BSc Mathematics SEM I ATKT MATHEMATICS PAPER 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: 20 marks
3. Q1 Additive inverse of a real number
   • (a) Exists and is unique (b) Does not exist
   • (c) If exists then is unique None of these li. If then
   • (c) Inf AE A,sup AEA (d) of these
   • iii. If0 <x <1 then
   • (d) None of these Iv. The sequence where x,=
   • (a) constant (b)
   • (c) divergent (d) none of these Every Cauchy. sequence in R is
   • (a) an increasing sequence (b) — divergent
   • (c) convergent (d) None of these
   • (d) none of these
   • (c) 0 (d) of these Vili. Tf sequence (X,) of real numbers satisfies , ,VneN then
   • (a) converges to 0 (b) diverges
   • (c) converges to 1 (d) none of these Ix, The graph of a function y= intersects x axis
   • (a) at (0,0) (b) nowhere
   • (c) at every point (d) none of these
4. Q10 The function f(x) = 2x + 3 is continuous
   • (a) Only if x >0 (b) only <0
   • (c) For each R (d) None of these
5. Q2 a) Attempt any ONE question from the following: 8 marks
6. Q1 State the arithmetic mean and geometric mean (AM-GM) inequality for real numbers. Apply it to prove that (a + b)(b + c)(c + a) = 8abc li. State and prove the Archimedean order property for IR. Hence prove that if real number x satisfies 0 < x < for every positive real then x = 0
   • b) Attempt any TWO questions from the following: 12
   • i. Prove that = |x||y| and < |x ER li. State the law of trichotomy of real numbers. Hence prove that the square of any non-zero real number is positive Find an upper bound, a lower bound , the supremum and the infimum for {x:x — 3] < 4} if they exist
   • iv. State only the Cauchy-Schwartz inequality for R. Apply it to prove that
7. Q3 a) Attempt any ONE question from the following: 8 marks
   • i. Let and be two real sequences such that > p and (y,) > q, then prove that (x, > Prove that the real sequence (1 + is convergent in R
   • b) Attempt any TWO questions from the following: Prove that every convergent sequence in R is Cauchy Use — Ng definition to prove that the sequence asn > Give an example of the following 12
   • a) A bounded sequence which is not convergent
   • b) Sequences and (y,) such that (x,) 0 but does not Prove that lim nn = 1
8. Q4 a) Attempt any ONE question from the following: 8 marks
9. Q1 Let f: R > R bea function and let 1. R.When do we say that lim f(x) = ? Prove that if lim f(x) exists then it li. Let f: Rbea function and R. If (f(xn)) converges to f(p) for any sequence (Xn) that converges to p then prove that f is continuous at p
   • b) Attempt any TWO questions from the following: 12
10. Q1 Draw the graph of the function f(x) = x* +1 ,for <2 ll. Using — 6 definition, show that the function f (x) Ris State Sandwich theorem for limit of functions. Use it to prove that
    • iv. If lim f(x) 1 then prove that lim
11. Q5 Attempt any FOUR questions from the following: 20 marks
    • a) Prove that the additive identity in R is unique
    • b) If A and B are bounded subsets of then prove that AUB is bounded in R Show that sequence (=) is Cauchy in R
    • d) Show that the sequence is monotonic and bounded
    • f) For which value of b would the function = bx* be continuous at x = 1?

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