IIT JAM Mathematics Test Series 2023 : Real Analysis – Uniform Convergence

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of topic Uniform Convergence of the Chapter Real Analysis. These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Shore with Your Friends.

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IIT JAM Mathematics Test Series 2023 : Real Analysis – Uniform Convergence


Q1. For n ≥1. Let gn(x) = sin2(x+1/n) , x∈[0,∞) and fn(x) = 0x gn(t) dt. Then

(a) {fn}converges point-wise to a function f on [0,∞), but does not converge uniformly on [0,∞)
(b) {fn} does not converge to any function on [0,∞)
(c) {fn} converges uniformly on [0,1)
(d) {fn} converges uniformly on [0,∞)

Q2. Let fn : [1,2] → [0,1] be given by fn(x) = (2-x)n for all non-negative integers n . Let f(x) = limn→∞ fn(x) for 1 ≤ x ≤ 2. Then which of the following is true ?

(a) f is a continuous function on [1,2]
(b) fn converges uniformly to f on [1,2] as n→∞
(c) limn→∞ 1∫2 fn(x)dx = 12 f(x) dx
(d) for any a∈(1,2) we have limn→∞ f’n(a) ≠ f'(a)

Q3. Let fn(x) = xn/(1+x) and gn(x) = xn/(1+nx) for x∈N. Then on the interval [0,1]

(a) both {fn} and {gn} converges uniformly
(b) neither {fn} nor {gn} converges uniformly
(c) {fn} converges uniformly but {gn} does not converge uniformly
(d) {gn} converges uniformly but {fn} does not converges uniformly

Q4. Which of the following sequence {fn}n=1 of functions does NOT converge uniformly on [0,1] ?

(a) fn(x) = (e-x)/n
(b) fn(x) = (1-x)n
(c) fn(x) = (x2 + nx)/n
(d) fn(x) = sin(nx+n)/n

Q5. Let fn(x) = x1/n for n∈[0,1]. Then

(a) limn→∞ fn(x) exists for all x∈[0,1]
(b) limn→∞ fn(x) defines a continuous function on [0,1]
(c) {fn} converges uniformly on [0,1]
(d) limn→∞ fn(x) = 0 for all x∈[0,1]

Q6. Let f(x) = Σn=1 [sin(nx)]/n2. Then

(a) limn→∞ f(x) = 0
(b) limn→∞ f(x) = 1
(c) limn→∞ f(x) = π2/6
(d) limn→∞ f(x) does not exist

Q7. Let {fn} be a sequence of real valued differentiable function on [a,b] such that fn(x) → f(x) as n→∞ for every x∈[a,b] and for some Riemann-integrable function f:[a,b] → R . Consider the statements :

P1 : {fn} converges uniformly
P2 : {f’n} converges unformly
P3 : ab fn(x) dx → ab f(x) dx
P4 : f is differentiable

Then which one of the following need NOT be true

(a) P1 implies P3
(b) P2 implies P1
(c) P2 implies P4
(d) P3 implies P1

Q8. Let fn(x)=(-x)n , x∈[0,1]. Then decide which of the following are true.

(a) there exist a point-wise convergent Subsequence of fn
(b) fn has non point-wise convergent subsequence.
(c) fn converges point-wise everywhere.
(d) fn has exactly one point-wise convergent subsequence.

Q9. Consider all sequences {fn} of real valued continuous functions on [0,∞). Identify which of the following statements are correct.

(a) If {fn} converges to f point-wise on [0,∞) then limn→∞ 0 fn(x) dx = 0 f(x) dx
(b) If {fn} converges to f uniformly on [0,∞) then limn→∞ 0 fn(x) dx = 0 f(x) dx
(c) If {fn} converges to f point-wise on [0,∞) then f is continuous on [0,∞)
(d) There exists a sequence of continuous functions {fn} on [0,∞) such that fn converges to f uniformly on [0,∞) but limn→∞ 0 fn(x) dx ≠ 0 f(x) dx

Q10. Which one of the following statements is true for the sequence of functions

fn(x) = 1/(n2+x2), n=1,2,3…. , x ∈[1/2,1]

(a) The sequence is monotonic and has 0 as the limit for all x ∈[1/2,1] as n→∞
(b) The sequence is not monotonic but has f(x) = 1/x2 as the limit as n→∞
(c) The sequençe is monotonic and has f(x) = 1/x2 as the limit as n→∞
(d) The sequence is not monotonic but has 0 as the limit.

Q11. Let fx(x) = sinx/√n, n = 1,2,3,… and x ∈ [-1,1]. Choose incorrect

(a) {fn(x)}n=1 does not converge uniformly in [-1,1]
(b) limn→∞ -11 fn(n) dx ≠ 0
(c) {fn(x)} does not converge uniformly in [-1,1]
(d) fn(x) , n = 1,2 is not uniformly continuous in [-1,1]

Q12. Consider two sequence {fn} and {gn} of functions where fn:[0,1] → R and gn: R → R are defined by fn(x) = xn and gn(x) = cos(n-x)π/2 if x∈[n-1, n+1] and gn(x) = 0 otherwise. Then

(a) Neither {fn} nor {gn} is uniformly convergent
(b) {fn} is not uniformly convergent but {gn} is
(c) {gn} is not uniformly convergent but {fn}
(d) Both {fn} and {gn} are uniformly convergent

Q13. Let {fn} be a sequence of continuous functions on R.

(a) If {fn} converges to f point-wise on R then limn→∞ -∞ fn(x) dx = -∞ f(x) dx
(b) If {fn} converges to f uniformly on R limn→∞ -∞ fn(x) dx = -∞ f(x) dx
(c) If {fn} converges to f uniformly on R then f is continuous on R
(d) There exists a sequence of continuous functions {fn} on R, such that {fn} converges to f uniformly on R , but limn→∞ -∞ fn(x) dx ≠ -∞ f(x) dx

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Answer Key

01.(c), (d)06.(a)11.(a), (b), (c), (d)
02.(c)07.(d)12.(a)
03.(d)08.(a), (b)13.(c), (d)
04.(b)09.(c), (d)
05.(a)10.(a)

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