MCQ
On the set $Q^+$ of all positive rational numbers a binary operation $*$ is defined by $\text{a}*\text{b}=\frac{\text{ab}}2\forall\text{ a, b}\in \text{Q}^+$. The inverse of $8$ is:
- A$\frac{1}{8}$
- ✓$\frac{1}2$
- C$2$
- D$4$
✓
Answer
Correct option: B.
$\frac{1}2$
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 $e * a = a$, $\forall\text{ a}\in\text{Q}^+$
Then,
$\frac{\text{ae}}{2}=\text{a}\text{ and }\frac{\text{ea}}{2}=\text{a},\forall\text{ a}\in\text{Q}^+$
$e = 2$, $\forall\text{ a}\in\text{Q}^+$
Thus, 2 is the identity element in $Q^+$ with respect to $*$.
Let $\text{b}\in\text{Q}^+$ be the inverse of $8$. Then,
$8 * b = e = b * 8$
$8 * b = e$ and $b * 8 = e$
$\frac{(8)\text{b}}2=2\text{ and }\frac{\text{b}(8)}2=2$ $[\because\ \text{e}=2]$
$b = 12$
Thus, $\frac{1}2$ is the inverse of $8$.
$a * e = a = e * a$, $\forall\text{ a}\in\text{Q}^+$
$a * e = a$ and $e * a = a$, $\forall\text{ a}\in\text{Q}^+$
Then,
$\frac{\text{ae}}{2}=\text{a}\text{ and }\frac{\text{ea}}{2}=\text{a},\forall\text{ a}\in\text{Q}^+$
$e = 2$, $\forall\text{ a}\in\text{Q}^+$
Thus, 2 is the identity element in $Q^+$ with respect to $*$.
Let $\text{b}\in\text{Q}^+$ be the inverse of $8$. Then,
$8 * b = e = b * 8$
$8 * b = e$ and $b * 8 = e$
$\frac{(8)\text{b}}2=2\text{ and }\frac{\text{b}(8)}2=2$ $[\because\ \text{e}=2]$
$b = 12$
Thus, $\frac{1}2$ is the inverse of $8$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The area enclosed between the parabolas $y^2=4 x$ and $x^2=4 y$ is
→If three coterminous edges of a parallelopiped are represented by $a - b , b - c$ and $c - a$, then its volume is
→The area bounded by the curve $y^2=x^2$, and the line $x=8$ is____
→The logical statement $(p \rightarrow q) \wedge(q \rightarrow \sim p)$ is equivalent to:
→Which of the following distribution of probabilities of a random variable X is the probability distribution?
The value of $\lambda$ for which the straight line $\frac{x-\lambda}{3}=\frac{y-1}{2+\lambda}=\frac{z-3}{-1}$ may lie on the plane $x-2 y=0$ is
→If α, β, γ are direction angles of a line and α = 60°, β = 45°, γ = ______.
The p.d.f. of a c.r.v. $X$ is
$ f(x)=\left\{\begin{array}{c} 6 x(1-x), \quad 0 \leq x \leq 1 \\ 0, \quad \text { elsewhere } \end{array}\right. $
If $P(X<a)=P(X>a)$, then $a=$
Let $\begin{vmatrix}\text{x}^2+3\text{x}&\text{x}-1&\text{x}+ 3\\\text{x}+1&-2\text{x}&\text{x}-4\\\text{x}-3&\text{x}+4&3\text{x}\end{vmatrix}=\text{ax}^4+\text{bx}^3+\text{cx}^2+\text{dx}+\text{e}$ be an identity in $x,$ where $a, b, c, d, e$ are independent of $x.$ Then the value of $e$ is:
→