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Objective Questions

Latest IIT JAM Mathematical Statistics Exam Question (Objective Questions), MCQ in English

Subjects : Mathematical Statistics

Question Bank IIT JAM Mathematical Statistics Exam (Only MCQ) - English
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Mathematical Statistics

Question 1Single Answer
What is the limit of the sequence \( an = \frac{2n + 3}{n + 5} \) as \( n \to \infty \)?
  1. A1
  2. BDiverges
  3. C3
  4. D2
Question 2Single Answer
What is the limit of the sequence \( an = \frac{\sin(n)}{n} \) as \( n \to \infty \)?
  1. A1
  2. B0
  3. C-1
  4. DDiverges
Question 3Single Answer
What is the limit of the sequence \( an = \sqrt{n + 1} - \sqrt{n} \)?
  1. A\(\infty\)
  2. B1
  3. C0
  4. D-1
Question 4Single Answer
What is the limit of the sequence \( an = \frac{\ln(n)}{n} \)?
  1. A1
  2. B0
  3. C\(\infty\)
  4. D-1
Question 5Single Answer
Does the sequence \( an = n \sin\left(\frac{1}{n}\right) \) converge?
  1. AYes, to 1
  2. BYes, to 0
  3. CNo, diverges
  4. DYes, to \(\pi\)
Question 6Single Answer
What can be said about a sequence \( an \) if it is Cauchy?
  1. AIt diverges
  2. BIt is monotonic
  3. CIt is unbounded
  4. DIt is bounded and convergent
Question 7Single Answer
Is the sequence \( an = n \) a Cauchy sequence?
  1. AYes
  2. BOnly for finite \( n \)
  3. CNo
  4. DDepends on \( n \)
Question 8Single Answer
What is true for a sequence \( an \) that is not Cauchy?
  1. AIt converges
  2. BIt diverges
  3. CIt is bounded
  4. DIt is monotonic
Question 9Single Answer
Does the sequence \( an = \frac{n}{n + 1} \) satisfy the Cauchy criterion?
  1. AYes
  2. BNo
  3. COnly for \( n > 1 \)
  4. DOnly for \( n < 1 \)
Question 10Single Answer
What is the Cauchy condition for a sequence \( an \)?
  1. AFor every \( \epsilon > 0 \), there exists \( N \) such that \( |an - am| < \epsilon \) for all \( n, m > N \)
  2. B\( |an| \to 0 \)
  3. C\( an \to \infty \)
  4. D\( an \) is bounded
Question 11Single Answer
Is the sequence \( an = \cos\left(\frac{n\pi}{2}\right) \) Cauchy?
  1. AYes
  2. BNo
  3. COnly for even \( n \)
  4. DOnly for odd \( n \)
Question 12Single Answer
What is the limit of the Cauchy sequence \( an = \frac{1}{n} \)?
  1. A\(\infty\)
  2. B1
  3. C0
  4. DDoes not exist
Question 13Single Answer
Is the sequence \( an = \frac{n}{n + 1} \) monotonic?
  1. AConstant
  2. BDecreasing
  3. CNeither
  4. DIncreasing
Question 14Single Answer
What is the limit of the monotonic sequence \( an = 1 - \frac{1}{n} \)?
  1. A0
  2. B1
  3. C-1
  4. D\(\infty\)
Question 15Single Answer
Does a bounded monotonic sequence always converge?
  1. AYes
  2. BNo
  3. COnly if increasing
  4. DOnly if decreasing
Question 16Single Answer
What is the limit of the sequence \( an = \frac{2n}{2n + 1} \), which is monotonic?
  1. A1
  2. B0
  3. C2
  4. D\(\infty\)
Question 17Single Answer
What happens to an unbounded increasing sequence?
  1. AConverges to 0
  2. BDiverges to \(\infty\)
  3. CConverges to 1
  4. DDiverges to \(-\infty\)
Question 18Single Answer
Is the sequence \( an = \frac{n + 1}{n} \) monotonic?
  1. AIncreasing
  2. BConstant
  3. CNeither
  4. DDecreasing
Question 19Single Answer
What is the limit of the sequence \( an = \frac{\sqrt{n}}{n} \)?
  1. A-1
  2. B1
  3. C\(\infty\)
  4. D0
Question 20Single Answer
What is the limit of the sequence \( an = \frac{n}{\ln(n)} \)?
  1. A0
  2. B\(\infty\)
  3. C1
  4. D-1
Question 21Single Answer
What is the limit superior of the sequence \( an = \frac{1}{n} \)?
  1. A\(\infty\)
  2. B1
  3. C0
  4. D-1
Question 22Single Answer
What is the limit inferior of the sequence \( an = \sin(n) \)?
  1. A0
  2. B1
  3. C-1
  4. D\(\infty\)
Question 23Single Answer
Does the sequence \( an = n \) have a limit superior?
  1. AYes, 0
  2. BYes, 1
  3. CNo, it is \(-\infty\)
  4. DNo, it is \(\infty\)
Question 24Single Answer
What is the limit inferior of the sequence \( an = \cos(n) \)?
  1. A0
  2. B1
  3. C\(\infty\)
  4. D-1
Question 25Single Answer
What is true if \( \limsup an = \liminf an \)?
  1. AThe sequence converges
  2. BThe sequence diverges
  3. CThe sequence is unbounded
  4. DThe sequence is Cauchy
Question 26Single Answer
What test is best for the series ( sum{n=1}infty frac{1}{2n + n} )?
  1. AComparison test
  2. BRatio test
  3. CRoot test
  4. DIntegral test
Question 27Single Answer
Does the series ( sum{n=1}infty frac{1}{n4} ) converge?
