Chapter 12 Algebraic Expressions (Additional Questions)

An algebraic expression is a mathematical phrase that contains numbers, variables, and mathematical operations (like addition, subtraction, multiplication, and division).

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Let's analyze each option:

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(A) $5 + 3 = 8$: This is an arithmetic equation. It contains numbers and an equality sign, but no variables.

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(B) $2x - 5$: This contains a number ($2$), a variable ($x$), another number ($5$), and mathematical operations (multiplication between $2$ and $x$, and subtraction). This fits the definition of an algebraic expression.

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(C) $10 > 7$: This is an inequality. It compares two numbers using a relational operator ($>$), but it is not an algebraic expression.

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(D) $6 \times 4$: This is an arithmetic expression or calculation. It contains numbers and multiplication, but no variables.

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Based on the definitions, only option (B) is an algebraic expression as it contains a variable ($x$).

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The correct answer is (B) $2x - 5$.

In an algebraic expression, a variable is a symbol, usually a letter, that represents an unknown or changing quantity.

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The given expression is $7x + 5$.

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In this expression:

$\bullet$ $7$ is the coefficient of $x$.

$\bullet$ $x$ is the variable.

$\bullet$ $5$ is the constant term.

$\bullet$ $+$ is the addition operator.

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Therefore, the symbol that represents the unknown quantity, and is thus the variable, is $x$.

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The correct answer is (B) x.

In an algebraic expression, a constant term is a term that has a fixed value because it does not contain any variables. It is the term without any literal part (variable).

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The given expression is $3y - 8$. This expression can be written as $3y + (-8)$.

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Let's look at the terms in the expression:

$\bullet$ The first term is $3y$. This term contains the variable $y$, so it is not a constant term.

$\bullet$ The second term is $-8$. This term is just a number and does not contain any variable. Therefore, it is the constant term.

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It is important to include the sign preceding the number when identifying the constant term. Since the expression is $3y - 8$, the constant term is $-8$, not just $8$.

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The correct answer is (C) -8.

In an algebraic term, the coefficient is the numerical factor that is multiplied by the variable(s).

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The given term is $-5m$.

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This term represents the product of the number $-5$ and the variable $m$.

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The numerical factor in the term $-5m$ is $-5$.

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Therefore, the coefficient of $m$ in the term $-5m$ is $-5$. It is important to include the sign.

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The correct answer is (B) -5.

In algebra, terms are classified based on their variable parts.

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Let's define the given options:

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(A) Unlike terms: Terms that have different algebraic factors, i.e., different variables or different powers of the same variable. For example, $2x$ and $3y$, or $4a^2$ and $5a$.

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(B) Like terms: Terms that have the same algebraic factors, meaning they have the same variables raised to the same powers. The numerical coefficients can be different. For example, $2x$ and $-5x$, or $3ab^2$ and $7ab^2$.

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(C) Constants: Terms that are only numbers and do not contain any variables. For example, $5$, $-10$, $1/2$.

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(D) Variables: Symbols, usually letters, that represent unknown or changing quantities. For example, $x$, $y$, $a$, $b$.

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The question asks for terms having the same algebraic factors. According to the definition, these are called like terms.

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The correct answer is (B) Like terms.

Like terms are terms that have the same variables raised to the same powers. The numerical coefficients can be different.

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We are given the terms: $2x, 3y, -5x, 7xy, x$.

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Let's examine the variables in each term:

$\bullet$ $2x$: The variable is $x$ with a power of $1$.

$\bullet$ $3y$: The variable is $y$ with a power of $1$.

$\bullet$ $-5x$: The variable is $x$ with a power of $1$.

$\bullet$ $7xy$: The variables are $x$ and $y$, both with a power of $1$.

$\bullet$ $x$: The variable is $x$ with a power of $1$ (since $x = 1x$).

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Now let's identify the terms that have exactly the same variable part:

$\bullet$ Terms with the variable $x$ (and power 1): $2x$, $-5x$, and $x$. These terms have the same algebraic factor ($x$).

$\bullet$ Terms with the variable $y$ (and power 1): $3y$.

$\bullet$ Terms with variables $xy$ (both power 1): $7xy$.

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Comparing these groups with the given options:

(A) $2x, 3y, x$: $2x$ and $x$ are like terms, but $3y$ is unlike.

(B) $2x, -5x, x$: All these terms have the variable $x$ with power 1. They are like terms.

(C) $3y, 7xy$: These terms have different variable parts ($y$ vs $xy$). They are unlike terms.

(D) $2x, 7xy$: These terms have different variable parts ($x$ vs $xy$). They are unlike terms.

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The set of like terms among the options is (B).

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The correct answer is (B) $2x, -5x, x$.

To find the sum of algebraic terms, we combine like terms. Like terms are terms that have the same variable(s) raised to the same power(s).

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The given terms are $2x$ and $3x$.

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Both terms have the variable $x$ raised to the power of $1$ ($x^1 = x$). Therefore, they are like terms.

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To add like terms, we add their numerical coefficients and keep the variable part the same.

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The coefficients are $2$ and $3$.

Sum of coefficients $= 2 + 3 = 5$.

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The variable part is $x$.

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So, the sum of $2x$ and $3x$ is $(2+3)x = 5x$.

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Let's look at the options:

(A) $5x^2$: Incorrect, the variable power should remain $1$.

(B) $6x$: Incorrect, this would be the result of multiplication ($2x \times 3 = 6x$ or $2 \times 3x = 6x$), not addition of coefficients.

(C) $5x$: Correct, sum of coefficients is $5$ and variable is $x$.

(D) $5$: Incorrect, the variable $x$ must be included as it is part of the terms being added.

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The correct answer is (C) $5x$.

To subtract algebraic terms, we can only combine like terms. Like terms have the same variable(s) raised to the same power(s).

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The problem asks to subtract $5a$ from $8a$, which can be written as: $8a - 5a$

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Both terms, $8a$ and $5a$, have the variable $a$ raised to the power of $1$. Thus, they are like terms.

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To subtract like terms, we subtract their numerical coefficients and keep the variable part the same.

The coefficients are $8$ and $5$.

Subtracting the coefficients: $8 - 5 = 3$.

The variable part remains $a$.

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So, the result of the subtraction is $3a$.

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Let's check the options:

(A) $3a$: This matches our calculated result.

(B) $13a$: This would be the result of adding $8a$ and $5a$ ($8a + 5a = 13a$).

(C) $-3a$: This would be the result of subtracting $8a$ from $5a$ ($5a - 8a = -3a$).

(D) $40a^2$: This would be the result of multiplying $8a$ and $5a$ ($8a \times 5a = 40a^2$).

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The correct answer is (A) $3a$.

To simplify an algebraic expression, we combine like terms. Like terms are terms that have the same variable(s) raised to the same power(s).

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The given expression is $4p - 7q + 2p + 3q$.

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Identify the like terms:

$\bullet$ Terms with variable $p$: $4p$ and $2p$.

$\bullet$ Terms with variable $q$: $-7q$ and $3q$.

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Group the like terms together:

$(4p + 2p) + (-7q + 3q)$

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Combine the coefficients of the like terms:

For the $p$ terms: $4p + 2p = (4 + 2)p = 6p$.

For the $q$ terms: $-7q + 3q = (-7 + 3)q = -4q$.

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Put the combined terms back together to get the simplified expression:

$6p - 4q$

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Let's check the options:

(A) $6p - 4q$: This matches our simplified expression.

(B) $6p + 4q$: Incorrect sign for the $q$ term.

(C) $2p - 10q$: Incorrect coefficients for both terms.

(D) $6pq$: Incorrect, this is the product of terms, not the sum/difference of like terms.

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The correct answer is (A) $6p - 4q$.

To find the value of an algebraic expression for a given value of the variable, we substitute the given value for the variable into the expression and then calculate the result.

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The given expression is $3x - 5$.

The given value for the variable is $x = 4$.

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Substitute $x=4$ into the expression:

$3(4) - 5$

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Now, perform the operations following the order of operations (multiplication before subtraction):

First, multiply $3$ by $4$: $3 \times 4 = 12$.

So the expression becomes:

$12 - 5$

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Next, perform the subtraction:

$12 - 5 = 7$

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So, when $x = 4$, the value of the expression $3x - 5$ is $7$.

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Let's check the options:

(A) 7: Correct.

(B) 12: Incorrect, this is just $3 \times 4$.

(C) 17: Incorrect, this would be $3(4) + 5$.

(D) -1: Incorrect, this would be $3(1) - 4$.

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The correct answer is (A) 7.

To evaluate an algebraic expression for a given value of the variable, we substitute the given value for the variable into the expression and then perform the necessary arithmetic operations according to the order of operations (PEMDAS/BODMAS).

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The given expression is $m^2 + 2m - 3$.

The value of the variable is given as $m = -1$.

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Substitute $m = -1$ into the expression:

$(-1)^2 + 2(-1) - 3$

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Now, follow the order of operations:

Step 1: Evaluate the exponent.

$(-1)^2 = (-1) \times (-1) = 1$

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The expression becomes:

$1 + 2(-1) - 3$

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Step 2: Perform multiplication.

$2(-1) = -2$

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The expression becomes:

$1 + (-2) - 3$

Which can be written as:

$1 - 2 - 3$

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Step 3: Perform addition and subtraction from left to right.

$1 - 2 = -1$

$-1 - 3 = -4$

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Thus, the value of the expression $m^2 + 2m - 3$ when $m = -1$ is $-4$.

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Comparing this result with the given options:

(A) 0

(B) -4

(C) -2

(D) 2

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The correct answer is (B) -4.

The question asks about the area of mathematics where the formula for the perimeter of a square, which is an algebraic expression, is applied.

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The formula for the perimeter of a square with side length $s$ is given by $P = 4s$.

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The expression $4s$ is an algebraic expression because it contains a number ($4$), a variable ($s$), and an operation (multiplication). This expression relates the perimeter of a square to its side length.

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Now, let's consider the given options:

(A) Geometry: Geometry is the branch of mathematics that deals with shapes, sizes, positions of figures, and the properties of space. Calculating the perimeter of a shape is a fundamental concept in geometry. Formulas for calculating perimeter, area, volume, etc., are standard tools used in geometry.

(B) Arithmetic: Arithmetic deals with basic numerical operations (addition, subtraction, multiplication, division) on numbers. While you use arithmetic to calculate the perimeter when you plug in a specific value for $s$ in the formula $4s$, the formula itself represents a general geometric property using algebraic terms. Arithmetic is used *within* the application, but the application area is not arithmetic itself.

(C) Algebra: Algebra is the study of mathematical symbols and the rules for manipulating them. The formula $4s$ is indeed an algebraic expression. However, the question asks about *using* this expression, which implies applying it to a specific context. While the formula is *from* algebra, its application in calculating the perimeter of a square is within another field.

(D) Data Handling: Data handling involves collecting, organizing, analyzing, and interpreting data. This is unrelated to the calculation of geometric properties.

