Algebraic Expressions

Identify monomials, binomials and trinomials from the following algebraic expressions.

(i) 7p² × a³b

(ii) 5 + 3x³y³z²

(iii) 3x² ÷ p

(iv) $\dfrac{3a + 2b - 5c}{7}$

(v) xy + yz - zx

(vi) ax² + bx × y²

Answer

(i) 7p² × a³b

Multiplication combines these into a single unit. There are no + or - signs separating them.

Number of terms: 1

Hence, it is a monomial.

(ii) 5 + 3x³y³z²

The + sign separates the constant '5' from the variable term.

Number of terms: 2

Hence, it is a binomial.

(iii) 3x² ÷ p

Division results in a single algebraic term : $\dfrac{3\text x^2}{\text p}$

So,

Number of terms: 1

Hence, it is a monomial.

(iv) $\dfrac{3a + 2b - 5c}{7}$

While the entire expression is over a single denominator, the numerator contains three distinct terms (3a, 2b, and -5c) separated by + and -.

⟹ Number of terms: 3

Hence, it is a trinomial.

(v) xy + yz - zx

There are three distinct products separated by + and -.

So,

Number of terms: 3

Hence, it is a trinomial.

(vi) ax² + bx × y²

The multiplication combines bx and y² into one term (bxy²). The only separator is the + sign.

Term 1: ax²

Term 2: bxy²

Number of terms: 2

Hence, it is a binomial.

Write down the numerical as well as literal coefficient of each of the following monomials:

(i) -2p³q²

(ii) $\dfrac{5}{9}xyz^2$

(iii) $-\dfrac{3ab^2}{2}$

(iv) $\dfrac{3}{a^2}$

(v) $\dfrac{2ab}{c}$

(vi) $-\dfrac{2x^2}{yz}$

Answer

(i) -2p³q²

Numerical coefficient = −2, Literal coefficient = p³q²

(ii) $\dfrac{5}{9}xyz^2$

Numerical coefficient = $\dfrac{5}{9}$, Literal coefficient = xyz²

(iii) $-\dfrac{3ab^2}{2}$

Numerical coefficient = $-\dfrac{3}{2}$, Literal coefficient = ab²

(iv) $\dfrac{3}{a^2}$

Numerical coefficient = 3, Literal coefficient = $\dfrac{1}{a^2}$

(v) $\dfrac{2ab}{c}$

Numerical coefficient = 2, Literal coefficient = $\dfrac{ab}{c}$

(vi) $-\dfrac{2x^2}{yz}$

Numerical coefficient = -2, Literal coefficient = $\dfrac{x^2}{yz}$

In -5p²q³r⁴, write down the coefficient of:

(i) p²

(ii) 5pq²

(iii) p²q²

(iv) pqr

(v) -pq²r³

(vi) 5q³r

Answer

(i) p²

Coefficient of p²:

Divide the original expression by p² then

-5p²q³r⁴ ÷ p²

= -5q³r⁴

∴ Coefficient of p² is -5q³r⁴

(ii) 5pq²

Coefficient of 5pq²:

Divide the original expression by 5pq² then

-5p²q³r⁴ ÷ 5pq²

= -pqr⁴

∴ Coefficient of 5pq² is -pqr⁴

(iii) p²q²

Coefficient of p²q²:

Divide the original expression by p²q² then

-5p²q³r⁴ ÷ p²q²

= -5qr⁴

∴ Coefficient of p²q² is -5qr⁴

(iv) pqr

Coefficient of pqr:

Divide the original expression by pqr then

-5p²q³r⁴ ÷ pqr

= -5pq²r³

∴ Coefficient of pqr is -5pq²r³

(v) -pq²r³

Coefficient of -pq²r³:

Divide the original expression by -pq²r³ then

-5p²q³r⁴ ÷ -pq²r³

= 5pqr

∴ Coefficient of -pq²r³ is 5pqr

(vi) 5q³r

Coefficient of 5q³r:

Divide the original expression by 5q³r then

-5p²q³r⁴ ÷ 5q³r

= -p²r³

∴ Coefficient of 5q³r is -p²r³

Identify like terms in the following:

(i) a², b², -2a², c², 4a

(ii) 3x, 4xy, -yz, $\dfrac{1}{2}$zy

(iii) -2xy², x²y, 5y²x, x²z

(iv) abc, ab²c, acb², c²ab, b²ac, a²bc, cab²

Answer

(i) a², b², -2a², c², 4a

Since a² and -2a² have the same literal part

∴ Like terms are a² and -2a².

