BTL What is the value of m if the vector is solenoidal. BTL What is the value of a, b, c if the vector may be irrotational.

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1 VALLIAMMAI ENGINEERING OLLEGE SRM NAGAR, KATTANDKULATHUR Department of Mathematics MA65 - MATHEMATIS II QUESTION BANK - 6 UNIT - I VETOR ALULUS Part - A. Find, if at (, -, ). BTL-. Find the Directional derivative of at (,,)in the direction. BTL-. Find Directional derivative of at (,-,) in the direction. BTL-. State Gauss Divergence theorem. BTL- 5. State Stokes theorem. BTL- 6. State Greens theorem. BTL- 7. Give the unit normal vector to the surface at (,, ). BTL- 8. Give the unit normal vector to the surface at (,,). BTL- 9. If., Give at (,,). BTL-. If is the position vector, Give. BTL-. Show that. BTL-. Show that the vector is solenoidal. BTL-. Show that. BTL-. If, evaluate e from (,,) to (,,) along the curve. BTL- 5. If, evaluate from (,) to (,) along the path BTL- 6. Using Green s theorem evaluate where is the boundary of the square enclosed by the lines. BTL- 7. Use Stokes theorem to evaluate where and is the curve BTL-5 8. Evaluate Using Gauss Divergence theorem for taken over the cube. BTL-5 9. What is the value of m if the vector is solenoidal. BTL-6. What is the value of a, b, c if the vector may be irrotational. BTL-6 Part - B. (a) Find the angle between the normals to the surface (b) Verify and Estimate using Gauss divergence theorem for cube bounded by the planes. (a) Find the value of such that the vector r n r xy = at points (,,) and (-,-, ). BTL- F = x i + y j+ k. BTL- is both solenoidal and irrotational. BTL- taken over the (b) Verify and Estimate using Stokes theorem for F = ( x y ) i + xy j in the rectangular region of plane bounded by the lines BTL-. (a) Find its scalar potential, if the vector field (b) Show that Stokes theorem is verified for F = ( x + xy ) i + ( y + x y) j is irrotational. BTL- where S is the surface

2 bounded by the planes. (a) Find the values of a and b so that the surfaces orthogonally at (,-,-). BTL- (b) Verify and Analyse Guass divergence theorem for bounded by the planes n n 5. (a) Prove that ( r ) = n( n + ) r BTL- above the xy-plane. BTL- ax by = ( a + ) x and x y = may cut F = x i y j+ y k taken over the cube BTL- (b) Using Stokes theorem, evaluate F. d r where F = y i + x j ( x + ) k where is the boundary of the triangle with vertices (,,),(,,) & (,,). BTL-5 6. (a) Verify and Estimate using divergence theorem for over the cube formed by the planes BTL- (b) Show that F = (xy ) i + ( x + y) j+ ( y x) k is irrotational and hence find its scalar potential. BTL- 7. (a) Verify Green s theorem in the plane for [(x 8y ) dx + (y 6xy) dy] where is the boundary of c the region bounded by x =, y =, x + y =. BTL- b. Show that F = ( y + x ) i + (xy ) j+ (x y + ) k formulate its scalar potential. BTL-6 is irrotational and hence find and 8. (a) Find by Stoke s theorem for over the open surfaces of the cube not included in the XOY plane. BTL- (b) If and, prove that and. BTL-5 9. (a) Verify by Green s theorem and find where is the square bounded by. BTL- (b) Evaluate where and S is the closed surface of the sphere. BTL- 6. (a) Find the work done in moving a particle in the vector field F = ( y + ) i + x j+ ( y x) k along the curve from (,,) to (,,). BTL- (b) Evaluate (x y ) dx + ( x + y ) dy where is the square bounded by the lines x =, x =, c y = and y = by Green s theorem. BTL- UNIT II ORDINARY DIFFERENTIAL EQUATIONS Part A. Find the P.I of ( D ) y = sinh x. BTL-. Find the P.I of ( D ) y = sin x sin x. + BTL-. Find the P.I of. BTL-. Find the particular Integral for ( D D ) y = x. 5. Find the P.I of ( D ) y = x. + BTL- x 6. Find the P.I of ( D D+ 5) y = e cosx. + BTL- + BTL- 7. Estimate the P.I of. BTL- 8. Estimate the P.I of. BTL- 9. Estimate the P.I of. BTL-. Estimate the P.I of. BTL-. Solve ( D ) y =. BTL-

