Relations And Functions

01

PYQ 2023

medium

mathematics ID: cuet-ug-

The relation R in the set A = {1, 2, 3, 4} is given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)} is :

1

Reflexive and symmetric but not transitive

2

Reflexive and transitive but not symmetric

3

Symmetric and transitive but not reflexive

4

an equivalence relation

02

PYQ 2023

medium

mathematics ID: cuet-ug-

For real numbers a and b, define aRb if b-a+\sqrt5 is an irrational number. Then the relation R is:

1

Symmetric

2

Reflexive

3

Transitive

4

Equivalence

03

PYQ 2023

medium

mathematics ID: cuet-ug-

Let f: R\toR defined by f(x)=2x 3 -7 for x\inR. Then: (A) f is one-one function (B) f is many to one function (C) f is bijective function (D) f is into function Choose the correct answer from the options given below:

1

(A) and (D) only

2

(B) and (D) only

3

(B) and (C) only

4

(A) and (C) only

04

PYQ 2023

easy

mathematics ID: cuet-ug-

Match List I with List II LIST I LIST II A . Range of y=cosec -1 x I . R-(-1, 1) B . Domain of sec -1 x II . (0, π) C . Domain of sin -1 x III . [-1, 1] D . Range of y=cot -1 x IV . -{0} Choose the correct answer from the options given below:

1

(A)-(I), (B)-(II), (C)-(IV), (D)-(III)

2

(A)-(IV), (B)-(I), (C)-(III), (D)-(II)

3

(А)-(III), (B)-(IV), (C)-(II), (D)-(I)

4

(A)-(II), (B)-(III), (C)-(I), (D)-(IV)

05

PYQ 2023

medium

mathematics ID: cuet-ug-

If set A has 5 elements and set B has 7 elements than number of one-one and onto mapping from A to B is:

1

2

2

0

3

1

4

3

06

PYQ 2023

medium

mathematics ID: cuet-ug-

If f(x) - 27x 3 and g(x) =, then gof(x) is :

1

x

2

2x

3

3x

4

0

07

PYQ 2023

medium

mathematics ID: cuet-ug-

The absolute maximum value of the function f(x)=sinx + cosx, x [0,] is:

1

2

2

3

1

4

08

PYQ 2023

medium

mathematics ID: cuet-ug-

The value of C in Rolles's theorem for the function,is :

1

2

3

4

09

PYQ 2023

medium

mathematics ID: cuet-ug-

The demand function for a certain product is such that, where is the number of units of the product demanded and is the price per unit. Marginal revenue when 10 units are sold is:

1

59

2

780

3

1080

4

4900

10

PYQ 2023

medium

mathematics ID: cuet-ug-

Domain of function is

1

[-1,1]

2

3

4

11

PYQ 2023

medium

mathematics ID: cuet-ug-

The inverse of the function f: R\toR given by f(x) = 2x +7 is:

1

2(x+7)

2

3

x-7

4

12

PYQ 2023

medium

mathematics ID: cuet-ug-

The domain of the function f(x) = log is:

1

[2,\infty)

2

(-\infty,-2) U (2,\infty)

3

(2,\infty)

4

(-\infty, -2] U (-2, -1)

13

PYQ 2023

medium

mathematics ID: cuet-ug-

The region R on the set, given by is a multiple of 4 is :

1

Reflexive but not Symmetric

2

Reflexive but not Transitive

3

Symmetric but not Transitive

4

Equivalence relation

14

PYQ 2023

medium

mathematics ID: cuet-ug-

The function f(x) - x 3, x \in R has :

1

Maximum value at x = 0

2

Minimum value at x = 0

3

Neither maximum and nor minimum value at x = 0

4

Maximum value and minimum value at x = 0

15

PYQ 2023

easy

mathematics ID: cuet-ug-

Let be defined by, for every . Then is:

1

one-one and onto

2

one-one and not onto

3

neither one-one nor onto

4

onto and not one-one

Official Solution

Correct Option: (3)