(ii) 3x, 4xy, -yz, $\dfrac{1}{2}$zy

Since -yz and $\dfrac{1}{2}$zy have the same literal part

∴ Like terms are -yz and $\dfrac{1}{2}$zy.

(iii) -2xy², x²y, 5y²x, x²z

Since -2xy² and 5y²x have the same literal part

∴ Like terms are -2xy² and 5y²x.

(iv) abc, ab²c, acb², c²ab, b²ac, a²bc, cab²

Since ab²c, acb², b²ac and cab² have the same literal part

∴ Like terms are ab²c, acb², b²ac and cab².

Generate algebraic expressions for the following:

(i) The product of a and b added to their sum.

(ii) The quotient of x by 8 is multiplied by y.

(iii) Thrice x added to y squared.

(iv) One-third of x multiplied by the sum of p and q.

(v) The number to be added to x + y to make it equal to p.

(vi) From a rod (p + q) units in length, n equal pieces are cut. Find the length of each piece.

(vii) Five times m subtracted from the sum of p and thrice x.

(viii) The number obtained when m times the difference of x and y is subtracted from n times the sum of x and y.

(ix) The sum of 7 numbers each equal to p.

(x) The product of three numbers a, b and c subtracted from the sum of x and y.

Answer

(i) The product of a and b added to their sum.

Product of a and b = ab

Sum of a and b = (a + b)

∴ The expression is (a + b) + ab

(ii) The quotient of x by 8 is multiplied by y.

Quotient of x by 8 = $\dfrac{x}{8}$

Multiplied by y = $\dfrac{x}{8} \times y$

∴ The expression is $\dfrac{x}{8} \times y$

(iii) Thrice x added to y squared.

Thrice x = 3x

y squared = y²

∴ The expression is y² + 3x

(iv) One-third of x multiplied by the sum of p and q.

One-third of x = $\dfrac{1}{3}x$

Sum of p and q = (p + q)

∴ The expression is $\dfrac{1}{3}x(p + q)$

(v) The number to be added to x + y to make it equal to p.

Let number = ?

x + y + ? = p

⇒ ? = p − (x + y)

∴ The expression is p - (x + y)

(vi) From a rod (p + q) units in length, n equal pieces are cut. Find the length of each piece.

Total length = p + q

Number of pieces = n

Length of each piece = $\dfrac{\text{Total length}}{\text{Number of pieces}}$

Length of each piece = $\dfrac{p + q}{n}$

∴ The expression is $\dfrac{p + q}{n}$

(vii) Five times m subtracted from the sum of p and thrice x.

Five times m = 5m

Sum of p and thrice x = (p + 3x)

∴ The expression is (p + 3x) - 5m

(viii) The number obtained when m times the difference of x and y is subtracted from n times the sum of x and y.

Difference of x and y = (x − y)

m times difference = m(x - y)

Sum of x and y = x + y

n times sum = n(x + y)

∴ The expression is n(x + y) - m(x - y)

(ix) The sum of 7 numbers each equal to p.

Adding p seven times is the same as 7 multiplied by p.

∴ The expression is 7p

(x) The product of three numbers a, b and c subtracted from the sum of x and y.

Product of a, b, c = abc

Sum of x and y = x + y

∴ The expression is (x + y) - abc

Identify which of the following expressions are polynomials. If so, write their degrees.

(i) $\dfrac{2}{3}x^4 - 7x + 15$

(ii) 8x² - 3x + 6√x + 1

(iii) 5x² - $\dfrac{2}{x}$ + 7

(iv) 9x²y² - 3xy² + 5x²y - 6x

(v) 6p⁴ - p³q² + pq³ + q⁴

(vi) 4x⁵ - 7x⁵y + 3xy⁴ + 8y⁵

(vii) ab² - $\dfrac{7}{a^2}$ + 5b² + 6

Answer

(i) $\dfrac{2}{3}x^4 - 7x + 15$

All powers of x are whole numbers.

