Binary Operations Practice Questions — JAMB UTME

Question 1Question →

A binary operation $\otimes$ on the set of real numbers $\mathbb{R}$ is defined by $a \otimes b = a^2 + b^2 - ab$ . If $x \otimes 3 = 19$ and $x > 0$ , find the value of $x$ .

Show answer & explanation

Answer: 5

Answer

The value of

x

is

5

.

Applying the operation rule gives

x^2 + 3^2 - 3x = 19

, which simplifies to

x^2 - 3x - 10 = 0

. Factoring this equation yields

(x - 5)(x + 2) = 0

. Since

x

is constrained to be positive (

x > 0

), the unique valid answer is

5

.

Step-by-Step Solution

1

Apply the definition of the binary operation to

x \otimes 3

x^2 + 3^2 - 3(x) = x^2 - 3x + 9

Substitute

a = x

and

b = 3

into

a \otimes b = a^2 + b^2 - ab

.

2

Equate the result to 19 and rearrange into standard quadratic form

x^2 - 3x - 10 = 0

Subtract 19 from both sides to set the quadratic equation to zero.

3

Factor the quadratic equation and solve for

x

(x - 5)(x + 2) = 0 \implies x = 5 \text{ or } x = -2

Find two numbers that multiply to

-10

and add up to

-3

.

4

Apply the restriction

x > 0

x = 5

Reject the negative solution

x = -2

because

x

must be strictly positive.

Key Concept

Evaluation of Binary Operations and Solving Quadratic Equations

Question 2Question →

A binary operation $\star$ on the set of real numbers is defined by $a \star b = 3a + 2b - 1$ . What is the value of $x$ such that $4 \star x = 25$ ?

6

7

11

14

Show answer & explanation

Answer: 7

Answer

The value of

x

is

7

.

Applying the binary operation definition

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

with

a = 4

and

b = x

gives

3(4) + 2x - 1 = 25

. Simplifying gives

2x + 11 = 25

, leading to

2x = 14

and

x = 7

.

Step-by-Step Solution

1

Substitute

a = 4

and

b = x

into the definition of the binary operation

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

.

4 \star x = 3(4) + 2x - 1

Applying the given rule for the binary operation.

2

Simplify the expression on the left-hand side.

4 \star x = 12 + 2x - 1 = 2x + 11

Performing basic arithmetic multiplication and addition of constants.

3

Set the simplified expression equal to

25

and solve for

x

.

2x + 11 = 25 \implies 2x = 14 \implies x = 7

Subtracting

11

from both sides and dividing by

2

.

Key Concept

Evaluation of non-commutative binary operations and solving algebraic equations involving operational rules.

Question 3Question →

A binary operation $\ast$ on the set of real numbers is defined by $a \ast b = a + b + 5$ . What is the identity element of this operation?

Show answer & explanation

Answer: -5

Answer

The identity element of the binary operation is

-5

.

By the definition of an identity element

e

, the relation

a \ast e = a

must hold for all real numbers

a

. Applying the rule

a \ast b = a + b + 5

gives

a + e + 5 = a

. Subtracting

a

from both sides leads to

e + 5 = 0

, which gives

e = -5

.

Step-by-Step Solution

1

Set up the identity element equation

a \ast e = a \implies a + e + 5 = a

By definition of an identity element, operating any element

a

with

e

yields

a

.

2

Solve the equation for

e

e = -5

Subtracting

a

from both sides gives

e + 5 = 0

, which yields

e = -5

.

Key Concept

Identity Element of a Binary Operation

Question 4Question →

A binary operation $\ast$ on the set of real numbers is defined by $a \ast b = a + b - 2ab$ . If $(x \ast 3) \ast 2 = 38$ , find the value of $x$ .

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

Applying the binary operation definition sequentially yields

x \ast 3 = 3 - 5x

for the inner expression, and

(3 - 5x) \ast 2 = 15x - 7

for the composite expression. Equating

15x - 7 = 38

leads to

15x = 45

, giving

x = 3

.

