#### Online Mock Tests

#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

## Chapter 2: Relations

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations Exercise 2.1 [Page 8]

(i) If \[\left( \frac{a}{3} + 1, b - \frac{2}{3} \right) = \left( \frac{5}{3}, \frac{1}{3} \right)\] find the values of *a* and *b*.

(ii) If (*x* + 1, 1) = (3, *y* − 2), find the values of *x* and *y*.

If the ordered pairs (*x*, −1) and (5, *y*) belong to the set {(*a*, *b*) : *b* = 2*a* − 3}, find the values of *x* and *y*.

If *a* ∈ [−1, 2, 3, 4, 5] and *b* ∈ [0, 3, 6], write the set of all ordered pairs (*a*, *b*) such that *a* + *b*= 5.

If *a* ∈ [2, 4, 6, 9] and *b* ∈ [4, 6, 18, 27], then form the set of all ordered pairs (*a*, *b*) such that a divides *b* and *a* < *b*.

If *A* = {1, 2} and *B* = {1, 3}, find *A* × *B* and *B* × *A*.

Let *A* = {1, 2, 3} and *B* = {3, 4}. Find *A* × *B* and show it graphically.

If *A* = {1, 2, 3} and *B* = {2, 4}, what are *A* × *B*, *B* × *A*, *A* × *A*, *B* × *B* and (*A* × *B*) ∩ (*B* × *A*)?

If *A* and *B* are two set having 3 elements in common. If *n*(*A*) = 5, *n*(*B*) = 4, find *n*(*A* × *B*) and *n*[(*A* × *B*) ∩ (*B* × *A*)].

Let *A* and *B* be two sets. Show that the sets *A* × *B* and *B* × *A* have elements in common iff the sets *A* and *B* have an elements in common.

Let *A* and *B* be two sets such that *n*(*A*) = 3 and *n*(*B*) = 2.

If (*x*, 1), (*y*, 2), (*z*, 1) are in *A* × *B*, find *A* and *B*, where *x*, *y*, *z* are distinct elements.

Let *A* = {1, 2, 3, 4} and R = {(*a*, *b*) : *a* ∈ *A*, *b* ∈ *A*, *a* divides *b*}. Write *R* explicitly.

If *A* = {−1, 1}, find *A* × *A* × *A*.

State whether of the statement is true or false. If the statement is false, re-write the given statement correctly:

If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}

True

False

State whether of the statement is true or false. If the statement is false, re-write the given statement correctly:

If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ B and y ∈ A.

True

False

State whether of the statement is true or false. If the statement is false, re-write the given statement correctly:

(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ ϕ) = ϕ.

True

False

If *A* = {1, 2}, from the set *A* × *A* × *A*.

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(i) A × B

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(ii) B × A

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(iii) A × A

If A = {1, 2, 4} and B = {1, 2, 3}, represent set graphically:

(iv) B × B

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations Exercise 2.2 [Page 12]

Given A = {1, 2, 3}, B = {3, 4}, C ={4, 5, 6}, find (A × B) ∩ (B × C ).

If *A* = {2, 3}, *B* = {4, 5}, *C* ={5, 6}, find *A* × (*B* ∪ *C*), A × (*B* ∩ *C*), (*A* × *B*) ∪ (*A* × *C*).

If *A* = {1, 2, 3}, *B* = {4}, *C* = {5}, then verify that:

(i) *A* × (*B* ∪ *C*) = (*A* × *B*) ∪ (*A* × *C*)

If *A* = {1, 2, 3}, *B* = {4}, *C* = {5}, then verify that:

(ii) *A* × (*B* ∩ *C*) = (*A* × *B*) ∩ (*A* × *C*)

If *A* = {1, 2, 3}, *B* = {4}, *C* = {5}, then verify that:

(iii) *A* × (*B* − *C*) = (*A* × *B*) − (*A* × *C*)

Let *A* = {1, 2}, *B* = {1, 2, 3, 4}, *C* = {5, 6} and *D* = {5, 6, 7, 8}. Verify that:

(i) *A* × *C* ⊂ *B* × *D*

Let *A* = {1, 2}, *B* = {1, 2, 3, 4}, *C* = {5, 6} and *D* = {5, 6, 7, 8}. Verify that:

(ii) *A* × (*B* ∩ *C*) = (*A* × *B*) ∩ (*A* × *C*)

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(i) A × (B ∩ C)

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(ii) (A × B) ∩ (A × C)

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(iii) A × (B ∪ C)

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find

(iv) (A × B) ∪ (A × C)

Prove that:

(i) (*A* ∪ *B*) × *C* = (*A* × *C*) ∪ (*B* × *C*)

(ii) (*A* ∩ B) × C = (*A* × *C*) ∩ (B×*C*)

If *A* × *B* ⊆ *C* × *D* and *A* × *B* ≠ ϕ, prove that *A* ⊆ *C* and *B* ⊆ *D*.

