Let A-1-1- B-1-1- C-0- - Let x- y - A - B- x2-y2-1 and R2-x- y - A - C- x2-y2-1

The correct option is D R_2 defines a function from A onto CR_2={(x,y)ϵ A × C : x^2+y^2=1} x^2+y^2=1 ⇒ y=±√(1 x^2) ⇒ y=√(1 x^2) (∵ y≥ 0) ∴ R_2 is a ...

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Byju's Answer

Standard XII

Mathematics

Definite Integral as Limit of Sum

Let A=[-1,1],...

Question

Let A = [-1, 1], B = [-1, 1], C = $[0, \infty)$ . Let ${(x, y) ϵ A \times B : x^{2} + y^{2} = 1}$ and $R_{2} = {(x, y) ϵ A \times C : x^{2} + y^{2} = 1}$

R_{1}

defines a function from A into B

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R_{2}

defines a function from A into C

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R_{1}

defines a function from A onto B

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R_{2}

defines a function from A onto C

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Solution

The correct option is D $R_{2}$ defines a function from A onto C
$R_{2} = {(x, y) ϵ A \times C : x^{2} + y^{2} = 1}$
$x^{2} + y^{2} = 1 \Rightarrow y = \pm \sqrt{1 - x^{2}}$
$\Rightarrow y = \sqrt{1 - x^{2}} (∵ y \geq 0)$
$∴ R_{2}$ is a function from A onto C.
$∴$ The correct answer is (d)

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Similar questions

Q. Let

A = [- 1, 1], B = [- 1, 1], C = [0, \infty)

. Define

R_{1} = {(x, y) \in A \times B : x^{2} + y^{2} = 1}

and

R_{2} = {(x, y) \in A \times C : x^{2} + y^{2} = 1}

. Then choose the correct option:

Q. Let

A = [- 1, 1], B = [- 1, 1], C = [0, \infty)

. Define

R_{1} = {(x, y) \in A \times B : x^{2} + y^{2} = 1}

and

R_{2} = {(x, y) \in A \times C : x^{2} + y^{2} = 1}

. Then choose the correct option:

Q. Let

A = \{x \in R : - 1 \leq x \leq 1\} = B and C = \{x \in R : x \geq 0\} and

let

S = \{(x, y) \in A \times B : x^{2} + y^{2} = 1\} and S_{0} = \{(x, y) \in A \times C : x^{2} + y^{2} = 1\}

. Then,
(a) S defines a function from A to B
(b) S₀ defines a function from A to C
(c) S₀ defines a function from A to B
(d) S defines a function from A to C

Q. A

= [- 1, 1]

. B

= [0, 1]

, C

= [- 1, 0]

S_{1} = {(x, y) | x^{2} + y^{2} = 1, x ϵ A, y ϵ A}

S_{2} = {(x, y) | x^{2} + y^{2} = 1, x ϵ A, y ϵ B}

S_{3} = {(x, y) | x^{2} + y^{2} = 1, x ϵ A, y ϵ C}

S_{4} = {(x, y) | x^{2} + y^{2} = 1, x ϵ B, y ϵ C}

Q. Let

C : x^{2} + y^{2} - x - y - 6 = 0

and point

P (a - 1, a + 1)

lies inside the circle

C

. If the line

x + y - 2 = 0

divides the circle in two segments, then

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Let A = [-1, 1], B = [-1, 1], C = [0,∞). Let {(x,y)ϵ A×B:x2+y2=1} and R2={(x,y)ϵ A×C:x2+y2=1}

The correct option is D R2 defines a function from A onto CR2={(x,y)ϵ A×C:x2+y2=1} x2+y2=1 ⇒y=±√1−x2 ⇒y=√1−x2 (∵y≥0) ∴ R2 is a function from A onto C. ∴ The correct answer is (d)

Let A = [-1, 1], B = [-1, 1], C = $[0, \infty)$ . Let ${(x, y) ϵ A \times B : x^{2} + y^{2} = 1}$ and $R_{2} = {(x, y) ϵ A \times C : x^{2} + y^{2} = 1}$

The correct option is D $R_{2}$ defines a function from A onto C
$R_{2} = {(x, y) ϵ A \times C : x^{2} + y^{2} = 1}$
$x^{2} + y^{2} = 1 \Rightarrow y = \pm \sqrt{1 - x^{2}}$
$\Rightarrow y = \sqrt{1 - x^{2}} (∵ y \geq 0)$
$∴ R_{2}$ is a function from A onto C.
$∴$ The correct answer is (d)