B.Tech / B.E. – Semester I / II Examination Subject: Higher Engineering Mathematics (MA-101) Code: [As per your scheme]
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) higher engineering mathematics b s grewal
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) Unit – C: Fourier Series &
Evaluate by Simpson’s 3/8 rule: [ \int_0^6 \fracdx1 + x^2 ] taking ( h = 1 ). (7 marks) (7 marks) Max
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
Solve using Laplace transform: [ y'' + 4y = 8t, \quad y(0) = 0, \quad y'(0) = 2 ] (7 marks)