Binary Operations Alıştırma Soruları — JAMB UTME

Soru 1Soru →

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 $x$ .

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Evaluation of Binary Operations and Solving Quadratic Equations

Soru 2Soru →

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

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Soru 3Soru →

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?

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Identity Element of a Binary Operation

Soru 4Soru →

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 $x$ .

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Nested composition of defined binary operations

Soru 5Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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.

Adım Adım Çözüm

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}

.

Anahtar Kavram

Identity and Inverse Elements in Binary Operations

Soru 6Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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}

.

Adım Adım Çözüm

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}

.

Anahtar Kavram

Identity and Inverse Elements of a Binary Operation

Soru 7Soru →

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 $2$ ?

Cevabı ve açıklamayı göster

Cevap: -0.4

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Identity and Inverse Elements in Binary Operations

Soru 8Soru →

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

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Identity and inverse elements of a binary operation

Soru 9Soru →

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}

Cevabı ve açıklamayı göster

Cevap: $-1$

Cevap

-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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Non-commutative nested binary operation evaluation

Tahmini Süre:2m 0s

Soru 10Soru →

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

Cevabı ve açıklamayı göster

Cevap: $-7$

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Identity element of a binary operation

Tahmini Süre:45s

Soru 11Soru →

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}

Cevabı ve açıklamayı göster

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

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

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}

.

Anahtar Kavram

Binary Operations: Identity and Inverse Elements with Unknown Parameters

Tahmini Süre:2m 0s

Soru 12Soru →

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

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Identity and Inverse Elements in Binary Operations

Soru 13Soru →

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 $(4 \oplus 1) \oplus 2$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Evaluation of Binary Operations

Soru 14Soru →

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}

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Identity and Inverse Elements of Binary Operations

Soru 15Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

\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}

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Non-commutative binary operation composition and algebraic equation solving

Soru 16Soru →

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 $x$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 17Soru →

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 $(x \ast 3) \ast 4^{-1} = 5$ .

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Identity and Inverse Elements in Binary Operations

Soru 18Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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}

.

Adım Adım Çözüm

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}

.

Anahtar Kavram

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

Tahmini Süre:2m 0s

Soru 19Soru →

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?

Cevabı ve açıklamayı göster

Cevap: -7

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Identity Element in Binary Operations

Soru 20Soru →

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 $p$ ?

Cevabı ve açıklamayı göster

Cevap: 3.25

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Solving linear equations derived from non-commutative binary operations

Binary Operations

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram