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 full marks

1. Q1 Choose correct alternative in each of the following: The series of real numbers 20 marks
   • (a) is not convergent (b) to
   • (c) Converges to 0 (d) the above ij, and yp, are two series of real numbers such that — Yn) and y, are both convergent then x,
   • (a) convergent (b) Is divergent
   • (c) Conditionally convergent (d) None of the above
   • iii. The series R is
   • (a) (b) Divergent for anyr R
   • (c) Convergent if |r| < (d) _None of the above The function y = has equal to
   • (c) log(x + 1) (d) None of the above The function f(x) = |x ER
   • (a) differentiable (b) at x =5 Is differentiable at every
   • (c) (d) None of the above
   • Q.P.Code: 51639 vi, The value of lim is
   • (a) loge (5) (b) 5
   • (c) 3 (d) None of the above
   • vii. The function f(x) = — x (5,6) is
   • (a) Continuous but not bounded
   • (c) Discontinuous (d) of the above
   • viii. The function f(x) = logx,x
   • (c) Decreasing function (d) None of the above
   • ix. The function f(x) = 3x? — 7x +2 is
   • (a) Increasing for allx (b) Decreasing for all x R
   • (c) Increasing for all x > (d) None of the above
   • x. If > Rare such that f is differentiable then
   • (a) Both f,g are differentiable (b) Atleast one of is
   • (c) is differentiable (d) None of the above
   • a) Attempt any ONE question from the following: Let is the sequence of partial sums for series Prove that is decreasing sequence and is increasing sequence. Further prove that 8
   • Q.P.Code: 51639 lim (=) = 0 and is absolutely convergent series then prove that series x, is also convergent
   • b) Attempt any TWO questions from the following: Prove that is conditionally convergent series li. State the ratio test and use it to test the convergence of 12
   • iii. | Prove that for each non-negative integer n, And deduce that the sequence of partial sums is
   • iv. Let =A and =B. Then prove that
2. Q3 a) Attempt any ONE question from the following: 8 marks
   • i. Let f be a real valued continuous function on [a, b] such that f(a) # f (b). Then for each k, f(a) < k < f(b), prove that there exists (a, b) such that f(c) =k li. and prove Chain Rule for the derivatives of a composite
   • b) Attempt any TWO questions from the following: Find if x3 + y? = 3xy 12
   • ii. Fory=e*sin x , prove that y2— 2y1 + 2y =0 . Hence prove
   • iii. Prove that — 15x + 1 =0 has at least one root in [- 4, 4]
   • iv. R be given by f(x) = cos x . Show that f is
   • Q.P.Code: 51639
3. Q4 a) Attempt any ONE question from the following: 8 marks
   • i. State and prove Rolle’s theorem
   • ii. If f is a differentiable function defined on an open interval (a, b) and f'(x) < then prove that f is decreasing on
   • b) Attempt any TWO questions from the following: 12
   • i. Find the local maximum and minimum of the function f(x) =x if they exist
   • ii. | Use Rolle’s theorem to show that the equation x? + has exactly one real root
   • iii. For what values of x is the curve y = 6x?+ 5x + 7 concave upwards and when is it concave downwards? Also find a point of inflection
   • iv. Verify Mean Value Theorem for f(x) = x* and
4. Q5 Attempt any FOUR questions from the following: Test for convergence of the series > stating the result used 20 marks
   • b) Prove that is convergent if and only if |x| < 1
   • c) Find derivative of y = x? cos x
   • d) If Ris an even function and differentiable on R then prove that an odd function
   • e) Expand in powers of (x—1)using

Read from the scan above, so a character or two may differ. The scan is the original.