Mathematics 2 PDF Past Paper 2022-23

# Even Semester Examination 2022 - 23 ## 1^st Year, B.Tech ### Mathematics - II #### Duration: 3:00 hrs #### Max Marks: 100 Note: - Attempt all questions. All Questions carry equal marks. In case of any ambiguity or missing data, the same may be assumed and state the assumption made in t...

# Even Semester Examination 2022 - 23 ## 1^st Year, B.Tech ### Mathematics - II #### Duration: 3:00 hrs #### Max Marks: 100 Note: - Attempt all questions. All Questions carry equal marks. In case of any ambiguity or missing data, the same may be assumed and state the assumption made in the answer. ## **BAST-105** ### Q1 Answer any four parts of the following: * Solve (x+1)dy - y = e^x(x+1)² * Solve d²y/dx²+dy/dx+ y = cos2x. * Find the value of λ for which the differential equation (xy²+2x²y)dx+(x+y)x²dy = 0 is exact. Solve the equation for this value of λ. * Solve x³d³y/dx³+2x²d²y/dx²+ 2y = 10(x + 2). * Solve d²y/dx²+4x²y = x⁴. * Apply the method of variation of parameter to solve d²y/dx² - y = 1/(1+e^x). ### Q2 Answer any four parts of the following: * Solve x²d²y/dx²+ 6xdy/dx + 6y = xlogx. * Apply the method of variation of parameter to solve d²y/dx²+2dy/dx + y = secx. * By the removal of first derivative, solve d²y/dx² - n²y/x = 0. * Using changing the independent variable method, solve d²y/dx²+ Cotx + 4yCosec²x = 0. * For Legendre's functions, prove that ∫_-1¹P_m(x)P_n(x)dx = 0 for m≠n. * For Bessel's function, prove that xJ'_n(x) = nJ_n(x) - xJ_{n+ 1}(x). ### Q3 Answer any two parts of the following: * Change the order of integration and evaluate ∫₀²∫₀^2-xxydxdy. * Evaluate ∫∫_sA.nds where A = (x+y²)î - 2xĵ + 2yzk and S is the surface of the plane 2x + y + 2z = 6 in the first octant. * Apply Gauss Divergence theorem to evaluate ∫∫_sF.nds, where F = 4x³i - x²yj + x²zk and S is the surface of the cylinder x²+y² = a² bounded by the planes z = 0 and z = b. ### Q4 Answer any two parts of the following: * If u = e^x(xcosy - ysiny) is a harmonic function, find an analytic function f(z) = u + iv such that f(1) = e. * Find the values of a and b such that the function f(z) = x² + ay² - 2xy + i(bx² - y² + 2xy ) is analytic. * Show that the function v= sinhx cosy is harmonic and find its harmonic conjugate. ### Q5 Answer any two parts of the following: * Find Laurent's series expansion of (z² - 6z - 1)/((z - 1)(z - 3)(z + 2)) valid in the region 3 < |z + 2| < 5. * Evaluate the integrals using Cauchy integral formula: * ∫_c(4 - 3z)/z(z - 1)(z - 2)dz where |z| = 3. * ∫_c(3z² + z + 1)/((z² - 1)(z + 3))dz where |z| = 2. * Using contour integration to solve ∫₀^2π(a/a+bcose)dθ, a> |b|. ## **BSCT-201** ### Q1 Answer any four parts of the following: * Solve the differential equation cos3x + y cosx = sinx. * Solve (e^x + 1)cosx dx + e^xsinx dy = 0. * Solve p² + 2pycotx - y² = 0. * Solve d/dx(y) + logy = (logy)². * Solve y = 2px + yp². ### Q2 Answer any four parts of the following: * Solve x²d²y/dx²-3xdy/dx + 5y = x²Sin(logx). * Using the method of variation of parameter, solve d²y/dx²+ y = secx. * Apply the method of changing the independent variable to solve d²y/dx²+ Cotx+4yCosec²x = 0. * Express f(x) = 4x³ + 6x² + 7x + 2 in terms of Legendre's polynomials. * Prove that d/dx[x^-nJ_n(x)] = -x^-nJ_{n+ 1}(x). ### Q3 Attempt any two parts of the following: * Change the order of integration and evaluate ∫₀²∫₀^2-xxydxdy. * Evaluate ∫∫_s(yz î + zxj + xyk).nds where S is the surface of the sphere x² + y² + z² = a² in the first octant. * Use Stoke’s theorem to evaluate ∫_c[(x + 2y)dx + (x - 2)dy + (y – z)dz, where C is the boundary of the triangle with vertices (2, 0, 0), (0,3, 0)and (0,0,6) oriented in the anti-clockwise direction. ### Q4 Attempt any two parts of the following: * Determine p such that the function f(z) = log(x² + y²) + i tan^-1(y/x) is an analytic function. * Determine the analytic function