Download App

RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS3 Marks
Question
Test whether the following relations $R_{3 }$ are:
  1. Reflexive.
  2. Symmetric.
  3. Transitive.
$R_3$ on $R$ defined by $(\text{a, b})\in\text{R}_3\Leftrightarrow\ \text{a}^2-4\text{ab}+3\text{b}^2=0$
✓

Answer

Reflexivity: Consider a be an arbitrary element of $R_3$
Then, $\text{a}\in\text{R}_3$
Implies that $a_2 - 4a_2 + 3a_2 = 0$
So, $R_3$ is reflexive.
Symmetry: Consider, $\text{a, b}\in\text{R}_3$
Implies that $a_2 - 4a_2b_2 + 3b_2 = 0$
But $\text{b}_2-4\text{b}_2\text{a}_2+3\text{a}_2\neq0$ for all $\text{a, b}\in\text{R}$
So, $R_3$ is not symmetric.
Transitivity: $1,2\in\text{R}_3$ and $2,3\in\text{R}_3$
Implies that $1 - 8 + 6 = 0$ and $4 - 24 + 27 = 0$
But $1 - 12 + 9 \neq0$
So, $R_3$ is not transitive.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from RELATIONS AND FUNCTIONS→RELATIONS AND FUNCTIONS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Differentiate the following functions from first principles: $e^{-x}.$
→
Form the differential equation corresponding to $\text{y}^2=\text{a}(\text{b}-\text{x}^2)$ bt eliminating a and b.
→
For the following matrices verify the distributivity of matrix, multiplication over matrix addtion i.e., A(B + C) = AB + AC.
$\text{A}=\begin{bmatrix}1&-1\\0&2\end{bmatrix},\text{B}=\begin{bmatrix}-1&0\\2&1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}0&1\\1&-1\end{bmatrix}$
→
Write a value of $\int\frac{1+\log\text{x}}{3+\text{x}\log\text{x}}\text{ dx}$
→
Find $f^{-1}$ if it exists: $f : A \rightarrow B,$ where, $A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81}$ and $f(x) = x^2.$
→
A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.
→
A function $f: R \rightarrow R$ given by $f(x)=2 x+3$. Prove that $f$ is invertible.
→
Write the coordinates of the point on the curve $\text{x}=\text{e}^\text{t}\cos\text{t},\text{y}=\text{e}^\text{t}$ where the tangent line makes an angle $\frac{\pi}{4}$ with x-axis.
→
Find the equation of the plane through the intersection of the planes 3x - y + 2z - 4 = 0 and x + y + z - 2 = 0 and the point (2, 2, 1).
→
Evaluate the following integrals:$\int\sqrt{\text{cosec}\text{x}-1}\text{ dx}$
→