To determine the characteristics of the function defined by, consider the following properties: One-One (Injective): A function is one-one if implies for all . For, assume which leads to . This implies either or . Since it's possible for (e.g.,), the function is not one-one. Onto (Surjective): A function is onto if for every, there exists such that . Consider . There is no real number such that because the square of any real number is non-negative. Hence, not all elements in have a preimage, so is not onto. Conclusion: The function is neither one-one nor onto. Options Conclusion one-one and onto No one-one and not onto No neither one-one nor onto Yes onto and not one-one No

Options	Conclusion
one-one and onto	No
one-one and not onto	No
neither one-one nor onto	Yes
onto and not one-one	No

16

PYQ 2024

hard

mathematics ID: cuet-ug-

Relation on the set defined as is:

1

Reflexive, symmetric and transitive

2

Reflexive and transitive but not symmetric

3

Neither reflexive nor symmetric but transitive

4

Neither reflexive nor symmetric nor transitive

17

PYQ 2024

medium

mathematics ID: cuet-ug-

Let be defined as, then:

1

is one-one and onto.

2

is one-one but not onto.

3

is neither one-one nor onto.

4

is onto but not one-one.

18

PYQ 2025

hard

mathematics ID: cuet-ug-

Let A = \{1, 2, 3\}. Then, the number of relations containing (1, 2) and (1, 3), which are reflexive and symmetric but not transitive, is

1

1

2

2

3

3

4

4

Official Solution

Correct Option: (1)

Step 1: Understanding the Concept: We need to construct a relation R on the set A = \{1, 2, 3\} that satisfies four conditions: 1. It contains (1, 2) and (1, 3). 2. It is reflexive. 3. It is symmetric. 4. It is not transitive. We then need to count how many such unique relations exist. Step 3: Detailed Explanation: Let R be the relation we are looking for. Condition 1 \& 2 (Reflexivity): For R to be reflexive on A, it must contain all pairs (a, a) for a A. So, R must contain \{(1, 1), (2, 2), (3, 3)\}. Condition 1 \& 3 (Symmetry): The relation must contain (1, 2) and (1, 3). For R to be symmetric, if (a, b) R, then (b, a) must also be in R. Since (1, 2) R, we must have (2, 1) R. Since (1, 3) R, we must have (3, 1) R. Combining these, the smallest possible relation R that satisfies the first three properties is: Condition 4 (Not Transitive): Now we must check if R\textsubscript{min} is transitive. A relation is transitive if (a, b) R and (b, c) R implies (a, c) R. Let's look for a counterexample. Consider (2, 1) R\textsubscript{min} and (1, 3) R\textsubscript{min}. For transitivity, the pair (2, 3) must be in R\textsubscript{min}. However, (2, 3) R\textsubscript{min}. Similarly, consider (3, 1) R\textsubscript{min} and (1, 2) R\textsubscript{min}. For transitivity, (3, 2) must be in R\textsubscript{min}. But (3, 2) R\textsubscript{min}. Since we found a case where transitivity fails, R\textsubscript{min} is not transitive. So, R\textsubscript{min} is one such relation. Are there any others? The only pairs not in R\textsubscript{min} are (2, 3) and (3, 2). If we add these to R\textsubscript{min}, we must add both to maintain symmetry. Let's define a new relation R' = R\textsubscript{min} \{(2, 3), (3, 2)\}. R' = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)\}. This is the universal relation A A. Is R' transitive? Let's check the pair that failed before: (2, 1) R' and (1, 3) R'. Is (2, 3) R'? Yes, it is now. The universal relation on a set is always an equivalence relation, meaning it is reflexive, symmetric, and transitive. So R' is transitive. This means we cannot add the pair \{(2, 3), (3, 2)\} because it makes the relation transitive. Step 4: Final Answer: The only relation that satisfies all the given conditions is R\textsubscript{min}. Therefore, there is only 1 such relation.

