GSEB Class 7 Maths Solutions Chapter 12 બીજગણિતીય પદાવલિ Exercise 12.1

Question 1. નીચે આપેલી બાબતોમાં ચલ, અચલ અને ગાણિતિક પ્રક્રિયાઓનો ઉપયોગ કરી બીજગણિતીય પદાવલિઓ બનાવોઃ
Answer:

Question 1. (i) yમાંથી z બાદ કરો.
Answer: \( y - z \)
In simple words: When asked to subtract a variable from another, just write the first variable followed by a minus sign and then the second variable.

Exam Tip: Pay attention to the order of subtraction; "subtract A from B" means \( B - A \), not \( A - B \).

Question 1. (ii) x અને yના સરવાળાના અડધા
Answer: \( \frac {1}{2}(x + y) \)
In simple words: To find half of a sum, first add the numbers together and then divide the whole result by two.

Exam Tip: Remember to put the sum in parentheses before multiplying by a fraction to ensure the entire sum is divided.

Question 1. (iii) સંખ્યા zનો તે જ સંખ્યા સાથેનો ગુણાકાર
Answer: \( z^2 \)
In simple words: When you multiply a number by itself, you are finding its square, which is shown by writing the number with a small '2' above it.

Exam Tip: Squaring a number means multiplying it by itself, so \( z \times z = z^2 \).

Question 1. (iv) p અને qના ગુણાકારનો ચતુર્થ ભાગ
Answer: \( \frac {1}{4}pq \)
In simple words: To get a quarter of a product, first multiply the variables, then either divide by four or multiply by one-fourth.

Exam Tip: The phrase "a quarter part" or "ચતુર્થ ભાગ" always means multiplying by \( \frac {1}{4} \).

Question 1. (v) x અને y બંને સંખ્યાનો વર્ગ અને તેમનો સરવાળો
Answer: \( x^2 + y^2 \)
In simple words: First find the square of each number, then add these squared values together.

Exam Tip: Squaring each term separately before adding is different from squaring their sum, so be careful with the wording.

Question 1. (vi) m અને n સંખ્યાના ગુણાકારના ત્રણ ગણામાં 5 ઉમેરતાં
Answer: \( 3mn + 5 \)
In simple words: First multiply m and n, then multiply that result by three, and finally add five to the total.

Exam Tip: Follow the order of operations: multiplication first, then addition.

Question 1. (vii) y અને zના ગુણાકારને 10માંથી બાદ કરતાં
Answer: \( 10 - yz \)
In simple words: When you subtract a product from a number, the number comes first, followed by a minus sign and then the product.

Exam Tip: The phrase "subtract from" means the number after "from" is written first in the expression.

Question 1. (viii) a અને bના ગુણાકારમાંથી તેમનો સરવાળો બાદ કરતાં
Answer: \( ab - (a + b) \)
In simple words: First find the product of 'a' and 'b', then find their sum, and finally take the sum away from the product.

Exam Tip: Use parentheses for the sum \( (a + b) \) to ensure that the entire sum is subtracted, not just the first term.

Question 2. (i) નીચે આપેલ પદાવલિમાંથી પદ અને તેમના અવયવ ઓળખી કાઢો. આ પદ અને અવયવને ટ્રી ચાર્ટ વડે દર્શાવોઃ
Answer: Find the individual parts and pieces that make up each expression shown below. Then, draw a tree diagram to illustrate these parts and their building blocks.

Question 2. (i) (a) x-3
Answer:

Exam Tip: In a tree chart, terms are branches from the expression, and factors are branches from the terms. Remember that a constant term like -3 has factors -1 and 3.

Question 2. (i) (b) 1 + x + x²
Answer:

Exam Tip: For terms involving powers like \( x^2 \), remember to show each instance of the variable as a separate factor, e.g., \( x^2 \) breaks down into \( x \) and \( x \).

Question 2. (i) (c) y-y³
Answer:

Exam Tip: Always remember to include the negative sign as a factor (-1) when a term is negative, such as with \( -y^3 \).

Question 2. (i) (d) 5xy² + 7x²y
Answer:

Exam Tip: For terms like \( 5xy^2 \), ensure each factor is listed separately: the numerical coefficient (5), and each variable (x, y, y).

Question 2. (i) (e) - ab + 2b² - 3a²
Answer:

Exam Tip: For expressions with multiple terms, draw separate branches for each term, and then further branches for the factors of each term.

Question 2. (ii) નીચે આપેલ પદાવલિમાંથી પદ અને અવયવ ઓળખી કાઢોઃ
Answer: Identify the terms and factors from the given expressions below.

Exam Tip: For each expression, clearly separate the terms, and then break down each term into its individual factors (numerical and variable).

Question 3. નીચે આપેલી પદાવલિમાં (અચલ સિવાયના) પદનો સંખ્યાત્મક સહગુણક શોધીને લખો:
Answer: Write the numerical coefficient of the term (excluding constants) in the given expressions.

Exam Tip: The numerical coefficient is the number multiplying the variable part of a term. If no number is explicitly written, the coefficient is 1 (or -1 if negative).

