BSc Mathematics SEM VI 2016 17 2016-17 Maths Paper I Old Course Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 All questions are compulsory
2. Q2 Figures to the right indicate marks
3. Q1 (a) Attempt any ONE of the following = L and if f(x,y) and limy., f(x,y) both exists, then prove that Give an example to show that the converse is not true 8 marks
   • (ii) Let S be an open subset ofR” and f,g:S If f(x) then using e — 6 definition prove
   • (b) Attempt any TWO of the following 12
   • (i) Let R? >: R* be defined by f(x, y) = (x? x? + y’). Using — 6 definition show that each component of f is continuous at (1, 2)
   • (ii) If f: be defined by f(x, y) = |x| + then show that and do not exist. Check whether f is continuous at (0, 0)
   • (iv) definition, discuss the continuity of f:IR* > R? given by
4. Q2 (a) Attempt any ONE of the following Let S be an open subset of R? and be such that exists on S. If and are continuous on S, then show that = 8 marks
   • (ii) Let S be an open subset of IR” and > be differentiable at with total derivative Show that f (a; y) exists for all y and
   • (b) Attempt any TWO of the following 12
   • (i) | State and prove the Mean Value Theorem for a scalar field
   • (ii) Let > R be a differentiable function. Let A(1,3), B(3,3), C(1,7), D(6,15). The directional derivative of f at A in the direction of AB is 3 and in the direction of AC is 26. Find the directional derivative of f atA in the direction of AD
   • (p)Compute the Jacobian matrices v), & f)(x,
   • (q)Verify that f)(1,1)=
   • (iv) Letz = where y Use chain rule
5. Q3 (a) Attempt any ONE of the following 8 marks
   • (i) State and prove Stoke’s Theorem for an oriented smooth simple parametrized surface in bounded by simple ,closed, curve traversed counter clockwise assuming general form of Green’s Theorem State Divergence Theorem for a solid in 3 — space bounded by an orientable closed surface with positive orientation and prove the Divergence theorem for cubical region
   • Q. P. Code: 04987
   • (b) Attempt any TWO of the following 12
   • (i) | Compute the surface area of the part of the paraboloid z = x* + that lies under the plane z = 9 Evaluate the surface integral [[ nas if and Sis (ili) Use Stokes’ Theorem to compute the integral Sf, curl F , where F(x,y,z)=yzit+xzj+xyk and S is the part of the sphere x* 4 that lies inside the cylinder x*+y*=1 above the xy —plane (iV) Verify Divergence Theorem for vector field +2xzk and V is the cube bounded by the planes x = 0, x = 1,y y=1,z=0,
6. Q4 Attempt any THREE of the following for (x, y) (0, 0) and f(0, 0) = 0, then find and Also find 15 marks
   • (ii) Let R be defined by Find and check whether they are equal
   • (iii) Find all differentiable vector fields > for which the Jacobian matrix = diag ( p(x) , q(y) , where p, g,r : are
   • (iv) f(x,y) has continuous partial derivatives with respect to x and
   • y. lf. x , then show that
   • (V) Evaluate the surface integral S is the triangle with the (vl) Define the Fundamental Vector Product for a surface S whose vector equation is = T in uv-plane

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