M2 Important Questions

MA6251 - MATHEMATICS – II QUESTION BANK

UNIT - I VECTOR CALCULUS

1. (a) Find the angle between the normals to the surface 2 xy = z at points (1,4,2) and (-3 ,-3, 3). BTL-1
(b) Verify and Estimate using Gauss divergence theorem for → → → → F = x i + y j+ z k 2 2 2 taken over the cube bounded by the planes . BTL-2
2. (a) Find the value of such that the vector → r r n is both solenoidal and irrotational. BTL-1
(b) Verify and Estimate using Stokes theorem for → → → F = (x − y ) i + 2xy j 2 2 in the rectangular region of plane bounded by the lines BTL-2
3. (a) Find its scalar potential, if the vector field → → → F = (x + xy ) i + ( y + x y) j 2 2 2 2 is irrotational. BTL-1
(b) Show that Stokes theorem is verified for where S is the surface bounded by the planes above the xy-plane. BTL-3
4. (a) Find the values of a and b so that the surfaces 3 2 2 ax − by z = (a + 3)x and 4 11 2 3 x y − z = may cut orthogonally at (2,-1,-3). BTL-2
(b) Verify and Analyse Guass divergence theorem for → → → → F = xz i − y j+ yz k 2 4 taken over the cube bounded by the planes BTL-4
5. (a) Prove that 2 2 ( ) ( 1) − ∇ = + n n r n n r BTL-3
(b) Using Stokes theorem, evaluate → → ∫ F.d r where → → → → F = y i + x j− (x + z) k 2 2 where C is the boundary of the triangle with vertices (0,0,0),(1,0,0) & (1,1,0). BTL-5
6. (a) Verify and Estimate using divergence theorem for over the cube formed by the planes BTL-2 (b) Show that → → → → F = (2xy − z ) i + (x + 2yz) j+ ( y − 2zx) k 2 2 2 is irrotational and hence find its scalar potential. BTL-3
7. (a) Verify Green’s theorem in the plane for [(3 8 ) (4 6 ) ] 2 2 x y dx y xy dy c − + − ∫ where C is the boundary of the region bounded by x = 0, y = 0, x + y = 1. BTL-4
b. Show that → → → → F = ( y + 2xz ) i + (2xy − z) j+ (2x z − y + 2z) k 2 2 2 is irrotational and hence find and formulate its scalar potential. BTL-6
8. (a) Find by Stoke’s theorem for over the open surfaces of the cube not included in the XOY plane. BTL-1 (b) If and , prove that and . BTL-5
9. (a) Verify by Green’s theorem and find where C is the square bounded by . BTL-1
(b) Evaluate where and S is the closed surface of the sphere . BTL- 6
10. (a) Find the work done in moving a particle in the vector field → → → → F = ( y + 3) i + xz j+ ( yz − x) k along the curve from (0,0,0) to (2,1,1). BTL-1
(b) Evaluate ∫ − + + c (2x y )dx (x y )dy 2 2 2 2 where C is the square bounded by the lines x = 0, x = 2, y = 0 and y =3 by Green’s theorem. BTL-4

