A binary operation \star defined on the set of real numbers \mathbb{R} \setminus \left\{\f… — JAMB UTME

A binary operation $\star$ defined on the set of real numbers $\mathbb{R} \setminus \left\{\frac{1}{2}\right\}$ is given by $a \star b = a + b - 2ab$ . If $x^{-1}$ represents the inverse of an element $x$ under $\star$ , what is the value of $x$ for which $x^{-1} = 3$ ?

$\frac{3}{5}$ Answer
B
$\frac{2}{5}$
C
$\frac{1}{3}$
D
$-\frac{3}{5}$

Answer

The value of

x

\frac{3}{5}

First find the identity element

e

using

a \star e = a

, which gives

a + e - 2ae = a \implies e(1 - 2a) = 0 \implies e = 0

. Next, use the inverse definition

x \star x^{-1} = e \implies x + x^{-1} - 2x x^{-1} = 0

. Substituting

x^{-1} = 3

yields

x + 3 - 2x(3) = 0 \implies -5x + 3 = 0 \implies x = \frac{3}{5}

Step-by-Step Solution

Find the identity element

e

under the operation

\star

e = 0

By definition,

a \star e = a \implies a + e - 2ae = a \implies e(1 - 2a) = 0 \implies e = 0

for all

a \neq \frac{1}{2}

Set up the inverse equation for element

x

x \star x^{-1} = 0 \implies x + x^{-1} - 2x x^{-1} = 0

An element operating on its inverse yields the identity element

e = 0

Substitute

x^{-1} = 3

into the inverse equation and solve for

x

x + 3 - 6x = 0 \implies -5x + 3 = 0 \implies 5x = 3 \implies x = \frac{3}{5}

Substitute the given inverse value and perform simple linear algebraic rearrangement.

Key Concept

Identity and Inverse Elements in Binary Operations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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