Step-by-Step Solution

1

Evaluate the inner binary operation expression

x \ast 3

x \ast 3 = 3 - 5x

Apply the definition

a \ast b = a + b - 2ab

with

a = x

and

b = 3

.

2

Evaluate the outer binary operation

(3 - 5x) \ast 2

(3 - 5x) \ast 2 = 15x - 7

Substitute the result from step 1 into the outer operation definition with

a = 3 - 5x

and

b = 2

.

3

Set the resulting expression equal to 38 and solve the linear equation

x = 3

Solve

15x - 7 = 38

by adding 7 to both sides to get

15x = 45

, then dividing by 15.

Key Concept

Nested composition of defined binary operations

Question 5Question →

A binary operation $\circ$ defined on the set of real numbers $\mathbb{R} \setminus \{1\}$ is given by $a \circ b = a + b - ab$ . What is the inverse of $3$ under this operation?

\frac{3}{2}

-\frac{3}{2}

1

\frac{1}{3}

Show answer & explanation

Answer: $\frac{3}{2}$

Answer

The inverse of

3

under the given operation is

\frac{3}{2}

.

To find the inverse of

3

under the operation

a \circ b = a + b - ab

, we must first determine the identity element

e

. Setting

a \circ e = a

yields

a + e - ae = a

, which simplifies to

e(1 - a) = 0

. For all

a \neq 1

, the identity element is

e = 0

. Next, using the definition of inverse

3 \circ x = 0

, we substitute into the operational formula to obtain

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

. Thus, the option specifying

\frac{3}{2}

is correct.

Step-by-Step Solution

1

Find the identity element

e

of the operation

\circ

.

e = 0

By definition of identity,

a \circ e = a

. Substituting into the operational formula gives

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

. Since

a \neq 1

,

e = 0

.

2

Set up the inverse equation for

3

, letting

x

be the inverse of

3

.

3 \circ x = 0

By definition of an inverse element,

a \circ a^{-1} = e

.

3

Expand

3 \circ x

using the operation rule and solve for

x

.

x = \frac{3}{2}

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

.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 6Question →

A binary operation $\Delta$ defined on the set of rational numbers $\mathbb{Q}$ is given by $a \Delta b = \frac{ab}{4}$ . What is the inverse element of $6$ under this operation?

\frac{2}{3}

\frac{8}{3}

\frac{1}{6}

\frac{3}{8}

Show answer & explanation

Answer: $\frac{8}{3}$

Answer

The inverse element of

6

under the given binary operation is

\frac{8}{3}

.

To find the inverse of an element under a binary operation, the identity element

e

must first be found using

a \Delta e = a

, which yields

\frac{ae}{4} = a \implies e = 4

. Then, setting

6 \Delta x = 4

gives

\frac{6x}{4} = 4

, which simplifies to

3x = 8

, giving the inverse

x = \frac{8}{3}

.

Step-by-Step Solution

1

Find the identity element

e

of the binary operation.

e = 4

By definition of identity element,

a \Delta e = a

. Substituting into the definition gives

\frac{ae}{4} = a \implies ae = 4a \implies e = 4

.

2

Set up the inverse equation for the element

6

.

6 \Delta x = 4

, where

x

is the inverse of

6

.

By definition of inverse element,

a \Delta a^{-1} = e

.

3

Solve for the inverse

x

.

x = \frac{8}{3}

Applying the operation rule:

\frac{6x}{4} = 4 \implies \frac{3x}{2} = 4 \implies 3x = 8 \implies x = \frac{8}{3}

.

Key Concept

Identity and Inverse Elements of a Binary Operation

Question 7Question →

A binary operation $\circ$ on the set of real numbers $\mathbb{R}$ is defined by $a \circ b = a + b + 2ab$ . If the identity element of the operation is $e$ , what is the value of $x$ such that the inverse of $x$ under $\circ$ is equal to $2$ ?

Show answer & explanation

Answer: -0.4

Answer

The value of

x

is

-0.4

.