Question 4. (a) zવાળાં પદો શોધો અને ઝના સહગુણક લખોઃ
Answer: Find terms containing x and write their coefficients.

Exam Tip: When finding the coefficient of a variable, consider everything else in that term (numbers and other variables) as its coefficient.

Question 4. (b) yવાળું પદ શોધી તેમનો સહગુણક લખો:
Answer: Find terms containing \( y^2 \) and write their coefficients.

Exam Tip: Be careful to only select terms that specifically contain the requested variable and its power (e.g., \( y^2 \)), and then identify the remaining part of that term as its coefficient.

Question 5. નીચેનાનું એકપદી, દ્વિપદી અને ત્રિપદીમાં વર્ગીકરણ કરો:
Answer: Classify the following expressions into monomial, binomial, and trinomial based on the number of terms they contain.

Question 5. (i) 4y - 7z
Answer: The expression \( 4y - 7z \) contains two distinct parts, specifically \( 4y \) and \( -7z \). Therefore, this is a binomial expression.
In simple words: This expression has two parts joined by a minus sign, so it's called a binomial.

Exam Tip: A binomial is an algebraic expression that consists of exactly two terms, separated by a plus or minus sign.

Question 5. (ii) y²
Answer: The expression \( y^2 \) has just one part, which is \( y^2 \). Therefore, this is a monomial expression.
In simple words: This expression has only one single part, so it's called a monomial.

Exam Tip: A monomial is an algebraic expression with only one term, which can be a number, a variable, or a product of numbers and variables.

Question 5. (iii) x + y - xy
Answer: The expression \( x + y - xy \) is made up of three terms, specifically \( x \), \( y \), and \( -xy \). Therefore, this is a trinomial expression.
In simple words: This expression has three parts joined by plus or minus signs, so it's called a trinomial.

Exam Tip: A trinomial is an algebraic expression that contains exactly three terms, separated by plus or minus signs.

Question 5. (iv) 100
Answer: The number \( 100 \) is only one term, which is \( 100 \). Therefore, this is a monomial expression.
In simple words: This number is just one part by itself, so it's called a monomial.

Exam Tip: Any constant number, without variables, is considered a single term and thus a monomial.

Question 5. (v) ab - a - b
Answer: The expression \( ab - a - b \) includes three separate terms, which are \( ab \), \( -a \), and \( -b \). Therefore, this is a trinomial expression.
In simple words: This expression contains three different parts, making it a trinomial.

Exam Tip: Be careful with signs; a minus sign separates terms, and the negative sign belongs to the term that follows it.

Question 5. (vi) 5-3t
Answer: The expression \( 5 - 3t \) contains two distinct parts, specifically \( 5 \) and \( -3t \). Therefore, this is a binomial expression.
In simple words: This expression has two parts, making it a binomial.

Exam Tip: Terms are separated by addition or subtraction operations. Here, 5 is one term and -3t is the second term.

Question 5. (vii) 4p²q - 4pq²
Answer: The expression \( 4p^2q - 4pq^2 \) contains two different terms, namely \( 4p^2q \) and \( -4pq^2 \). Therefore, this is a binomial expression.
In simple words: This expression has two distinct parts separated by a minus sign, so it is a binomial.

Exam Tip: Even though the terms look similar, they are distinct because the powers of p and q are different in each term.

Question 5. (viii) 7mn
Answer: The expression \( 7mn \) has just one term, which is \( 7mn \). Therefore, this is a monomial expression.
In simple words: This expression is a single part formed by multiplication, so it is a monomial.

Exam Tip: A monomial can involve multiple variables multiplied together, as long as there are no addition or subtraction operations within the term.

Question 5. (ix) z² - 3z + 8
Answer: The expression \( z^2 - 3z + 8 \) is made up of three parts, which are \( z^2 \), \( -3z \), and \( 8 \). Therefore, this is a trinomial expression.
In simple words: With three different parts, this expression is called a trinomial.

Exam Tip: Always count the terms carefully, noting that each term is a product of factors, and terms are separated by plus or minus signs.

Question 5. (x) a² + b²
Answer: The expression \( a^2 + b^2 \) contains two different terms, specifically \( a^2 \) and \( b^2 \). Therefore, this is a binomial expression.
In simple words: This expression has two parts joined by a plus sign, making it a binomial.

Exam Tip: The terms \( a^2 \) and \( b^2 \) are separate because they involve different variables raised to a power.

Question 5. (xi) z² + z
Answer: The expression \( z^2 + z \) has two parts, specifically \( z^2 \) and \( z \). Therefore, this is a binomial expression.
In simple words: This expression has two distinct parts, so it's a binomial.

Exam Tip: Terms like \( z^2 \) and \( z \) are unlike terms because they have different powers of the same variable, which keeps them as separate terms.

Question 5. (xii) 1 + x + x²
Answer: The expression \( 1 + x + x^2 \) is made up of three distinct parts, namely \( 1 \), \( x \), and \( x^2 \). Therefore, this is a trinomial expression.
In simple words: With three distinct parts, this expression is classified as a trinomial.

