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Let L be the set of all lines in XY plane and R be the relation in L d...

Let L be the set of all lines in XY plan | Class 12 Mathematics Chapter Relations and Functions, Relations and Functions NCERT Solutions

Welcome to the NCERT Solutions for Class 12 Mathematics - Chapter Relations and Functions. This page offers a step-by-step solution to the specific question from Exercise 1, Question 14: . With detailed answers and explanations for each chapter, students can strengthen their understanding and prepare confidently for exams. Ideal for CBSE and other board students, this resource will simplify your study experience.

Question 14: Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Answer:

R = {(L₁, L₂): L₁ is parallel to L₂}

R is reflexive as any line L₁ is parallel to itself i.e., (L₁, L₁) ∈ R.

Now,

Let (L₁, L₂) ∈ R.

⇒ L₁ is parallel to L_2.

⇒ L₂ is parallel to L_1.

⇒ (L₂, L₁) ∈ R

∴ R is symmetric.

Now,

Let (L₁, L₂), (L₂, L₃) ∈R.

⇒ L₁ is parallel to L₂. Also, L₂ is parallel to L_3.

⇒ L₁ is parallel to L_3.

∴R is transitive.

Hence, R is an equivalence relation.

The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x + 4.

Slope of line y = 2x + 4 is m = 2

It is known that parallel lines have the same slopes.

The line parallel to the given line is of the form y = 2x + c, where c ∈R.

Hence, the set of all lines related to the given line is given by y = 2x + c, where c ∈ R.

Other Questions of Chapter Relations and Functions

Q: Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.

Show that the function f : R_* → R_* defined by f(x) = 1/x is one-one and onto,where R_* is the set of all non-zero real numbers. Is the result true, if the domain R_* is replaced by N with co-domain being same as R_*?

Q: Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by
(i) R = { (a,b) : |a - b| is a multiple of 4}
(ii) R = {(a,b):a = b} is an equivalence relation.
Find the set of all elements related to 1 in each case.

Q: Given an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.

Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Letbe a function defined as. The inverse of f is map g: Range

(A)

(B)

(C)

(D)

Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f^–1 and show that (f^–1)^–1 = f.

Q: Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b²} is neither reflexive nor symmetric nor transitive.

Q: Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

Popular questions of class 12 Mathematics

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(A) 10π (B) 12π (C) 8π (D) 11π

Q: If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

The total revenue in Rupees received from the sale of x units of a product is given by

R (x) = 13x² + 26x + 15

Find the marginal revenue when x = 7.

Maximise Z = 3x + 4y

Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0

Q: If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Q: Find the rate of change of the area of a circle with respect to its radius r when
(a) r = 3 cm
(b) r = 4 cm

Answer the following as true or false.

\begin{align}(i) \overrightarrow{a}\; and\; \overrightarrow{-a}\; are\; collinear.\end{align}

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

Q: Integrals (ax + b)²

Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 metres north-west (iii) 40°

(iv) 40 watt (v) 10^–19 coulomb (vi) 20 m/s²

Recently Viewed Questions of Class 12 Mathematics

Q: \begin{align} \int \frac {sec^2 x}{Coses^2 x} . dx\end{align}

Q: \begin{align} \int \frac{x^3 - x^2 + x - 1}{x-1} . dx\end{align}

In Figure, identify the following vectors.

(i) Coinitial (ii) Equal (iii) Collinear but not equal

Q: Find the principal value of \begin{align} cos^{-1}\left(-\frac{1}{2}\right)\end{align}

Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Maximise Z = 3x + 4y

Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0

Q: Find the principal value of \begin{align} cos^{-1}\left(\frac{1}{2}\right) + 2sin^{-1}\left(\frac{1}{2}\right)\end{align}

Q: Find the principal value of \begin{align} tan^{-1}\left(-\sqrt3\right)\end{align}

Q: If A = \(\begin{bmatrix}1 & 1 & -2\\2 & 1 & -3\\5 & 4 & -9\end{bmatrix}\), Find |A|

Study Tips for Answering NCERT Questions:

NCERT questions are designed to test your understanding of the concepts and theories discussed in the chapter. Here are some tips to help you answer NCERT questions effectively:

Read the question carefully and focus on the core concept being asked.
Reference examples and data from the chapter when answering questions about Relations and Functions.
Review previous year question papers to get an idea of how such questions may be framed in exams.
Practice answering questions within the time limit to improve your speed and accuracy.
Discuss your answers with your teachers or peers to get feedback and improve your understanding.

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Comments

Juhi Kumari
Jun 29, 2021

I understood the question I was searching very easily here The methods are as simple as possible .....

Sandra
Jun 07, 2020

It is very useful..