First find the identity element

e

by solving

a \circ e = a

, which yields

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

. Next, by definition of an inverse,

x \circ x^{-1} = e

. Substituting

x^{-1} = 2

and

e = 0

gives

x \circ 2 = 0

. Expanding this using the binary operation rule yields

x + 2 + 2(x)(2) = 0 \implies 5x + 2 = 0 \implies x = -0.4

.

Step-by-Step Solution

1

Find the identity element

e

of the operation

\circ

e = 0

By definition of identity element,

a \circ e = a \implies a + e + 2ae = a

, which simplifies to

e(1 + 2a) = 0

, giving

e = 0

.

2

Set up the inverse equation using

x^{-1} = 2

x \circ 2 = 0

The inverse of

x

satisfies

x \circ x^{-1} = e

. Since

x^{-1} = 2

and

e = 0

,

x \circ 2 = 0

.

3

Solve for

x

x = -0.4

Expanding

x \circ 2 = 0

gives

x + 2 + 4x = 0 \implies 5x = -2 \implies x = -0.4

.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 8Question →

A binary operation $\ast$ is defined on the set of real numbers $\mathbb{R}$ by $a \ast b = a + b - k$ , where $k$ is a constant. If the identity element of the operation is $4$ , what is the inverse of $7$ under this operation?

1

-1

-3

7

Show answer & explanation

Answer: 1

Answer

The inverse of 7 under the binary operation is 1.

First, use the identity property a * e = a with e = 4 to determine that k = 4. Then, set 7 * x = 4 using the operational definition 7 + x - 4 = 4. Solving the linear equation x + 3 = 4 gives the correct inverse value of 1.

Step-by-Step Solution

1

Determine the value of the constant k using the identity element property.

k = 4

By definition of identity element e, a * e = a. Given e = 4, substituting into the definition yields a + 4 - k = a, which simplifies to k = 4.

2

Write the full operational formula.

a * b = a + b - 4

Substitute k = 4 into the original rule a * b = a + b - k.

3

Solve for the inverse element of 7.

x = 1

Let x be the inverse of 7. By definition of inverse, 7 * x = e, so 7 + x - 4 = 4. Simplifying gives x + 3 = 4, hence x = 1.

Key Concept

Identity and inverse elements of a binary operation

Question 9Question →

A binary operation $\star$ defined on the set of real numbers $\mathbb{R}$ is given by $a \star b = 2a - 3b + ab$ . If $(2 \star x) \star 3 = 16$ , what is the value of $x$ ?

\frac{11}{4}

-1

-\frac{1}{5}

-\frac{3}{5}

Show answer & explanation

Answer: $-1$

Answer

-1

Applying the binary operation rule

a \star b = 2a - 3b + ab

step-by-step gives

2 \star x = 4 - x

. Operating on this with

3

yields

(4 - x) \star 3 = 2(4 - x) - 3(3) + 3(4 - x) = 11 - 5x

. Setting

11 - 5x = 16

gives

x = -1

.

Step-by-Step Solution

1

Evaluate the inner expression

2 \star x

using the given definition

a \star b = 2a - 3b + ab

.

2 \star x = 2(2) - 3(x) + (2)(x) = 4 - 3x + 2x = 4 - x

The expression inside the parentheses must be simplified first.

2

Substitute

4 - x

into the outer operation

(4 - x) \star 3

.

(4 - x) \star 3 = 2(4 - x) - 3(3) + (4 - x)(3) = 8 - 2x - 9 + 12 - 3x = 11 - 5x

Apply the operation rule with first element

a = 4 - x

and second element

b = 3

.

3

Equate the resulting expression to 16 and solve for

x

.

11 - 5x = 16 \implies -5x = 5 \implies x = -1

Solve the linear equation to determine the value of

x

.

Key Concept

Non-commutative nested binary operation evaluation

Estimated Time:2m 0s

Question 10Question →

A binary operation $\star$ is defined on the set of real numbers $\mathbb{R}$ by $a \star b = a + b + 7$ . What is the identity element of the operation?

-7

7

0

-1

Show answer & explanation

Answer: $-7$

Answer

The identity element of the operation is

-7

.