f(z) = u + iv such that u v = e^x(cosy - siny). * State and prove cauchy - Riemann equations for cartesian coordinate system. ### Q5 Attempt any two parts of the following: * Evaluate ∫_c(1/z(z+iπ))dz where C is |z+3i| = 1. * Evaluate ∫_c(1/z(z-1)(z-2))dz along the path y = x², for |z - 1| < 1. * Expand the function f(z) = 1/(z² - 1) in a Laurent’s series. * Using Cauchy Residue theorem, evaluate ʃ_c(1/((z - 1)²(z + 2)))dz where C is |z| = 3. ## **TMA-201** ### Q1. Answer any four parts of the following: * Solve (x+1)dy/dx-y = e^x(x+1)². * Solve d²y/dx²+dy/dx+ y = cos2x. * Find the value of λ for which the differential equation (xy²+2x²y)dx+(x+y)x²dy = 0 is exact. Solve the equation for this value of λ. * Solve x³d³y/dx³+2x²d²y/dx²+ 2y = 10(x + 2). * Solve d²y/dx²+4x²y = x⁴. * Apply the method of variation of parameter to solve d²y/dx² - y = 1/(1+e^x). ### Q2 Answer any four parts of the following: * Solve x²d²y/dx² + 6xdy/dx + 6y = xlogx. * Apply the method of variation of parameter to solve d²y/dx²+2dy/dx + y = secx. * By the removal of first derivative, solve d²y/dx² - n²y/x = 0. * Using changing the independent variable method, solve d²y/dx²+ Cotx + 4yCosec²x = 0. * For Legendre’s functions, prove that ∫_-1¹P_m(x)P_n(x)dx = 0 for m≠n. * For Bessel's function, prove that xJ'_n(x) = nJ_n(x) - xJ_{n+ 1}(x). ### Q3. Answer any two parts of the following: * Change the order of integration and evaluate ∫₀²∫₀^2-xxydxdy. * Evaluate ∫∫_sA.nds where A = (x+y²)î - 2xĵ + 2yzk and S is the surface of the plane 2x + y + 2z = 6 in the first octant. * Apply Gauss Divergence theorem to evaluate ∫∫_sF.nds, where F = 4x³i - x²yj + x²zk and S is the surface of the cylinder x²+y² = a² bounded by the planes z = 0 and z = b. ### Q4. Answer any two parts of the following: * If u = e^x(xcosy - ysiny) is a harmonic function, find an analytic function f(z) = u + iv such that f(1) = e. * Find the values of a and b such that the function f(z) = x² + ay² - 2xy + i(bx² - y² + 2xy ) is analytic. * Show that the function v= sinhx cosy is harmonic and find its harmonic conjugate. ### Q5 Answer any two parts of the following: * Solve (D³ – 7DD'² – 6 D'³)z = sin(x + 2y) + e^2x+y. * If a string of length I is initially at rest in equilibrium position and each of its point is given the velocity b sin(πx/l) find the displacement y(x,t). * An insulated rod of length l has its ends A and B maintained at 0°C and 100°C respectively until steady state conditions prevail. If B is suddenly reduced to 0°C and maintained at 0°C, find the temperature at a distance x from A at time t. ## **Special** ## **BAST-105** ### Q1 Attempt any four parts of the following: * Solve xdy/dx+y/xlogx = 2logx. * Find the value of λ for which the differential equation (xy²+2x²y)dx+(x+y)x²dy = 0 is exact. Solve the equation for this value of λ. * Solve the differential equation (D²-5D+6)y = e^x. * Solve x²dy/dx-3x+5y = xlogx. * Determine the solution of the differential equation p² + 2/y = 0. ### Q2 Attempt any four parts of the following: * Form the partial differential equation from 2z = (x²+y²)/x. * Solve ∂²z/∂x² = ex+2y. * Solve the partial differential equation p² + q² = 2. * Form the partial differential equation from z = f(x² - y²). * Determine the solution of p² + q² = x + y. ### Q3 Attempt any two parts of the following: * State and prove the orthogonal properties of Legendres polynomials. * Apply the method of variation of parameter to solve d²y/dx² +y = cosecx. * Apply the method of changing the independent variable to solve d²y/dx² +4x²y = x⁴. ### Q4 Attempt any two parts of the following: * Test the convergence of the series 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ... . * Test the convergence of the series ∑_n=1^∞(2n²+3n)/(n⁵+5n). * Find the Fourier series expansion for the function f(x) = xsinx, -π < x < π. * If f(x) = {x, 0≤x≤π, then show that: * f(x) = (4/π)[sinx - sin3x + sin5x + ...]