19

PYQ 2025

medium

mathematics ID: cuet-ug-

Let f: R R be defined as f(x) = 10x. Then (Where R is the set of real numbers)

1

f is both one-one and onto

2

f is onto but not one-one

3

f is one-one but not onto

4

f is neither one-one nor onto

20

PYQ 2025

medium

mathematics ID: cuet-ug-

Let f: R R be defined as f(x) = 10x. Then (Where R is the set of real numbers)

1

f is both one-one and onto

2

f is onto but not one-one

3

f is one-one but not onto

4

f is neither one-one nor onto

21

PYQ 2025

medium

mathematics ID: cuet-ug-

Let A = \{1, 2, 3\}. Then, the number of relations containing (1, 2) and (1, 3), which are reflexive and symmetric but not transitive, is

1

1

2

2

3

3

4

4

Official Solution

Correct Option: (1)

Step 1: Understanding the Concept: We need to construct a relation R on the set A = \{1, 2, 3\} that satisfies four conditions: 1. It contains (1, 2) and (1, 3). 2. It is reflexive. 3. It is symmetric. 4. It is not transitive. We then need to count how many such unique relations exist. Step 3: Detailed Explanation: Let R be the relation we are looking for. Condition 1 \& 2 (Reflexivity): For R to be reflexive on A, it must contain all pairs (a, a) for a A. So, R must contain \{(1, 1), (2, 2), (3, 3)\}. Condition 1 \& 3 (Symmetry): The relation must contain (1, 2) and (1, 3). For R to be symmetric, if (a, b) R, then (b, a) must also be in R. Since (1, 2) R, we must have (2, 1) R. Since (1, 3) R, we must have (3, 1) R. Combining these, the smallest possible relation R that satisfies the first three properties is: Condition 4 (Not Transitive): Now we must check if R\textsubscript{min} is transitive. A relation is transitive if (a, b) R and (b, c) R implies (a, c) R. Let's look for a counterexample. Consider (2, 1) R\textsubscript{min} and (1, 3) R\textsubscript{min}. For transitivity, the pair (2, 3) must be in R\textsubscript{min}. However, (2, 3) R\textsubscript{min}. Similarly, consider (3, 1) R\textsubscript{min} and (1, 2) R\textsubscript{min}. For transitivity, (3, 2) must be in R\textsubscript{min}. But (3, 2) R\textsubscript{min}. Since we found a case where transitivity fails, R\textsubscript{min} is not transitive. So, R\textsubscript{min} is one such relation. Are there any others? The only pairs not in R\textsubscript{min} are (2, 3) and (3, 2). If we add these to R\textsubscript{min}, we must add both to maintain symmetry. Let's define a new relation R' = R\textsubscript{min} \{(2, 3), (3, 2)\}. R' = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)\}. This is the universal relation A A. Is R' transitive? Let's check the pair that failed before: (2, 1) R' and (1, 3) R'. Is (2, 3) R'? Yes, it is now. The universal relation on a set is always an equivalence relation, meaning it is reflexive, symmetric, and transitive. So R' is transitive. This means we cannot add the pair \{(2, 3), (3, 2)\} because it makes the relation transitive. Step 4: Final Answer: The only relation that satisfies all the given conditions is R\textsubscript{min}. Therefore, there is only 1 such relation.

	LIST I		LIST II
A.	Range of y=cosec^-1x	I.	R-(-1, 1)
B.	Domain of sec^-1x	II.	(0, π)
C.	Domain of sin^-1x	III.	[-1, 1]
D.	Range of y=cot^-1x	IV.	$%5B%5Cfrac%7B-%CF%80%7D%7B2%7D%2C%5Cfrac%7B%CF%80%7D%7B2%7D%5D$ -{0}

High-Yield Trend

Chapter Questions
21 MCQs

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Injectivity (One-One):

Surjectivity (Onto):

Conclusion:

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Relations And Functions

High-Yield Trend

Chapter Questions 21 MCQs

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Injectivity (One-One):

Surjectivity (Onto):

Conclusion:

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Chapter Questions
21 MCQs