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations Exercise 2.3 [Pages 20 - 21]

If A = [1, 2, 3], B = [4, 5, 6], which of the following are relations from A to B? Give reasons in support of your answer.

(i) [(1, 6), (3, 4), (5, 2)]

(ii) [(1, 5), (2, 6), (3, 4), (3, 6)]

(iii) [(4, 2), (4, 3), (5, 1)]

(iv) A × B.

A relation R is defined from a set A = [2, 3, 4, 5] to a set B = [3, 6, 7, 10] as follows:

(*x*, *y*) ∈ R ⇔ *x* is relatively prime to *y*

Express R as a set of ordered pairs and determine its domain and range.

Let A be the set of first five natural numbers and let R be a relation on A defined as follows:

(*x*, *y*) ∈ R ⇔ *x* ≤ *y*

Express R and R^{−1} as sets of ordered pairs. Determine also (i) the domain of R^{−1} (ii) the range of R.

Find the inverse relation R^{−1} in each of the cases:

(i) R = {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}

Find the inverse relation R^{−1} in each of the cases:

(ii) R = {(*x*, *y*), : *x*, *y* ∈ N, *x* + 2*y* = 8}

Find the inverse relation R^{−1} in each of the cases:

(iii) R is a relation from {11, 12, 13} to (8, 10, 12] defined by *y* = *x* − 3.

Write the relation as the sets of ordered pairs:

(i) A relation R from the set [2, 3, 4, 5, 6] to the set [1, 2, 3] defined by *x* = 2*y*.

Write the relation as the sets of ordered pairs:

(ii) A relation R on the set [1, 2, 3, 4, 5, 6, 7] defined by (*x*, *y*) ∈ R ⇔ *x* is relatively prime to *y*.

Write the relation as the sets of ordered pairs:

(iii) A relation R on the set [0, 1, 2, ....., 10] defined by 2*x* + 3*y* = 12.

Write the relation as the sets of ordered pairs:

(iv) A relation R from a set A = [5, 6, 7, 8] to the set B = [10, 12, 15, 16,18] defined by (*x*, *y*) ∈ R ⇔ *x* divides *y*.

Let *R* be a relation in N defined by (*x*, *y*) ∈ R ⇔ *x* + 2*y* =8. Express R and R^{−1} as sets of ordered pairs.

Let A = (3, 5) and B = (7, 11). Let R = {(*a*, *b*) : *a* ∈ A, *b* ∈ B, *a* − *b* is odd}. Show that R is an empty relation from A into B.

Let A = [1, 2] and B = [3, 4]. Find the total number of relation from A into B.

Determine the domain and range of the relation R defined by

(i) R = [(*x*, *x* + 5): *x* ∈ (0, 1, 2, 3, 4, 5)]

Determine the domain and range of the relation R defined by

(ii) R = {(*x*, *x*^{3}) :* x* is a prime number less than 10}

Determine the domain and range of the relations:

(i) R = {(*a*, *b*) : *a* ∈ N, *a* < 5, *b* = 4}

Determine the domain and range of the relations:

(ii) \[S = \left\{ \left( a, b \right) : b = \left| a - 1 \right|, a \in Z \text{ and} \left| a \right| \leq 3 \right\}\]

Let A = {*a*, *b*}. List all relations on A and find their number.

Let A = (*x*, *y*, *z*) and B = (*a*, *b*). Find the total number of relations from A into B.

Let R be a relation from N to N defined by R = [(*a*, *b*) : *a*, *b* ∈ N and *a* = *b*^{2}]. Are the statement true?

(i) (*a*, *a*) ∈ R for all *a* ∈ N

Let R be a relation from N to N defined by R = [(*a*, *b*) : *a*, *b* ∈ N and *a* = *b*^{2}]. Are the statement true?

(ii) (*a*, *b*) ∈ R ⇒ (*b*, *a*) ∈ R

Let R be a relation from N to N defined by R = [(*a*, *b*) : *a*, *b* ∈ N and *a* = *b*^{2}]. Are the statement true?

