B.Tech / B.E. – Semester I / II Examination Subject: Higher Engineering Mathematics (MA-101) Code: [As per your scheme]
Verify Cauchy-Riemann equations for ( f(z) = e^z ) and find ( f'(z) ). (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) (7 marks) Verify Green’s theorem for ( \oint_C
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) (7 marks) Unit – B: Multiple Integrals &
Evaluate by Simpson’s 3/8 rule: [ \int_0^6 \fracdx1 + x^2 ] taking ( h = 1 ). (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)