BSc Mathematics SEM IV 2018 19 Apr 2018-19 MATHEMATICS PAPER I Question Paper - Mumbai University | munotes

Questions asked in this paper

• (i1) Figures to the right indicate marks for respective parts

1. Q1 Choose correct alternative in each of the following : 20 marks
   • i. > IR be bounded function and P ,Q be partitions of [a,b] then
   • (c) L(P,f)) (d) None of the above li. The norm of a partition P = {0 < <1< < <
   • (c) 1 (d) None of the above
   • iii. If IR is R- integrable then which of the following is true
   • (a) f must be continuous (b) f must be differentiable
   • (c) f must be monotonic (d) None of the above Rbeacontinuous function. Then f(t)dt = 0,Va > 0 if and only if
   • (a) f=0 (b) fis an odd function
   • (c) f #0 for only finitely many (d) None of the above If R are continuous functions such that = g(x)dx then
   • (a) f (b) f(x) = g(x) is
   • (c) such that None of the above
   • vi. The type 2 integral f dx Diverges (b) Converge to 0 Converge to =In 3 Converges to Integral f dx converges if
   • (c) p=l (d) None of the above
   • ix. dx=
   • (c) va (d) None of these
   • x. Identify the definite integral that computes the volume of the solid generated by revolving the region bounded by the graph of and the line y=x, between x=0 and x=1 about the line x = 1
2. Q2 Attempt any ONE question from the following : 8 marks
   • a) i. — Let f:[a,b] > IR be a bounded function. Prove that f is Riemann integrable on [a, b] if and only if for any 0 there exist a partition P of [a, b] such that
   • ii. If f;g:[a,b] are R- integrable then prove that f + g is R- integrable and Attempt any TWO questions from the following : (12)
   • b) i. Let f be a bounded function on [a, b]. Let P and P’ are two partitions of [a, b] with P Show that L f) = ul. If f is an R-integrable function on [a,b] then prove that | f | is R-integrable on
   • iii. Using Riemann Criterion, prove that the function f : [0, 1] — R defined by f(x) = x is Riemann integrable
   • iv. If R are integrable functions such that f(x) S [a, b] then prove that dx < g(x) dx
3. Q3 Attempt any ONE question from the following : 8 marks
   • a) i. State and prove the Fundamental Theorem of Calculus li. State and prove Comparison Test for improper integrals of type-I
4. Q3 Attempt any TWO questions from the following : Let F : [0,1] Rbe defined by F(x) = f0<xs1 Show that is differentiable over [0, 1].Let [0,1] > given by that dx converges if and only if p < 1 12 marks
   • iv. State Abel’s and Dirichlet’s Tests for the conditional convergence of type 1 improper integral and discuss convergence of I = sin x? dx
5. Q4 Attempt any ONE question from the following : 8 marks
   • a) Prove that converges if and only if m and n are both With usual notations for beta and gamma functions prove that
6. Q4 Attempt any TWO questions from the following : Prove that dy 12 marks
   • iii. Find the volume of the solid whose base is the disk x* <1. and the cross sections by the planes perpendicular to the y — axis between y = —1 and y = 1 are isosceles right triangles with one leg in disk by the method of slicing
   • iv. Find the volume of the solid generated by revolving the regions bounded by the lines y = 2x ,y = 1and about x — axis. by the Washer method
7. Q5 Attempt any FOUR questions from the following : 20 marks
   • a) Iff(x) = 1+ 2x,x P bea partition such that
   • b) Riemann integrable on then for any k IR prove that kf is also Riemann
   • c) Show that if F’(x) = 0,Vx [a, b] then f is a constant function
   • d) Identify the type and discuss the convergence of each of the following integrals Prove that Vsin x dx J? x =
   • f) the area of the surface generated by revolving the curves about x = 2,/4 — y,

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