In Exercise 2.1 of Chapter 2, we shall discuss problems based on ordered pairs and cartesian products of sets. Problems in this exercise are well formulated to help gain more knowledge about the concepts. Students can access RD Sharma Class 11 Solutions which are developed by our expert tutors in accordance with the latest syllabus. Here, the solutions to this exercise are provided in PDF format, which can be downloaded easily from the links given below.
RD Sharma Solutions for Class 11 Maths Exercise 2.1 Chapter 2 – Relations
Access answers to RD Sharma Solutions for Class 11 Maths Exercise 2.1 Chapter 2 – Relations
(1) (i) If (a/3 + 1, b – 2/3) = (5/3, 1/3), find the values of a and b.
(ii) If (x + 1, 1) = (3y, y – 1), find the values of x and y.
Solution:
Given:
(a/3 + 1, b – 2/3) = (5/3, 1/3)
By the definition of equality of ordered pairs,
Let us solve for a and b
a/3 + 1 = 5/3 and b – 2/3 = 1/3
a/3 = 5/3 – 1 and b = 1/3 + 2/3
a/3 = (5-3)/3 and b = (1+2)/3
a/3 = 2/3 and b = 3/3
a = 2(3)/3 and b = 1
a = 2 and b = 1
∴ Values of a and b are a = 2 and b = 1
(ii) If (x + 1, 1) = (3y, y – 1), find the values of x and y.
Given:
(x + 1, 1) = (3y, y – 1)
By the definition of equality of ordered pairs,
Let us solve for x and y
x + 1 = 3y and 1 = y – 1
x = 3y – 1 and y = 1 + 1
x = 3y – 1 and y = 2
Since y = 2, we can substitute in
x = 3y – 1
= 3(2) – 1
= 6 – 1
= 5
∴ Values of x and y are, x = 5 and y = 2
2. If the ordered pairs (x, – 1) and (5, y) belong to the set {(a, b): b = 2a – 3}, find the values of x and y.
Solution:
Given:
The ordered pairs (x, – 1) and (5, y) belong to the set {(a, b): b = 2a – 3}
Solving for the first ordered pair
(x, – 1) = {(a, b): b = 2a – 3}
x = a and -1 = b
By taking b = 2a – 3
If b = – 1 then 2a = – 1 + 3
= 2
a = 2/2
= 1
So, a = 1
Since x = a, x = 1
Similarly, solving for the second ordered pair
(5, y) = {(a, b): b = 2a – 3}
5 = a and y = b
By taking b = 2a – 3
If a = 5 then b = 2×5 – 3
= 10 – 3
= 7
So, b = 7
Since y = b, y = 7
∴ Values of x and y are, x = 1 and y = 7
3. 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.
Solution:
Given: a ∈ {- 1, 2, 3, 4, 5} and b ∈ {0, 3, 6},
To find: the ordered pair (a, b) such that a + b = 5
Then the ordered pair (a, b) such that a + b = 5 are as follows
(a, b) ∈ {(- 1, 6), (2, 3), (5, 0)}
4. 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.
Solution:
Given:
a ∈ {2, 4, 6, 9} and b ∈{4, 6, 18, 27}
Here,
2 divides 4, 6, 18 and is also less than all of them
4 divides 4 and is also less than none of them
6 divides 6, 18 and is less than 18 only
9 divides 18, 27 and is less than all of them
∴ Ordered pairs (a, b) are (2, 4), (2, 6), (2, 18), (6, 18), (9, 18) and (9, 27)
5. If A = {1, 2} and B = {1, 3}, find A x B and B x A.
Solution:
Given:
A = {1, 2} and B = {1, 3}
A × B = {1, 2} × {1, 3}
= {(1, 1), (1, 3), (2, 1), (2, 3)}
B × A = {1, 3} × {1, 2}
= {(1, 1), (1, 2), (3, 1), (3, 2)}
6. Let A = {1, 2, 3} and B = {3, 4}. Find A x B and show it graphically
Solution:
Given:
A = {1, 2, 3} and B = {3, 4}
A x B = {1, 2, 3} × {3, 4}
= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}
Steps to follow to represent A × B graphically,
Step 1: One horizontal and one vertical axis should be drawn
Step 2: Element of set A should be represented on the horizontal axis, and on the vertical axis, elements of set B should be represented
Step 3: Draw dotted lines perpendicular to horizontal and vertical axes through the elements of sets A and B
Step 4: The point of intersection of these perpendicular represents A × B
7. If A = {1, 2, 3} and B = {2, 4}, what are A x B, B x A, A x A, B x B, and (A x B) ∩ (B x A)?
Solution:
Given:
A = {1, 2, 3} and B = {2, 4}
Now let us find: A × B, B × A, A × A, (A × B) ∩ (B × A)
A × B = {1, 2, 3} × {2, 4}
= {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}
B × A = {2, 4} × {1, 2, 3}
= {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}
A × A = {1, 2, 3} × {1, 2, 3}
= {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
B × B = {2, 4} × {2, 4}
= {(2, 2), (2, 4), (4, 2), (4, 4)}
The intersection of two sets represents common elements of both the sets
So,
(A × B) ∩ (B × A) = {(2, 2)}
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