UNIT – II ORDINARY DIFFERENTIAL EQUATIONS

1.(a) Identify the solution of BTL-1
(b) Using the method of variation of parameter to Evaluate (D2 +1) y = x sinx. BTL-2
2. (a) Identify the solution of ( 4 13) sin 3 ( 9). 2 2 2 D − D + y = e x + x + x + x BTL-1
(b) Using the method of variation of parameter to Evaluate (D2 + 25) y = sec5x. BTL-2
3. (a) Identify the solution of ( ) x D D y x e 3 2 − 7 − 6 = (1+ ) . BTL-1
(b) Solve y y y e x x ''−2 '+ = log , Using the method of variation of parameters. BTL-3
4. (a) Give the complimentary function, particular integral of( 3 2) cos . 2 D − D + y = x x . BTL-2 (b) Using method of variation of parameters find the solution of BTL-4
5. (a) Solve ( ) ( ) . 1 1 sin log 2 2 x x D − xD+ y = x x + BTL-3
(b) Evaluate the simultaneous equations. + 2 −3 = 5 , −3x + 2y = 0 dt dy x y t dt dx BTL-5 given that x() () 0 = 0, y 0 = −1. BTL-1
6. (a) Give the general solution of sin( ) log . 2 2 2 2 y x dx dy x dx d y x + + = BTL-2 (b) Solve: 2 sin2 , 2x cos 2t. dt dy y t dt dx + = − = BTL-3
7. (a) Find the solution of ( ) 2 3 ( ) 2 3 12 6 . 2 2 2 y x dx dy x dx d y x+ − + − = BTL-4 (b) Formulate the ODE and hence solve BTL-6
8. (a) Identify the solution of ( ) 1 ( ) 1 4 cos[ ] log( ) 1 . 2 2 2 y x dx dy x dx d y + x + + + = +
(b) Evaluate the general solution of y = BTL-5
9. (a) Identify the solution of D2 x - 5x + 3y = sin t, D2 y +5y -3x = t BTL-1 .
(b) Formulate the ODE and hence solve (5 + 2x)2 y ’’ - 6 (2x + 5) y’ + 8y = 6x. BTL-6
10.(a) Identify the solution of Dx – y = t, Dy + x =1. BTL-1
(b) Find the solution of ((x + 3) 2 D2 - 4(x+3) D + 6 )y = log (x+3) BTL-4

UNIT – III LAPLACE TRANSFORMS

UNIT- IV ANALYTIC FUNCTIONS

1. (a) Describe the real and imaginary parts of an analytic function w = u+iv satisfy the Laplace equation in two dimension. via 0 0 2 2 ∇ u = and ∇ u = . BTL-1
(b) Given that , Estimate the analytic function whose real part is u. BTL-2
2. (a) Describe an analytic function with constant modulus is constant. BTL-1 (b) Estimate the analytic function w=u + iv if = BTL-2
3. (a) Identify the image of the infinite strip 2 1 4 1 ≤ y ≤ under the transformation w = 1/z . BTL-1 (b) If w = f(z) is analytic then Show that log = 0. BTL-3
4. (a) Estimate the analytic function f(z) = u+ iv given the imaginary part is v= x2 - y 2 . BTL-2 (b) Point out the bilinear transformation that maps the point =-1 into the points respectively. BTL-4
5. (a) If f (z) is a regular function of z, Show that 2 2 2 ∇ f (z) = 4 f ′(z) . BTL-3
(b) Test whether w = maps the upper half of the z – plane to the upper half of the w-plane and also find the image of the unit circle of the z- plane. BTL-5
6. (a) Give the bilinear transformation which maps z=1,0,-1 into w=0,-1, respectively. What are the invariant points of the transformation? BTL-2
(b) Show that the function u(x,y)= is harmonic . Fins also the conjugate harmonic function v. BTL-3
7. (a) If w = u(r, θ) + i v(r, θ) an analytic function , the curves of the family u(r, θ) = a cut orthogonally the curves of the family v(r,θ) = b where a and b are arbitrary constants. BTL-4
(b) Formulate the image of |z +1| = 1 under the map w = 1/z. BTL-6
8. (a) Identify the bilinear transformation that maps 1+I, -i, 2-i, at the z-plane into the points 0, 1, i, of the w-plane. BTL-1
(b) Test that under the mapping i z i z w + − = the image of the circle x2 + y2 < 1 is the entire half of the w – plane to the right of the imaginary axis. BTL-5
9. (a) Identify the bilinear mapping which maps -1,0,1 of the z-plane onto -1,i, 1 of the w-plane . Show that under this mapping the upper half of z- plane maps onto the interior of unit circle .BTL-1
(b) If f (z) = u + iv is an analytic function of z, then formulate that [log ( )] 0 2 ∇ f ′ z = . BTL-6
10. (a) The harmonic function ‘u’ satisfies the formal differential equation 0 2 _ 2 = ∂ ∂ ∂ z z u and identify that log | f ’(z) | is harmonic, where f(z) is a regular function. BTL-1
(b) Point out that 2 2 2 ∇ Re f (z) = 2 f ′(z) BTL-4

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