BSc Mathematics SEM II 2018 19 May 2018-19 MATHEMATICS PAPER I Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate marks for respective parts

1. Q3 Use of Calculator is not allowed
2. Q1 Choose correct alternative in each of the following: 20 marks
   • i. If liman an is
   • (c) alternating (d) None of the above
   • ii. If then y4 =
   • (c) 32 (d) none of these iil. Rolle’s theorem is applicable to f(x) =sin x in the interval
   • (c) (d) none of these
   • iv. The series of real numbers is
   • (c) geometric series (d) none of these The function f(x) = |x—41,x Ris
   • (a) differentiable atx =4 not differentiable at x =4
   • (c) differentiable at any x in R (d) of these Vi. Which of the following functions is increasing in [-1,
   • vii. of real numbers is a
   • (c) alternating series (d) none of these
   • viii. Amongst the following, the function which has a local minimum at the origin is
   • (c) y=Ixl (d)
   • (c) e (d) limit cannot be determined Xx, The function f(x) = +5x R is
   • (a) increasing on R (b) increasing when x
   • (c) decreasing on R (d) none of these Attempt any ONE question from the following: (08)
   • i. Prove that if an is convergent then the sequence converges to zero. Is converse true? Justify your answer
   • ii. Prove that the alternating series is convergent if it satisfies the following conditions:
   • (I) ay = for all n N i.e. sequence is non-increasing
   • b) Attempt any TWO questions from the following: 12
   • i. Prove that the series is divergent
   • ii. Let an and by be series of non-negative real numbers. Assume that there exists n, N such that.a, < b, for alln Then prove that, if Dy is convergent then is convergent
   • iii. Is the series convergent? If yes, find it’s limit Iv. Check whether the following series are convergent stating the results used
3. Q3 a) Attempt any ONE question from the following: 8 marks
4. Q1 Let f, g: R > R be two functions which are differentiable at p R Prove that fg is differentiable at p R ll. Letn N and u, v: > R be n differentiable functions. Prove that = UnVo + where the suffixes denote the order of derivatives and ug = vg = v
   • b) Attempt any TWO questions from the following: Let f: R > R be a continuous function. Define H(x) = x ux 12
   • i. If f: [a,b] Riis acontinuous function then prove that f attains its ili. Find the derivative of the following functions using chain rule:
   • iv. Find for siny + x*y? —cosx = 2y where y is a function of x
5. Q4 a) Attempt any ONE question from the following: 8 marks
6. Q1 For a real valued function f define local minimum at a point If f: (a,b) R has a local minimum at a point p (a, b) and if f is differentiable at p then prove that f = 0 il. State and prove Rolle’s theorem
   • b) Attempt any TWO questions from the following: 12
7. Q1 Find the intervals on which f(x) = 4x3 — 12x” — 36x + 1 is increasing or li. State L’ Hospital’s rule and evaluate. lim Expand x? + 2x + 1 in powers of Iv. Determine the intervals of concavity and the inflection points of
8. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) Check the following series for absolute and conditional convergence of
   • b) Check the convergence of the series Check if the following function is differentiable at x
   • d) If y + then prove that =
   • f) Find maximum value of in (0,0)

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