S Grewal - Higher Engineering Mathematics B

Find the radius of curvature for the curve ( y = a \log \sec\left(\fracxa\right) ) at any point. (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

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) Find the radius of curvature for the curve

Find the inverse Laplace transform of: [ \fracs^2 + 2s + 3(s^2 + 2s + 2)(s^2 + 2s + 5) ] (7 marks) (7 marks) Find the inverse Laplace transform of:

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)

If ( u = \log(x^3 + y^3 + z^3 - 3xyz) ), prove that: [ \left(\frac\partial\partial x + \frac\partial\partial y + \frac\partial\partial z\right)^2 u = -\frac9(x+y+z)^2 ] (7 marks)

Evaluate by Simpson’s 3/8 rule: [ \int_0^6 \fracdx1 + x^2 ] taking ( h = 1 ). (7 marks)