Higher Engineering Mathematics B S Grewal ((install)) Site

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

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

Find the half-range cosine series for ( f(x) = x(\pi - x) ) in ( (0,\pi) ). (7 marks)

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)

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)

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)

Max. Marks: 70

Find the radius of curvature for the curve ( y = a \log \sec\left(\fracxa\right) ) at any point. (7 marks)