For an identity element

e

, the condition

a \star e = a

must hold for all real numbers

a

. Substituting the operation definition gives

a + e + 7 = a

. Subtracting

a

from both sides results in

e + 7 = 0

, which solves to

e = -7

.

Step-by-Step Solution

1

Set up the defining equation for the identity element

e

.

a \star e = a

By definition, an identity element

e

leaves any element

a

unchanged under the operation.

2

Apply the given rule for the binary operation.

a + e + 7 = a

The binary operation is defined as

a \star b = a + b + 7

.

3

Solve for

e

.

e = -7

Subtracting

a + 7

from both sides gives

e = -7

.

Key Concept

Identity element of a binary operation

Estimated Time:45s

Question 11Question →

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}

-\frac{3}{5}

\frac{12}{17}

-\frac{5}{12}

Show answer & explanation

Answer: $-\frac{12}{23}$

Answer

-\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}

.

Step-by-Step Solution

1

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

.

2

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

.

3

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}

.

4

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}

.

Key Concept

Binary Operations: Identity and Inverse Elements with Unknown Parameters

Estimated Time:2m 0s

Question 12Question →

A binary operation $\star$ on the set of real numbers $\mathbb{R} \setminus \{1\}$ is defined by $a \star b = a + b - ab$ . If $y^{-1}$ denotes the inverse of an element $y$ under $\star$ , find the value of $x$ such that $(x \star 3)^{-1} = 2$ .

Show answer & explanation

Answer: 0.5

Answer

The value of

x

is

0.5

.

To solve for

x

, first calculate the identity element

e = 0

using

a \star e = a

. Next, use

y \star y^{-1} = 0

to get the inverse formula

y^{-1} = \frac{y}{y - 1}

. Substituting

y = x \star 3 = 3 - 2x

into the inverse formula yields

\frac{3 - 2x}{2 - 2x} = 2

, which simplifies to

2x = 1

, giving

x = 0.5

.

Step-by-Step Solution

1

Find the identity element

e

of the operation

\star

.

e = 0

By definition of the identity element,

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

, so

e = 0

for all

a \neq 1

.

2

Derive the general expression for the inverse element

y^{-1}

of

y

.

y^{-1} = \frac{y}{y - 1}

By definition of inverse element,

y \star y^{-1} = e \implies y + y^{-1} - y y^{-1} = 0 \implies y^{-1}(1 - y) = -y \implies y^{-1} = \frac{y}{y - 1}

.

3

Evaluate the inner operation

x \star 3

.

x \star 3 = 3 - 2x

Using the operational rule

a \star b = a + b - ab

, we obtain

x \star 3 = x + 3 - 3x = 3 - 2x

.

4

Set up and solve the equation

(x \star 3)^{-1} = 2

.

x = 0.5

Letting

y = 3 - 2x

, its inverse is

y^{-1} = \frac{3 - 2x}{(3 - 2x) - 1} = \frac{3 - 2x}{2 - 2x}

. Equating this to

2

gives

\frac{3 - 2x}{2 - 2x} = 2 \implies 3 - 2x = 4 - 4x \implies 2x = 1 \implies x = 0.5

.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 13Question →

A binary operation $\oplus$ is defined on the set of real numbers $\mathbb{R}$ by $a \oplus b = 2a + 3b - 5$ . What is the value of $(4 \oplus 1) \oplus 2$ ?

Show answer & explanation

Answer: 13

Answer

The value of

(4 \oplus 1) \oplus 2

is 13.

First, evaluate the inner expression

4 \oplus 1

using

a = 4

and

b = 1

, which gives

2(4) + 3(1) - 5 = 8 + 3 - 5 = 6

. Then, substitute this result into the outer expression to evaluate

6 \oplus 2

using

a = 6

and

b = 2

, yielding

2(6) + 3(2) - 5 = 12 + 6 - 5 = 13

.

Step-by-Step Solution

1

Evaluate the inner operation

4 \oplus 1

6

Substitute

a = 4

and

b = 1

into the operational rule

a \oplus b = 2a + 3b - 5

.

