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Pune University - FY BSc MATHEMATICS - II, April 2010

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Total No. of Questions : 5] [Total No. of Pages : 4 [3717] - 2 P180 F.Y. B.Sc. MATHEMATICS Calculus (Paper - II) (New Course) Time : 3 Hours] Instructions to the candidates : 1) All questions are compulsory. 2) Figures to the right indicate full marks. [Max. Marks : 80 Q1) Attempt all the subquestions : a) Find the rational number between b) Show that cos n [16] 2 and 3. is divergent. n=0 c) Define Cauchy sequence. d) Find the left hand and right hand limits of the function f(x) at x = 4, where f ( x ) = e) x 4 , x 4. x 4 If the function f : R R is defined as f(x) = 1, if x < C = 1 , if x C then show that f is continuous at x = C. f) ax 1 Evaluate, lim x . x 0 b 1 g) Test whether Rolle s theorem is applicable for the function f(x) = x on [ 1, 1]. h) State Taylor s theorem with Lagrange s form of remainder. P.T.O. Q2) Attempt any four of the following : [16] a) For any two distinct, positive real numbers a a nd b , prove that 1 ab < ( a + b ) . 2 b) If <xn>, <yn> and <zn> are three sequences such that x n yn z n , n N and lim x n = lim z n = l then prove that lim yn = l . n c) d) n n Show that the sequence <xn> of real numbers whose nth term is defined 1 1 1 1 by x n = + + +..... + , n N is convergent. 1.2 2.3 3.4 n. ( n + 1) Evaluate lim x 0 e 1 x 1+ e 1 x , if it exist. e) Examine for convergence of the series f) If x1 = 2 and x n +1 = 3 + 3n ( n + 1) n =1 n . 1 , n 1 then show that <xn> is contractive 2 xn sequence. Q3) Attempt any two of the following : a) n =1 series n =1 b) 1 Prove that P n [16] is convergent, if P > 1. Hence for what value of P is the n +1 n convergent? nP Show that the sequence < x n > defined by x 1 = 1 a nd x n +1 = 2 + x n , n N is convergent. ii) c) i) Solve i) Prove that limit of function f(x) as x C is unique, if it exists. Show that the sequence <xn> of reals whose nth term is defined by 1 x n = is a Cauchy sequence. n ii) [3717] -2 3 x < 1 x R and x 2 . 2+x 2 d) i) State the field axioms for set of real numbers. ii) Prove that lim x . sin x 0 1 = 0. x Q4) Attempt any four of the following : [16] a) State and prove Lagrange s mean value theorem. b) Verify Rolle s theorem for the function f ( x ) = c) Verify Cauchy s mean value theorem for the function f(x) = cosx and sin x on [ 0, ]. x e g(x) = sinx on 0, . 2 d) Find , , if the function f(x) is continuous on ( 3, 5) where f(x) = x + , 3 < x < 1 = 3x + 2, 1 x < 3 = + x, 3 x < 5. e) x3 x5 x7 + +........ By using Maclaurin s series prove that, tan x = x 3 5 7 f) 1 1 . Evaluate lxim 2 x 2 log( x 1) 1 Q5) Attempt any two of the following : [16] a) State and prove Leibnitz s theorem. Hence find y5, if y = x3. ex. b) i) If f : [a, b] R is continuous function on [a, b] and f(a) < k < f(b), then prove that there exists a point C (a, b) satisfying f(C) = k, where k R. ii) Discuss the continuity of the function f ( x ) = ( x 2 ) ( x 4 ) , x R . [3717] -2 3 c) Prove that, if f(x) is differentiable at point x = a, then it is continuous at x = a. Is the converse true? ii) d) i) If y = sin 1x, then show that, (1 x2)yn+2 (2n + 1)x.yn+1 n2yn = 0. i) Prove that b a b a < tan 1 b tan 1 a < , if a < b. 1 + b2 1 + a2 1 ii) [3717] -2 Evaluate, lim ( cos x ) x 2 . x 0 4

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Additional Info : F.Y. B.Sc. MATHEMATICS : Calculus (Paper - II) (New Course), Pune University
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