A Firstranker's choice
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 ?18 dx/xp , 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) G(m + n) (b) G(m)G(n)
(c) G(m)G(n) / G(m + n) (d) G(m)G(n) / G(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) uxx - uyy = 0 (b) uxx + uyy = 0
(c) uxy - uyx = 0 (d) uxx + uyy = 0
(vii) The function f(z) = vx2 + 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 ez = 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 a and ß having Normal form
(w - a) / (w - ß) = k (z - a) / (z - ß) 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) < ?ab 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 ?ab f(x) dx = 0. 3
UNIT—II
4. (a) Prove that the integral ?18 dx/xp converges if p < 1 and diverges if p = 1. 4
(b) Show that ?18 sin(x)/x2 dx converges absolutely. 3
(c) Show that ?08 e-x2 dx converges. 3
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5. (p) Prove that B(m, n) = G(m)G(n) / G(m + n) 4
(q) Prove that ?0p/2 sinp? cosq? d? = (G((p+1)/2)G((q+1)/2)) / (2G((p+q+2)/2)) 3
(r) Prove that G(n + 1) = nG(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 a, ß is of the form (w - a) / (w - ß) = k (z - a) / (z - ß) where k is constant. 5
(b) Under the transformation w = v2 eip/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 {xn} and {yn} are sequences in X such that xn ? x and yn ? y then, prove that d(xn, yn) ? d(x, y). 5
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