2

Evaluate the outer operation using the result from Step 1:

6 \oplus 2

13

Substitute

a = 6

and

b = 2

into the operational rule

a \oplus b = 2a + 3b - 5

.

Key Concept

Evaluation of Binary Operations

Question 14Question →

A binary operation $\circ$ on the set of real numbers $\mathbb{R}$ is defined by $a \circ b = a + b - \frac{1}{2}ab$ . What is the inverse of $3$ under this operation?

6

-6

-3

\frac{1}{3}

Show answer & explanation

Answer: $6$

Answer

The inverse of

3

under the defined binary operation is

6

.

To find the inverse of

3

, we first find the identity element

e

using

a \circ e = a \implies a + e - \frac{1}{2}ae = a

, which gives

e = 0

. Then, setting

3 \circ x = 0

gives

3 + x - \frac{3}{2}x = 0 \implies 3 = \frac{1}{2}x \implies x = 6

.

Step-by-Step Solution

1

Find the identity element

e

of the operation.

e = 0

By definition of identity element,

a \circ e = a

. Substituting into the operation formula gives

a + e - \frac{1}{2}ae = a \implies e\left(1 - \frac{1}{2}a\right) = 0 \implies e = 0

for all

a \neq 2

.

2

Set up the inverse equation for the element

3

.

3 \circ x = 0

Let

x

be the inverse of

3

. By definition of inverse element,

3 \circ x = e

, where

e = 0

.

3

Apply the binary operation definition to the left-hand side.

3 + x - \frac{1}{2}(3)(x) = 0

Substitute

a = 3

and

b = x

into

a \circ b = a + b - \frac{1}{2}ab

.

4

Solve the linear equation for

x

.

x = 6

Simplify the equation:

3 + x - \frac{3}{2}x = 0 \implies 3 - \frac{1}{2}x = 0 \implies \frac{1}{2}x = 3 \implies x = 6

.

Key Concept

Identity and Inverse Elements of Binary Operations

Question 15Question →

A binary operation $\Delta$ is defined on the set of real numbers $\mathbb{R}$ by $a \Delta b = a + 3b - 2ab$ . If $(3 \Delta x) \Delta 1 = 7$ , what is the value of $x$ ?

\frac{7}{3}

\frac{3}{5}

-\frac{7}{9}

-\frac{4}{3}

Show answer & explanation

Answer: $\frac{7}{3}$

Answer

\frac{7}{3}

Evaluating

3 \Delta x

gives

3 - 3x

. Then operating

(3 - 3x) \Delta 1

yields

(3 - 3x) + 3 - 2(3 - 3x) = 3x

. Equating

3x = 7

gives

x = \frac{7}{3}

.

Step-by-Step Solution

1

Evaluate the inner expression

3 \Delta x

using the definition

a \Delta b = a + 3b - 2ab

.

3 \Delta x = 3 + 3x - 2(3)(x) = 3 + 3x - 6x = 3 - 3x

.

Substitute

a = 3

and

b = x

into the operation rule.

2

Substitute the result

(3 - 3x)

as the first operand in the outer expression

(3 - 3x) \Delta 1

.

(3 - 3x) \Delta 1 = (3 - 3x) + 3(1) - 2(3 - 3x)(1)

.

Apply the binary operation definition with

a = 3 - 3x

and

b = 1

.

3

Expand and simplify the algebraic expression.

(3 - 3x) \Delta 1 = 3 - 3x + 3 - 6 + 6x = 3x

.

Distribute

-2

across

(3 - 3x)

to get

-6 + 6x

, then collect like terms.

4

Set the simplified expression equal to

7

and solve for

x

.

3x = 7 \implies x = \frac{7}{3}

.

Divide both sides by

3

to isolate

x

.

Key Concept

Non-commutative binary operation composition and algebraic equation solving

Question 16Question →

A binary operation $*$ defined on the set of real numbers $\mathbb{R}$ is given by $a * b = a^2 + 2b - 5$ . If $3 * x = 12$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

Applying the binary operation definition

a * b = a^2 + 2b - 5

to

3 * x

gives

3^2 + 2x - 5 = 2x + 4

. Equating

2x + 4 = 12

yields

2x = 8

, so

x = 4

.