3 . Solve Dx = -wy ; Dy = wx. BTL-. Solve Dx + y =, x Dy = t. BTL-. Point out the P.I of. BTL- 5. Point out the general solution for the method of variation of parameters for second order differential equation. BTL- 6. Point out the P.I of BTL- 7. Test whether the equation is linear equation with constant coefficients if not convert. BTL-5 8. Use the method of variation of parameters and evaluate. BTL-5 9. Rewrite the equation into the linear equation with constant coefficients. BTL-6. Rewrite the equation into the linear equation with constant coefficients. BTL-6 Part B. (a) Identify the solution of BTL- (b) Using the method of variation of parameter to Evaluate (D +) y = x sinx. BTL- x. (a) Identify the solution of ( D D + ) y = e sin x + ( x + x + 9). BTL- (b) Using the method of variation of parameter to Evaluate (D + 5) y = sec5x. BTL- 7 D 6 y = ( + x) e x. (a) Identify the solution of ( ) (b) Solve D. BTL- y' ' y' + y = e x logx, Using the method of variation of parameters. BTL-. (a) Give the complimentary function, particular integral of ( D D + ) y = x cos x.. BTL- (b) Using method of variation of parameters find the solution of BTL- BTL- x 5. (a) Solve ( x D xd+ ) y = xsin( logx) +. dx dy (b) Evaluate the simultaneous equations. + x y = 5 t, x + y = dt dt given that x ( ) =, y( ) =. BTL- BTL-5 d y dy 6. (a) Give the general solution of x + x + y = sin( log x ). BTL- dx dx dx dy (b) Solve: + y = sint, x = cos t. BTL- dt dt d y dy + BTL- dx dx 7. (a) Find the solution of ( x ) ( x + ) y = 6x. (b) Formulate the ODE and hence solve BTL-6 d y dy + [ ]. BTL- dx dx 8. (a) Identify the solution of ( x ) + ( + x) + y = cos log( + x)

4 (b) Evaluate the general solution of y = BTL (a) Identify the solution of D x - 5x + y = sin t, D y +5y -x = t BTL- (b) Formulate the ODE and hence solve (5 + x) y - 6 (x + 5) y + 8y = 6x. BTL-6.(a) Identify the solution of Dx y = t, Dy + x =. BTL- (b) Find the solution of ((x + ) D - (x+) D + 6 )y = log (x+) BTL- UNIT III LAPLAE TRANSFORMS Part - A. State the sufficient conditions for the existence of Laplace transform. BTL-. State first and second shifting theorem. BTL-. State and prove change of scale property BTL-. State Initial value and final value theorem. BTL- 5. State onvolution theorem BTL- t 6. Tell whether L cos exist? Justify. BTL- t 7. Give the proof of L[t f(t)]= -φ (s) if L[f(t)]= φ(s). BTL- 8. Estimate L[t cost] BTL- at 9. Estimate L sin BTL- t cos at cosbt. Estimate L BTL- t. Apply the initial value theorem for the Verification of the function f(t) = e -t BTL-. Apply the final value theorem for the Verification of the function f(t) = e -t BTL-. If L[f(t)]= φ(s) and f(t) has a limit t then show that L[ f(t)] = ψ ( s ) ds BTL- t t. Find L [f (t)], if f (t)=, for t b and f ( t + π ) = f ( t) ), for all t. BTL-, for b < t < b s + 5. Find L BTL- s s + 6. Find L log s BTL- 7. Using Laplace transform, evaluate te t sin tdt BTL-5 s 8. Evaluate L BTL-5 s + 9 s + 9. Formulate L BTL-6 s. Formulate L s( s ) BTL-6 s