. * f(x) = (4/π)[cos2x + (1/3)cos6x + (1/5)cos10x + ...]. ### Q5 Attempt any two parts of the following: * Determine p such that the function f(z) = log(x² + y²) + i tan^-1(y/x) is an analytic function. * Prove that the function u = e^x(xcosy - ysiny) satisfies Laplace equation. * Differentiate between Cauchy Goursat Theorem and Cauchy Integral Formula. * Evaluate ∫_c(1/(z² + 1))dz, where C is the circle |z| = 2. * Using contour integration method to evaluate ∫₀^2π(1/(a+bcose))dθ, a> |b|. ## **Special** ## **BSCT-201** ### Q1. Attempt any four parts of the following: * Solve the differential equation cos3x + y cosx = sinx. * Solve (e^x + 1)cosx dx + e^xsinx dy = 0. * Solve p² + 2pycotx - y² = 0. * Solve d/dx(y) + logy = (logy)². * Solve y = 2px + yp². ### Q2. Attempt any four parts of the following: * Solve x²d²y/dx²-3xdy/dx + 5y = x²Sin(logx). * Using the method of variation of parameter, solve d²y/dx²+ y = secx. * Apply the method of changing the independent variable to solve d²y/dx²+ Cotx+4yCosec²x = 0. * Express f(x) = 4x³ + 6x² + 7x + 2 in terms of Legendre’s polynomials. * Prove that d/dx[x^-nJ_n(x)] = -x^-nJ_{n+ 1}(x). ### Q3. Attempt any two parts of the following: * Change the order of integration and evaluate ∫₀²∫₀^2-xxydxdy. * Evaluate ∫∫_s(yz î + zxj + xyk).nds where S is the surface of the sphere x² + y² + z² = a² in the first octant. * Use Stoke’s theorem to evaluate ∫_c[(x + 2y)dx + (x - 2)dy + (y – z)dz, where C is the boundary of the triangle with vertices (2, 0, 0), (0,3, 0)and (0,0,6) oriented in the anti-clockwise direction. ### Q4. Attempt any two parts of the following: * Determine p such that the function f(z) = log(x² + y²) + i tan^-1(y/x) is an analytic function. * Determine the analytic function f(z) = u + iv such that u v = e^x(cosy - siny). * State and prove cauchy - Riemann equations for cartesian coordinate system. ### Q5. Attempt any two parts of the following: * Evaluate ∫_c(1/z(z+iπ))dz where C is |z+3i| = 1. * Evaluate ∫_c(1/z(z-1)(z-2))dz along the path y = x², for |z - 1| < 1. * Expand the function f(z) = 1/(z² - 1) in a Laurent’s series. * Using Cauchy Residue theorem, evaluate ʃ_c(1/((z - 1)²(z + 2)))dz where C is |z| = 3. ## **Special** ## **TMA-201** ### Q1. Answer any four parts of the following: * Solve (x+1)dy/dx-y = e^x(x+1)². * Solve d²y/dx²+dy/dx+ y = cos2x. * Find the value of λ for which the differential equation (xy²+2x²y)dx+(x+y)x²dy = 0 is exact. Solve the equation for this value of λ. * Solve x³d³y/dx³+2x²d²y/dx²+ 2y = 10(x + 2). * Solve d²y/dx²+4x²y = x⁴. * Apply the method of variation of parameter to solve d²y/dx² - y = 1/(1+e^x). ### Q2 Answer any four parts of the following: * Solve x²d²y/dx² + 6xdy/dx + 6y = xlogx. * Apply the method of variation of parameter to solve d²y/dx²+2dy/dx + y = secx. * By the removal of first derivative, solve d²y/dx² - n²y/x = 0. * Using changing the independent variable method, solve d²y/dx²+ Cotx + 4yCosec²x = 0. * For Legendre’s functions, prove that ∫_-1¹P_m(x)P_n(x)dx = 0 for m≠n. * For Bessel's function, prove that xJ'_n(x) = nJ_n(x) - xJ_{n+ 1}(x). ### Q3. Answer any two parts of the following: * Change the order of integration and evaluate ∫₀²∫₀^2-xxydxdy. * Evaluate ∫∫_sA.nds where A = (x+y²)î - 2xĵ + 2yzk and S is the surface of the plane 2x + y + 2z = 6 in the first octant. * Apply Gauss Divergence theorem to evaluate ∫∫_sF.nds, where F = 4x³i - x²yj + x²zk and S is the surface of the cylinder x²+y² = a² bounded by the planes z = 0 and z = b. ### Q4. Answer any two parts of the following: * If u = e^x(xcosy - ysiny) is a harmonic function, find an analytic function f(z) = u + iv such that f(1) = e. * Find the values of a and b such that the function f(z) = x² + ay² - 2xy + i(bx² - y² + 2xy ) is analytic. * Show that the function v= sinhx cosy is harmonic and find its harmonic conjugate.