Step-by-Step Solution

1

Substitute

a = 3

and

b = x

into the operational rule

a * b = a^2 + 2b - 5

.

3 * x = 3^2 + 2x - 5 = 9 + 2x - 5 = 2x + 4

To express the operation

3 * x

as an algebraic expression in terms of

x

.

2

Set the simplified algebraic expression equal to the given value of 12.

2x + 4 = 12

The question states that

3 * x = 12

.

3

Solve the linear equation for

x

.

2x = 8 \implies x = 4

Subtract 4 from both sides and divide by 2.

Key Concept

Evaluating binary operations and solving algebraic equations involving defined operational rules.

Question 17Question →

A binary operation $\ast$ on the set of real numbers $\mathbb{R} \setminus \{2\}$ is defined by $a \ast b = 2a + 2b - ab - 2$ . If $y^{-1}$ represents the inverse of an element $y$ under the operation $\ast$ , find the value of $x$ such that $(x \ast 3) \ast 4^{-1} = 5$ .

Show answer & explanation

Answer: 8

Answer

The value of x is 8.

To solve for x, first find the identity element by solving a * e = a, yielding e = 1. Next, compute 4^{-1} from 4 * 4^{-1} = 1, which gives 4^{-1} = 2.5. Then simplify x * 3 to 4 - x. Finally, substitute into (4 - x) * 2.5 = 5 and solve for x to get 8.

Step-by-Step Solution

1

Find the identity element e of the operation

e = 1

By definition, a * e = a. Substituting into the operation gives 2a + 2e - ae - 2 = a, which simplifies to (a - 2)(1 - e) = 0. Since a != 2, e must equal 1.

2

Calculate the inverse element 4^{-1}

4^{-1} = 2.5

By definition of an inverse element, 4 * 4^{-1} = e = 1. Applying the operation formula yields 2(4) + 2(4^{-1}) - 4(4^{-1}) - 2 = 1, which simplifies to 6 - 2(4^{-1}) = 1, so 4^{-1} = 2.5.

3

Express x * 3 in terms of x

x * 3 = 4 - x

Evaluating x * 3 using the operational definition gives 2x + 2(3) - 3x - 2 = 4 - x.

4

Solve the main equation (x * 3) * 4^{-1} = 5 for x

x = 8

Substituting x * 3 = 4 - x and 4^{-1} = 2.5 into the equation yields (4 - x) * 2.5 = 5. Applying the operation gives 2(4 - x) + 2(2.5) - 2.5(4 - x) - 2 = 5, which simplifies to 1 + 0.5x = 5, giving x = 8.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 18Question →

A binary operation $\odot$ defined on the set of real numbers $\mathbb{R}$ is given by $a \odot b = a + b + kab$ , where $k$ is a non-zero constant. If the inverse of $2$ under $\odot$ is $-4$ , what is the value of $(3 \odot 1)^{-1}$ ?

-\frac{52}{3}

\frac{13}{4}

-\frac{28}{5}

-\frac{13}{3}

Show answer & explanation

Answer: $-\frac{52}{3}$

Answer

The value of

(3 \odot 1)^{-1}

is

-\frac{52}{3}

.

First, the identity element is determined by solving

a \odot e = a

, which gives

a + e + kae = a \implies e = 0

. Next, using the inverse property

x \odot x^{-1} = 0

, we get

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

. Given

2^{-1} = -4

, substituting gives

\frac{-2}{1 + 2k} = -4

, leading to

k = -\frac{1}{4}

. Evaluating

3 \odot 1

yields

3 + 1 - \frac{3}{4} = \frac{13}{4}

. Finally, applying the inverse formula to

\frac{13}{4}

gives

\frac{-\frac{13}{4}}{1 - \frac{13}{16}} = -\frac{52}{3}

.

