Solve using Laplace transform: [ y'' + 4y = 8t, \quad y(0) = 0, \quad y'(0) = 2 ] (7 marks)
Find the volume of the sphere ( x^2 + y^2 + z^2 = a^2 ) using triple integration in spherical coordinates. (7 marks)
Max. Marks: 70
Find the radius of curvature for the curve ( y = a \log \sec\left(\fracxa\right) ) at any point. (7 marks)
Verify Green’s theorem for ( \oint_C (xy , dx + x^2 , dy) ), where ( C ) is the triangle with vertices (0,0), (1,0), and (0,1). (7 marks)
Verify Cauchy-Riemann equations for ( f(z) = e^z ) and find ( f'(z) ). (7 marks)
Using convolution theorem, evaluate: [ \mathcalL^-1 \left \frac1s(s^2 + a^2) \right ] (7 marks) Unit – E: Numerical Methods & Complex Variables Q9 (a) Using Newton-Raphson method, find a real root of ( x \log_10 x = 1.2 ) correct to 4 decimal places. (7 marks)