R a test series praveen chikara

[#] Topics: Set theory, Countable Sets, Sequences and Series.. [#] Topics: Continuity, Uniform Continuity, Differentiability.. [#] Topics:: Set theory, Countable sets, Sequences and Se

TABLE OF CONTENTS

Real Analysis Test 1……….…… [#]

Topics: Set theory, Countable Sets, Sequences and Series

Real Analysis Test 2……….…… [#]

Topics: Continuity, Uniform Continuity, Differentiability

Real Analysis Test 3……….…… [#]

Topics:Uniform Convergence, Riemann Integration

Real Analysis Test 4……….…… [#]

Topics:Metric Spaces, Functions of Several Variables

Real Analysis Test 5……….…… [#]

Topics:: Set theory, Countable sets, Sequences and Series, Point-set topology, Continuity, Uniform Continuity, Differentiability, Uniform Convergence,Riemann Integration, Metric Spaces

12 thousand via teaching, social social m

Real Analysis Test-1

Praveen Chhikara

Real Analysis Test-1

“Be transparent Let’s build a community that allows hard questions andhonest conversations so we can stir up transformation in one another.” -Germany Kent

Time duration: 80 minutes

Max points: 99 -Praveen ChhikaraTopics: Set theory, Countable Sets, Sequences and Series

(b) is bounded but not convergent

(c) is convergent and its limit is e

(d) is convergent and its limit is 1/e.Praveen Chhikara

3 Consider a sequence (an) in R Then

(a) if all convergent subsequences of (an) converge to 0, then (an) → 0.(b) if all monotone subsequences of (an) are convergent, then (an)must be bounded

(c) (an) is divergent if, and only if all of its subsequences are divergent.(d) If (an) diverges, then all of its subsequences diverge

4 Which of the following is an incorrect statement?

(b) every monotone convergent sequence is of bounded variation.(c) if (an) and (bn) are of bounded variation, then (an− bn) may not

(b) for every x ∈ V , there must exists an n0 ∈ N such that xn 0

=k x k1

(c) for every x ∈ V , there must exists an n0 ∈ N such that xn 0

=k x k2

(d) the dimension of V is uncountable infinite

7 Let A, B ⊆ R be nonempty sets Which of the following is wrong?

(a) the density of the set {2, 4, 6, } is 2

(b) there exists a finite subset of N with positive density

(c) if the density of a set is zero, then it must be a finite set

(d) for each 0 ≤ α ≤ 1, there exists a set with density equal to α

9 Let A := mn

(a) sup A < 1 and inf A = 1/3

(b) sup A ∈ A and inf A /∈ A

(c) sup A = 1 and inf A = 0

(d) sup A does not exist and inf A ∈ A

10 Let A be the collection of all circles in the complex plane having rationalradii and centers with rational coordinates Let B be the collection ofdisjoint intervals of positive length Then

(a) A is countable but B is uncountable

(b) B is countable but A is uncountable

(c) both A and B are countable

(d) both A and B are uncountable

11 Let f : X → Y be a function Pick the odd statement

(a) f (A ∩ B) = f (A) ∩ f (B), for all A, B ∈ X.Praveen Chhikara

(b) f−1(f (A)) = A, for all A ⊆ X.

(c) there exists a, b ∈ X such that a 6= b, f (a) = f (b)

(d) for all disjoint A, B ⊆ X, f (A) ∩ f (B) = ∅

12 The range of the function f : R → R, defined by f (x) = 1+x−[x]x−[x] , where[.] denotes the greatest integer function,

(a) has its supremum equal to 1

(b) does not have its minimum element

(c) is unbounded below

(d) does not have its maximum element

13 Pick the series which converge

(a) P sin 1

n.(b) P n2

2 )n Define a sequence (bn)

by bn = d(an, Z) where d(an, Z) = inf{d(an, x) : x ∈ Z} Then

(a) the sequence (an) is not convergent

(b) the sequence (bn) is convergent

(c) the sequence (bn) is bounded

(d) the sequence (a1

n) is convergent

16 Let (aPraveen Chhikaran) be a sequence given by an= nαsin n, where α ∈ R Then

(a) (an) has a convergent subsequence, for α = 1/2.

