Question

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.

Answer the following using the above information.

1. Let relation R be defined by R = {(L₁, L₂): L₁║L₂ where L₁,L₂ € L} then R is______ relation

a. Equivalence

b. Only reflexive

c. Not reflexive

d. Symmetric but not transitive

2. Let R = { (L₁, L₂) ∶ L₁ ┴ L₂ where L₁, L₂ € L } which of the following is true?

a. R is Symmetric but neither reflexive nor transitive

b. R is Reflexive and transitive but not symmetric

c. R is Reflexive but neither symmetric nor transitive

d. R is an Equivalence relation

3. The function f: R→R defined by f(x) = x − 4 is___________

a. Bijective

b. Surjective but not injective

c. Injective but not Surjective

d. Neither Surjective nor Injective

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

a. R

b. Z

c. W

d. Q

5. Let f : R = {(L ₁ , L ₂ ) : L ₁ is parallel to L ₂ and L ₁ : y = x – 4} then which of the following can be taken as L ₂ ?

a. 2x - 2y + 5 = 0

b. 2x + y = 5

c. 2x + 2y + 7 = 0

d. x + y = 7

Answer 1

1. (a) Equivalence

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

Check Reflexive

Since L, and L, are always parallel to each other

So, (L₁, L₁) ∈ R for all L₁

∴ R is reflexive

Check symmetric

If L, and L₂ are parallel to each other

Then, L₂ and L₁ are also parallel to each other

Check transitive

To check whether transitive or not,

If (x, y) ∈ R & (y, z) ∈ R, then (x, z) ∈ R

If L₁ and L₂ are parallel to each other,

And L₂ and L₃ are parallel to each other

Then, L and L₃ will also be parallel to each other

Thus, for all values of L₁, L₂, L₃

(L₁, L₂) ∈ R & (L₂, L₃) ∈ R, then (L₁, L₃) ∈ R

∴ R is transitive

Since R is reflexive, symmetric and transitive

∴ R is an Equivalence relation

2. (a) R is Symmetric but neither reflexive nor transitive

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

Check Reflexive

Since a line can never be perpendicular to itself

(L₁, L₁) ∉ & R for all L₁

∴ R is not reflexive

Check symmetric

If L, and L₂ are perpendicular to each other

Then, L₂ and L, are also perpendicular to each other

Thus (L₁, L₂) ∈ R, and (L₂, L₁) ∈ R

∴ R is symmetric

Check transitive

To check whether transitive or not,

If (x, y) ∈ R & (y, z) E∈ R, then (x, z) ∈ R

If L₁ and L₂ are perpendicular to each other, 

And L₂ and L₃ are also perpendicular to each other 

Then, L₁ and L₃ are not perpendicular to each other

∴ R is not transitive

Thus, R is Symmetric but neither reflexive nor transitive.

3. (a) Bijective

A linear function, defined from R to R is always one-one and onto 

∴ f(x) is Bijective

4. (a) R

For f(x) = x - 4

For all real values of x, we can get a real number f(x)

∴ Range of f(x) is R

5. (a) 2x - 2y + 5 = 0

Since L₂ must be parallel to L₁,

their slope must be equal

Slope of L₁: y = x - 4 is = 1

Checking Slope of given options

Slope of Part (a): 2x - 2y + 5 = 0

Slope = 1

Slope of Part (b): 2x + y = 5

Slope = -2

Slope of Part (c): 2x + 2y + 7 = 0

Slope = -1

Slope of Part (d): x + y = 7

Slope = -1

Since slope of option (a) is same as slope of L₁

Answer 2

1. (a) Equivalence

2. (a) R is Symmetric but neither reflexive nor transitive

3. (a) Bijective

4. (a) R

5. (a) 2x - 2y + 5 = 0