How would you do these three questions from my worksheet? Worked solutions would be really helpful to double check with thanks

1. The centre of mass of a rigid plate is that point at which the plate can be supported so it will not experience any unbalanced torques that will cause it to rotate. Centre of mass Plate If a plate D in the any plane has mass per unit area p(1',y), the centre of mass is (32C, ya) where = If!) 33,9(33, y) dmdy 6 mass of D = Hg sprint; y) drdy a mass ofD Consider a triangular metal plate D with vertices at (0,0), (1,0) and (1, 3), and mass per unit area p(:r:,y) = :c + y. (a) Sketch D. (b) Find the centre of mass of D. 3. Consider the force field F(x, y, z) = ar sin(72)i + bae j + (x2 cos(nz) - 2ze-")k. Let C be the curve parametristd by (x(t), y(t), z(t)) = (cos(2t), sin(3t), cos(4t)), Out (a) Using MATLAB, sketch C. (b) For what values of a and b is F a conservative vector field? (c) Assume that a and b are chosen so that F is a conservative vector field. (i) Find a scalar function o(r, y, =) such that (ii) Calculate the work done by F to move a particle along C. 4. Let S be the surface of the cone 2=2 - vx? ty for => -1, oriented with the outward unit normal. Let F(x, y, =) = xi + zj - yk be the velocity field of a fluid. (a) Sketch S. (b) Determine the flux of F across S