Download App

Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
A binary operation * is defined on the set R of all real numbers by the rule $\text{a}\times\text{b}=\sqrt{\text{a}^2+\text{b}^2}\ \forall\text{ a, b}\in\text{R}$.
Write the identity element for * on R.
✓

Answer

Let e be the identity element in R with respect to * such thata * e = a = e * a, $\forall\text{ a}\in\text{R}$
a * e = a and e * a = a, $\forall\text{ a}\in\text{R}$
Then,
$\sqrt{\text{a}^2+\text{e}^2}=\text{a}$ and $\sqrt{\text{e}^2+\text{a}^2}=\text{a},\forall\text{ a}\in\text{R}$
Implies that $\sqrt{\text{a}^2+\text{e}}=\text{a}$ and $\sqrt{\text{e}+\text{a}^2}=\text{a},\forall\text{ a}\in\text{R}$ [$\because$ $e^2 = e$]
Implies that $a^2 + e = a^2 and e + a^2 = a^2$, $\forall\text{ a}\in\text{R}$
Implies that $\text{e}=0\in\text{R},\forall\text{ a}\in\text{R}$
Thus, 0 is the identity element in R with respect to *.

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 Binary Operations→Binary Operations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equations of the tangent and normal to the given curves at the indicated points:
$y = x^2 $ at $(0, 0)$
→
Solve the following: $\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$.
→
From a deck of cards, three cards are drawn on by one without replacement. Find the probability that each time it is a card of spade.
→
Find 'a' for which $\text{f}(\text{x})=\text{a}(\text{x}+\sin\text{x})+\text{a}$ is increasing on R.
→
If $\text{A}=\begin{bmatrix}1&2\\0&3 \end{bmatrix}$ is written as B + C, where B is a symmetric matrix and C is a skew- symmetric matrix, then B is equal to.
→
Show that the relation $R$ defined in the set $A$ of all triangles as $R=\left\{\left(T_1, T_2\right): T_1\right.$ is similar to $\left.T_2\right\}$, is equivalence relation. Consider three right angle triangles $T_1$ with sides $3,4,5, T_2$ with sides $5,12,13$ and $T_3$ with sides $6,8,10$. Which triangles among $T_1, T_2$ and $T_3$ are related?
→
Find the distance of the plane 2x- 3y + 4z - 6 = 0 from the origin.
→
$\int(\text{e}^{\text{x}}+\frac{1}{\text{e}^{\text{x}}})^2\text{dx}$
→
Two dice are rolled once. Find the probability that:
  1. the numbers on two dice are different.
  2. the total of numbers on the two dice is at least.
→
Solve the following differential equation
$5\frac{\text{dy}}{\text{dx}}=\text{e}^\text{x}\text{y}^4$
→