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Since the formula $4s$ is used to find the perimeter of a square, which is a geometric figure, this is an example of using an algebraic expression in Geometry.

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The correct answer is (A) Geometry.

In algebra, expressions are often classified by the number of terms they contain.

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Here are some common classifications:

$\bullet$ Monomial: An expression with only one term. Examples: $5x$, $m^2$, $7$, $-3ab^2$.

$\bullet$ Binomial: An expression with exactly two terms. The terms are separated by addition or subtraction signs. Examples: $2x + 3y$, $a - b$, $x^2 + 5$, $p^3 - q^3$.

$\bullet$ Trinomial: An expression with exactly three terms. Examples: $a + b + c$, $x^2 + 2x - 3$, $p^2 - q^2 + r^2$.

$\bullet$ Polynomial: An expression with one or more terms. Monomials, binomials, and trinomials are all types of polynomials.

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Now let's look at the given options:

(A) $5x$: This expression has only one term ($5x$). It is a monomial.

(B) $2x + 3y$: This expression has two terms ($2x$ and $3y$), separated by an addition sign. It is a binomial.

(C) $a + b + c$: This expression has three terms ($a$, $b$, and $c$), separated by addition signs. It is a trinomial.

(D) $m^2$: This expression has only one term ($m^2$). It is a monomial.

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The question asks which of the options is a binomial.

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The correct answer is (B) $2x + 3y$.

Algebraic expressions are classified based on the number of terms they contain.

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The terms in an expression are parts that are added or subtracted. In the expression $4x^2 - 3xy + 7$, the terms are separated by the subtraction and addition signs.

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The terms are:

$\bullet$ $4x^2$

$\bullet$ $-3xy$

$\bullet$ $7$

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Let's count the number of terms in the expression $4x^2 - 3xy + 7$. There are exactly three terms.

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Expressions are classified as follows based on the number of terms:

$\bullet$ Monomial: An expression with one term.

$\bullet$ Binomial: An expression with two terms.

$\bullet$ Trinomial: An expression with three terms.

$\bullet$ Polynomial: An expression with one or more terms (a general term covering monomials, binomials, trinomials, and expressions with more terms).

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Since the expression $4x^2 - 3xy + 7$ has three terms, it is a trinomial.

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Let's consider the options:

(A) Monomial: Incorrect (has 3 terms).

(B) Binomial: Incorrect (has 3 terms).

(C) Trinomial: Correct (has 3 terms).

(D) Quadrinomial: Incorrect (means 4 terms).

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The correct answer is (C) Trinomial.

We classify algebraic expressions based on the number of terms they contain:

$\bullet$ A Monomial has 1 term.

$\bullet$ A Binomial has 2 terms.

$\bullet$ A Trinomial has 3 terms.

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Let's classify each given expression:

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(i) $3p$: This expression has only one term ($3p$). Therefore, it is a Monomial.

Matching classification: (b) Monomial. So, (i) - (b).

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(ii) $a - b$: This expression has two terms ($a$ and $-b$). Therefore, it is a Binomial.

Matching classification: (c) Binomial. So, (ii) - (c).

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(iii) $x^2 + y^2 + z^2$: This expression has three terms ($x^2$, $y^2$, and $z^2$). Therefore, it is a Trinomial.

Matching classification: (a) Trinomial or (d) Trinomial. So, (iii) - (a) or (d).

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(iv) $2x - 5y + 3$: This expression has three terms ($2x$, $-5y$, and $3$). Therefore, it is a Trinomial.

Matching classification: (a) Trinomial or (d) Trinomial. So, (iv) - (a) or (d).

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Now let's look at the options provided and check which one matches our findings:

We require: (i) - (b), (ii) - (c), and (iii) and (iv) matched with (a) and (d) (in some order).

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Option (A): (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)

$\bullet$ (i)-(b): $3p$ is Monomial. Correct.

$\bullet$ (ii)-(c): $a-b$ is Binomial. Correct.

$\bullet$ (iii)-(a): $x^2 + y^2 + z^2$ is Trinomial. Correct.

$\bullet$ (iv)-(d): $2x - 5y + 3$ is Trinomial. Correct.

This option correctly matches all expressions to their classifications.

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Option (B): (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)

$\bullet$ (i)-(b): $3p$ is Monomial. Correct.

$\bullet$ (ii)-(c): $a-b$ is Binomial. Correct.

$\bullet$ (iii)-(d): $x^2 + y^2 + z^2$ is Trinomial. Correct.

$\bullet$ (iv)-(a): $2x - 5y + 3$ is Trinomial. Correct.

This option also correctly matches all expressions to their classifications. However, given the structure of standard multiple-choice questions, option (A) likely presents the intended pairing where (iii) is matched with the first 'Trinomial' option listed, and (iv) with the second.

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Option (C): (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)

$\bullet$ (i)-(c): $3p$ is Monomial, not Binomial. Incorrect.

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Option (D): (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)

$\bullet$ (ii)-(d): $a-b$ is Binomial, not Trinomial. Incorrect.

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Both (A) and (B) provide valid matches. Assuming the standard convention where the first occurrence of a repeated option is matched first, option (A) is the most likely intended answer.

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The correct answer is (A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d).

Let's analyze the Assertion and the Reason separately.

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Assertion (A): The terms $5x$ and $-7x$ are like terms.

To determine if two terms are like terms, we check their variable parts. The numerical coefficients do not matter, but the variables and their powers must be identical.

In the term $5x$, the variable part is $x$.

In the term $-7x$, the variable part is $x$.

Both terms have the same variable ($x$) raised to the same power ($1$, since $x = x^1$). Therefore, the terms $5x$ and $-7x$ are indeed like terms.

So, Assertion (A) is True.

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Reason (R): Like terms have the same variables raised to the same power.

This is the standard definition of like terms in algebra. Terms are considered "like" if their variable components are identical, including the exponents of each variable.

So, Reason (R) is True.

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Now, let's consider if the Reason is the correct explanation for the Assertion.

Assertion (A) states that $5x$ and $-7x$ are like terms. Reason (R) gives the condition for terms to be like terms (same variables, same powers). Since $5x$ and $-7x$ meet this condition (both have $x$ to the power of 1), the reason correctly explains why the assertion is true.

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Therefore, both Assertion (A) and Reason (R) are true, and Reason (R) provides the correct explanation for Assertion (A).

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Based on this analysis, we look at the options:

(A) Both A and R are true, and R is the correct explanation of A. (Matches our conclusion)

(B) Both A and R are true, but R is not the correct explanation of A. (Incorrect)

(C) A is true, but R is false. (Incorrect)

(D) A is false, but R is true. (Incorrect)

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The correct answer is (A) Both A and R are true, and R is the correct explanation of A.

Let's analyze the Assertion and the Reason.

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Assertion (A): The value of $2x+1$ is 5 when $x=2$.

To check this, we substitute $x=2$ into the expression $2x+1$:

$2(2) + 1$

According to the order of operations, perform multiplication first:

$4 + 1$

Then, perform addition:

$4 + 1 = 5$

The value of the expression $2x+1$ when $x=2$ is indeed $5$.

So, Assertion (A) is True.

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Reason (R): Substituting the value of the variable into the expression gives the value of the expression.

This statement describes the general process for evaluating an algebraic expression for a specific value of its variable(s). By replacing the variable(s) with the given number(s) and performing the indicated operations, we find the numerical value of the expression.

So, Reason (R) is True.

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Now, let's consider if the Reason explains the Assertion.

Assertion (A) gives a specific example of evaluating an expression. Reason (R) describes the general method used for such evaluation. The reason (substituting the value of the variable) is exactly the method used to verify the assertion (that the value of $2x+1$ is 5 when $x=2$). Therefore, the Reason is the correct explanation for the Assertion.

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Based on this analysis, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).

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Let's check the options:

(A) Both A and R are true, and R is the correct explanation of A. (Matches our conclusion)

(B) Both A and R are true, but R is not the correct explanation of A. (Incorrect)

(C) A is true, but R is false. (Incorrect)

(D) A is false, but R is true. (Incorrect)

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The correct answer is (A) Both A and R are true, and R is the correct explanation of A.

Algebraic expressions are classified based on the number of terms they contain.

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The given expression is $10m + 5n$.

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The terms in an expression are parts separated by addition or subtraction signs.

In the expression $10m + 5n$, the terms are:

$\bullet$ $10m$

$\bullet$ $5n$

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Counting the terms, we find there are exactly two terms in the expression $10m + 5n$.

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The classification of expressions based on the number of terms is:

$\bullet$ Monomial: 1 term.

$\bullet$ Binomial: 2 terms.

$\bullet$ Trinomial: 3 terms.

$\bullet$ Polynomial: 1 or more terms.

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Since the expression $10m + 5n$ has two terms, it is classified as a Binomial.

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Let's check the options:

(A) Monomial: Incorrect (has 2 terms).

(B) Binomial: Correct (has 2 terms).

(C) Trinomial: Incorrect (has 2 terms).

(D) Constant: Incorrect (contains variables $m$ and $n$).

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The correct answer is (B) Binomial.

From the case study in Question 18, the expression for the total cost of buying $m$ apples at $\textsf{₹ }10$ each and $n$ oranges at $\textsf{₹ }5$ each is given by:

Cost $= 10m + 5n$

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In this question, the customer buys 3 apples and 4 oranges.

So, we have $m = 3$ (number of apples) and $n = 4$ (number of oranges).

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To find the total cost, we need to evaluate the expression $10m + 5n$ by substituting $m = 3$ and $n = 4$ into it.

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Substitute the values:

Cost $= 10(3) + 5(4)$

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Now, perform the multiplication:

$10 \times 3 = 30$

$5 \times 4 = 20$

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The expression becomes:

Cost $= 30 + 20$

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Finally, perform the addition:

Cost $= 50$

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The total cost is $\textsf{₹ }50$.

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Let's check the options:

(A) $\textsf{₹ }30$: Incorrect (cost of apples only).

(B) $\textsf{₹ }20$: Incorrect (cost of oranges only).

(C) $\textsf{₹ }50$: Correct.

(D) $\textsf{₹ }70$: Incorrect ($10 \times 4 + 5 \times 3 = 40 + 15 = 55$).

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The correct answer is (C) $\textsf{₹ }50$.

Algebraic expressions are classified based on the number of terms they contain.

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Here are the classifications based on the number of terms:

$\bullet$ An expression with one term is called a Monomial.

$\bullet$ An expression with two terms is called a Binomial.

$\bullet$ An expression with three terms is called a Trinomial.

$\bullet$ An expression with one or more terms is called a Polynomial.

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The sentence asks for the term used for an expression containing "only one term".

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Based on the definitions above, an expression containing only one term is called a Monomial.

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Let's check the options:

(A) Binomial: Incorrect (2 terms).

(B) Monomial: Correct (1 term).

(C) Trinomial: Incorrect (3 terms).

(D) Polynomial: Correct (1 or more terms), but Monomial is the specific term for exactly one term.

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While a monomial is a type of polynomial, the question asks for the specific term for an expression with *only* one term, which is Monomial.

