Verify Cauchy-Riemann equations for ( f(z) = e^z ) and find ( f'(z) ). (7 marks)
Evaluate by Simpson’s 3/8 rule: [ \int_0^6 \fracdx1 + x^2 ] taking ( h = 1 ). (7 marks)
Solve the wave equation ( \frac\partial^2 y\partial t^2 = 4 \frac\partial^2 y\partial x^2 ) with boundary conditions ( y(0,t)=0, y(3,t)=0, y(x,0)=0, \frac\partial y\partial t(x,0) = 5 \sin 2\pi x ). (7 marks) higher engineering mathematics b s grewal
Find the volume of the sphere ( x^2 + y^2 + z^2 = a^2 ) using triple integration in spherical coordinates. (7 marks)
Trace the curve ( r = a(1 + \cos\theta) ) (Cardioid) and find the area enclosed. (7 marks) Unit – B: Multiple Integrals & Vector Calculus Q3 (a) Evaluate: [ \int_0^1 \int_0^\sqrt1-x^2 \int_0^\sqrt1-x^2-y^2 \fracdz , dy , dx\sqrt1-x^2-y^2-z^2 ] (7 marks) Verify Cauchy-Riemann equations for ( f(z) = e^z
Find the half-range cosine series for ( f(x) = x(\pi - x) ) in ( (0,\pi) ). (7 marks)
Find the radius of curvature for the curve ( y = a \log \sec\left(\fracxa\right) ) at any point. (7 marks) (7 marks) Find the volume of the sphere
Solve using Laplace transform: [ y'' + 4y = 8t, \quad y(0) = 0, \quad y'(0) = 2 ] (7 marks)