Step-by-Step Solution

1

Find the identity element

e

under the operation

\odot

e = 0

By definition of identity element,

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

for all real numbers

a

.

2

Derive the formula for the inverse

x^{-1}

of an element

x

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

By definition of inverse,

x \odot x^{-1} = e \implies x + x^{-1} + kxx^{-1} = 0 \implies x^{-1}(1 + kx) = -x \implies x^{-1} = \frac{-x}{1 + kx}

.

3

Use the given inverse condition

2^{-1} = -4

to find the constant

k

k = -\frac{1}{4}

Substituting

x = 2

into the inverse formula gives

\frac{-2}{1 + 2k} = -4 \implies -2 = -4(1 + 2k) \implies -2 = -4 - 8k \implies 8k = -2 \implies k = -\frac{1}{4}

.

4

Evaluate the operation

3 \odot 1

3 \odot 1 = \frac{13}{4}

Using the operation definition with

k = -\frac{1}{4}

:

3 \odot 1 = 3 + 1 + \left(-\frac{1}{4}\right)(3)(1) = 4 - \frac{3}{4} = \frac{13}{4}

.

5

Calculate the inverse of

\frac{13}{4}

under

\odot

-\frac{52}{3}

Using the inverse formula

y^{-1} = \frac{-y}{1 + ky}

for

y = \frac{13}{4}

:

y^{-1} = \frac{-\frac{13}{4}}{1 + \left(-\frac{1}{4}\right)\left(\frac{13}{4}\right)} = \frac{-\frac{13}{4}}{1 - \frac{13}{16}} = \frac{-\frac{13}{4}}{\frac{3}{16}} = -\frac{13}{4} \times \frac{16}{3} = -\frac{52}{3}

.

Key Concept

Binary Operations: Finding Identity Elements, Unknown Parameters, and Inverse Elements

Estimated Time:2m 0s

Question 19Question →

A binary operation $\ast$ defined on the set of real numbers $\mathbb{R}$ is given by $a \ast b = a + b + 7$ . What is the identity element under this operation?

Show answer & explanation

Answer: -7

Answer

The identity element under the operation is

-7

.

The identity element

e

satisfies

a \ast e = a

for any real number

a

. Substituting into the definition gives

a + e + 7 = a

, which simplifies to

e = -7

.

Step-by-Step Solution

1

Set up the identity element equation using the definition

a \ast e = a

.

a + e + 7 = a

By definition, operating any element

a

with the identity element

e

yields

a

.

2

Subtract

a

from both sides of the equation.

e + 7 = 0

Isolating terms involving

e

.

3

Subtract

7

from both sides to solve for

e

.

e = -7

Determining the numerical value of the identity element.

Key Concept

Identity Element in Binary Operations

Question 20Question →

A binary operation $\star$ defined on the set of real numbers $\mathbb{R}$ is given by $x \star y = \frac{x + y}{x - y + 1}$ for $x - y \neq -1$ . If $5 \star p = 3$ , what is the value of $p$ ?

Show answer & explanation

Answer: 3.25

Answer

The value of

p

is

3.25

(or

\frac{13}{4}

).

Applying the operation rule

x \star y = \frac{x + y}{x - y + 1}

with

x = 5

and

y = p

gives

\frac{5 + p}{6 - p} = 3

. Cross-multiplying gives

5 + p = 18 - 3p

, which simplifies to

4p = 13

or

p = 3.25

.

Step-by-Step Solution

1

Substitute

x = 5

and

y = p

into the given binary operation definition

\frac{5 + p}{5 - p + 1} = 3

This sets up the equation for the given condition

5 \star p = 3

.

2

Simplify the denominator in the algebraic fraction

\frac{5 + p}{6 - p} = 3

Combining the constants

5 + 1 = 6

simplifies the denominator.

3

Multiply both sides by

(6 - p)

and expand

5 + p = 18 - 3p

Eliminating the denominator allows linear terms in

p

to be collected.

4

Rearrange terms to solve for

p

4p = 13 \implies p = 3.25

Adding

3p

to both sides and subtracting

5

gives

4p = 13

.

Key Concept

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