  1. AOnly for ( n > 4 )
  2. BNo
  3. CYes
  4. DOnly for ( n < 4 )
Question 28Single Answer
Is the series ( sum{n=1}infty frac{n}{n2 + 1} ) convergent?
  1. AYes
  2. BOnly for ( n > 1 )
  3. CNo
  4. DOnly for ( n < 1 )
Question 29Single Answer
What does the comparison test state for the series \( \sum an \) and \( \sum bn \) with \( 0 \leq an \leq bn \)?
  1. AIf \( \sum bn \) diverges, then \( \sum an \) converges
  2. BIf \( \sum an \) converges, then \( \sum bn \) converges
  3. CIf \( \sum bn \) converges, then \( \sum an \) converges
  4. DIf \( \sum an \) diverges, then \( \sum bn \) diverges
Question 30Single Answer
Does the series ( sum{n=1}infty frac{1}{n2 + n} ) converge by the comparison test?
  1. AOnly for ( n > 1 )
  2. BNo
  3. CYes
  4. DOnly for ( n < 1 )
Question 31Single Answer
What is the limit comparison test result for ( sum{n=1}infty frac{1}{n2 + 1} ) compared to ( sum frac{1}{n2} )?
  1. AConverges
  2. BDiverges
  3. CInconclusive
  4. DDepends on ( n )
Question 32Single Answer
Does the series ( sum{n=1}infty frac{n}{2n} ) converge by the ratio test?
  1. ANo
  2. BYes
  3. CInconclusive
  4. DDepends on ( n )
Question 33Single Answer
What is the result of applying D’Alembert’s ratio test to ( sum{n=1}infty frac{n!}{2n} )?
  1. AConverges
  2. BInconclusive
  3. CDiverges
  4. DDepends on ( n )
Question 34Single Answer
Does the series ( sum{n=1}infty frac{1}{n!} ) converge by the ratio test?
  1. ANo
  2. BYes
  3. CInconclusive
  4. DDepends on ( n )
Question 35Single Answer
What is the result of Cauchy’s nth root test for ( sum{n=1}infty left( frac{1}{2} right)n )?
  1. ADepends on ( n )
  2. BDiverges
  3. CInconclusive
  4. DConverges
Question 36Single Answer
Does the series ( sum{n=1}infty frac{n2}{2n} ) converge by the root test?
  1. AInconclusive
  2. BNo
  3. CYes
  4. DDepends on ( n )
Question 37Single Answer
What is the result of the integral test for ( sum{n=2}infty frac{1}{n ln(n)} )?
  1. ADiverges
  2. BConverges
  3. CInconclusive
  4. DDepends on ( n )
Question 38Single Answer
Does the series ( sum{n=1}infty frac{1}{n3} ) converge by the integral test?
  1. ADepends on ( n )
  2. BNo
  3. CInconclusive
  4. DYes
Question 39Single Answer
Does the series ( sum{n=1}infty frac{1}{n} ) converge by Cauchy’s condensation test?
  1. ANo
  2. BYes
  3. CInconclusive
  4. DDepends on ( n )
Question 40Single Answer
What is the result of Cauchy’s condensation test for ( sum{n=1}infty frac{1}{n2} )?
  1. AConverges
  2. BDiverges
  3. CInconclusive
  4. DDepends on ( n )
Question 41Single Answer
Does the series ( sum{n=2}infty frac{1}{n ln(n)} ) converge by Cauchy’s condensation test?
  1. ANo
  2. BYes
  3. CInconclusive
  4. DDepends on ( n )
Question 42Single Answer
What is the result of Cauchy’s condensation test for ( sum{n=1}infty frac{1}{n3} )?
  1. ADepends on ( n )
  2. BDiverges
  3. CInconclusive
  4. DConverges
Question 43Single Answer
Does the series ( sum{n=1}infty frac{ln(n)}{n2} ) converge by Cauchy’s condensation test?
  1. ANo
  2. BYes
  3. CInconclusive
  4. DDepends on ( n )
Question 44Single Answer
What test is Cauchy’s condensation test most useful for?
  1. ASeries with alternating terms
  2. BSeries with exponential terms
  3. CSeries with non-negative, decreasing terms
  4. DSeries with factorial terms
Question 45Single Answer
Does the series ( sum{n=1}infty frac{1}{sqrt{n}} ) converge by Cauchy’s condensation test?
  1. AYes
  2. BNo
  3. CInconclusive
  4. DDepends on ( n )
Question 46Single Answer
What is the result of Cauchy’s condensation test for ( sum{n=1}infty frac{1}{n4} )?
  1. AConverges
  2. BDiverges
  3. CInconclusive
  4. DDepends on ( n )
Question 47Single Answer
Does the series ( sum{n=2}infty frac{1}{n (ln(n))2} ) converge by Cauchy’s condensation test?
  1. AInconclusive
  2. BNo
  3. CYes
  4. DDepends on ( n )
Question 48Single Answer
What is the result of Cauchy’s condensation test for ( sum{n=1}infty frac{n}{2n} )?
  1. AInconclusive
  2. BDiverges
  3. CConverges
  4. DDepends on ( n )
Question 49Single Answer
Is the series ( sum{n=1}infty frac{(-1)n}{n} ) absolutely convergent?
  1. AYes
  2. BOnly for ( n > 1 )
  3. CNo
  4. DOnly for ( n < 1 )
Question 50Single Answer
Does the series ( sum{n=1}infty frac{(-1)n}{n2} ) converge absolutely?
  1. AOnly for ( n < 1 )
  2. BNo
  3. COnly for ( n > 1 )
  4. DYes
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