A binary operation \oplus defined on the set of real numbers is given by a \oplus b = a + … — JAMB UTME

A binary operation $\oplus$ defined on the set of real numbers is given by $a \oplus b = a + b + kab$ , where $k$ is a non-zero real constant. If the inverse of $3$ under this operation is $-\frac{1}{2}$ , what is the inverse of $4$ under the same operation?

$-\frac{12}{23}$ Cevap
B
$-\frac{3}{5}$
C
$\frac{12}{17}$
D
$-\frac{5}{12}$

Cevap

-\frac{12}{23}

To find the inverse of

4

, first determine the identity element

e

from

a \oplus e = a

, which gives

e = 0

. Next, set

x \oplus x^{-1} = 0

to get

x^{-1} = -\frac{x}{1 + kx}

. Using

3^{-1} = -\frac{1}{2}

, solve

-\frac{3}{1 + 3k} = -\frac{1}{2}

to find

k = \frac{5}{3}

. Finally, substituting

x = 4

yields

4^{-1} = -\frac{4}{1 + 4(\frac{5}{3})} = -\frac{12}{23}

Adım Adım Çözüm

Determine the identity element

e

of the operation.

e = 0

By definition of identity element,

a \oplus e = a \implies a + e + kae = a \implies e(1 + ka) = 0 \implies e = 0

for all valid

a

Derive the general formula for the inverse

x^{-1}

of an element

x

x^{-1} = -\frac{x}{1 + kx}

An element and its inverse yield the identity element under the operation:

x \oplus x^{-1} = 0 \implies x + x^{-1} + kxx^{-1} = 0 \implies x^{-1}(1 + kx) = -x

Use the given inverse of

3

to solve for the constant

k

k = \frac{5}{3}

Given

3^{-1} = -\frac{1}{2}

, substitute

x = 3

into the inverse formula:

-\frac{3}{1 + 3k} = -\frac{1}{2} \implies 1 + 3k = 6 \implies 3k = 5 \implies k = \frac{5}{3}

Calculate the inverse of

4

using

k = \frac{5}{3}

4^{-1} = -\frac{12}{23}

Substitute

x = 4

and

k = \frac{5}{3}

into the inverse formula:

4^{-1} = -\frac{4}{1 + 4\left(\frac{5}{3}\right)} = -\frac{4}{1 + \frac{20}{3}} = -\frac{4}{\frac{23}{3}} = -\frac{12}{23}

Anahtar Kavram

Binary Operations: Identity and Inverse Elements with Unknown Parameters

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