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The correct answer is (B) Monomial.

To add algebraic expressions, we combine like terms.

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The given expressions are $(2x + 3y)$ and $(5x - y)$. We need to find their sum:

$(2x + 3y) + (5x - y)$

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First, remove the parentheses. Since we are adding, the signs of the terms inside the second parenthesis remain unchanged.

$2x + 3y + 5x - y$

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Next, identify and group the like terms:

$\bullet$ Terms with variable $x$: $2x$ and $5x$.

$\bullet$ Terms with variable $y$: $3y$ and $-y$. (Remember that $-y$ is the same as $-1y$).

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Group the like terms:

$(2x + 5x) + (3y - y)$

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Combine the coefficients of the like terms:

For the $x$ terms: $2x + 5x = (2 + 5)x = 7x$.

For the $y$ terms: $3y - y = 3y - 1y = (3 - 1)y = 2y$.

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Combine the simplified terms:

$7x + 2y$

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So, the sum of the expressions is $7x + 2y$.

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Let's check the options:

(A) $7x + 2y$: Correct.

(B) $7x + 4y$: Incorrect (coefficient of $y$ is $3+1=4$, instead of $3-1=2$).

(C) $7x - 2y$: Incorrect sign for the $y$ term.

(D) $7x - 4y$: Incorrect coefficients and sign for the $y$ term.

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The correct answer is (A) $7x + 2y$.

To subtract one algebraic expression from another, we write the expression to be subtracted after the "from" expression, with a subtraction sign between them, and enclose the expression being subtracted in parentheses.

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We need to subtract $(3a - 2b)$ from $(5a + b)$. This can be written as:

$(5a + b) - (3a - 2b)$

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To remove the parentheses when a subtraction sign precedes them, we change the sign of each term inside the parentheses.

$(5a + b) - (3a - 2b) = 5a + b - (3a) - (-2b)$

$= 5a + b - 3a + 2b$

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Now, we combine the like terms. Like terms have the same variables raised to the same powers.

The like terms are:

$\bullet$ Terms with $a$: $5a$ and $-3a$

$\bullet$ Terms with $b$: $b$ (which is $1b$) and $2b$ (which is $+2b$)

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Group the like terms together:

$(5a - 3a) + (b + 2b)$

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Combine the coefficients of the like terms:

For the $a$ terms: $5a - 3a = (5 - 3)a = 2a$

For the $b$ terms: $b + 2b = 1b + 2b = (1 + 2)b = 3b$

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Combine the results:

$2a + 3b$

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Thus, subtracting $3a - 2b$ from $5a + b$ gives $2a + 3b$.

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Let's check the options:

(A) $2a - 3b$: Incorrect (the sign of the $b$ term is wrong).

(B) $2a + 3b$: Correct.

(C) $8a - b$: Incorrect (result of $(5a+3a) + (b-2b)$).

(D) $8a + 3b$: Incorrect (result of $(5a+3a) + (b+2b)$).

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The correct answer is (B) $2a + 3b$.

To find the value of an algebraic expression for given values of the variables, we substitute the given values into the expression and then perform the arithmetic calculations following the order of operations (PEMDAS/BODMAS).

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The given expression is $a^2 + b^2 - ab$.

The given values are $a = 2$ and $b = 3$.

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Substitute $a = 2$ and $b = 3$ into the expression:

$(2)^2 + (3)^2 - (2)(3)$

---

Now, we evaluate the expression step-by-step:

Step 1: Evaluate the exponents.

$(2)^2 = 2 \times 2 = 4$

$(3)^2 = 3 \times 3 = 9$

---

The expression becomes:

$4 + 9 - (2)(3)$

---

Step 2: Perform multiplication.

$(2)(3) = 2 \times 3 = 6$

---

The expression becomes:

$4 + 9 - 6$

---

Step 3: Perform addition and subtraction from left to right.

$4 + 9 = 13$

$13 - 6 = 7$

---

Thus, the value of the expression $a^2 + b^2 - ab$ when $a = 2$ and $b = 3$ is $7$.

---

Comparing this result with the given options:

(A) 7

(B) 13

(C) 19

(D) 5

---

The correct answer is (A) 7.

The perimeter of a polygon is the total distance around its boundary. A rectangle has four sides, with opposite sides being equal in length.

---

Let the length of the rectangle be $L$ and the width be $W$.

A rectangle has two sides of length $L$ and two sides of length $W$.

---

To find the perimeter, we add the lengths of all four sides:

Perimeter $= \text{Length} + \text{Width} + \text{Length} + \text{Width}$

Perimeter $= L + W + L + W$

---

Combine the like terms:

Perimeter $= (L + L) + (W + W)$

Perimeter $= 2L + 2W$

---

We can factor out the common factor of $2$ from the expression:

Perimeter $= 2(L + W)$

---

This algebraic expression $2(L+W)$ represents the perimeter of a rectangle with length $L$ and width $W$.

---

Let's examine the given options:

(A) $L + W$: This represents the sum of the length and width, not the perimeter.

(B) $LW$: This represents the area of the rectangle, not the perimeter.

(C) $2(L+W)$: This matches the formula we derived for the perimeter.

(D) $L^2 + W^2$: This expression is not related to the perimeter of a rectangle. (Note that $L^2 + W^2$ is the square of the diagonal length of the rectangle, by the Pythagorean theorem).

---

The correct answer is (C) $2(L+W)$.

We need to translate the given verbal phrase into an algebraic expression.

---

Let's break down the phrase "twice a number decreased by 5".

---

First, we need to represent "a number". Let's use a variable, say $x$, to represent this unknown number.

---

Next, consider "twice a number". "Twice" means multiplied by 2. So, "twice a number" is $2 \times x$, which is written as $2x$.

---

Finally, consider "decreased by 5". "Decreased by" means subtracting 5. So, "twice a number decreased by 5" means subtracting 5 from "twice a number".

This translates to $2x - 5$.

---

The algebraic expression for "twice a number decreased by 5" is $2x - 5$.

---

Let's look at the options:

(A) $2 + x - 5$: This represents "2 plus a number minus 5". Incorrect.

(B) $2x + 5$: This represents "twice a number increased by 5". Incorrect.

(C) $2x - 5$: This matches our translation "twice a number decreased by 5". Correct.

(D) $5 - 2x$: This represents "5 decreased by twice a number". This is different from "twice a number decreased by 5". For example, if the number is 10, $2(10) - 5 = 20 - 5 = 15$, but $5 - 2(10) = 5 - 20 = -15$.

---

The correct answer is (C) $2x - 5$.

Unlike terms are terms that have different algebraic factors. This means they have different variables or the same variables raised to different powers.

Like terms have the same variables raised to the same powers.

---

Let's examine each pair of terms:

---

(A) $7x$ and $-2x$: Both terms have the variable $x$ raised to the power of $1$. The variable parts are the same. Thus, they are like terms.

---

(B) $5ab$ and $8ba$: Both terms have variables $a$ and $b$, each raised to the power of $1$. The order of multiplication of variables does not change the term's algebraic factor ($ab = ba$). Thus, they are like terms.

---

(C) $3m^2$ and $3m$: The first term has the variable $m$ raised to the power of $2$ ($m^2$). The second term has the variable $m$ raised to the power of $1$ ($m^1 = m$). Since the powers of the variable $m$ are different ($2$ vs $1$), these terms have different algebraic factors ($m^2$ vs $m$). Thus, they are unlike terms.

---

(D) $pq$ and $-qp$: Both terms have variables $p$ and $q$, each raised to the power of $1$. The order of multiplication of variables does not change the term's algebraic factor ($pq = qp$). Thus, they are like terms.

---

The question asks to identify the pair of unlike terms.

---

The correct answer is (C) $3m^2$ and $3m$.

In mathematics, the product of two numbers or variables is the result of multiplying them together.

---

We are asked to find the expression for the product of $p$ and $q$.

---

To find the product of $p$ and $q$, we multiply $p$ by $q$. This can be written using the multiplication symbol as $p \times q$ or $p \cdot q$.

---

In algebra, when multiplying variables, or a number and a variable, the multiplication symbol is usually omitted and the terms are written next to each other. So, $p \times q$ is written as $pq$.

---

Let's look at the given options and their meanings:

(A) $p+q$: This represents the sum of $p$ and $q$.

(B) $p-q$: This represents the difference between $p$ and $q$ (subtracting $q$ from $p$).

(C) $pq$: This represents the product of $p$ and $q$ (multiplying $p$ by $q$).

(D) $p/q$: This represents the quotient of $p$ divided by $q$.

---

The expression that represents the product of $p$ and $q$ is $pq$.

---

The correct answer is (C) $pq$.

To evaluate an algebraic expression for a given value of the variable, we substitute the given value for the variable into the expression and then perform the arithmetic calculations following the order of operations (PEMDAS/BODMAS).

---

The given expression is $10 - 3y$.

The value of the variable is given as $y = -2$.

---

Substitute $y = -2$ into the expression:

$10 - 3(-2)$

---

Now, follow the order of operations:

Step 1: Perform multiplication.

Multiply the coefficient $3$ by the value of $y$, which is $-2$.

$3 \times (-2) = -6$

---

The expression becomes:

$10 - (-6)$

---

Step 2: Perform subtraction.

Subtracting a negative number is equivalent to adding the corresponding positive number.

$10 - (-6) = 10 + 6$

$10 + 6 = 16$

---

Thus, the value of the expression $10 - 3y$ when $y = -2$ is $16$.

---

Comparing this result with the given options:

(A) 4

(B) 16

(C) 10

(D) 13

---

The correct answer is (B) 16.

To add algebraic expressions (polynomials), we combine the like terms. Like terms are terms that have the same variables raised to the same powers.

---

The given expressions are $(5x^2 + 3x - 1)$ and $(2x^2 - x + 7)$. We need to find their sum:

$(5x^2 + 3x - 1) + (2x^2 - x + 7)$

---

Remove the parentheses. Since we are adding, the signs of the terms inside the second parenthesis remain unchanged.

$5x^2 + 3x - 1 + 2x^2 - x + 7$

---

Identify and group the like terms:

$\bullet$ Terms with $x^2$: $5x^2$ and $2x^2$

$\bullet$ Terms with $x$: $3x$ and $-x$ (which is $-1x$)

$\bullet$ Constant terms (no variable): $-1$ and $7$

---

Group the like terms:

$(5x^2 + 2x^2) + (3x - x) + (-1 + 7)$

---

Combine the coefficients of the like terms:

For the $x^2$ terms: $5x^2 + 2x^2 = (5 + 2)x^2 = 7x^2$.

For the $x$ terms: $3x - x = 3x - 1x = (3 - 1)x = 2x$.

For the constant terms: $-1 + 7 = 6$.

---

Combine the simplified terms to get the sum:

$7x^2 + 2x + 6$

---

So, the sum of the expressions is $7x^2 + 2x + 6$.

---

Let's check the options:

(A) $7x^2 + 2x + 6$: Correct.