(iii) (*a*, *b*) ∈ R and (*b*, *c*) ∈ R ⇒ (*a*, *c*) ∈ R

Let A = [1, 2, 3, ......., 14]. Define a relation on a set A by

R = {(*x*, *y*) : 3*x* − *y* = 0, where *x*, *y* ∈ A}.

Depict this relationship using an arrow diagram. Write down its domain, co-domain and range.

Define a relation R on the set N of natural number by R = {(*x*, *y*) : *y* = *x* + 5, *x* is a natural number less than 4, *x*, *y* ∈ N}. Depict this relationship using (i) roster form (ii) an arrow diagram. Write down the domain and range or R.

A = [1, 2, 3, 5] and B = [4, 6, 9]. Define a relation R from A to B by R = {(*x*, *y*) : the difference between *x* and *y* is odd, *x* ∈ A, *y* ∈ B}. Write R in Roster form.

Write the relation R = {(*x*, *x*^{3}) : *x* is a prime number less than 10} in roster form.

Let A = [1, 2, 3, 4, 5, 6]. Let R be a relation on A defined by {(*a*, *b*) : *a*, *b* ∈ A, *b* is exactly divisible by *a*}

(i) Writer R in roster form

(ii) Find the domain of R

(ii) Find the range of R.

The adjacent figure shows a relationship between the sets P and Q. Write this relation in (i) set builder form (ii) roster form. What is its domain and range?

Let R be the relation on Z defined by

R = {(*a*, *b*) : *a*, *b* ∈ Z, *a* − *b* is an integer}

Find the domain and range of R.

For the relation R_{1} defined on R by the rule (*a*, *b*) ∈ R_{1} ⇔ 1 + *ab* > 0. Prove that: (*a*, *b*) ∈ R_{1} and (*b* , *c*) ∈ R_{1} ⇒ (*a*, *c*) ∈ R_{1} is not true for all *a*, *b*, *c* ∈ R.

Let R be a relation on N × N defined by

(*a*, *b*) R (*c*, *d*) ⇔ *a* + *d* = *b* + *c* for all (*a*, *b*), (*c*, *d*) ∈ N × N

Show that:

(i) (*a*, *b*) R (*a*, *b*) for all (*a*, *b*) ∈ N × N

Let R be a relation on N × N defined by

(*a*, *b*) R (*c*, *d*) ⇔ *a* + *d* = *b* + *c* for all (*a*, *b*), (*c*, *d*) ∈ N × N

Show that:

(ii) (*a*, *b*) R (*c*, *d*) ⇒ (*c*, *d*) R (*a*, *b*) for all (*a*, *b*), (*c*, *d*) ∈ N × N

Let R be a relation on N × N defined by

(*a*, *b*) R (*c*, *d*) ⇔ *a* + *d* = *b* + *c* for all (*a*, *b*), (*c*, *d*) ∈ N × N

(iii) (*a*, *b*) R (*c*, *d*) and (*c*, *d*) R (*e*, *f*) ⇒ (*a*, *b*) R (*e*, *f*) for all (*a*, *b*), (*c*, *d*), (*e*, *f*) ∈ N × N

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations Exercise 2.4 [Pages 24 - 25]

If A = {1, 2, 4}, B = {2, 4, 5} and C = {2, 5}, write (A − C) × (B − C).

If *n*(A) = 3, *n*(B) = 4, then write *n*(A × A × B).

If R is a relation defined on the set Z of integers by the rule (*x*, *y*) ∈ R ⇔ *x*^{2} + *y*^{2} = 9, then write domain of R.

If R = {(*x*, *y*) : *x*, *y* ∈ Z, *x*^{2} + *y*^{2} ≤ 4} is a relation defined on the set Z of integers, then write domain of R.

If R is a relation from set A = (11, 12, 13) to set B = (8, 10, 12) defined by *y* = *x* − 3, then write R^{−1}.

Let A = {1, 2, 3} and\[R = \left\{ \left( a, b \right) : \left| a^2 - b^2 \right| \leq 5, a, b \in A \right\}\].Then write R as set of ordered pairs.

Let R = [(*x*, *y*) : *x*, *y* ∈ Z, *y* = 2*x* − 4]. If (*a*, -2) and (4, *b*^{2}) ∈ R, then write the values of *a* and *b*.

If R = {(2, 1), (4, 7), (1, −2), ...}, then write the linear relation between the components of the ordered pairs of the relation R.

If A = [1, 3, 5] and B = [2, 4], list of elements of R, if

R = {(*x*, *y*) : *x*, *y* ∈ A × B and *x* > *y*}

If R = [(*x*, *y*) : *x*, *y* ∈ W, 2*x* + *y* = 8], then write the domain and range of R.

