Free Printable Real Analysis Quiz By Safdar Cheema

1.

Which of the functions given below satisfy all three conditions of Rolle's theorem?

a)

f(x) = cos(x) , for x in [0 , 2π]

b)

f(x) = |x - 2| , for x in [0 , 4]

c)

fx) = 1 / x² , for x in [-1 , 1]

d)

f(x)=lxl, for x in [-1,1]

2.

$f\left(x\right)=x\$ for this rolles theorem

a)

fails

b)

it satisfies conditions of rolles theorem

3.

a)

yes

b)

no

4.

if f(x) is continuous then

a)

it is differentiable

b)

not differentiable

5.

$f(x)=\frac{\sin(x)}{x}$

How many points exist such that f'(c) = 0 in the interval [0, 18π]?

a)

17

b)

18

c)

19

d)

20

6.

Is the function $f\left(x\right)=\left|x\right|$ continuous

a)

yes

b)

no

7.

A closed set A in a metric space M is nowhere dense if and only if complement of A is

a)

closed

b)

open

c)

nowhere dense

d)

everywhere dense

8.

True or False: A metric space which is of second category is complete

a)

True

b)

False

9.

Which of the following spaces are second category

a)

R with usual metric

b)

R with discrete metric

c)

Q with discrete metric

d)

Q with usual metric

10.

A subset A of a complete metric space is complete if and only if A is

a)

bonded

b)

dense in M

c)

closed

d)

open

11.

Which of the following spaces are complete?

a)

R with usual metric

b)

R with discrete metric

c)

Q with discrete metric

d)

$l_2$ space with usual metric

12.

True or False: 1. Any subset of a discrete metric space has no limit points

a)

True

b)

False

13.

Let E be a closed set and p is a limit point of E. Then E\{p} is

a)

a closed set

b)

an open set

c)

a bounded set

d)

neither an open set nor a closed set

14.

Let M be a metric space. Then the only set which is both closed and dense is

a)

empty set

b)

finite set

c)

M

d)

None of these

15.

Let A be a set which intersects every open set. Then A must be

a)

a bounded set in M

b)

dense set in M

c)

an open set in M

d)

a closed set in M

16.

Which of the following is closed set in R

a)

[a, b]

b)

{a}

c)

(1, 2) $\cup$ {1}

d)

[1/n, 1] $\cup$ {0}

17.

In R with usual metric, B(1, 1) =

a)

[1, 1]

b)

(-1, 1)

c)

(-2, 0)

d)

(0, 2)

18.

True or False: 1. For every distinct points x, y, there always exists disjoint open sets containing x and y respectively

a)

True

b)

False

19.

Which of the following is true?

a)

$A\subseteq Int\ A$

b)

$Int\ A\subseteq A$

c)

$A\subseteq\overline{A}$

d)

$\overline{A}\ \subseteq A$

20.

Interior of (1, 1) is

a)

{1}

b)

(1, 1)

c)

Empty set

d)

None of these

21.

Every open ball in R is of the form

a)

(a, b)

b)

[a, b)

c)

(a, b]

d)

None of these

22.

True or False: 1. Any set M can have a metric in it

a)

True

b)

False

23.

Any subset of a discrete metric space is

a)

Bounded

b)

Open

c)

Closed

d)

Complete

24.

True or False: Let d(x, y) = |x - 2y|, where x, y $\in$ M and M is a non-empty set. Is d is a metric on M?

a)

True

b)

False

25.

A set which has a metric in it is called

a)

Metric space

b)

Complete space

c)

Complete metric space

d)

None of these

26.

Based on the chart, how many subsets are within the integers set?

a)

1

b)

2

c)

3

d)

none

27.

Let N be the set of Natural numbers and E be the set of Even numbers, then N and E are equivalent sets.

a)

True

b)

False

28.

Which of the following number is not irrational.

a)

$\sqrt{5}$

b)

$\sqrt{2}$

c)

$\sqrt{7}$

d)

$\sqrt{4}$

29.

Set of natural number is

a)

bounded above

b)

Not bounded above

c)

finite

d)

None of these

30.

Which of the following statements is incorrect?

a)

E ,set of even numbers is countable.

b)

Z is countable

c)

N is countable

d)

R is countable.

31.

Two sets A and B are said to be equivalent,if there exists a ______from A to B

a)

injection

b)

bijection

c)

surjection

d)

homomorphism

32.

Every set A is equivalant to itself

a)

True

b)

False

33.

Infinite countable set can be equivalant to its proper subset

a)

True

b)

False

34.

There does not exist a set of length zero which is uncountable

a)

True

b)

False

35.

Which of following sets are countable

a)

Z

b)

N

c)

R

d)

Q

Real Analysis Quiz By Safdar Cheema

Total questions: 35

Worksheet time: 2hrs 46mins

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