Higher Engineering Mathematics B S Grewal [work] -

B.Tech / B.E. – Semester I / II Examination Subject: Higher Engineering Mathematics (MA-101) Code: [As per your scheme]

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 volume of the sphere ( x^2 + y^2 + z^2 = a^2 ) using triple integration in spherical coordinates. (7 marks) higher engineering mathematics b s grewal

Solve the Laplace equation ( \frac\partial^2 u\partial x^2 + \frac\partial^2 u\partial y^2 = 0 ) for a rectangular plate with boundary conditions: ( u(0,y)=0, u(a,y)=0, u(x,0)=0, u(x,b) = \sin\left(\frac\pi xa\right) ). (7 marks) Unit – D: Laplace Transforms Q7 (a) Find the Laplace transform of: (i) ( t^2 e^-3t \sin 2t ) (ii) ( \frac1 - \cos att ) (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) (7 marks) Solve the Laplace equation ( \frac\partial^2

Max. Marks: 70

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) (7 marks) Prove that ( \nabla \times (

Prove that ( \nabla \times ( \nabla \times \vecF ) = \nabla(\nabla \cdot \vecF) - \nabla^2 \vecF ). Hence find ( \nabla \times (\nabla \times \vecr) ) where ( \vecr = x\hati + y\hatj + z\hatk ). (7 marks) Unit – C: Fourier Series & Partial Differential Equations Q5 (a) Find the Fourier series expansion of ( f(x) = x^2 ) in ( (-\pi, \pi) ). Hence deduce that: [ \frac11^2 + \frac12^2 + \frac13^2 + \cdots = \frac\pi^26 ] (7 marks)