(b) (an) does not have a bounded subsequence, for α = 3/2

(c) (an) has a convergent subsequence, for all α < 0

(d) for α = 0, and c ∈ (−12 ,12), there exists a subsequence (ank) of(an), such that ank → c

17 Let P an be a series with real-valued terms Define

pn:= |an| + an

2 , qn:=

|an| − an

2 , n ∈ N

Which of the following hold(s) true?

(a) If P an is conditionally convergent, then both P pn and P qndiverge

(b) If P an is conditionally convergent, then one of P pn and P qnmay converge

(c) IfP |an| is convergent, then both P pn and P qn converge.(d) IfP |an| is convergent, then one ofP pn and P qn may diverge

18 Let (an) be a bounded sequence in R Suppose S denotes the range of(an) Then

(a) there exist convergent and monotone subsequences (ank) and (amk)such that lim

(c) has countable index

(d) has uncountable index.Praveen Chhikara

20 Which of the following series is(are) convergent?

21 Let a, b ∈ Z The functions f : N × N → N and g : Z × Z → Z given

by f (x, y) = 2x−1(2y − 1) and ga,b(x, y) = ax + by respectively Then(a) f is one-to-one

(b) f is onto

(c) there exists a, b ∈ Z such that ga,b is onto

(d) there exists a, b ∈ Z such that ga,b is one-to-one

22 Which of the following statement(s) is(are) true in relation to theCauchy sequences in R?

(a) There exists a Cauchy sequence with rational terms converging to

(b) P an has the bounded sequence of partial sums

(c) P bn has the bounded sequence of partial sums

(d) P bn has a convergent sub-series

24 Which of the following sets is(are) uncountable?

(c) a nonempty open set in R.

k 2 ) does not converge

26 Which of the following hold(s) for real sequences (an) and (bn)?(a) If (an) and (anbn) are convergent, then so is (bn)

(b) If (an) and (bn) are unbounded, then so is (an+ bn)

1 D 2 C 3 B 4 C 5 B 6 B 7 C 8 D

9 D 10 C 11 C 12 D 13 C 14 C

Multi-Correct Questions

15 A,B,C,D 16 A,C,D 17 A,C 18 A,B,C,D 19 B,C

20 C 21 A,B,C 22 A,B,D 23 A,B,D 24 C 25 A,B,D 26 C

Best Wishes from Praveen Chhikara

Praveen Chhikara

Real Analysis Test-2

Praveen Chhikara

Real Analysis Test-2

“The sole meaning of life is to serve humanity” - Leo Tolstoy

Time duration: 80 minutes

Max points: 115.50 -Praveen Chhikara

Topics: Continuity, Uniform Continuity, Differentiability

(d) If f is continuous, and F 6= ∅, then F is uncountable

2 Which of the following statements implies the continuity of the function

f : R → R?

(a) f2 is continuous

(b) f (f (x)) is continuous.Praveen Chhikara

(b) equal to that of the open interval (0, 1).

(c) equal to that of Z[√2]

(d) equal to that of N × N

4 Suppose y ∈ R Let fy : R → R be a function defined by

fy(x) = d(x, y), where d(x, y) = |x − y| Then

(a) fy is not continuous for some y ∈ R

(b) fy is uniformly continuous for all y ∈ R

(c) there exists a y ∈ R such that fy is differentiable

(d) fy0 is a bounded function for all y ∈ R

5 Which of the following statements is true?

(a) A continuous function f : R → R that is one-to-one must beunbounded

(b) A continuous function f : R → R that is one-to-one must bedifferentiable

(c) A continuous function f : R+ → R that is onto must satisfylim

n→∞|f (x)| = ∞

(d) A continuous function f : R → R that is one-to-one must bestrictly monotone

6 Suppose f : (0, ∞) → R is differentiable and |f0(x)| < 1 Then

(a) (an), where an= f (23)n, is not convergent

(b) (bn), where bn = f (n+1n ), is not convergent

(c) (cn), where cn= f (1n), is convergent

(d) there exists an x0 > 0, such that f (x0) = x0

Praveen Chhikara

7 Suppose A := {x ∈ [0, 1] : x = 2mn for some m ∈ Z, n ∈ N} and

B := Z[√2] ∩ [0, 1] Let f : [0, 1] → R be a function defined by

Then

(a) f is continuous at all points x /∈ A ∪ B

(b) f is discontinuous at each point of its domain

(c) f is differentiable at exactly one point of its domain

(d) f is continuous at exactly one point but differentiable nowhere

8 Suppose f : (a, b) → R is differentiable and |f0(x)| 6= 0 for x ∈ (a, b).Then

(a) f need not be one-to-one

(b) f is strictly monotone

(c) f0 is bounded

(d) there exists some c, d ∈ (a, b) such that f0(c)f0(d) < 0

9 Suppose f : [0, 1] → R is a function and S is the range of f Then(a) If f is strictly increasing, then S may be a subset of Q

