Download SGBAU BSc 2019 Summer 5th Sem Mathematics Mathematical Analysis Question Paper

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B.Sc. (Part—III) Semester—V Examination

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MATHEMATICS (New)

(Mathematical Analysis)

Paper—IX

Time : Three Hours] [Maximum Marks : 60

N.B. :— (1) Question No. 1 is compulsory.

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(2) Attempt ONE question from each Unit.

1. Choose the correct alternatives :

Let f: [0, 1] — R be Riemann integrable. Which of the following is always true :

(a) f is continuous

(b) f is monotone

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(c) f has only finite number of discontinuities

(d) the set of discontinuities of f may be infinite ?

(i) An improper integral ∫₁^∞ dx/x^p , a ∈ R is convergent if :

(a) p < 1 (b) p > 1

(c) p ≥ 1 (d) p = 1

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(ii) B(m, n) is:

(a) Γ(m + n) (b) Γ(m)Γ(n)

(c) Γ(m)Γ(n) / Γ(m + n) (d) Γ(m)Γ(n) / Γ(m - n)

(iv) In the real line R. which of the following is true ?

(a) Every bounded sequence converges (b) Every sequence converges

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(c) Every Cauchy sequence converges (d) None of the above

(v) Every neighbourhood is a/an :

(a) Closed set (b) Open set

(c) Open closed set (d) None of the above

(vi) A function u(x, y) is harmonic in region D if :

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(a) u_xx - u_yy = 0 (b) u_xx + u_yy = 0

(c) u_xy - u_yx = 0 (d) u_xx + u_yy = 0

(vii) The function f(z) = √x² + y² is:

(a) Harmonic function (b) Analytic function

(c) Conjugate function (d) Not analytic function

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(viii) If f(z) and f'(z) are both analytic functions then f(z) is :

(a) Identically zero (b) Constant

(c) Unbounded (d) None of the above

(ix) The points z where e^z = 10 form a :

(a) Circle (b) Straight line

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(c) Hyperbola (d) Parabola

(x) A bilinear transformation with two non-infinite fixed points α and β having Normal form

(w - α) / (w - β) = k (z - α) / (z - β) Elliptic if :

(a) |k| = 1, k is real (b) k ≠ 1, k is not real

(c) |k| = 1 (d) None of the above

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UNIT—I

2. (a) Prove that every continuous function is integrable. 4

(b) Let the function f be defined as

f(x) = 1, when x is rational

= -1, when x is irrational

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Show that f is not R-integrable over [0, 1] but |f| ∈ R [0, 1]. 3

(c) Show that any constant function defined on a bounded closed interval is integrable. 8

3. (p) If f is a bounded and integrable function over [a, b] and M, m are bounds of f over [a, b], prove that :

m(b — a) < ∫_a^b f(x)dx < M(b - a). 3

(q) Prove that 2/5 < ∫₀¹ dx/(1+x²) < 1/2 3

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(r) If f is continuous and non-negative on [a, b], then show that ∫_a^b f(x) dx = 0. 3

UNIT—II

4. (a) Prove that the integral ∫₁^∞ dx/x^p converges if p < 1 and diverges if p ≥ 1. 4

(b) Show that ∫₁^∞ sin(x)/x² dx converges absolutely. 3

(c) Show that ∫₀^∞ e^-x2 dx converges. 3

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5. (p) Prove that B(m, n) = Γ(m)Γ(n) / Γ(m + n) 4

(q) Prove that ∫₀^π/2 sin^pθ cos^qθ dθ = (Γ((p+1)/2)Γ((q+1)/2)) / (2Γ((p+q+2)/2)) 3

(r) Prove that Γ(n + 1) = nΓ(n). 3

UNIT—III

6. (a) If f(z) = u(x, y) + iv(x, y) be analytic in a region D, then prove that u(x, y) and v(x, y) satisfy Cauchy-Riemann equations. 4

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(b) If f(z) and f'(z) are analytic functions, prove that f(z) is constant. 3

(c) Show that u = 2x — x³ + 3xy² is harmonic and find its harmonic conjugate function. Hence find f(z) = u + iv. 3

7. (p) If u and v are harmonic in region R, prove that (∂u/∂y - ∂v/∂x) + i(∂u/∂x + ∂v/∂y) is analytic in R. 4

(q) If the function f(z) = u + iv be analytic in domain D then prove that, the family of curves u(x, y) = c₁ and v(x, y) = c₂ form an orthogonal system, where c₁ and c₂ are arbitrary constants. 3

(r) Determine a, b, c, d so that the function f(z) = (x² + axy + by²) + i (cx² + dxy + y²) is analytic. 3

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UNIT—IV

8. (a) Prove that, every bilinear transformation with two non infinite fixed points α, β is of the form (w - α) / (w - β) = k (z - α) / (z - β) where k is constant. 5

(b) Under the transformation w = √2 e^iπ/4 z, find the image of the rectangle bounded by x=0, y=0, x=2 and y = 3. 5

9. (p) Prove that the cross ratio remains invariant under a bilinear transformation. 5

(q) Prove that under the transformation w = (z-i)/(z+i) the region I(z) > 0 is mapped into the region |w| < 1. 5

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UNIT—V

10. (a) Show that d(x, y) = |x — y|, ∀x, y ∈ R defines a metric on R. 5

(b) Define:

(i) Limit point

(ii) Boundary point. 2

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(c) Prove that every neighbourhood is an open set. 3

11. (p) Define:

(i) Complete metric space

(ii) Open set. 2

(q) Prove that every convergent sequence in a metric space is a Cauchy sequence. 3

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(r) Let X be a metric space. If {x_n} and {y_n} are sequences in X such that x_n → x and y_n → y then, prove that d(x_n, y_n) → d(x, y). 5

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