(B) $7x^2 + 4x + 6$: Incorrect (coefficient of $x$ is $3-1=2$, not $3+1=4$).

(C) $7x^2 + 2x - 8$: Incorrect (constant term is $-1+7=6$, not $-1-7=-8$).

(D) $3x^2 + 4x - 8$: Incorrect coefficients for $x^2$ and $x$, and incorrect constant term.

---

The correct answer is (A) $7x^2 + 2x + 6$.

To subtract one algebraic expression (polynomial) from another, we write the expression to be subtracted after the "from" expression, placing the expression being subtracted in parentheses and preceding it with a subtraction sign.

---

We need to subtract $(x^2 - 4x + 5)$ from $(3x^2 + 2x - 1)$. This is written as:

$(3x^2 + 2x - 1) - (x^2 - 4x + 5)$

---

To remove the parentheses when they are preceded by a subtraction sign, we change the sign of each term inside the parentheses.

$3x^2 + 2x - 1 - (x^2) - (-4x) - (+5)$

$= 3x^2 + 2x - 1 - x^2 + 4x - 5$

---

Now, we combine the like terms. Like terms have the same variables raised to the same powers.

Identify the like terms:

$\bullet$ Terms with $x^2$: $3x^2$ and $-x^2$ (remember $-x^2 = -1x^2$)

$\bullet$ Terms with $x$: $2x$ and $4x$

$\bullet$ Constant terms: $-1$ and $-5$

---

Group the like terms together:

$(3x^2 - x^2) + (2x + 4x) + (-1 - 5)$

---

Combine the coefficients of the like terms:

For the $x^2$ terms: $3x^2 - x^2 = (3 - 1)x^2 = 2x^2$

For the $x$ terms: $2x + 4x = (2 + 4)x = 6x$

For the constant terms: $-1 - 5 = -6$

---

Combine the simplified terms to get the result of the subtraction:

$2x^2 + 6x - 6$

---

Let's check the options:

(A) $2x^2 - 6x + 6$: Incorrect signs.

(B) $2x^2 + 6x - 6$: Correct.

(C) $-2x^2 - 6x + 6$: Incorrect coefficients and signs.

(D) $2x^2 - 2x + 4$: Incorrect coefficients and constant term.

---

The correct answer is (B) $2x^2 + 6x - 6$.

The area of a two-dimensional shape is the amount of space it covers. For a square, which has four equal sides and four right angles, the area is calculated by multiplying the length of one side by itself.

---

The formula for the area of a square with side length $s$ is:

Area $= \text{side} \times \text{side} = s \times s = s^2$

---

In this question, the side length of the square is given as $x$.

---

Using the formula, substitute $s = x$:

Area $= x \times x = x^2$

---

So, the algebraic expression for the area of a square with side length $x$ is $x^2$.

---

Let's examine the given options:

(A) $2x$: This expression represents twice the side length, which is not the area (it might relate to the sum of two sides).

(B) $4x$: This expression represents four times the side length, which is the perimeter of the square.

(C) $x^2$: This matches the formula for the area of a square with side length $x$.

(D) $x^3$: This expression represents the side length cubed, which would be the volume of a cube with side length $x$, not the area of a square.

---

The correct answer is (C) $x^2$.

Like terms are terms that have the same variables raised to the same powers. The order of the variables in the product does not matter (e.g., $xy$ is the same as $yx$).

---

Let's examine each set of terms to see if they are like terms:

---

(A) $3x^2y, -5yx^2, \frac{1}{2}x^2y$

$\bullet$ First term: $3x^2y$. Variables are $x^2$ and $y^1$.

$\bullet$ Second term: $-5yx^2$. We can rearrange the variables as $-5x^2y$. Variables are $x^2$ and $y^1$.

$\bullet$ Third term: $\frac{1}{2}x^2y$. Variables are $x^2$ and $y^1$.

All terms have the same variable parts ($x^2y$). Therefore, this is a set of like terms.

---

(B) $ab, a^2b, ab^2$

$\bullet$ First term: $ab$. Variables are $a^1$ and $b^1$.

$\bullet$ Second term: $a^2b$. Variables are $a^2$ and $b^1$. The power of $a$ is different from the first term.

$\bullet$ Third term: $ab^2$. Variables are $a^1$ and $b^2$. The power of $b$ is different from the first term.

The variable parts are different ($ab$, $a^2b$, $ab^2$). Therefore, these are unlike terms.

---

(C) $p, -p^2, p^3$

$\bullet$ First term: $p$. Variable is $p^1$.

$\bullet$ Second term: $-p^2$. Variable is $p^2$. The power of $p$ is different.

$\bullet$ Third term: $p^3$. Variable is $p^3$. The power of $p$ is different.

The variable parts are different ($p$, $p^2$, $p^3$). Therefore, these are unlike terms.

---

(D) $xy, yz, zx$

$\bullet$ First term: $xy$. Variables are $x$ and $y$.

$\bullet$ Second term: $yz$. Variables are $y$ and $z$. Different variables from the first term.

$\bullet$ Third term: $zx$. Variables are $z$ and $x$. Different variables from the first term.

The variable parts are different ($xy$, $yz$, $zx$). Therefore, these are unlike terms.

---

The only set that contains like terms is (A).

---

The correct answer is (A) $3x^2y, -5yx^2, \frac{1}{2}x^2y$.

The given formula for converting Celsius ($C$) to Fahrenheit ($F$) is:

$F = \frac{9}{5}C + 32$

---

We are given that the temperature in Celsius is $25^\circ\text{C}$. So, $C = 25$.

---

To find the temperature in Fahrenheit, we substitute $C = 25$ into the formula and evaluate the expression:

$F = \frac{9}{5}(25) + 32$

---

Now, perform the calculations following the order of operations (multiplication/division before addition):

First, calculate $\frac{9}{5}(25)$. We can simplify this by cancelling the $5$ in the denominator with the $25$:

$F = \frac{9}{\cancel{5}_{1}} \times \cancel{25}^{5} + 32$

$F = 9 \times 5 + 32$

$F = 45 + 32$

---

Next, perform the addition:

$F = 45 + 32 = 77$

---

So, a temperature of $25^\circ\text{C}$ is equal to $77^\circ\text{F}$.

---

Let's check the options:

(A) $77^\circ\text{F}$: Correct.

(B) $45^\circ\text{F}$: Incorrect (this is just $\frac{9}{5} \times 25$).

(C) $57^\circ\text{F}$: Incorrect.

(D) $32^\circ\text{F}$: Incorrect (this is the freezing point in Fahrenheit, corresponding to $0^\circ\text{C}$).

---

The correct answer is (A) $77^\circ\text{F}$.

Let's examine each statement based on the standard definitions of algebraic expressions:

---

(A) A binomial has exactly two terms. This is the correct definition of a binomial. Examples: $a+b$, $2x-5y$. This statement is True.

---

(B) A trinomial has exactly three terms. This is the correct definition of a trinomial. Examples: $a+b+c$, $x^2+2x-3$. This statement is True.

---

(C) A monomial has exactly one term. This is the correct definition of a monomial. Examples: $5x$, $-7y^2$, $10$. This statement is True.

---

(D) A polynomial can have any number of terms, including zero terms.

A polynomial is generally defined as an expression consisting of one or more terms, where each term is a constant or a product of a constant and variables raised to non-negative integer powers. The number of terms in a simplified polynomial expression must be a finite non-negative integer.

The zero polynomial, which is the number $0$, is a polynomial and is considered to have zero terms.

So, the number of terms a polynomial can have is any non-negative integer (0, 1, 2, 3, ...).

The statement says "A polynomial can have any number of terms, including zero terms". This implies that any non-negative integer number of terms is possible for a polynomial, and explicitly mentions that zero terms are included in the possibilities. This aligns with the nature of polynomials having a finite, non-negative integer number of terms, including the case of the zero polynomial.

However, in some introductory contexts, the definition of a polynomial might be restricted to expressions with one or more terms. If that restrictive definition is assumed, then the statement "including zero terms" would make (D) false.

Given that options (A), (B), and (C) are precisely true according to standard definitions, and option (D) uses slightly ambiguous phrasing ("any number of terms" could be interpreted in different ways), but the phrase "including zero terms" refers to the zero polynomial, which is a polynomial with zero terms, statement (D) appears True under a comprehensive definition of polynomials.

Let's re-evaluate the possibility that D is false. If the definition of a polynomial being used here means "an expression with one or more terms" (excluding the zero polynomial from the general case), then stating it can have zero terms would be false in that specific context.

Assuming A, B, and C are intended to be definitively true based on standard definitions of Monomial, Binomial, and Trinomial as expressions with exactly 1, 2, and 3 terms respectively, it is most likely that statement (D) is intended to be false based on a restrictive definition of "polynomial" in this context, possibly implying "polynomials covered by the classifications above", which all have at least one term, or that the phrase "any number of terms" is meant to imply something other than just non-negative integers. However, interpreting "any number of terms" as "any possible finite non-negative integer number of terms" makes D true.

Let's reconsider the possibility that the phrasing "any number of terms" might implicitly suggest possibilities beyond finite non-negative integers (like infinitely many terms), which polynomials do not have. If interpreted this way, statement (D) would be false.

Given the common usage in introductory algebra where Monomial, Binomial, and Trinomial imply 1, 2, and 3 terms, and 'Polynomial' often encompasses these plus expressions with more terms, the statement that seems potentially misleading or false among the options is (D), possibly due to the interpretation of "any number" or a restrictive definition of polynomial being used implicitly.

However, if we strictly follow mathematical definitions, Monomial, Binomial, and Trinomial count the number of non-zero terms in a simplified expression. A polynomial is a finite sum of monomials. The number of non-zero terms can be any non-negative integer. Thus, statement (D) "A polynomial can have any number of terms, including zero terms" is technically TRUE (referring to the count of non-zero terms being any non-negative integer). This suggests there might be an issue with the question or options if A, B, and C are also intended to be true.

Let's assume the most likely intent in an introductory context: A, B, C are true definitions. D is false because "any number of terms" is too broad if it implies non-integer or infinite counts, OR because "polynomial" is implicitly defined as having 1 or more terms in contrast to the zero polynomial.

---

Based on the typical way these concepts are introduced and contrasted, the statement that is most likely intended to be false is (D).

---

The correct answer is (D) A polynomial can have any number of terms, including zero terms. (This statement is often considered false in introductory texts by implying possibilities beyond finite non-negative integers or by assuming a definition of polynomial excluding the zero polynomial from the general case).

In an algebraic term, the coefficient is the numerical factor that is multiplied by the variable(s).

---

The given term is $-12xy$.

---

This term represents the product of the number $-12$ and the variables $x$ and $y$. It can be written as $(-12) \times (x) \times (y)$.

---

We are asked for the coefficient of the variable part $xy$.

In the term $-12xy$, the part that is multiplying $xy$ is $-12$.

---

Therefore, the coefficient of $xy$ in the term $-12xy$ is $-12$. It is important to include the sign.