(b) If f is increasing, then S may be a unbounded

(c) If f is strictly increasing and continuous, then S may be a subset

of Q

(d) If f is strictly increasing and continuous, then S cannot be a subset

of R \ Q

10 Let f : R → R be a differentiable function Suppose f (0) = 1 and

f0(x) ≤ 5 for all x ∈ (0, 7) Then f (2) is atmost equal to

(a) 9

(b) 10

(c) 11

(d) 12Praveen Chhikara

2 Multi-correct Questions

11 For E ⊂ R, consider the following statements:

P : Every continuous function f : E → R is uniform continuous.Q: E is compact

R: Every continuous function f : E → R is bounded

Which of the following is(are) false?

(a) R ; P

(b) P ⇒ Q

(c) Q ⇒ P

(d) R ⇒ Q

12 The equation x sin x + cos x − x2 = 0 has

(a) at least two solutions in R

(b) at most two solutions in R

(c) exactly one positive real solution

(d) exactly one negative real solution

13 Let f : [0, ∞) → R be a continuous function Suppose that f isdifferentiable for all x > 0 and that lim

x→∞f0(x) = 0 Further supposelim

x→∞(f (x) + f0(x)) exists finitely Then

(d) f must be a bounded function

14 For A ⊆ R, the function f : A → R, f (x) = x2 is a uniform continuousfunction if

(a) A is bounded

(b) A is unbounded

(c) A is compact

(d) A is the set Z of all integers

15 Let T : RPraveen Chhikaran → Rm be a linear transformation Then

(a) T need not be continuous.

(b) T is differentiable

(c) T is differentiable but not continuously differentiable

(d) the partial derivatives of T need not exist

16 Which of the following is(are) correct inequality(ies)?

17 Let f : R → R be a continuous and periodic function Then

(a) f attains its supremum

(b) f need not be bounded

(c) there do not exist sequences (xn) and (yn) in R such that

(a) f attains infimum but not supremum

(b) f attains supremum but not infimum

(c) f attains neither an infimum nor a supremum

(d) f attains its supremum and infimum on any nonempty open terval A ⊂ [0, 1]

in-19 Let f : R → R be the characteristic function of the Cantor set C.Then,

20 A function f : R → R is said to be symmetrically continuous at x ∈ R

if lim

h→0[f (x + h) − f (x − h)] = 0 Which of the following is(are) true?(a) There exists a continuous function at x which is not symmetricallycontinuous at x

(b) A symmetrically continuous function at x must be continuous atx

(c) If f is symmetrically continuous at x, then lim

(a) if f is strictly monotone, then A must be empty

(b) if f has IVP on (a, b) and is discontinuous, then A must benonempty

(c) if f has IVP, then A can be non-compact

(d) if f has IVP, then A can be compact

22 Let f : R → R be a continuous function Then

(a) if f is uniformly continuous on both (−∞, 0] and [0, ∞), then f

X1, X2, , Xn, , then f is uniformly continuous on

∞

S

i=1

Xi

23 Which of the following is(are) true about real-valued functions?

(a) There exists a strictly monotone function f on [0, 1] that is continuous at each point of (0, 1) ∩ Q and continuous at each point

dis-of (0, 1) ∩ (R \ Q)

Praveen Chhikara

(b) If f is continuous on an open interval and has no local maximum

or local minimum, then f must be monotone

(c) There exists a monotone function on [0, 1] that is discontinuousonly at points of (0, 1) ∩ (R \ Q)

(d) If f : [a, b] is monotone, then f must be a continuous on an countable dense set of [0, 1]

un-24 Let f : I → R be a function where I be a nondegenerate interval in R

We define the oscillation of f on I by ωf (I) = sup

x,y∈I

|f (x) − f (y)| andoscillation of f at x0 ∈ I by ωf(x0) = inf

25 Pick the correct statement(s)

(a) There exists a function f : R → R such that

f0(x) =

(0; if x < 01; if x > 0

(b) If f : R → R is differentiable and there is an M > 0 such that

26 Let f : R → R be a differentiable function If f0(x) 6= 1 for all x ∈ R,then

(a) f has at least one fixed point

(b) f may have more than one fixed point

(c) f has a unique fixed point

(d) f may have no fixed point.Praveen Chhikara

27 Let f : [a, b] → R be a differentiable function Suppose there does notexist any x ∈ [a, b] for which f (x) = f0(x) = 0.