5 Part - B. (a) Identify the Laplace Transform of the function [t Sint os t] BTL- (b) Give the general solution of (D π + 9) y = cost, given that y () =, y =-. BTL- cos t. (a) Identify the Laplace Transform of the function t BTL- (b) Give the general solution of (D + D + ) y = e -t, given that y () =,y () =. BTL- t sin t t. (a) Identify the Laplace Transform of the function dt + t e cos t BTL- t (b) Solve y -y +y= t + when y () = -and y ()= using Laplace transforms. BTL- π sinωt, for < t <. (a) Estimate L[f(t)], if f(t)= ω π and f t + = f ( t), for all t. BTL- π π, for < t < ω ω ω (b) Using convolution theorem, find s L BTL- ( s + s + 5) 5. (a) Apply initial,final value theorem for the verification of the function f(t) = + e -t (sint + cost). BTL- (b) Using onvolution theorem, Evaluate L BTL-5 s( s + ) 6. (a) Give L [f(t)], if f(t)= t, for t c and f ( t + c) = f ( t) ), for all t. BTL- c t, for c< t < c (b) Find L [e -t t cost cost + cosht] BTL- 7. (a) Using convolution theorem, find L BTL- ( s + s + 5) (b) Using onvolution theorem calculate the inverse Laplace transform of s L ( s + )( s + ) BTL-6 8. (a) Identify the Laplace transform of the square- wave function of period a defined as f(t)=, when < t < a / BTL-, whena / < t < a (b) Evaluate cos at cosbt L BTL-5 t s 9. (a) Identify the Inverse Laplace transform of tan + cot BTL- s (b) Formulate,solve using Laplace transforms, (D +D) y = t +t, given that y=, y = - when t= BTL-6 s + 6s + 6. (a) Identify the Inverse Laplace transform of BTL- s( s + s + ) 5s + (b) Find L BTL- ( s + s + 5) ( s ) 5

6 UNIT- IV ANALYTI FUNTIONS Part-A. Examine if f()= analytic? Justify your Answer. BTL-. Identify the constants a,b,c if is analytic. BTL-. Define conformal mapping. BTL-. Identify the real and imaginary parts of. BTL- 5. State necessary and sufficient condition for artesian coordinates in auchy-riemann Equation BTL- 6. Identify the invariant point of the bilinear transformation BTL- 7. Estimate the invariant points of the transformation w = BTL Estimate the invariant point of the bilinear transformation BTL- 9. Give the image of the circle under the transformation w = 5. BTL-. Under the transformation give the image of the circle in the complex plane. BTL-. Show that is not analytic at any point. BTL-. Show that an analytic function in a region R with constant modulus is constant. BTL-. Show that is harmonic and determine its harmonic conjugate. BTL-. If f() is an analytic function whose real part is constant, Point out f() is a constant function. BTL- 5. Explain that a bilinear transformation has at most fixed points. BTL- 6. Find the fixed points of the transformation BTL- 7. Test the analyticity of the function BTL-5 8. Evaluate the image of hyperbola under the transformation BTL-5 9. Formulate the critical points of the transformation BTL-6. Formulate the bilinear transformation which maps =,-i,- into w = i,, respectively. BTL-6 Part-B. (a) Describe the real and imaginary parts of an analytic function w = u+iv satisfy the Laplace equation in two dimension. via u = and u =. BTL- (b) Given that, Estimate the analytic function whose real part is u. BTL-. (a) Describe an analytic function with constant modulus is constant. BTL- (b) Estimate the analytic function w=u + iv if = BTL-. (a) Identify the image of the infinite strip y under the transformation w = /. BTL- (b) If w = f() is analytic then Show that log =. BTL-. (a) Estimate the analytic function f() = u+ iv given the imaginary part is v= x - y. BTL- (b) Point out the bilinear transformation that maps the point respectively. BTL- 5. (a) If f () is a regular function of, Show that 6 f ( ) = f ( ). BTL- =- into the points

