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Test: Relations and Functions- Case Based Type Questions- 2 - JEE MCQ


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15 Questions MCQ Test Mathematics (Maths) Class 12 - Test: Relations and Functions- Case Based Type Questions- 2

Test: Relations and Functions- Case Based Type Questions- 2 for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The Test: Relations and Functions- Case Based Type Questions- 2 questions and answers have been prepared according to the JEE exam syllabus.The Test: Relations and Functions- Case Based Type Questions- 2 MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Relations and Functions- Case Based Type Questions- 2 below.
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Test: Relations and Functions- Case Based Type Questions- 2 - Question 1

Direction: Read the following text and answer the following questions on the basis of the same:

An organization conducted bike race under 2 different categories–boys and girls. Totally there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.

Let B = {b1, b2, b3} G = {g1, g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions

Ravi wishes to form all the relations possible from B to G. How many such relations are possible?

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 1
Given

B = {b1, b2, b3} G = {g1, g2}

Numbers of Relation from B to G

= 2Numbers of elements of B x Number of elements of G

= 23 x 2

= 26

Test: Relations and Functions- Case Based Type Questions- 2 - Question 2

Direction: Read the following text and answer the following questions on the basis of the same:

An organization conducted bike race under 2 different categories–boys and girls. Totally there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.

Let B = {b1, b2, b3} G = {g1, g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions

Let R : B → G be defined by R = {(b1, g1), (b2, g2), (b3, g1)}, then R is__________

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 2
R : B → G be defined by R = {(b1, g1), (b2, g2), (b3, g1)}

R is surjective, since, every element of G is the image of some element of B under R, i.e., For g1, g2 ∈ G, there exists an elements b1, b2, b3 ∈ B, (b1 g1) (b2, g2), (b3, g1) ∈ R.

Test: Relations and Functions- Case Based Type Questions- 2 - Question 3

Direction: Read the following text and answer the following questions on the basis of the same:

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Let relation R be defined by R = {(L1, L2) : L1 || L2 where L1, L2 ∈ L} then R is______ relation

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 3
Let relation R be defined by

R = {(L1, L2) : L1 || L2 where L1, L2 ∈ L}.

R is reflexive, since every line is parallel to itself.

Further, (L1, L2) ∈ R

⇒ L1 is parallel to L2

⇒ L2 is parallel to L1

⇒ (L2, L1) ∈ R

Hence, R is symmetric.

Moreover, (L1, L2), (L2, L3) ∈ R

⇒ L1 is parallel to L2 and L2 is parallel to L3

⇒ L1 is parallel to L3

⇒ (L1, L3) ∈ R

Therefore, R is an equivalence relation

Test: Relations and Functions- Case Based Type Questions- 2 - Question 4

Direction: Read the following text and answer the following questions on the basis of the same:

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Let f : R → R be defined by f(x) = x − 4. Then the range of f(x) is _______

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 4
Range of f(x) is R
Test: Relations and Functions- Case Based Type Questions- 2 - Question 5

Direction: Read the following text and answer the following questions n the basis of the same:

Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.

Let f : R → R be defined by f(x) = x2 is_________

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 5
f : R ® R be defined by f(x) = x2

∵ f(–1) = f(1) = 1, but –1 ≠ 1

∴ f is not injective

Now, –2 ∈ R. But, there does not exist any element x ∈ R such that f(x) = –2 or x2 = –2

∴ f is not surjective.

Hence, function f is neither injective nor surjective.

Test: Relations and Functions- Case Based Type Questions- 2 - Question 6

Direction: Read the following text and answer the following questions n the basis of the same:

Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.

Let f : {1, 2, 3, ….} → {1, 4, 9, ….} be defined by f(x) = x2 is _________

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 6
f : {1, 2, 3, ....} → {1, 4, 9, ...} be defined by f(x) = x2

x1 ∈ {1, 2, 3, ...} and x2 ∈ {1, 2, 9, ....}

f(x1) = f(x2)

⇒ x12 = x22

⇒ x1 = x2

∴ f is injective

Now, 4 ∈ {1, 4, 9...}, there exist 2 in {1, 2, 3 ...} such that f(x) = 22 = 4, Hence, f is surjective Therefore f is bijective.

Test: Relations and Functions- Case Based Type Questions- 2 - Question 7

Direction: Read the following text and answer the following questions n the basis of the same:

Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.

The function f : Z → Z defined by f(x) = x2 is ______

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 7
f : z → z defined by f(x) = x2

So, f(–1) = f(1), but 1 ≠ –1

∴ f is not injective

Now, –2 ∈ Z, but, there does not exist any element x ∈ z such that

f(x) = –2

or x2 = –2

∴ f is not surjective

Hence, f is neither injective nor surjective.