Let A and B be two sets such that *n*(A) = 3 and *n*(B) = 2. If (*x*, 1), (*y*, 2), (*z*, 1) are in A × B, write A and B

Let A = [1, 2, 3, 5], B = [4, 6, 9] and R be a relation from A to B defined by R = {(*x*, *y*) : *x* − *y*is odd}. Write R in roster form.

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 2 Relations Exercise 2.5 [Pages 25 - 26]

If A = {1, 2, 4}, B = {2, 4, 5}, C = {2, 5}, then (A − B) × (B − C) is

(a) {(1, 2), (1, 5), (2, 5)}

(b) [(1, 4)]

(c) (1, 4)

(d) none of these

If R is a relation on the set A = [1, 2, 3, 4, 5, 6, 7, 8, 9] given by *x* R *y* ⇔ *y* = 3*x*, then R =

(a) [(3, 1), (6, 2), (8, 2), (9, 3)]

(b) [(3, 1), (6, 2), (9, 3)]

(c) [(3, 1), (2, 6), (3, 9)]

(d) none of these

Let A = [1, 2, 3], B = [1, 3, 5]. If relation R from A to B is given by = {(1, 3), (2, 5), (3, 3)}, Then R^{−1} is

(a) {(3, 3), (3, 1), (5, 2)}

(b) {(1, 3), (2, 5), (3, 3)}

(c) {(1, 3), (5, 2)}

(d) None of these

If A = [1, 2, 3], B = [1, 4, 6, 9] and R is a relation from A to B defined by '*x*' is greater than *y*. The range of R is

(a) {1, 4, 6, 9}

(b) (4, 6, 9)

(c) [1]

(d) none of these.

If R = {(*x*, *y*) : *x*, *y* ∈ Z, *x*^{2} +* **y*^{2} ≤ 4} is a relation on Z, then the domain of R is

(a) [0, 1, 2]

(b) [0, −1, −2]

(c) {−2, −1, 0, 1, 2]

(d) None of these

A relation R is defined from [2, 3, 4, 5] to [3, 6, 7, 10] by : *x* R *y* ⇔ *x* is relatively prime to *y*. Then, domain of R is

(a) [2, 3, 5]

(b) [3, 5]

(c) [2, 3, 4]

(d) [2, 3, 4, 5]

A relation ϕ from C to R is defined by *x* ϕ *y* ⇔ |x| = *y*. Which one is correct?

(a) (2 + 3

*i*) ϕ 13(b) 3ϕ (−3)

(c) (1 +

*i*) ϕ 2(d)

*i*ϕ 1

Let R be a relation on N defined by *x* + 2*y* = 8. The domain of R is

(a) [2, 4, 8]

(b) [2, 4, 6, 8]

(c) [2, 4, 6]

(d) [1, 2, 3, 4]

R is a relation from [11, 12, 13] to [8, 10, 12] defined by* y* = *x* − 3. Then, R^{−1} is

(a) [(8, 11), (10, 13)]

(b) [(11, 8), (13, 10)]

(c) [(10, 13), (8, 11), (12, 10)]

(d) none of these

If the set A has *p* elements, B has *q* elements, then the number of elements in A × B is

(a) p + q

(b)

*p*+*q*+ 1(c) pq

(d)

*p*^{2}

Let R be a relation from a set A to a set B, then

(a) R = A ∪ B

(b) R = A ∩ B

(c) R ⊆ A × B

(d) R ⊆ B × A

If R is a relation from a finite set A having *m* elements of a finite set B having *n* elements, then the number of relations from A to B is

(a) 2

^{mn}(b) 2

^{mn}− 1(c) 2

*mn*(d)

*m*^{n}

If R is a relation on a finite set having *n* elements, then the number of relations on A is

(a) 2

^{n}(b) \[2^{n^2}\]

(c)

*n*^{2}(d)

*n*^{n}

## Chapter 2: Relations

## RD Sharma solutions for Class 11 Mathematics Textbook chapter 2 - Relations

RD Sharma solutions for Class 11 Mathematics Textbook chapter 2 (Relations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics Textbook chapter 2 Relations are Cartesian Product of Sets, Brief Review of Cartesian System of Rectanglar Co-ordinates, Relation, Concept of Functions, Some Functions and Their Graphs, Algebra of Real Functions, Ordered Pairs, Equality of Ordered Pairs, Pictorial Diagrams, Graph of Function, Pictorial Representation of a Function, Exponential Function, Logarithmic Functions.

Using RD Sharma Class 11 solutions Relations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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