---

Let's look at the options:

(A) 12: Incorrect, the sign is negative.

(B) -12: Correct, this is the numerical factor multiplying $xy$.

(C) x: Incorrect, this is a variable part, not the coefficient.

(D) y: Incorrect, this is a variable part, not the coefficient.

---

The correct answer is (B) -12.

We need to translate the verbal phrase "5 less than a number $p$" into an algebraic expression.

---

The phrase "less than" indicates subtraction.

---

"5 less than a number $p$" means that we start with the number $p$ and subtract $5$ from it.

---

So, the expression is $p - 5$.

---

Let's check the options:

(A) $5 - p$: This represents "5 decreased by a number $p$" or "a number $p$ less than 5". This is different from "5 less than a number $p$". For example, if $p=10$, $10-5=5$, but $5-10=-5$.

(B) $p - 5$: This matches our translation of "5 less than a number $p$".

(C) $5p$: This represents "5 times a number $p$" or the product of 5 and $p$. Incorrect.

(D) $p/5$: This represents "a number $p$ divided by 5" or the quotient of $p$ and 5. Incorrect.

---

The correct answer is (B) $p - 5$.

To find the value of an algebraic expression for given values of the variables, we substitute the given values into the expression and then perform the arithmetic calculations.

---

The given expression is $a + b + c$.

The given values are $a = 1$, $b = -1$, and $c = 0$.

---

Substitute these values into the expression:

$1 + (-1) + 0$

---

Now, perform the addition from left to right:

$1 + (-1) = 1 - 1 = 0$

The expression becomes:

$0 + 0$

$0 + 0 = 0$

---

Thus, the value of the expression $a + b + c$ when $a = 1, b = -1, c = 0$ is $0$.

---

Comparing this result with the given options:

(A) 1

(B) -1

(C) 0

(D) 2

---

The correct answer is (C) 0.

To simplify the expression, we need to combine the like terms. Like terms have the same variables raised to the same powers.

---

The given expression is $(5x^2) + (-2x^2) + (3x^2)$.

---

All three terms have the variable part $x^2$. Thus, they are like terms.

---

To add like terms, we add their numerical coefficients and keep the variable part the same.

---

The coefficients are $5$, $-2$, and $3$.

Sum of coefficients $= 5 + (-2) + 3$

$= 5 - 2 + 3$

$= 3 + 3$

$= 6$

---

The variable part is $x^2$.

---

So, the simplified expression is $6x^2$.

---

Let's look at the options:

(A) $6x^2$: Correct.

(B) $10x^2$: Incorrect (Sum of absolute values of coefficients: $|5|+|-2|+|3|=5+2+3=10$).

(C) $6x^6$: Incorrect, the powers of the variable are added only during multiplication of terms, not addition/subtraction.

(D) $10x^6$: Incorrect, both the coefficient and the power of the variable are wrong.

---

The correct answer is (A) $6x^2$.

A constant term in an algebraic expression is a term that has a fixed value because it does not contain any variables. It is a term that is just a number.

---

Let's examine each option:

---

(A) $2x$: This term contains the variable $x$. Its value changes depending on the value of $x$. Therefore, it is not a constant term.

---

(B) $-5$: This term is just a number. It does not contain any variables, so its value is fixed at $-5$. Therefore, it is a constant term.

---

(C) $3ab$: This term contains the variables $a$ and $b$. Its value changes depending on the values of $a$ and $b$. Therefore, it is not a constant term.

---

(D) $y^2$: This term contains the variable $y$. Its value changes depending on the value of $y$. Therefore, it is not a constant term.

---

The only term among the options that is a constant term is $-5$.

---

The correct answer is (B) -5.

The formula for the area of a circle with radius $r$ is given by:

$\text{Area} = \pi r^2$

---

We are given the radius $r = 7$ cm and the value of $\pi = \frac{22}{7}$.

---

To find the area, we substitute these values into the formula:

$\text{Area} = \frac{22}{7} \times (7 \text{ cm})^2$

---

First, calculate the square of the radius:

$(7 \text{ cm})^2 = 7 \text{ cm} \times 7 \text{ cm} = 49 \text{ cm}^2$

---

Now, substitute this value back into the area formula:

$\text{Area} = \frac{22}{7} \times 49 \text{ cm}^2$

---

Perform the multiplication. We can cancel the 7 in the denominator with the 49:

$\text{Area} = \frac{22}{\cancel{7}_{1}} \times \cancel{49}^{7} \text{ cm}^2$

$\text{Area} = 22 \times 7 \text{ cm}^2$

---

Calculate the final product:

$22 \times 7 = 154$

---

So, the area of the circle is $154 \text{ cm}^2$.

---

Let's check the options:

(A) $22\text{ cm}^2$: Incorrect.

(B) $44\text{ cm}^2$: Incorrect (This is $2 \times \pi \times r$ if using $r=7$ and $\pi=22/7$, which is the circumference).

(C) $154\text{ cm}^2$: Correct.

(D) $616\text{ cm}^2$: Incorrect (This would be $4 \times 154$, perhaps $4 \times$ Area or something else).

---

The correct answer is (C) $154\text{ cm}^2$.

An algebraic expression is a combination of constants and variables, connected by one or more mathematical operations (addition, subtraction, multiplication, division, etc.).

---

For the given expression, which is: $5x^2 - 3y + 8$

The variables are the symbols that represent quantities that can take on different values. In this expression, the variables are $x$ and $y$.

The constants are the terms that have a fixed numerical value. These include coefficients of the variables and the term without a variable. In this expression, the constants are $5$ (coefficient of $x^2$), $-3$ (coefficient of $y$), and $8$ (the constant term).

The terms of an algebraic expression are the parts of the expression that are separated by addition or subtraction signs.

---

For the given expression: $3ab - 7c + 5d - 9$

The terms are:

$3ab$

$-7c$

$5d$

$-9$

Like terms are terms in an algebraic expression that have the same variables raised to the same powers.

Only the coefficients of the variables can be different.

---

Example of a pair of like terms:

$7x$ and $-2x$ (Both have the variable $x$ raised to the power of $1$).

Another example: $5a^2b$ and $12a^2b$ (Both have variables $a^2b$).

---

Example of a pair of unlike terms:

$3x$ and $3y$ (The variables are different, $x$ vs $y$).

Another example: $4ab$ and $4a^2b$ (The variable $a$ has different powers, $1$ vs $2$).

To identify like terms, we look for terms that have the same variables raised to the same powers.

---

Looking at the list: $2x, -3y, 5x^2, 7y, -4x, 8x^2, -9z, 11y$

---

Terms with variable $x$ (to the power 1):

$2x$ and $-4x$ are like terms.

---

Terms with variable $y$ (to the power 1):

$-3y$, $7y$, and $11y$ are like terms.

---

Terms with variable $x^2$:

$5x^2$ and $8x^2$ are like terms.

---

Term with variable $z$ (to the power 1):

$-9z$ is a term on its own (no other like term in the list).

---

Therefore, the sets of like terms are:

$(2x, -4x)$

$(-3y, 7y, 11y)$

$(5x^2, 8x^2)$

The coefficient of a term in an algebraic expression is the numerical factor that multiplies the variable or variables in that term.

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The given expression is: $4x^2y - 5xy + 2y^2$

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The term containing $x^2y$ is $4x^2y$. The numerical factor is $4$.

So, the coefficient of $x^2y$ is $4$.

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The term containing $xy$ is $-5xy$. The numerical factor is $-5$.

So, the coefficient of $xy$ is $-5$.

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The term containing $y^2$ is $2y^2$. The numerical factor is $2$.

So, the coefficient of $y^2$ is $2$.

We need to add the two expressions $(2x + 3y)$ and $(5x - y)$.

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$(2x + 3y) + (5x - y)$

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Remove the parentheses:

$2x + 3y + 5x - y$

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Group the like terms together:

$(2x + 5x) + (3y - y)$

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Combine the coefficients of the like terms:

$(2 + 5)x + (3 - 1)y$

$7x + 2y$

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The sum of the expressions is $7x + 2y$.

We need to subtract the expression $(4a - 2b)$ from $(7a + 5b)$.

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This means we write: $(7a + 5b) - (4a - 2b)$

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Remove the parentheses. Remember that subtracting an expression is equivalent to adding the opposite of each term in the expression being subtracted:

$7a + 5b - 4a + 2b$

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Group the like terms together:

$(7a - 4a) + (5b + 2b)$

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Combine the coefficients of the like terms:

$(7 - 4)a + (5 + 2)b$

$3a + 7b$

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The result of the subtraction is $3a + 7b$.

We need to simplify the expression: $(3m - 4n) + (2m + 6n) - (m - n)$

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First, remove the parentheses. Remember to change the signs of the terms inside the parentheses being subtracted:

$3m - 4n + 2m + 6n - m + n$

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Next, group the like terms together. Group the terms with $m$ and the terms with $n$:

$(3m + 2m - m) + (-4n + 6n + n)$

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Now, combine the coefficients of the like terms:

For the terms with $m$: $(3 + 2 - 1)m = (5 - 1)m = 4m$

For the terms with $n$: $(-4 + 6 + 1)n = (2 + 1)n = 3n$

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Combine the simplified terms:

$4m + 3n$

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The simplified expression is $4m + 3n$.

The given expression is $2x - 5$.

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We are asked to find its value when $x = 3$.

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Substitute the value $x=3$ into the expression:

$2(3) - 5$

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Perform the multiplication:

$6 - 5$

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Perform the subtraction:

$1$

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The value of the expression $2x - 5$ when $x = 3$ is $1$.

The given expression is $a^2 + 2ab + b^2$.

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We are given the values $a = 2$ and $b = 1$.

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Substitute these values into the expression:

$(2)^2 + 2(2)(1) + (1)^2$

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Calculate the values:

$(2)^2 = 4$

$2(2)(1) = 4$

$(1)^2 = 1$

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Now substitute these calculated values back into the expression:

$4 + 4 + 1$

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Perform the addition:

$4 + 4 + 1 = 9$

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The value of the expression $a^2 + 2ab + b^2$ when $a = 2$ and $b = 1$ is $9$.

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Alternate Approach:

Recognize that the expression $a^2 + 2ab + b^2$ is the expansion of the perfect square $(a+b)^2$.

So, $a^2 + 2ab + b^2 = (a+b)^2$

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Substitute the values $a=2$ and $b=1$ into the simplified expression:

$(2+1)^2$

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Perform the addition inside the parentheses:

$(3)^2$

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Calculate the square:

$(3)^2 = 9$

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The value of the expression is $9$.

The expression for the perimeter of a square is given as $4s$, where $s$ is the side length.

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We are given that the side length $s = 7$ cm.

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To find the perimeter, we substitute the value of $s$ into the expression:

Perimeter $= 4 \times s$

Perimeter $= 4 \times 7$

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Perform the multiplication:

$4 \times 7 = 28$

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So, the perimeter of the square when the side length is 7 cm is $28$ cm.

Given:

Length of the rectangle, $l = 10$ meters.

