Bachelor of Science (B.Sc.) Sem I 2023 2024 Dec 2024 MATHS I Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 (a)Attempt any one. [each 8Mks]
   • 1) State the properties of real numbers with respect to
   • 2)Let S be anonempty subset of R real numbers then prove that a real number M is supremum of S < MVx ti)for every 04 x
   • (b)Attempt any two. [each 6Mks]
   • 1)State and prove Hausdorff Property real numbers R
   • 2)State and prove Archimedean Property of real numbers R
   • 3)Prove that i)Every real number has a unique Additive inverse.ii)Additive identity in R is
2. Q2 (a)Attempt any one. [each 8Mks] 1}Prove that (1 + convergent
   • 2)Prove that Every monotonic increasing sequence of R converges to its upper bound) if bounded above
   • (b)Attempt any two. [each 6Mks]
   • 1)Prove that a convergent sequence of real numbers R has a unique limit
   • 2) Prove that every convergent sequence of Real numbers R is about the
   • 3)State and prove Sandwich Theorem for limit of a sequence in R
3. Q3 (a) Attempt any one. [each 8Mks]
   • 1)Prove that The differential equation Q(x)y"where P(x), Q(x)are functions of x and n is real number, n 1, is always reducible to a suitable Linear Differential Equation, of the =
   • 2)State to find Integrating factor and solve the following differential equation Attempt any two. [each 6
   • 1)A bacteria culture is known to grow at a rate proportional to the amount present.After one bacteria are observed in the culture, and after four hours 3000.Find an expression for the number of bacteria present in the culture at any time t.Also determine the number of bacteria originally in the culture
   • 2)Find Orthogonal trajecturics of y = cx?
   • 3) Solve the following linear equation = + =
4. Q4 Attempt any three, [each 5 Mks]
   • 1)If a,b,c prove that (a+ b)(a+c)(b + c) 8abc
   • 2) Write the expression [5 — interva! form
   • 3) whether the following sequences are monoionic x, = —,
   • 4) Use Sandwich Theorem to show following sequence is convergent x, = = N
   • 5) Solve the differential equation + xy =
   • 6) Solve the differential equation (x? — y)dx — xdy = 0

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