Binary Operations Practice Questions — JAMB UTME — Page 2

Question 21Question →

A binary operation $\Delta$ defined on the set of real numbers $\mathbb{R}$ is given by $a \Delta b = a + b - \frac{ab}{4}$ . What is the inverse of $2$ under this operation?

-4

4

-2

-\frac{4}{3}

Show answer & explanation

Answer: $-4$

Answer

The inverse of

2

under the binary operation is

-4

.

To find the inverse of

2

, we first determine the identity element

e

using

a \Delta e = a

, which yields

e = 0

. Setting

2 \Delta x = 0

gives

2 + x - \frac{2x}{4} = 0

, which simplifies to

2 + \frac{x}{2} = 0

, resulting in

x = -4

.

Step-by-Step Solution

1

Determine the identity element

e

of the operation.

e = 0

By definition,

a \Delta e = a

. Thus,

a + e - \frac{ae}{4} = a \implies e\left(1 - \frac{a}{4}\right) = 0

, which gives

e = 0

for all real

a

.

2

Set up the inverse equation for the element

2

.

2 + x - \frac{2x}{4} = 0

Let

x

be the inverse of

2

. By definition,

2 \Delta x = e

, where

e = 0

.

3

Simplify and solve for

x

.

x = -4

Simplify

2 + x - \frac{x}{2} = 0 \implies 2 + \frac{x}{2} = 0 \implies \frac{x}{2} = -2 \implies x = -4

.

Key Concept

Inverse Element in Binary Operations

Question 22Question →

A binary operation $\ast$ is defined on the set of real numbers $\mathbb{R}$ by $a \ast b = a + b - 4$ . If $x^{-1}$ represents the inverse of an element $x$ under the operation $\ast$ , what is the value of $x$ such that $x \ast 3 = 10^{-1}$ ?

-1

-5

-9

\frac{11}{10}

Show answer & explanation

Answer: $-1$

Answer

The value of

x

is

-1

.

First, the identity element

e

is determined using

a \ast e = a

, which gives

a + e - 4 = a \implies e = 4

. Next, the inverse

10^{-1}

is calculated from

10 \ast 10^{-1} = 4

, yielding

10 + 10^{-1} - 4 = 4 \implies 10^{-1} = -2

. Finally, setting

x \ast 3 = -2

gives

x + 3 - 4 = -2 \implies x - 1 = -2 \implies x = -1

.

Step-by-Step Solution

1

Find the identity element

e

under the operation

\ast

.

e = 4

By definition of an identity element,

a \ast e = a \implies a + e - 4 = a \implies e = 4

.

2

Determine the inverse of

10

, denoted

10^{-1}

, under

\ast

.

10^{-1} = -2

By definition of an inverse element,

10 \ast 10^{-1} = e \implies 10 + 10^{-1} - 4 = 4 \implies 6 + 10^{-1} = 4 \implies 10^{-1} = -2

.

3

Substitute

10^{-1} = -2

into the given equation

x \ast 3 = 10^{-1}

and solve for

x

.

x = -1

Using the definition of the operation,

x \ast 3 = x + 3 - 4 = x - 1

. Setting

x - 1 = -2

yields

x = -1

.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 23Question →

A binary operation $\circ$ defined on the set of real numbers $\mathbb{R}$ is given by $a \circ b = 3a + 2b - 1$ . What is the value of $x$ for which $(2 \circ x) \circ 3 = 26$ ?

1

\frac{11}{6}

0

\frac{5}{6}

Show answer & explanation

Answer: $1$

Answer

The value of

x

for which

(2 \circ x) \circ 3 = 26

is

1

.

Applying the binary operation rule

a \circ b = 3a + 2b - 1

to the inner expression gives

2 \circ x = 2x + 5

. Then applying the rule to

(2x + 5) \circ 3

gives

3(2x + 5) + 2(3) - 1 = 6x + 20

. Equating

6x + 20 = 26

yields

6x = 6

, giving

x = 1

.

Step-by-Step Solution

1

Evaluate the inner binary operation expression

2 \circ x

using the rule

a \circ b = 3a + 2b - 1

.

2 \circ x = 3(2) + 2(x) - 1 = 6 + 2x - 1 = 2x + 5

The definition of the binary operation replaces

a

with

2

and

b

with

x

.

2

Substitute the result

(2x + 5)

into the outer binary operation expression

(2 \circ x) \circ 3

.

(2x + 5) \circ 3 = 3(2x + 5) + 2(3) - 1 = 6x + 15 + 6 - 1 = 6x + 20

Apply the rule

a \circ b = 3a + 2b - 1

where

a = 2x + 5

and

b = 3

.

3

Equate the simplified expression to

26

and solve for

x

.

6x + 20 = 26 \implies 6x = 6 \implies x = 1

Setting the calculated expression equal to the given value allows isolation of

x

.

Key Concept

Evaluation of non-commutative composite binary operations

Question 24Question →

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}

\frac{2}{5}

\frac{1}{3}

-\frac{3}{5}

Show answer & explanation

Answer: $\frac{3}{5}$

Answer

The value of

x

is

\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

1

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}

.

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

.

3

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

Binary Operations

Answer

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