Let A := {x ∈ [a, b] : f (x) = 0} be nonempty Then A

Then

(a) if a ∈ C, lim

x→af (x) exists

(b) f is continuous at uncountable number of points

(c) f is discontinuous at most countable infinite number of points.(d) f is differentiable no-where

Answer KeySingle-Correct Questions

1 D 2 C 3 B 4 B 5 D 6 C 7 D 8 B

9 D 10 C

Multi-Correct Questions

11 A,B 12 A,B,C,D 13 A,B,D 14 A,B,C,D 15 B

16 A,B,C,D 17 A,C,D 18 A,B,C,D 19 B,D 20 D

21 A,C,D 22 A,B,C 23 A,B 24 A,B,D 25 B,C 26.A,C 27 A,B,C 28 B,D

Best Wishes from Praveen Chhikara

Praveen Chhikara

Real Analysis Test-3

Praveen Chhikara

Real Analysis Test-3

“Every great dream begins with a dreamer Always remember, you havewithin you the strength, the patience, and the passion to reach for the stars

to change the world.” - Harriet Tubman

Time duration: 55 minutes

Max points: 81 -Praveen ChhikaraTopics: Uniform Convergence, Riemann Integration

1 Let (fn) be a sequence of functions on [0, 1] defined by

fn(x) =

(n; if 0 < x < 1n0; if x = 0 or x ≥ n1.Suppose (fn) → f Then

2 Suppose f : [−1, 1] → R is a function given by

f (x) =

(1; if x 6= 1,0; if x = 1

If g is the indefinite integral of f , then

(a) g is not of bounded variation on [−1, 1]

(a) converges to the zero function on [0, 1]

(b) converges uniformly on [0, 1]

(c) converges to a function which is discontinuous at x = 1

(d) converges to a non-differentiable function on [0, 1]

4 For the series

(b) The sum of the series is continuous on R

(c) The sum of the series is monotone in some nonempty interval ofR

(d) The sum of the series is not differentiable at any point of R

5 Suppose that f : [a, b] → R is Riemann integrable Let F (x) =

x

R

a

f (t)dt Which of the following is correct?

(a) If f is discontinuous at x0 ∈ [a, b], then F (x) is discontinuous at

x0

Praveen Chhikara

(b) Even if f is discontinuous at x0 ∈ [a, b], F (x) is continuous at x0but is not differentiable at x0.

(c) The function F (x) is continuous but not necessarily a Lipschitzfunction on [a, b]

(d) F0(x), if it exists, need not be equal to f (x)

6 Let {α1, α2, } be an enumeration of the set Q of all rational numbers

If (fn) is a sequence of functions given by

(a) (fn) converges uniformly on [0, 1]

(b) (fn) converges to a Riemann integrable function on [0, 1]

(c) f (x) = lim

m→∞cos2m(n!πx) for all x ∈ R

(d) f (x) = lim

n→∞fn(x) is a function of bounded variation on [0, 1]

7 Let f : [0, 1] → R be a bounded function Suppose ℘[0, 1] denotesthe collection of all partitions of [0, 1] Further suppose U (f, P ) andL(f, P ) denote respectively the upper sum and the lower sum of f for

P ∈ ℘[0, 1] If there exist sequences (an) in R and (Pn) in ℘[0, 1] suchthat

U (f, Pn) − L(f, Pn) < an,then which of the following guarantees the Riemann integrability of f

(a) f is not continuous at all x ∈ (−1, 1).

(b) f is continuous but need not be differentiable at all x ∈ (−1, 1)

0 for each continuous function g : [0, 1] → R Then

(a) f must be differentiable everywhere

(b) f need not be Riemann integrable on [0, 1]

(c) f must be Riemann integrable but need not be of bounded tion on [0, 1]

(d) f0(a) 6= g(a) for some nonzero real number a

11 Let fn, n ∈ N and f be differentiable functions with fn → f on [0, 1].Suppose f is bounded and Riemann integrable Then which of thefollowing ensure(s) that lim