Exam Tip: A constant term (like 1) is a valid term, and terms with the same variable but different powers (like \( x \) and \( x^2 \)) are considered distinct terms.

Question 6. નીચે આપેલી જોડ સજાતીય કે વિજાતીય પદોની છે તે કહો:
Answer: State whether the given pairs are like or unlike terms.

Question 6. (i) 1, 100
Answer: The numbers \( 1 \) and \( 100 \) are like terms since they both contain the same variable part, which is \( x^0 \). (For example, \( 1 = 1x^0 \) and \( 100 = 100x^0 \) can be written).
In simple words: Both are just plain numbers without any variables, so they are like terms.

Exam Tip: All constant terms are considered like terms because their variable part (if any) is \( x^0 \), which equals 1.

Question 6. (ii) -7x, \( \frac {5}{2}x \)
Answer: The terms \( -7x \) and \( \frac {5}{2}x \) are like terms because they both have the same variable part, \( x \).
In simple words: Since both terms have the same variable 'x' with the same power, they are like terms.

Exam Tip: For terms to be "like terms," their variable parts (including powers) must be exactly the same; only the numerical coefficients can differ.

Question 6. (iii) -29x, -29y
Answer: The terms \( -29x \) and \( -29y \) are unlike terms since their variable parts, \( x \) and \( y \), are not the same.
In simple words: One term has 'x' and the other has 'y', so they are unlike terms.

Exam Tip: If the variables themselves are different (e.g., x vs. y), the terms are automatically unlike, regardless of their coefficients or powers.

Question 6. (iv) 14xy, 42yx
Answer: The terms \( 14xy \) and \( 42yx \) are like terms because, despite the variable order, they have the identical variable combination (\( xy \) is equal to \( yx \)).
In simple words: Even though the letters are swapped, 'xy' is the same as 'yx', so these are like terms.

Exam Tip: The order of variables in a product does not matter (commutative property), so \( xy \) is equivalent to \( yx \).

Question 6. (v) 4m²p, 4mp²
Answer: The terms \( 4m^2p \) and \( 4mp^2 \) are unlike terms because the variable powers (for \( m \) and \( p \) ) are not the same in each term.
In simple words: In the first term, 'm' is squared; in the second, 'p' is squared. Because the powers are different, these are unlike terms.

Exam Tip: For terms to be like terms, not only must the variables be the same, but their corresponding powers must also be identical.

Question 6. (vi) 12xz, 12x²z²
Answer: The terms \( 12xz \) and \( 12x^2z^2 \) are unlike terms because the variable powers (for \( x \) and \( z \) ) are not the same in each term.
In simple words: The powers of 'x' and 'z' are different in each term, so they are unlike terms.

Exam Tip: Carefully compare the powers of each variable in every term. A mismatch in even one variable's power makes the terms unlike.

Question 7. નીચેનામાંથી સજાતીય પદ શોધી કાઢો :
Answer: Identify the like terms from the following expressions.

Question 7. (a) -xy², – 4yx², 8x², 2xy², 7y, -11x², – 100x, -11yx, 20x²y, -6x², y, 2xy, 3x
Answer: Like terms are found by checking for matching variable parts, including their exponents. For instance, \( -xy^2 \) and \( 2xy^2 \) are like terms as they both contain \( xy^2 \) as their variable piece. Likewise, \( -4yx^2 \) and \( 20x^2y \) are like terms because \( yx^2 \) is the same as \( x^2y \).
The like terms are:
\( -xy^2 \) and \( 2xy^2 \)
\( -4yx^2 \) and \( 20x^2y \)
\( 8x^2 \), \( -11x^2 \), and \( -6x^2 \)
\( 7y \) and \( y \)
\( -100x \) and \( 3x \)
\( -11yx \) and \( 2xy \)
In simple words: We find groups of terms that have the exact same letters (variables) and the same small numbers (powers) attached to those letters. The numbers in front can be different.

Exam Tip: Systematically go through each term and find all other terms that have the exact same variable combination and powers, regardless of their numerical coefficients.

Question 7. (b) 10pq, 7p, 8q, -p²q², -7qp, – 100q, -23, 12q²p², – 5p², 41, 2405p, 78qp, 13p²q, qp², 701p²
Answer: To find like terms, check the variable sections of each term, including their exponents. Terms that have identical variable parts are known as like terms. For example, \( 10pq \), \( -7qp \), and \( 78qp \) are grouped together as \( pq \) is equivalent to \( qp \).
The like terms are:
\( 10pq \), \( -7qp \), \( 78qp \)
\( 7p \), \( 2405p \)
\( 8q \), \( -100q \)
\( -p^2q^2 \), \( 12q^2p^2 \)
\( -23 \), \( 41 \)
\( -5p^2 \), \( 701p^2 \)
\( 13p^2q \), \( qp^2 \)
In simple words: We group terms where all the variable letters and their small power numbers match exactly, even if the order of letters is different or the big number in front is different.

Exam Tip: Be mindful of the commutative property for multiplication (e.g., \( pq \) is the same as \( qp \)) and ensure variables with the same base but different powers are not grouped as like terms (e.g., \( p^2 \) and \( p \)).