Test: Relations and Functions- Case Based Type Questions- 2 - Question 8

Direction: Read the following text and answer the following questions on the basis of the same:

An organization conducted bike race under 2 different categories–boys and girls. Totally there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.

Let B = {b1, b2, b3} G = {g1, g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions

Ravi wants to know among those relations, how many functions can be formed from B to G?

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 8
Given

B = {b1, b2, b3} G = {g1, g2}

So, B hase 3 elements, G has 2 elements

Numbers of functions from B to G = 2 x 2 x 2

= 23

Test: Relations and Functions- Case Based Type Questions- 2 - Question 9

Direction: Read the following text and answer the following questions on the basis of the same:

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

The function f : R → R defined by f(x) = x − 4 is______

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 9
The function f is one-one,

for f(x1) = f(x2)

⇒ x1 – 4 = x2 – 4

⇒ x1 = x2

Also, given any real number y in R, there exists y + 4 in R

Such that f(y + 4) = y + 4 – 4 = y

Hence, f is onto

Hence, function is both one-one and onto, i.e., bijective.

Test: Relations and Functions- Case Based Type Questions- 2 - Question 10

Direction: Read the following text and answer the following questions on the basis of the same:

An organization conducted bike race under 2 different categories–boys and girls. Totally there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.

Let B = {b1, b2, b3} G = {g1, g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions

Let R : B → B be defined by R = {(x, y) : x and y are students of same sex}, Then this relation R is_______

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 10
R : B → B be defined by R = {(x, y) : x and y are students of same sex}

R is reflexive, since, (x, x) ∈ R

R is symmetric, since, (x, y) ∈ R and (y, x) ∈ R

R is transitive. For a, b, c ∈ B

∃ (a, b) (b, c) ∈ R

and (a, c) ∈ R.

Therefore R is equivalence relation.

Test: Relations and Functions- Case Based Type Questions- 2 - Question 11

Direction: Read the following text and answer the following questions on the basis of the same:

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Let R = {(L1, L2) : L1 ⊥ L2 where L1, L2 ∈ L} which of the following is true?

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 11
R is not reflexive, as a line L1 can not be perpendicular to itself, i.e., (L1, L1) ∉ R.

R is symmetric as (L1, L2) ∈ R

As, L1 is perpendicular to L2

and L2 is perpendicular to L1

(L2, L1) ∈ R

R is not transitive. Indeed, it L1 is perpendicular to L2 and L2 is perpendicular to L3, then L1 can never be perpendicular to L3.

In fact L1 is parallel to L3,

i.e., (L1, L2) ∈ R, (L2, L3) ∈ R but (L1, L3) ∉ R

i.e., symmetric but neither reflexive nor transitive.

Test: Relations and Functions- Case Based Type Questions- 2 - Question 12

Direction: Read the following text and answer the following questions n the basis of the same:

Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.

Let : N → R be defined by f(x) = x2. Range of the function among the following is _______

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 12
Range of f = {1, 4, 9, 16, ...}

∵ N = {1, 2, 3, ....}

Test: Relations and Functions- Case Based Type Questions- 2 - Question 13

Direction: Read the following text and answer the following questions on the basis of the same:

An organization conducted bike race under 2 different categories–boys and girls. Totally there were 250 participants. Among all of them finally three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project.

Let B = {b1, b2, b3} G = {g1, g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions

Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 13
In a function,

Every element of set B will have an image.

Every element of set B will only one image in set

For injective functions

All elements of set G should have a unique pre-image

Which is not possible

Test: Relations and Functions- Case Based Type Questions- 2 - Question 14

Direction: Read the following text and answer the following questions on the basis of the same:

Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.

Let R = {(L1, L2) : L1 || L2 and L1 : y = x – 4} then which of the following can be taken as L2?

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 14
Since, L1 || L2 then slope of both the lines should be same.

Slope of L1 = 1

⇒ Slope of L2 = 1

And 2x – 2y + 5 = 0

–2y = –2x – 5

y = x + (5/2)

Slope of 2x – 2y + 5 = 0 is 1

So, 2x – 2y + 5 = 0 can be taken as L2.

Test: Relations and Functions- Case Based Type Questions- 2 - Question 15

Direction: Read the following text and answer the following questions n the basis of the same:

Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.

Let f : N → N be defined by f(x) = x2 is ________

Detailed Solution for Test: Relations and Functions- Case Based Type Questions- 2 - Question 15
f : N → N be defined by f(x) = x2 for x, y ∈ N, f(x) = f(y)

⇒ x2 = y2

⇒ x = y

∴ f is injective

Now; 2 ∈ N, But, there does not exist any x in n such that f(x) = x2 = 2

∴ f is not surjective

Hence, function is injective but not surjective.

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