BSc Mathematics SEM IV ATKT 2018-19 ATKT Maths 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: Which of the following equation has a root in (1,2)? 20 marks
   • (c) 2x + (d) None of these If A is countable set and B is uncountable set then the most we can say about A 0 B
   • (a) Finite (b) At most countable
   • (c) Uncountable (d) Countable If the decimal representation of a number is non terminating and non repeating then ili. the number is A whole number An irrational number
   • iv. The norm of the partition P = { —6, —5.8, —5.2, —4.8, —3.2, —3 } is
   • (c) 1.6 (d) None of these Let P and Q be any two Partitions of interval Then the statement that is always true is
   • (c) U(P,f) (d) None of these ED29AE1247B4DC6F8A832BE3E86F0767
   • vi. Let f and g be functions such that the function f + g is integrable on J , then Both f and g must be integrable (b) Atleast one of f and g must be f and g may or may not be (d) None of these
   • (C) on I
   • vii. Let > R such that F(x) = tdt then F’(x) = Let f(x) and g(x) be two positive R-integrable functions on such that
   • (a) | g(x)dxis convergent if | f(x)dx is convergent
   • (b) | g(x)dx is divergent if | is divergent
   • (c) f (x)dx is divergent if g(x)dx is divergent
   • (d) both(a) and (b) are true
   • ix. + 1) is
3. Q10 The equation 1 represents
   • (a) ellipsoid (b) sphere
   • (c) Hyperboloid of one sheet (d) Hyperboloid of two sheets Attempt any ONE question from the following: (08)
4. Q1 State and prove Nested Interval Theorem Using Nested Interval Theorem prove that if f: [a,b] > R is a continuous function with f(a)f(b) < 0 then there existsc =0
   • b) Attempt any TWO questions from the following: is If = (0, for all nEN then prove that I, = @ ED29AE1247B4DC6F8A832BE3E86F0767 Show that a real number is rational iff it has repeating decimal Show that the equation + 1 = 0 has three solutions in the interval Iv. Find convergent subsequence of sequence 12
5. Q3 a) Attempt any ONE question from the following: Let f: [a,b] R bea bounded function. Prove that f is R-integrable on [a, b| iff for any > there partition P- of [a,b] such that 8 marks
   • ii. f: [a,b] R be acontinuous function. Then show that f is Riemann
   • b) Attempt any TWO questions from the following: 12
6. Q1 Let f : [a,b] be a bounded function with m = Inf (f) and M = Sup(f) on [a, b]. With usual notations, define L(P, f) and U(P, f) where P is a partition of [a,b]. Hence prove that
   • ii. Prove that the function : [0,4]—> R defined by f(x) = 2x* +1 is Riemann integrable and evaluate f
   • iii. Let f R be defined by for3<x<6 Prove that f is Riemann integrable and f(x)dx = 36 Using Riemann Criterion, show that the function f : [0, 1]— R defined by f (x) = is Riemann integrable Attempt any ONE question from the following: (08) Let f: R be R-integrable on [a,b] and F(x) = } f(t)dt,Vx If f is continuous on [a,b] then show that F is differentiable and F (x) = f (x) ED29AE1247B4DC6F8A832BE3E86F0767
   • ii. Show that (1 — exists iff m > 0,n > 0
   • b) Attempt any TWO questions from the following: 12
7. Q1 Solve the improper integral } ii, By using “integration by parts” solve the integral | xe “dx lil. Express f in terms of beta function, iv Evaluate ff p YaA where Dis the region bounded by the line y = x and the
8. Q5 Attempt any FOUR questions from the following: Show that G = covers A = (0,1) but has no finite sub-cover for 20 marks
   • b) Show that set of real numbers R is uncountable
   • c) bea partition of [0,4] and [0,4]— R is a function such that f(x) = 3 — x? then find the lower sum L(P, f ) and upper sum U(P, f )
   • d) Show that the function f : [1,3]— Ris Riemann integrable, where Check the convergence of improper integral } dx by comparison test
   • f) the order of integration and evaluate xdydx ED29AE1247B4DC6F8A832BE3E86F0767

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