Breadth of the rectangle, $b = 6$ meters.

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To Find:

The area of the rectangle.

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Solution:

The formula for the area of a rectangle is given as:

Area $= l \times b$

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Substitute the given values of $l$ and $b$ into the formula:

Area $= 10$ meters $\times 6$ meters

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Perform the multiplication:

$10 \times 6 = 60$

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The units for the area will be meters $\times$ meters, which is square meters ($m^2$).

Area $= 60$ $m^2$

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The area of the rectangle is $60$ square meters.

The phrase "sum of $p$ and $q$" means we add $p$ and $q$ together, which is written as $p+q$.

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The phrase "divided by $5$" means the result of the sum is divided by $5$.

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So, the algebraic expression for "sum of $p$ and $q$ divided by $5$" is:

$(p+q) \div 5$

or more commonly written as a fraction:

$\frac{p+q}{5}$

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The algebraic expression is $\frac{p+q}{5}$.

The phrase "thrice $x$" means three times $x$, which can be written as $3 \times x$ or simply $3x$.

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The phrase "subtract $y$ from thrice $x$" means we are taking $y$ away from the quantity "thrice $x$".

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So, we start with "thrice $x$" ($3x$) and subtract $y$ from it.

The expression is $3x - y$.

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The algebraic expression for "subtract $y$ from thrice $x$" is $3x - y$.

The given expression is $4xy + 5x - 7$.

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The terms of this expression are the parts separated by addition or subtraction signs:

$4xy$

$5x$

$-7$

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Now, let's write the factors for each term.

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For the term $4xy$:

This term is formed by multiplying $4$, $x$, and $y$.

The factors are $4$, $x$, and $y$.

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For the term $5x$:

This term is formed by multiplying $5$ and $x$.

The factors are $5$ and $x$.

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For the term $-7$:

This is a constant term. Its factors are $-7$ (or $-1$ and $7$).

(a) The expression $5x^2$ has only one term ($5x^2$).

An algebraic expression with one term is called a monomial.

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(b) The expression $a + b + c$ has three terms ($a$, $b$, and $c$).

An algebraic expression with three terms is called a trinomial.

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(c) The expression $m - n$ has two terms ($m$ and $-n$).

An algebraic expression with two terms is called a binomial.

We need to find the sum of the expressions $2x^2$, $-3x^2$, and $5x^2$.

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Write the sum:

$2x^2 + (-3x^2) + 5x^2$

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Simplify the expression. Since these are all like terms (they all have $x^2$), we can add their coefficients:

$(2 + (-3) + 5)x^2$

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Calculate the sum of the coefficients:

$2 - 3 + 5 = -1 + 5 = 4$

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Combine the result with the variable part:

$4x^2$

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The sum of the expressions is $4x^2$.

We need to find the difference between $8y$ and $3y$. This means we subtract $3y$ from $8y$.

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The expression for the difference is:

$8y - 3y$

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Both terms, $8y$ and $3y$, are like terms because they have the same variable ($y$) raised to the same power ($1$).

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To find the difference, we subtract the coefficients of the like terms and keep the variable part the same:

$(8 - 3)y$

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Perform the subtraction of the coefficients:

$8 - 3 = 5$

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Combine the result with the variable:

$5y$

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The difference between $8y$ and $3y$ is $5y$.

Given:

The cost of one pen is $\textsf{₹}p$.

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To Find:

An expression for the cost of one dozen pens.

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Solution:

We know that one dozen contains $12$ items.

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If the cost of one pen is $\textsf{₹}p$, then the cost of $12$ pens will be $12$ times the cost of one pen.

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Cost of one dozen pens = $12 \times (\text{cost of one pen})$

Cost of one dozen pens = $12 \times p$

Cost of one dozen pens = $12p$

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The expression for the cost of one dozen pens is $\textsf{₹}12p$.

The given expression is $3x^2 - x + 5$.

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We need to evaluate this expression when the value of $x$ is $-1$.

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Substitute $x = -1$ into the expression:

$3(-1)^2 - (-1) + 5$

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First, calculate the exponent $(-1)^2$:

$(-1)^2 = (-1) \times (-1) = 1$

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Substitute this value back into the expression:

$3(1) - (-1) + 5$

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Perform the multiplication $3(1)$ and handle the double negative $-(-1)$:

$3 + 1 + 5$

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Perform the addition:

$3 + 1 + 5 = 4 + 5 = 9$

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The value of the expression $3x^2 - x + 5$ when $x = -1$ is $9$.

First, we find the sum of the expressions $2x^2 + 3x - 5$ and $-x^2 + 2x + 1$.

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Sum $= (2x^2 + 3x - 5) + (-x^2 + 2x + 1)$

Remove parentheses:

Sum $= 2x^2 + 3x - 5 - x^2 + 2x + 1$

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Group like terms:

Sum $= (2x^2 - x^2) + (3x + 2x) + (-5 + 1)$

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Combine like terms:

Sum $= (2-1)x^2 + (3+2)x + (-5+1)$

Sum $= 1x^2 + 5x - 4$

Sum $= x^2 + 5x - 4$

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Next, we need to subtract the expression $x^2 - x - 4$ from this sum ($x^2 + 5x - 4$).

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Difference $= (x^2 + 5x - 4) - (x^2 - x - 4)$

Remove parentheses, changing the sign of each term in the second expression:

Difference $= x^2 + 5x - 4 - x^2 + x + 4$

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Group like terms:

Difference $= (x^2 - x^2) + (5x + x) + (-4 + 4)$

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Combine like terms:

Difference $= (1-1)x^2 + (5+1)x + (-4+4)$

Difference $= 0x^2 + 6x + 0$

Difference $= 6x$

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The result is $6x$.

We need to simplify the expression: $5(p+q) - 2(p-q)$

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First, apply the distributive property to remove the parentheses.

Distribute $5$ into the first set of parentheses:

$5 \times p + 5 \times q = 5p + 5q$

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Distribute $-2$ into the second set of parentheses:

$-2 \times p + (-2) \times (-q) = -2p + 2q$

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Rewrite the expression with the parentheses removed:

$5p + 5q - 2p + 2q$

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Now, group the like terms together. Group the terms containing $p$ and the terms containing $q$:

$(5p - 2p) + (5q + 2q)$

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Combine the coefficients of the like terms:

For the terms with $p$: $(5 - 2)p = 3p$

For the terms with $q$: $(5 + 2)q = 7q$

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Combine the simplified terms:

$3p + 7q$

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The simplified expression is $3p + 7q$.

The given formula for converting Celsius (C) to Fahrenheit (F) is:

$F = \frac{9}{5}C + 32$

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We are given that the temperature in Celsius is $25^\circ\text{C}$. So, $C = 25$.

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Substitute $C = 25$ into the formula:

$F = \frac{9}{5}(25) + 32$

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First, calculate the multiplication $\frac{9}{5} \times 25$. We can cancel out the 5 from the denominator and 25:

$F = \frac{9}{\cancel{5}_{1}} \times \cancel{25}^{5} + 32$

$F = 9 \times 5 + 32$

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Perform the multiplication:

$F = 45 + 32$

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Perform the addition:

$F = 77$

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So, when the temperature is $25^\circ\text{C}$, the temperature in Fahrenheit is $77^\circ\text{F}$.

Given:

The side length of the square is $s = 2x + 1$.

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To Find:

An expression for the perimeter of the square.

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Solution:

The formula for the perimeter of a square is $P = 4 \times \text{side length}$.

So, $P = 4 \times s$.

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Substitute the given expression for the side length ($2x + 1$) into the formula:

$P = 4 \times (2x + 1)$

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Apply the distributive property (multiply $4$ by each term inside the parentheses):

$P = (4 \times 2x) + (4 \times 1)$

$P = 8x + 4$

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The expression for the perimeter of the square is $8x + 4$.

The given expression is $7m^2n - 3mn^2 + m^2n^2 - 5$.

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The terms in the expression are:

$7m^2n$

$-3mn^2$

$m^2n^2$

$-5$

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The numerical coefficient of a term is the numerical factor multiplying the variable part.

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For the term $7m^2n$, the numerical coefficient is $7$.

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For the term $-3mn^2$, the numerical coefficient is $-3$.

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For the term $m^2n^2$, the term can be written as $1 \times m^2n^2$. The numerical coefficient is $1$.

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For the term $-5$, this is a constant term. The numerical coefficient is the term itself, which is $-5$.

An algebraic expression is a mathematical phrase that contains variables, constants, and algebraic operations (like addition, subtraction, multiplication, division). It does not contain an equality sign (=).

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Let's consider the given example: $5x^2y - 3xy + 7$

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Terms:

The terms of an algebraic expression are the parts that are separated by addition or subtraction signs.

In the expression $5x^2y - 3xy + 7$, the terms are:

$5x^2y$

$-3xy$

$7$

Each term is a product of factors, which can be numerical (coefficients) or algebraic (variables raised to powers).

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Variables:

Variables are symbols (usually letters) that represent quantities that can change or take on different values. They are unknown values that we might want to solve for or examine how the expression changes as their values change.

In the expression $5x^2y - 3xy + 7$, the variables are $x$ and $y$.

The powers to which the variables are raised ($x^2$, $y$ to the power of $1$) are part of the variable component of a term.

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Constants:

A constant is a term in an algebraic expression that has a fixed numerical value and does not contain any variables. It represents a quantity that does not change.

In the expression $5x^2y - 3xy + 7$, the constant term is $7$.

Coefficients of terms with variables are also constant numerical factors, but "constant term" usually refers to the term without any variables.

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Coefficients:

The coefficient of a term is the numerical factor that is multiplied by the variable part of the term. It tells us how many times the variable part is taken.

In the expression $5x^2y - 3xy + 7$:

For the term $5x^2y$, the variable part is $x^2y$. The numerical factor multiplying it is $5$. So, the coefficient of $x^2y$ is $5$.

For the term $-3xy$, the variable part is $xy$. The numerical factor multiplying it is $-3$. So, the coefficient of $xy$ is $-3$.

For the term $7$, it is a constant term. If we consider it as having a variable part $x^0y^0$ (since any non-zero number raised to the power 0 is 1), the numerical factor is $7$. The coefficient of the constant term is the term itself, which is $7$.

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In summary, the expression $5x^2y - 3xy + 7$ is an algebraic expression composed of three terms ($5x^2y$, $-3xy$, and $7$). It involves variables ($x$ and $y$), numerical coefficients ($5$ and $-3$ for the variable terms, and $7$ as the constant term's coefficient), and the operations of multiplication, subtraction, and addition. The variables $x$ and $y$ can take on various values, while the coefficients and the constant term remain fixed.

Explanation of Adding Algebraic Expressions:

To add algebraic expressions, we combine the like terms. Like terms are terms that have the same variables raised to the same powers. The process involves:

1. Write the sum of the given expressions, usually enclosed in parentheses.

2. Remove the parentheses. When adding expressions, the signs of the terms within the parentheses remain unchanged.

3. Identify the like terms. These are terms with the same variable parts (e.g., $x^2$ terms, $xy$ terms, $y^2$ terms, constant terms).

