BSc Mathematics SEM III 2019 2020 Oct 2020 MATHS PAPER I Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 Figures to right indicate full marks
2. Q1 Choose correct alternative in each of the following (2 marks each) ) The set {(x,y) <x + y < 4} is -------in R2
   • a) both open and b) open neither open nor closed d) none of these 2)The sequence (xm) a in
   • b) — all > 0 in llxm >
   • d) none of these
3. Q3 The sequence x = (m,m+1,m+ 2) is
   • a) bounded but not convergent in R? b) convergent but not bounded in R? neither bounded nor convergent in none of these
4. Q4 > Ras f(x,y) =1 = (0,0)
   • a) fis continuous but |f] is not continuous at (0,0)
   • b) fis not continuous but is continuous at (0,0) neither f nor |f] is continuous at (0,0)
   • d) none of these is -------------for a differentiable function 4
   • a) a linear transformation but not an element of R
   • b)a real number not both linear transformation as well as an element of R d) none of
5. Q6 For differentiable function f:S > Rat directional derivative of f at a in direction of
   • a) b) Vf(a) c) 0 d) none of these
6. Q7 Fora function: R (@) Is
   • a) directional derivative of f at a in direction of any nonzero vector u
   • b) directional derivative of f at a in direction of = @; directional derivative of fat in direction of (1,0,.,0,.0) =
   • d) none of these let f(x,y) = sinxy + log(x + y) then taylor’s polynomials of degree 2 about (1,0) is
   • a) b) —x? c) —x* d) none of these
7. Q9 For >: as g(u,v,w) = (uvw,u2 +v*+w?) =
   • a) b) c) d) none of these Ox dy Ox dy or Ox Oy Or
   • d) none of these
8. Q2 a) Attempt any ONE question from the following.(8marks each)
9. Q1 Let the sequence Xm = then prove that x,, is convergentin R" each of the coordinate sequence x; is convergent in R 2)State and prove Mean Value theorem for scalar field
   • b) Attempt any TWO question from the each) ) Find the directional derivative of following function at indicated points and direction if exists (0,0) in the direction of u = 2)Find the real value of @ (0,1) such that f(a + v) — f(a) =D, flat Where u unit vector in direction of v
10. Q3 Let S bea nonempty open subset of R”.Let a=
    • (a) exists for i = 1,2,.n then prove that
11. Q4 In the following find the partial derivatives of f at (0,0) if exists
12. Q3 a) Attempt any ONE question from the following.(8marks each)
13. Q1 State and prove Chain rule for scalar
14. Q2 Let S be a nonempty open subset of a S and suppose Vf, Vg, f) exists at
    • a.Then prove that f)(a) where is real constant. ii) provided g(a) # 0 and g(x) # 0 in neighbourhood of a vu, Attempt any TWO question frem the each) ) State and prove Euler’s Theorem for function of three variables
15. Q2 State and prove Mean Value theorem for differentiable scalar fields 3)i)Use chain rule to find total derivative f(x,y) = 3x3y? + 5x?y3
    • ii) Find the level curve of following f for given k 4)Prove that following functions are continuous but not differentiable at origin f(x,y) = |xy
16. Q4 a) Attempt any ONE question from the foilowing.(8marks each) )State and prove the relation between total derivative and jacobian matrix of vector valued
17. Q2 State and prove Mean Value Inequality for vector field f differentiable over nonempty subset
    • b) Attempt any TWO question from the following.(6marks each) ) the Jacobian matrix of a vector field at the given point
18. Q2 Let S be a nonempty open subset of R”.Let f:5 > bea vector field. If fis differentiable S then it is continuous at a S. What about converse?
19. Q3 Let S bea nonempty open subset of R?Let f:5 bea vector field. If fis at M > 0,5 > 0 such that — f(a)|| < al
20. Q4 Let S be a nonempty open subset of f:5 > bea vector field differentiable over
    • S. Let a,b S and the line segment joining a and b which is the set
21. Q5 Attempt any FOUR question from the each) ) Prove that every linear T: R" > is continuous on R”
22. Q2 i) Evaluate f(x, y) and f (x, y) for following and check whether both are equal or not. f(x,y) = (x,y) (0,0)
    • ii) In following find @ so that R is continuous at (0,0)
23. Q3 i)Find total derivative of following function at indicated point
    • ii)Find directional derivative of following function at indicated point using gradient function +) Prove that following functions are differentiable at origin
24. Q5 Use Lagranges multiplier method to find maximum and minimum values of given function subject to specified constraints.f (x,y,z) = xyz subject to x2 + 2y* =6 Define Hessian Matrix for > R scalar field and find hessian matrix for > Rgiven by f(%y,2) + + y*z at(1,0,1) Define the following terms for f:S > R for nonempty open subset S of R”
    • a) stationary point of f b) absolute maximum at a of f saddle point of f

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