Section A
Q1.
(a) Let p be a prime number. Then show that (p-1)! + 1 equiv 0 mod (p). Also, find the remainder when 6^{44} cdot (22)! + 3 is divided by 23.
(b)
(i) If u = u(y-z, z-x, x-y), then find the value of rac{partial u}{partial x} + rac{partial u}{partial y} + rac{partial u}{partial z}.
(ii) If u(x, y, z) = rac{x}{y + z} + rac{y}{z + x} + rac{z}{x + y}, then find the value of xrac{partial u}{partial x}+yrac{partial u}{partial y}+zrac{partial u}{partial z}.
(c) Evaluate the integral iint_R (x-y)^2 cos2(x+y) dx dy, where R is the rhombus with successive vertices at (pi, 0), (2pi, pi), (pi, 2pi) and (0, pi).
(d)
Solve graphically the following LPP :
Max z = 5x1 - 3x2
subject to
3x1 + 2x2 le 12
-x1 + x2 ge 1
-x1 + x2 le 2
x1, x2 ge 0
If the objective function z is changed to Max z = 6x1 + 4x2, while the constraints remain the same, then comment on the number of solutions. Will (4, 0) be also a solution?
(e) Evaluate the integral int_C ext{Re}(z2) dz from 0 to 2+4i along the curve C: y = x2.
