Q^+ is the set of all positive rational numbers with the binary o… - Vidyadip

MCQ

$Q^+$ is the set of all positive rational numbers with the binary operation $*$ defined by $\text{a}^*\text{b}=\frac{\text{ab}}2\ \forall\text{ a, b}\in\text{Q}^+$. The inverse of an element $\text{a}\in\text{Q}^+$ is:

A
$\text{a}$
B
$\frac{1}{\text{a}}$
C
$\frac{2}{\text{a}}$
✓
$\frac{4}{\text{a}}$

✓

Answer

Correct option: D.

$\frac{4}{\text{a}}$

Let e be the identity element in $Q^+$ with respect to $*$ such that
$a^ * e = a = e^ * a, \forall\text{ a}\in\text{Q}^+$
$a^ * e = a$ and $e^ * a = a, \forall\text{ a}\in\text{Q}^+$
$\frac{\text{ae}}2=\text{a}$ and $\frac{\text{ea}}2=\text{a}$, $\forall\text{ a}\in\text{Q}^+$
$\text{e}=2\in\text{Q}^+, \forall\text{ a}\in\text{Q}^+$
Thus$, 2$ is the identity element in $Q^+$ with respect to $*.$
Let $\text{ a}\in\text{Q}^+$ and $\text{ b}\in\text{Q}^+$ be the inverse of $a.$
Then,
$a^ * e = a = e^ * a$
$a^ * b = e$ and $b^ * a = e$
$\frac{\text{ab}}2=2$ and $\frac{\text{ba}}2=2$
$\text{b}=\frac{4}{\text{a}}\in\text{Q}^+$
Thus, $\frac{4}{\text{a}}$ is the inverse of $\text{ a}\in\text{Q}^+$.

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 "M.C.Q (1 Marks)" 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

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $-\hat{i}+2 \hat{j}-2 \hat{k}$ is $30 ,$ then $\alpha$ is equal to

The probability distribution of a discrete random variable $X$ is given below:

$\text{X}:$	$1$	$2$	$3$	$4$
$\text{P}(\text{X}):$	$\frac{1}{10}$	$\frac{1}{5}$	$\frac{3}{10}$	$\frac{2}{5}$

The value of $E(X^2)$ is:

Three integers are chosen at random from the first $20$ integers. The probability that their product is even is,

If $X$ follows a binomial distribution with parameter $\text{n}=8$ and $\text{p}=\frac{1}{2},$ then $\text{P(|X}-4|\leq2)$ equals :

What is integrating factor of $\frac{\text{dy}}{\text{dx}}+\text{y}\sec\text{x}=\tan\text{x}?$

If O is the origin, OP = 3 with direction ratios proportional to -1, 2, -2 then the coordinates of P are:

If $\int \frac{\mathrm{d} \theta}{\cos ^{2} \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{\mathrm{e}}|\mathrm{f}(\theta)|+\mathrm{C}$ where $\mathrm{C}$ is a constant of integration, then the ordered pair $(\lambda, f(\theta))$ is equal to

Calculate: $\int(\text{x}^3-\frac{1}{\text{x}}+{3\text{x}})\text{dx:}$

If $\text{A}=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix},$ then $A^5 =$

If $c_{i j}$ is the cofactor of the element $a_{i j}$ of the determinant $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$, then write the value of $a_{32} \cdot c_{32}$.