4. Group the like terms together. It's often helpful to rearrange the terms so that like terms are adjacent.

5. Combine the like terms by adding their numerical coefficients. The variable part remains the same.

6. Write the resulting simplified expression.

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Addition Example:

We need to add the expressions $(3x^2 + 5xy - 4y^2 + 2)$ and $(-7x^2 + 2xy + 6y^2 - 5)$.

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Step 1: Write the sum of the expressions.

$(3x^2 + 5xy - 4y^2 + 2) + (-7x^2 + 2xy + 6y^2 - 5)$

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Step 2: Remove the parentheses.

$3x^2 + 5xy - 4y^2 + 2 - 7x^2 + 2xy + 6y^2 - 5$

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Step 3 & 4: Identify and group the like terms.

Group terms with $x^2$: $3x^2, -7x^2$

Group terms with $xy$: $5xy, 2xy$

Group terms with $y^2$: $-4y^2, 6y^2$

Group constant terms: $2, -5$

$(3x^2 - 7x^2) + (5xy + 2xy) + (-4y^2 + 6y^2) + (2 - 5)$

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Step 5: Combine the coefficients of the like terms.

For $x^2$ terms: $(3 - 7)x^2 = -4x^2$

For $xy$ terms: $(5 + 2)xy = 7xy$

For $y^2$ terms: $(-4 + 6)y^2 = 2y^2$

For constant terms: $(2 - 5) = -3$

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Step 6: Write the resulting simplified expression.

$-4x^2 + 7xy + 2y^2 - 3$

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The sum of the given expressions is $-4x^2 + 7xy + 2y^2 - 3$.

Explanation of Subtracting Algebraic Expressions:

To subtract one algebraic expression from another, we essentially add the opposite of the expression being subtracted. This involves changing the sign of each term in the expression being subtracted and then combining the like terms.

The process involves:

1. Write the subtraction problem, putting the expression being subtracted second (after the "from" expression) and enclosing both expressions in parentheses.

2. Remove the parentheses. The parentheses around the first expression can be removed without changing signs. For the second expression (the one being subtracted), remove the parentheses and change the sign of every term inside. A positive term becomes negative, and a negative term becomes positive.

3. Identify and group the like terms from the resulting expression.

4. Combine the like terms by adding their numerical coefficients. The variable part remains the same.

5. Write the resulting simplified expression.

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Subtraction Example:

We need to subtract $(-2a^2 + 5ab - 3b^2)$ from $(6a^2 - 3ab + 5b^2)$.

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Step 1: Write the subtraction problem.

$(6a^2 - 3ab + 5b^2) - (-2a^2 + 5ab - 3b^2)$

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Step 2: Remove parentheses, changing the signs of the terms being subtracted.

The terms in the first parenthesis keep their signs: $6a^2 - 3ab + 5b^2$.

The terms in the second parenthesis change their signs:

$-(-2a^2) = +2a^2$

$-(+5ab) = -5ab$

$-(-3b^2) = +3b^2$

So the expression becomes:

$6a^2 - 3ab + 5b^2 + 2a^2 - 5ab + 3b^2$

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Step 3 & 4: Identify and group the like terms.

Group terms with $a^2$: $6a^2, +2a^2$

Group terms with $ab$: $-3ab, -5ab$

Group terms with $b^2$: $5b^2, +3b^2$

$(6a^2 + 2a^2) + (-3ab - 5ab) + (5b^2 + 3b^2)$

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Step 5: Combine the coefficients of the like terms.

For $a^2$ terms: $(6 + 2)a^2 = 8a^2$

For $ab$ terms: $(-3 - 5)ab = -8ab$

For $b^2$ terms: $(5 + 3)b^2 = 8b^2$

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Step 6: Write the resulting simplified expression.

$8a^2 - 8ab + 8b^2$

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The result of the subtraction is $8a^2 - 8ab + 8b^2$.

Explanation of Finding the Value of an Algebraic Expression:

To find the value of an algebraic expression for given values of the variables, we substitute the given numerical value for each variable wherever it appears in the expression. After substitution, we perform the indicated mathematical operations (exponents, multiplication, division, addition, and subtraction) following the order of operations (PEMDAS/BODMAS).

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Example:

The given expression is $2x^3 - 4x^2 + 5x - 1$.

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Case 1: Find the value when $x = 2$.

Substitute $x = 2$ into the expression:

$2(2)^3 - 4(2)^2 + 5(2) - 1$

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Calculate the powers:

$(2)^3 = 2 \times 2 \times 2 = 8$

$(2)^2 = 2 \times 2 = 4$

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Substitute these values back and perform multiplications:

$2(8) - 4(4) + 5(2) - 1$

$16 - 16 + 10 - 1$

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Perform addition and subtraction from left to right:

$16 - 16 = 0$

$0 + 10 = 10$

$10 - 1 = 9$

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The value of the expression when $x = 2$ is $9$.

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Case 2: Find the value when $x = -2$.

Substitute $x = -2$ into the expression:

$2(-2)^3 - 4(-2)^2 + 5(-2) - 1$

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Calculate the powers:

$(-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8$

$(-2)^2 = (-2) \times (-2) = 4$

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Substitute these values back and perform multiplications:

$2(-8) - 4(4) + 5(-2) - 1$

$-16 - 16 - 10 - 1$

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Perform addition and subtraction from left to right:

$-16 - 16 = -32$

$-32 - 10 = -42$

$-42 - 1 = -43$

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The value of the expression when $x = -2$ is $-43$.

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Comparison of Results:

When $x = 2$, the value of the expression is $9$.

When $x = -2$, the value of the expression is $-43$.

The results are different. This demonstrates that the value of an algebraic expression depends on the specific value(s) assigned to its variable(s).

We need to simplify the expression: $(5x^2 - 2xy + 3y^2) - (-2x^2 + xy - 4y^2) + (x^2 - 3xy - y^2)$.

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Step 1: Remove the parentheses.

For the first set of parentheses, we just remove them:

$5x^2 - 2xy + 3y^2$

For the second set of parentheses, which is preceded by a minus sign, we change the sign of each term inside:

$-(-2x^2) = +2x^2$

$-(+xy) = -xy$

$-(-4y^2) = +4y^2$

For the third set of parentheses, which is preceded by a plus sign, we just remove them:

$+x^2 - 3xy - y^2$

Combining these, the expression becomes:

$5x^2 - 2xy + 3y^2 + 2x^2 - xy + 4y^2 + x^2 - 3xy - y^2$

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Step 2: Identify and group the like terms.

Group terms with $x^2$: $5x^2, +2x^2, +x^2$

Group terms with $xy$: $-2xy, -xy, -3xy$

Group terms with $y^2$: $3y^2, +4y^2, -y^2$

$(5x^2 + 2x^2 + x^2) + (-2xy - xy - 3xy) + (3y^2 + 4y^2 - y^2)$

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Step 3: Combine the coefficients of the like terms.

For $x^2$ terms: $(5 + 2 + 1)x^2 = 8x^2$

For $xy$ terms: $(-2 - 1 - 3)xy = -6xy$

For $y^2$ terms: $(3 + 4 - 1)y^2 = 6y^2$

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Step 4: Write the resulting simplified expression.

$8x^2 - 6xy + 6y^2$

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The simplified expression is $8x^2 - 6xy + 6y^2$.

Algebraic Expressions in Geometric Formulas:

Algebraic expressions are used in formulas for geometric figures to represent the measurements (like perimeter, area, volume) using variables. These variables often stand for dimensions like length, breadth, height, radius, or side length. Using variables allows us to write a general formula that can be applied to any figure of that type, regardless of its specific dimensions. We can then substitute specific numerical values for the variables to calculate the measurements for a particular figure.

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Perimeter and Area of a Rectangle:

Given:

Length of the rectangle, $l = (3x+2)$

Breadth of the rectangle, $b = (x-1)$

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To Find:

Expressions for the perimeter and area.

Value of perimeter and area when $x = 5$.

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Solution:

The formula for the perimeter of a rectangle is $P = 2 \times (\text{length} + \text{breadth})$.

$P = 2 \times ((3x+2) + (x-1))$

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First, simplify the expression inside the parentheses by combining like terms:

$(3x+2) + (x-1) = 3x + 2 + x - 1$

Group like terms: $(3x + x) + (2 - 1)$

Combine like terms: $4x + 1$

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Now, substitute this back into the perimeter formula:

$P = 2 \times (4x + 1)$

Apply the distributive property:

$P = 2 \times 4x + 2 \times 1$

$P = 8x + 2$

The algebraic expression for the perimeter is $8x + 2$.

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The formula for the area of a rectangle is $A = \text{length} \times \text{breadth}$.

$A = (3x+2) \times (x-1)$

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Apply the distributive property (or FOIL method) to multiply the binomials:

$A = 3x(x-1) + 2(x-1)$

$A = (3x \times x) + (3x \times -1) + (2 \times x) + (2 \times -1)$

$A = 3x^2 - 3x + 2x - 2$

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Combine the like terms ($-3x$ and $2x$):

$A = 3x^2 + (-3 + 2)x - 2$

$A = 3x^2 - x - 2$

The algebraic expression for the area is $3x^2 - x - 2$.

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Now, find the value of the perimeter and area when $x = 5$.

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Value of Perimeter when $x=5$:

Substitute $x=5$ into the perimeter expression $P = 8x + 2$:

$P = 8(5) + 2$

$P = 40 + 2$

$P = 42$

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Value of Area when $x=5$:

Substitute $x=5$ into the area expression $A = 3x^2 - x - 2$:

$A = 3(5)^2 - (5) - 2$

First, calculate the power: $(5)^2 = 25$

$A = 3(25) - 5 - 2$

Perform multiplication: $3 \times 25 = 75$

$A = 75 - 5 - 2$

Perform subtraction from left to right:

$75 - 5 = 70$

$70 - 2 = 68$

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When $x = 5$, the value of the perimeter is $42$ units and the value of the area is $68$ square units.

Given:

Large rectangle length ($L$) = $5x - 2$

Large rectangle breadth ($B$) = $3x + 1$

Small rectangle length ($l$) = $2x - 1$

Small rectangle breadth ($b$) = $x$

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To Find:

Expression for the area of the remaining paper.

Area of the remaining paper when $x = 3$.

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Solution:

The area of a rectangle is given by the formula Area = length $\times$ breadth.

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Area of the large rectangular sheet ($A_{large}$) = $L \times B$

$A_{large} = (5x - 2)(3x + 1)$

Multiply the binomials using the distributive property (FOIL):

$A_{large} = 5x(3x+1) - 2(3x+1)$

$A_{large} = (5x \times 3x) + (5x \times 1) + (-2 \times 3x) + (-2 \times 1)$

$A_{large} = 15x^2 + 5x - 6x - 2$

Combine like terms ($5x$ and $-6x$):

$A_{large} = 15x^2 - x - 2$

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Area of the smaller rectangular piece ($A_{small}$) = $l \times b$

$A_{small} = (2x - 1)(x)$

Apply the distributive property:

$A_{small} = (2x \times x) - (1 \times x)$

$A_{small} = 2x^2 - x$

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The area of the remaining paper is the area of the large rectangle minus the area of the small rectangle cut out.

