Download SGBAU BSc 2019 Summer 3rd Sem Mathematics Advanced Calculus Question Paper

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B.Sc. Examination

MATHEMATICS

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(Advanced Calculus)

Paper—V

Time : Three Hours] [Maximum Marks : 60

Note :— (1) Question No. 1 is compulsory. Attempt once.

(2) Attempt ONE question from each unit.

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1. Choose the correct alternative :—
   1. Every Cauchy sequence of real number is ;
      1. unbounded
      2. bounded
      3. bounded as well as unbounded
      4. None of these

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   2. The sequence <s_n> where s_n = n/(n+1) is
      1. monotonically increasing
      2. monotonically decreasing
      3. constant sequence
      4. None of these

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   3. The harmonic series Σ(1/n) is
      1. Convergent
      2. Oscillatory
      3. Divergent
      4. None of these

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   4. Let Σa_n be a series with positive terms and lim (a_n)^1/n = l, then the series Σa_n is convergent if :
      1. l=1
      2. l>1
      3. l<1
      4. None of these

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   5. If lim_{(x,y)->(x0,y0)} f(P)=f(P₀); where P, P₀ ∈ R² then
      1. f is discontinuous at P
      2. f is continuous at P₀
      3. f is continuous at P
      4. None of these

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   6. If lim_x->x0 lim_y->y0 f(x, y) = l exist then repeated limits are
      1. equal
      2. not equal
      3. not exist
      4. None of these

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   7. The function f(P) has absolute minima at P₀ in D if
      1. f(P) < f(P₀); ∀ P ∈ D
      2. f(P) > f(P₀); ∀ P ∈ D
      3. f(P) = f(P₀); ∀ P ∈ D
      4. None of these

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   8. The value of ∂(x,y) / ∂(u,v) is equal to
      1. I_xV_x - I_yV_y
      2. I_xV_y - I_yV_x
      3. I_yV_x - I_xV_y
      4. None of these

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   9. The value of ∫₀¹ ∫₁² x²ydydx is
      1. -1
      2. 5/4
      3. 1/3
      4. 1/2

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   10. The value of ∫₀¹ ∫₀¹ ∫₀¹ e^xdydz is
       1. 0
       2. 2
       3. -1
       4. e - 1

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UNIT—I

2. (a) Prove that a convergent sequence of a real numbers is bounded. 5
   (b) Show that the sequence <S_n>, S_n = 1/1! + 1/2! + ...... +1/n! is convergent. 5
3. (p) Prove that every convergent sequence of real numbers is a Cauchy sequence. 5
   (q) Show that the sequence <S_n>, where S_n = (1 + 1/n)ⁿ is convergent and that lim (1 + 1/n)ⁿ lies between 2 and 3. 5

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UNIT—II

4. (a) Prove that the series Σx_n converges if and only if for every € > 0, ∃ a M(€) ∈ N such that m≥n≥M=|x_n+1 +x_n+2 + .. +x_m| < € 5
   (b) Test the convergence of the series x/(x+2) + (x+2)/(x+4) + (x+4)/(x+6) + ..., x ∈ R, x≠0. 5
5. (p) Prove that p-series Σ(1/n^p) is convergent for p > 1 and divergent for p < 1. 6
   (q) Test the convergence of the series Σ (√(n+a) - √n)/(√(n+a²) + √n), n ∈ N. 4

UNIT—III

6. (a) Prove that lim_(x,y)->(1,1) (3x—2y)= 1 by using € — δ definition of a limit of a function. 4
   

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   (b) Expand x⁴ + y⁴ — 3xy in powers of x — 2 and y — 3. 3
   (c) Let real valued functions f and g be continuous in an open set D ⊆ R² then prove that f + g is continuous in D. 3
7. (p) Prove that the function f(x, y) = x + y is continuous ∀ (x, y) ∈ R² 4
   (q) Expand e^xy at the point (2, 1) upto first three terms. 3
   (r) Let f(x, y) = xy/(x²+y²), show that the simultaneous limit does not exist at the origin in spite of the fact that the repeated limits exist at the origin and each equals to zero. 3

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UNIT—IV

8. (a) Find the maximum and minimum values of x³ + y³ — 3axy. 5
   (b) Find the least distance of the origin from the plane x — 2y + 2z = 9 by using Lagrange’s method of multipliers. 5
9. (p) If x, y are differentiable functions of u, v and u, v are differentiable functions of r, s then prove that
   ∂(x,y)/∂(r,s) = ∂(x,y)/∂(u,v) * ∂(u,v)/∂(r,s)
   (q) If xu = yz, yv = xz and zw = xy find the value of ∂(u,v,w)/∂(x,y,z)

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UNIT—V

10. (a) Evaluate by changing the order of integration : ∫01 ∫x2√x dy dx 5
    (b) Evaluate ∫∫∫(2x+y)dv, where v is the closed region bounded by the cylinder z=4—x2 and the planes x=0, x=2, y=0, y=2, z=0. 5
11. (p) Evaluate by Stoke’s theorem ∫(e^xdx+2ydy—dz) , where c is the curve x² + y² = 4, z=2. 5
    (q) Evaluate by Gauss Divergence theorem ∬ f.nds; where f=(x³ —yz)i+(y³ —2x)j+ (z³ —xy)k and s is the surface of rectangular parallelepiped 0

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