BSc Mathematics SEM I 2022 2023 Nov 2023 MATHEMATICS 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 multiplication and prove that multiplicative identity is unique
   • 2) State and prove Cauchy Schwarz Inequality
   • (b) Attempt any two. [each
   • 1) State and prove Archimedian Property of real numbers. Using it prove that if exist such that <x
   • 2) Define Infimum of a Nonempty subset S of R and Prove that Let S be a subset of R then Infimum of S whenever exist is unique
   • 3) Prove that the square of a real number is always non negative
2. Q2 (a) Attempt any one. [each 8Mks]
   • 1) Prove that Every convergent sequence in R is bounded and give an example of a bounded sequence but not convergent
   • 2) Prove that: If x, # 0 is a sequence in R for anyn N andx # 0 >
   • (b) Attempt any two. [each 6Mks]
   • 1) Prove that A convergent sequence of non-negative real number has a non
   • 2) Prove that If (x,) is a sequence in R such that (x,) x and R, then
   • 3) Use — no) definition to show that has limit 1 as n tends to infinity
3. Q3 (a) Attempt any one. [each 8Mks]
   • 1) Define homogeneous Differential equation & solve: x sin =y sin +X VCD/ FYBSC- SEM I MATHEMATICS I- 75MARKS
   • 2) Define Linear Differential Equation (LDE) & solve LDE: x = 4 with initial condition
   • (b) Attempt any two. [each 6 Mks]
   • 1) Find the current in R-C circuit with R = 202, C = 0.01F, E(t) = 200e7**,1(0) =0
   • 2) Check exactness and solve DE, (2x log x — xy) dy + 2y dx = 0
   • 3) Find a family of orthogonal trajectories for the family of curves =c,c#0
4. Q4 Attempt any three. [each 5 Mks]
   • 1) Find the and of the set S={xeR/x?-x-6<0}
   • 2) Find disjoint neighbourhood of a =1.5, b =1.6
   • 3) For the sequence where terms are given below, state whether they are bounded above or below and also whether they are monotonic increasing or monotonic decreasing
   • 4)Discuss the convergence of (a,) if (ay) = = inR
   • 5)If the population of a country doubles in 50 years, in how many years will it triple under the assumption that the rate of increase is proportional to the number of people in that country?
   • 6) Reduce to first order and solve, y + y” =0

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