Area$_{remaining}$ = $A_{large} - A_{small}$

Area$_{remaining}$ = $(15x^2 - x - 2) - (2x^2 - x)$

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Remove the parentheses, changing the signs of the terms being subtracted:

Area$_{remaining}$ = $15x^2 - x - 2 - 2x^2 + x$

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Group like terms:

Area$_{remaining}$ = $(15x^2 - 2x^2) + (-x + x) - 2$

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Combine like terms:

Area$_{remaining}$ = $(15 - 2)x^2 + (-1 + 1)x - 2$

Area$_{remaining}$ = $13x^2 + 0x - 2$

Area$_{remaining}$ = $13x^2 - 2$

The algebraic expression for the area of the remaining paper is $13x^2 - 2$.

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Now, find the area of the remaining paper when $x = 3$.

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Substitute $x = 3$ into the expression for the remaining area:

Area$_{remaining}$ = $13(3)^2 - 2$

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First, calculate the power: $(3)^2 = 9$

Area$_{remaining}$ = $13(9) - 2$

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Perform the multiplication:

$13 \times 9 = 117$

Area$_{remaining}$ = $117 - 2$

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Perform the subtraction:

$117 - 2 = 115$

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When $x = 3$, the area of the remaining paper is $115$ square units.

Given:

Total population of the town $= (2x + 3y)$

Number of males in the town $= (x - y)$

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To Find:

An expression for the number of females.

The total population and the number of females when $x = 1000$ and $y = 500$.

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Solution:

We know that the total population of a town consists of the number of males and the number of females.

Total Population = Number of Males + Number of Females

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To find the number of females, we rearrange this relationship:

Number of Females = Total Population - Number of Males

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Substitute the given expressions into this equation:

Number of Females $= (2x + 3y) - (x - y)$

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Remove the parentheses. Remember to change the signs of the terms in the second expression:

Number of Females $= 2x + 3y - x + y$

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Group the like terms together:

Number of Females $= (2x - x) + (3y + y)$

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Combine the coefficients of the like terms:

Number of Females $= (2 - 1)x + (3 + 1)y$

Number of Females $= 1x + 4y$

Number of Females $= x + 4y$

The algebraic expression for the number of females is $x + 4y$.

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Now, find the total population and the number of females when $x = 1000$ and $y = 500$.

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Total Population:

Total Population $= 2x + 3y$

Substitute $x = 1000$ and $y = 500$:

Total Population $= 2(1000) + 3(500)$

Total Population $= 2000 + 1500$

Total Population $= 3500$

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Number of Females:

Using the derived expression: Number of Females $= x + 4y$

Substitute $x = 1000$ and $y = 500$:

Number of Females $= 1000 + 4(500)$

Number of Females $= 1000 + 2000$

Number of Females $= 3000$

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Alternatively, we can check this using the original relationship:

Total Population = Number of Males + Number of Females

Number of Males $= x - y = 1000 - 500 = 500$

Total Population $= 3500$

Number of Females $= 3000$

$3500 = 500 + 3000$

$3500 = 3500$ (This confirms our calculation for the number of females is correct)

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When $x = 1000$ and $y = 500$, the total population is $3500$ and the number of females is $3000$.

Given:

Principal (P) = $\textsf{₹}5000$

Rate of Interest (R) = $8\%$ per annum

Time (T) = $3$ years

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To Find:

Simple Interest (SI)

Total Amount (A) after 3 years

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Solution:

The formula for Simple Interest is given by:

$SI = \frac{P \times R \times T}{100}$

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Substitute the given values of P, R, and T into the formula:

$SI = \frac{5000 \times 8 \times 3}{100}$

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Calculate the Simple Interest. We can simplify by cancelling out the common factors. The two zeros in 5000 and the two zeros in 100 can be cancelled:

$SI = \frac{50\cancel{00} \times 8 \times 3}{\cancel{100}_{1}}$

$SI = 50 \times 8 \times 3$

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Perform the multiplication:

$SI = (50 \times 8) \times 3$

$SI = 400 \times 3$

$SI = 1200$

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The Simple Interest is $\textsf{₹}1200$.

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Now, calculate the Total Amount after 3 years.

The formula for the Total Amount is the sum of the Principal and the Simple Interest:

Total Amount (A) = Principal (P) + Simple Interest (SI)

$A = P + SI$

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Substitute the values of P and SI:

$A = 5000 + 1200$

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Perform the addition:

$A = 6200$

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The Total Amount after 3 years is $\textsf{₹}6200$.

We need to simplify the expression: $3(a^2 - 2ab + b^2) - 2(a^2 + ab - b^2) - (a^2 + 3ab + 2b^2)$.

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Step 1: Apply the distributive property to remove the parentheses.

For the first set of parentheses, multiply each term by $3$:

$3 \times a^2 = 3a^2$

$3 \times (-2ab) = -6ab$

$3 \times b^2 = 3b^2$

So, $3(a^2 - 2ab + b^2) = 3a^2 - 6ab + 3b^2$.

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For the second set of parentheses, multiply each term by $-2$:

$-2 \times a^2 = -2a^2$

$-2 \times ab = -2ab$

$-2 \times (-b^2) = +2b^2$

So, $-2(a^2 + ab - b^2) = -2a^2 - 2ab + 2b^2$.

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For the third set of parentheses, which is preceded by a minus sign, we multiply each term by $-1$ (or simply change the sign of each term):

$-(a^2) = -a^2$

$-(+3ab) = -3ab$

$-(+2b^2) = -2b^2$

So, $-(a^2 + 3ab + 2b^2) = -a^2 - 3ab - 2b^2$.

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Rewrite the expression by combining the results from removing the parentheses:

$3a^2 - 6ab + 3b^2 - 2a^2 - 2ab + 2b^2 - a^2 - 3ab - 2b^2$

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Step 2: Identify and group the like terms.

Group terms with $a^2$: $3a^2, -2a^2, -a^2$

Group terms with $ab$: $-6ab, -2ab, -3ab$

Group terms with $b^2$: $3b^2, +2b^2, -2b^2$

$(3a^2 - 2a^2 - a^2) + (-6ab - 2ab - 3ab) + (3b^2 + 2b^2 - 2b^2)$

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Step 3: Combine the coefficients of the like terms.

For $a^2$ terms: $(3 - 2 - 1)a^2 = (1 - 1)a^2 = 0a^2 = 0$

For $ab$ terms: $(-6 - 2 - 3)ab = (-8 - 3)ab = -11ab$

For $b^2$ terms: $(3 + 2 - 2)b^2 = (5 - 2)b^2 = 3b^2$

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Step 4: Write the resulting simplified expression.

$0 - 11ab + 3b^2$

$-11ab + 3b^2$

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The simplified expression is $-11ab + 3b^2$.

First, we find the sum of the first two expressions: $(4x + 3y - 7z)$ and $(-2x - 5y + 2z)$.

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Sum 1 $= (4x + 3y - 7z) + (-2x - 5y + 2z)$

Remove parentheses:

Sum 1 $= 4x + 3y - 7z - 2x - 5y + 2z$

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Group the like terms:

Sum 1 $= (4x - 2x) + (3y - 5y) + (-7z + 2z)$

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Combine the coefficients of the like terms:

Sum 1 $= (4 - 2)x + (3 - 5)y + (-7 + 2)z$

Sum 1 $= 2x - 2y - 5z$

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Next, we find the sum of the last two expressions: $(x - 2y - 3z)$ and $(3x + y - z)$.

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Sum 2 $= (x - 2y - 3z) + (3x + y - z)$

Remove parentheses:

Sum 2 $= x - 2y - 3z + 3x + y - z$

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Group the like terms:

Sum 2 $= (x + 3x) + (-2y + y) + (-3z - z)$

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Combine the coefficients of the like terms:

Sum 2 $= (1 + 3)x + (-2 + 1)y + (-3 - 1)z$

Sum 2 $= 4x - y - 4z$

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Finally, we subtract the second sum (Sum 2) from the first sum (Sum 1).

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Result $= \text{Sum 1} - \text{Sum 2}$

Result $= (2x - 2y - 5z) - (4x - y - 4z)$

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Remove the parentheses, changing the sign of each term in the expression being subtracted:

Result $= 2x - 2y - 5z - 4x + y + 4z$

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Group the like terms:

Result $= (2x - 4x) + (-2y + y) + (-5z + 4z)$

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Combine the coefficients of the like terms:

Result $= (2 - 4)x + (-2 + 1)y + (-5 + 4)z$

Result $= -2x - y - z$

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The simplified expression is $-2x - y - z$.

Given:

Perimeter of the triangle (P) = $5x - 3$

Length of the first side ($s_1$) = $2x + 1$

Length of the second side ($s_2$) = $x - 4$

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To Find:

An expression for the length of the third side ($s_3$).

The perimeter when $x=5$.

The length of the third side when $x=5$.

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Solution:

The perimeter of a triangle is the sum of the lengths of its three sides.

Perimeter = Side 1 + Side 2 + Side 3

$P = s_1 + s_2 + s_3$

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To find the length of the third side ($s_3$), we can rearrange the formula:

$s_3 = P - (s_1 + s_2)$

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First, find the sum of the lengths of the two given sides ($s_1 + s_2$):

Sum of two sides $= (2x + 1) + (x - 4)$

Remove parentheses and group like terms:

Sum of two sides $= 2x + 1 + x - 4$

Sum of two sides $= (2x + x) + (1 - 4)$

Sum of two sides $= 3x - 3$

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Now, subtract this sum from the perimeter expression to find the third side:

$s_3 = (5x - 3) - (3x - 3)$

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Remove the parentheses, changing the signs of the terms being subtracted:

$s_3 = 5x - 3 - 3x + 3$

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Group and combine like terms:

$s_3 = (5x - 3x) + (-3 + 3)$

$s_3 = (5 - 3)x + (-3 + 3)$

$s_3 = 2x + 0$

$s_3 = 2x$

The expression for the length of the third side is $2x$.

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Now, find the perimeter when $x = 5$.

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Substitute $x = 5$ into the perimeter expression $P = 5x - 3$:

$P = 5(5) - 3$

$P = 25 - 3$

$P = 22$

The perimeter when $x=5$ is $22$ units.

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Finally, find the length of the third side for this value of $x=5$.

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Substitute $x = 5$ into the third side expression $s_3 = 2x$:

$s_3 = 2(5)$

$s_3 = 10$

The length of the third side when $x=5$ is $10$ units.

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We can verify this: When $x=5$, $s_1 = 2(5)+1 = 11$, $s_2 = 5-4 = 1$, and $s_3 = 10$. The sum of the sides is $11+1+10 = 22$, which matches the perimeter we calculated when $x=5$.