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Test: Real Analysis - 5 - Mathematics MCQ


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20 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Test: Real Analysis - 5

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Test: Real Analysis - 5 - Question 1

If S is a set of real numbers, and c1 and c2 are two least upper bounds of S, then

Test: Real Analysis - 5 - Question 2

Every integer n > 1 is

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Test: Real Analysis - 5 - Question 3

If E is a non-empty set, then

Test: Real Analysis - 5 - Question 4

Let R = {(1,3 ), (2,2), (3 ,2 )} and S = {(2 ,1 ) (3,2), (2 ,3 )} be two relations of set ,4 = {1 , 2 ,3 } . Then , RoS is equal to

Test: Real Analysis - 5 - Question 5

​Statement (A): Set S of real number is bounded above, if sup S is finite.
Statement (B): Set S of real numbers is unbounded above, if sup S = ∞

Test: Real Analysis - 5 - Question 6

What does the shaded region in the following diagram represent?

Test: Real Analysis - 5 - Question 7

An integer m is said to be related to another integer n, if m is a multiple of n. Then, the relation is

Detailed Solution for Test: Real Analysis - 5 - Question 7

For any integer n, we have n | n => nRn
So, nRn for all n ∈ Z implies R is reflexive
Now, 2|6 but 6 + 2,
implies (2 ,6) ∈ R but (6,2) ∉ R
So, R is not symmetric.
Let (m, n) ∈ R and (n , p) ∈ R
Then,

implies m | p => (m,p) ∈ R
So, R is transitive.
Hence, R is reflexive and transitive but it is not symmetric.

Test: Real Analysis - 5 - Question 8

The order of a set A is 3 and that of a set B is 2. What is the number of relations from A to B?

Detailed Solution for Test: Real Analysis - 5 - Question 8

since, n(A) = 3 and n(B) = 2 Therefore, Number of relations from

= 64

Test: Real Analysis - 5 - Question 9

Which one of the following is the empty set?

Detailed Solution for Test: Real Analysis - 5 - Question 9

Since, x2 + 1 = 0, given x2 = -1 
x = ± i
Therefore, x is not real but x is real (given)
Hence, No value of x is possible.

Test: Real Analysis - 5 - Question 10

Which one of the following is correct?
Here, P(A) denotes the power set of a set A.

Detailed Solution for Test: Real Analysis - 5 - Question 10

A-P(A)=A,
which is correct as A and P(A) are disjoint sets.

Test: Real Analysis - 5 - Question 11

Let A = {x : x ∈ R, | x | < 1};
B = { x : x ∈ R, |x — 1|  ≥ 1} and A ∪ B = R - D , then the set D is

Detailed Solution for Test: Real Analysis - 5 - Question 11

A =[x : x ∈ R, -1 < x < 1]
B =[x : x ∈ R : x - 1 ≤ - 1 or x - 1 ≥ 1] =[x :x ∈ R : x ≤ 0 o r x ≥ 2]
Hence, A ∪ B = R - D , where
D = [x : x ∈ R, 1 ≤ x < 2]

Test: Real Analysis - 5 - Question 12

The set of intelligent students in a class is

Detailed Solution for Test: Real Analysis - 5 - Question 12

Since, intelligency is not defined for students in a class i. e., not a well defined collection.

Test: Real Analysis - 5 - Question 13

“Every non-empty set S of real numbers which is bounded above has supremum” is

Detailed Solution for Test: Real Analysis - 5 - Question 13

Test: Real Analysis - 5 - Question 14

For real numbers x and y, we write xRy <=> x - y + √2 is an irrational number. Then, the relation R is

Detailed Solution for Test: Real Analysis - 5 - Question 14

For any x ∈ R, we have x - x +√2 = √2 an irrational number.
implies xRx for all x. So, R is reflexive.
R is not symmetric, because and R is not transitive also because and but .

Test: Real Analysis - 5 - Question 15

If A and B are two sets satisfying A - B = B - A, then which one of the following is correct?

Detailed Solution for Test: Real Analysis - 5 - Question 15

Since, A and B are two sets satisfying A - B = B - A , which is possible only if A = B.

Test: Real Analysis - 5 - Question 16

Sets A and B have n elements in common. How many elements will (A x B) and (B x A) have in common?

Detailed Solution for Test: Real Analysis - 5 - Question 16

The total number of elements common in (A x B) and (B x A) is n2.(by property)

Test: Real Analysis - 5 - Question 17

If (A - B) ∪ (B - A) = A for subsets A and B of the universal set U, then which one of the following is correct?

Detailed Solution for Test: Real Analysis - 5 - Question 17

Here,

or A = φ
Here, B = φ satisfy the given condition (i)

Test: Real Analysis - 5 - Question 18

The equation of the curve for which the square of the ordinate is twice the rectangle contained by the abscissa and the intercept of the normal on x-axis and passing through (2, 1) is

Detailed Solution for Test: Real Analysis - 5 - Question 18

∵ Equation of normal at (x, y) is
x − y = dx/dy (X − x)
Put, y = 0
Then, X = x + y dy/dx
Given, y2 = 2x X
⇒ y2 = 2x ( x + y dy/dx)
⇒ dy/dx = (y2 − 2x2)/(2xy) = ((y/x)2 − 2)/ (2y/x)
Put y = vx, we get dy/dx = v + x dv/dx
Then, v + x dv/dx = v2−2/2v
On integrating both sides, we get
ln (2 + v2 ) + ln|x| = ln c
⇒ ln (|x|(2 + v2 )) = ln c
⇒ |x| ( 2 + y2/x2) = c
∵ It passes through (2, 1), then 2 (2 + 1/4 ) = c
⇒ c = 9/2
Then, |x| ( 2 + y2/x2 ) = 9/2
⇒ 2x2 + y2 = 9/2 |x|
⇒ 4x2 + 2y2 = 9|x|

Test: Real Analysis - 5 - Question 19

Let A = {(x, y) : y = ex, x ∈ R}, B = {(x, y) : y = e-x,x ∈ R}. Then

Detailed Solution for Test: Real Analysis - 5 - Question 19

Since, y = ex, y = e-x will meet,
when ex = e-x
implies e2x = 1,
So, x = 0,y = 1
Hence, A and B meet on (0,1),
Hence, A ∩ B = φ

Test: Real Analysis - 5 - Question 20

If A, B and C are three sets and U is the universal set such that n(U) = 700, n(A) = 200, n(B) = 300 and h (A ∩ B) = 100, then what is the value of (A' ∩ B')?

Detailed Solution for Test: Real Analysis - 5 - Question 20

Given that, n(U) = 700, n(A) = 200, n(B) = 300 and n(A ∩ B)=100
We know that,


= 700-400 = 300

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