region (i)|z+1| < 1,(ii)1 < |z + 1 < 2| and (iii)|z + 1| > 2.

2. Find the Laurents series of f(z) z21

z2+5z+6valid in the region (i)|z| < 2,

(ii)2 < |z| < 3| and (iii)|z| > 3.3. Find the Taylors series expansion of f(z) = 1z(1z) about z = 1.

State the region of convergence of the series.

4. Find the Taylors series expansion of f(z) = z(z+1)(z+2) about z = i andstate the region of convergence.

5. Using Cauchys residue theorem evaluateC

zdz(z1)(z2)2 where C is the

circle |z 2| = 12 .

6. Using Cauchys residue theorem evaluateC

(3z2+z)dz(z1)(z2+9) where C is the

circle |z 2| = 2.




7. Evaluate

x2dx(x2+a2)(x2+b2)

using contour integration, where a > b > 0.

8. Evaluate0

dx(x2+a2)3

using contour integration, where a > 0.

9. Evaluate

cosxdx(x2+a2)(x2+b2)

using contour integration, where a > b > 0.

10. Evaluate0

x sinxdx(x2+a2)

using contour integration.




UNIT V - LAPLACE TRANSFORM

PART A

1. Define Laplace transform.

2. State the change of scale property in Laplace transformation.

3. State the first shifting property in Laplace transformation.

4. State the second shifting property in Laplace transformation.

5. Find the Laplace of unit step function.

6. Find the Laplace of unit impulse function.

7. Find the Laplace transforms of

(a) sin 2t sin 3t

(b) cos2 2t

(c) sin3 2t

(d) e3t2 cos 5t 3 sin 5t(e) e3t sin2 t

(f) t cos at

(g) t2 sin at

(h) 1et

t

(i) t3e3t

(j) tet sin 3t(k) sin kt kt cos kt(l) sin att

(m) sin2 tt

8. Find L{f(t)}, if f(t) ={

sin 2t, 0 < t < 2pi ;0, t > pi.

9. Find L(t) and L 1

pit.

10. Find L1(F (s)) if F (s) is :

(a) 1(s+1)

32




(b) s2+2s+3

s3

(c) s(s4)5

(d) 1(3s4)4

(e) 2s+3s2+4

(f) 1s(s+a)(g) s3

s26s+10(h) 1

s2+2s+5

(i) log s+1s1(j) cot1 s

(k) log s2+1s2+4

PART B

1. Find the Laplace transforms of

(a) e3t(2t+ 3)3

(b) et sin 3t cos t

(c) tet cos t

(d) te2t sin 3t

(e) t cos 2t

(f) (t sin t)2

(g) (1 + tet)3

(h) cosh at cos at

2. Find the inverse Laplace transforms of

(a) ss2+4

(b) 1s3+1

(c) 1s4+4a4

(d) s2

s4+64

(e) s2+1

s4+s2+1

(f) s+1s2+s+1

(g) log(1 + a2

s2)




(h) tan1(s2)

(i) tan1( s+23 )

3. Use convolution theorem to find the inverse Laplace transforms of thefollowing functions:

(a) 1(s+1)(s+2)(b) s

(s2+a2)2

(c) s2

(s2+a2)2(s2+b2)2

(d) 1(s4+4)

4. Solve y 4y + 8y = e2t, y(0) = 2, y(0) = 25. Solve y + 4y = sint, y(0) = 0, y(0) = 0

6. Solve y + 9y = cos 2t, y(0) = 1, y(pi2 ) = 1

7. Solve the simultaneous differential equations dydt + 2x = sin 2t anddydt + 2x = cos 2t, x(0) = 1, y(0) = 0.

8. Solve y + y = t cos 2t, y(0) = y(0) = 0

9. Solve y + 4y = cos 2t, y(pi) = y(pi) = 0

10. Solve x 2x + x = t2et, x(0) = 2, x(0) = 3



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