BSc Mathematics SEM III ATKT MATHEMATICS MATHS PAPER II Question Paper - Mumbai University | munotes

Questions asked in this paper

• (ii) Figures to the right indicate marks for respective parts

1. Q1 Choose correct alternative in each of the following 20 marks
   • i. Theset S={(x,y) /0 < y* <3} is
   • (a) aclosed set. (b) well as closed set
   • (c) an open set. (d) . None of these
   • ii. Let g: R* > R defined as And if f is continuous on the whole plane, then g(x, y) is
   • (c) 4y (d) None of these
   • iii. f(x,y) + Then the direction along which the directional derivative of f at
   • (c) (2,2) (d) None of these
   • iv. Let A: Total derivative is a linear transformation B: Every differentiable scalar field is continuous Then which of the following is true?
   • (a) A is true, B is false. (b) A is false, B is true
   • (c) A & B are true. (d) Both A & Bare false
   • v. lf f(x,y) = y) R? then
   • (a) f is differentiable at (0,0). (b) f is continuous at (0,0) and D,,f (0,0) exist for any vector u
   • (c) The partial derivatives f,, f, (d) None of these does not exist at (0,0)
   • vi. If Ris a differentiable function such that = 0 ,then
   • (a) f is independent of x and z. (b) f depends on x and z only
   • (c) f is constant. (d) None of these
   • vii. Which of the following is the level set of f(x, y,z) = x? fork
   • (a) Sphere of radius | centered at (b) Circle of radius | centered at origin
   • (c) Sphere of radius 2 centered at (d) Sphere of radius | centered at (1,0,0) vil. If u(x,y) = x2 + y2,x =r+e%, y = log(s) then is
   • ix. point of the function = x*y —x
   • (c) (1, (d) None of these
   • x. point is a point where
   • (a) — the function has maximum b) the function has minimum value
   • (c) the function has zero value. (a) the function has neither maximum nor
2. Q2 any ONE question from the following: Let > R be a real valued function. Let | R such that (a,b) f (x,y) = l. Also assume that the one dimensional limits 8 marks
   • y) and f (x, y) exists, then prove that
   • ii. If (x,) and (yp) are convergent sequences in a, are real constant, show that (aX, + Byn) is also convergent in any TWO questions from the following: (12)
   • b) i. Using — 6 definition show that f is continuous at (0,0), where
   • ii. Prove that every linear transformation T: R” > is continuous on R”
   • iii. Let f:R” > R anda R”. Define D;f the i-th partial derivative of f at a, 1 <i <n. Determine whether the partial derivatives of f exist at (0,0) for the following function. In case they exist, find them
   • iv. Let f:R? > R,a= (-1,2),u = = (12,5) and w = (15,1)
3. Q3 Attempt any ONE question from the following: 8 marks
   • a) i. bean open set in R” and f:U > R be differentiable at a U. Prove that
   • (a) exists for each i = with an example that converse of this is not true State and prove sufficient condition for the equality of mixed partial derivatives Attempt any TWO questions from the following: (12)
   • b) Find total derivative as linear transformation T for the function = x* + 2xy + y? at the point a = (—1,—2)
   • ii. Find directional derivative of f(x,y) = x? — 3xy along the parabola
   • ii. Find the equation of the tangent plane and normal line to the surface
   • iv. Evaluate the total derivative of z = + where x = 4 + 4t* and
4. Q4 Attempt any ONE question from the following: 8 marks
   • a) i. State and prove Taylor’s Theorem for a real valued function of two variables
   • ii. Let Q(x,y) = Ax* + 2Bxy + Cy” be a function of two variables and A= AC — B? Then prove that
   • (1) if A> 0 and A > 0 then > 0 V # (0,0)
   • (2) if A> 0 and A < 0 then < 0 V y) (x,y) (0,0) if A< 0 , then in every open ball around origin there exist points (x,y) such that Q(x, y) < 0 and there exist points (x, y) such that Q(x, y) > 0
5. Q4 Attempt any TWO questions from the following: 12 marks
   • b) = f(x,y) where f has continuous partial derivatives of second order, x=utyv, y = u— v,show that
   • ii. = xi + yj + zk then prove that the Jacobian matrix Df (x, y, is the identity matrix of order 3
   • b) Find all differentiable vector fields — for which the Jacobian matrix Df is a diagonal matrix of form diag (p(x), where p, are
   • iii. Find the critical points, saddle points and local extrema if any for the function
   • iv. Find the points on the surface z? = xy + 1 nearest to the origin. Also find the any FOUR questions from the following: (20) Let > R be defined by f(x,y) = _ li if (x,y) # (0,0)
   • a) et f: e defined by f(x,y ,0) Define f (0,0) so that f is continuous at origin
   • b) Find the real value of (0,1) if it exists, satisfying for the following function at the given points
   • c) Find level surfaces of f(x, y,z) = + y? for the constants K = 1,9
   • e) Using Taylor’s formula find the quadratic approximation for the quantities
   • f) Find the Hessian matrix of > R given by

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