7 (b) Test whether w = maps the upper half of the plane to the upper half of the w-plane and also find the image of the unit circle of the - plane. BTL-5 6. (a) Give the bilinear transformation which maps =,,- into w=,-, respectively. What are the invariant points of the transformation? BTL- (b) Show that the function u(x,y)= is harmonic. Fins also the conjugate harmonic function v. BTL- 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- (b) Formulate the image of + = under the map w = /. BTL-6 8. (a) Identify the bilinear transformation that maps +I, -i, -i, at the -plane into the points,, i, of the w-plane. BTL- i (b) Test that under the mapping w = the image of the circle x + y < is the entire half of the i + w plane to the right of the imaginary axis. BTL-5 9. (a) Identify the bilinear mapping which maps -,, of the -plane onto -,i, of the w-plane. Show that under this mapping the upper half of - plane maps onto the interior of unit circle.btl- (b) If f () = u + iv is an analytic function of, then formulate that [ log f ( ) ] =. BTL-6. (a) The harmonic function u satisfies the formal differential equation u = _ and identify that log f () is harmonic, where f() is a regular function. BTL- (b) Point out that Re f ( ) = f ( ) BTL- UNIT- V OMPLEX INTEGRATION Part - A. State auchy s integral theorem. BTL-. Define isolated singularity BTL-. Identify the type of singularity of function Sin. BTL-. State Taylor s theorem. BTL- 5. State auchy s residue theorem and auchy s integral formula BTL- 6. Identify the value of e d, where is =? BTL- 7. Estimate the residue of the function f ( ) = at a simple pole. BTL- ( ) 8. Give the Laurent s series of f ( ) = valid in the region + < ( ). BTL- 9. Give the Laurent s series expansion of ( ) = e f ( ). Give the Taylor s series for f ( ) = Sin about 7 about =. BTL- π =. BTL- e. alculate the residue at = of f ( ) =. BTL-

8 . alculate the residue of the function f ( ) = at a simple pole. BTL- ( ) +. Determine the residues at poles of the function f ( ) =. BTL- ( )( ). Expand as Laurent s series about = in the annulus < <. BTL- ( ) 5. Obtain the expansion of log( + ) when <. BTL Expand the principal part and residue at the pole of the function f ( ) = BTL- ( + ) + 7. Evaluate d where is the circle = in the plane. BTL Evaluate d where is = using auchy s integral formula. BTL-5 πi 9. Integrate d + where is the circle Z =. BTL-6. Integrate BTL- 6 Part-B. (a) Identify the Laurent s series expansion for the function < < and > BTL- (b) Using contour integration estimate ( x a )( x +. (a) Identify the Laurent s series expansion for the function region < + <. BTL- dx + a ) (b) Estimate,( a > ) ( x f ( ) = in the regions ( )( ) x dx, a>, b >. BTL- + b ) Using ontour Integration. BTL-. (a) Identify the Laurent s series expansion of f() = f ( ) = in the ( + ) ( ) in the region Z < and < Z <. BTL- (b) Apply auchy s integral formula solve. (a) using auchy s residue theorem give the value of cosπ + sinπ d, is the circle =. BTL- ( )( ) + ( )( ) = + + (b) Find the Taylor s series to represent the function in <. ( )( ) d, BTL- BTL- 5. (a) Apply auchy s residue theorem, alculate the value of cos π + sin π ) ( + )( + = d, BTL- 8

9 π dθ (b) Evaluate a cosθ + a a <, Using ontour Integration. BTL-5 6. (a) Using Laurent s series, find f() = valid in ( i ) valid in Z + < ( ) (ii) < Z + < (iii) Z+ > BTL- + (b) Using auchy s integral formula calculate d where is the circle (i) + + i = (ii) + - i =. BTL- 7. (a) Point out the poles of ( ) ( + ) and hence evaluate ( ) ( + ) = e (b) Formulate ( ) d, using auchy s residue theorem. BTL-6 + π = 8. (a) Identify the Laurent s series of f() = d. BTL- + Z > and < Z - < BTL- ( )( ) d (b) Evaluate where is =. BTL-5 ( ) ( + ) (a) Identify the Taylor s series to represent the function. ( )( ) (b) Formulate π in dθ, using the method of contour integration. BTL-6 + 5sinθ (a) If f(a) = d where is the circle =, Identify f(), f(), ( a) f '( i), f ''( i). BTL- (b) Evaluate π + cosθ 5 cos d θ, using the method of contour integration. BTL- + θ < BTL- **** ALL THE BEST **** 9