The highest power of x is 4.

Yes, it is a polynomial. Degree = 4

(ii) 8x² - 3x + 6√x + 1

The term 6√x can be written as 6x^1/2, 1/2 is not a whole number.

∴ It is not a polynomial.

(iii) 5x² - $\dfrac{2}{x}$ + 7

The term $\dfrac{2}{x}$ can be written as 2x^-1, power of x is a negative integer.

∴ It is not a polynomial.

(iv) 9x²y² - 3xy² + 5x²y - 6x

All powers of x and y are whole numbers.

Degrees of terms:

9x²y² → 2 + 2 = 4

3xy² → 1 + 2 = 3

5x²y → 2 + 1 = 3

6x → 1

The highest degree is 4.

Yes, it is a polynomial. Degree = 4

(v) 6p⁴ - p³q² + pq³ + q⁴

All powers of p and q are whole numbers.

Degrees of terms:

6p⁴ → 4

p³q² → 3 + 2 = 5

pq³ → 1 + 3 = 4

q⁴ → 4

The highest degree is 5.

Yes, it is a polynomial. Degree = 5

(vi) 4x⁵ - 7x⁵y + 3xy⁴ + 8y⁵

All powers of x and y are whole numbers.

Degrees of terms:

4x⁵ → 5

7x⁵y → 5 + 1 = 6

3xy⁴ → 1 + 4 = 5

8y⁵ → 5

The highest degree is 6.

Yes, it is a polynomial. Degree = 6

(vii) ab² - $\dfrac{7}{a^2}$ + 5b² + 6

The term $\dfrac{7}{a^2}$ means 7a^-2, power of a is a negative integer.

∴ It is not a polynomial.

Add the following expressions:

(i) 2x², -5x², -x², 6x²

(ii) x² - 2xy + 3y², 5y² + 3xy - 6x²

(iii) 2x + 9y - 7z, 3y + z - 3x, 2z - 4y - x

(iv) 2ab + 3bc - 5ca, 4bc - 3ab + 7ca, 2ca - ab - 5bc

(v) 3x³ + 2x² - 6x + 3, 2x³ - 3x² - x - 4, 1 + 2x - 3x² - 4x³

(vi) 3n² + 5mn - 6m², 2m² - 3mn - 4n², 2mn - 3m² - 7n²

(vii) 3z³ - z² + 5, 1 - 2z + z², 3 + 2z - z³

Answer

(i) 2x², -5x², -x², 6x²

Since these are all like terms, we can stack them in a single column:

$\begin{array}{r} 2x^2 \\ -5x^2 \\ -\phantom{5} x^2 \\ +6x^2 \\ \hline 2x^2 \\ \hline \end{array}$

Hence, the answer is 2x²

(ii) x² - 2xy + 3y², 5y² + 3xy - 6x²

Arranging the expressions so that x² is under x², xy is under xy, and y² is under y²:

$\begin{array}{rcccc} x^2 & - & 2xy & + & 3y^2 \\ -6x^2 & + & 3xy & + & 5y^2 \\ \hline -5x^2 & + & xy & + & 8y^2 \\ \hline \end{array}$

Hence, the answer is -5x² + xy + 8y²

(iii) 2x + 9y - 7z, 3y + z - 3x, 2z - 4y - x

Arranging the expressions so that x is under x, y is under y, and z is under z:

$\begin{array}{rcccc} 2x & + & 9y & - & 7z \\ -3x & + & 3y & + & z \\ -\phantom{3} x & - & 4y & + & 2z \\ \hline -2x & + & 8y & - & 4z \\ \hline \end{array}$

Hence, the answer is -2x + 8y - 4z

(iv) 2ab + 3bc - 5ca, 4bc - 3ab + 7ca, 2ca - ab - 5bc

Arranging the expressions so that ab is under ab, bc is under bc, and ca is under ca:

$\begin{array}{rcccc} 2ab & + & 3bc & - & 5ca \\ -3ab & + & 4bc & + & 7ca \\ -\phantom{3} ab & - & 5bc & + & 2ca \\ \hline -2ab & + & 2bc & + & 4ca \\ \hline \end{array}$

Hence, the answer is -2ab + 2bc + 4ca

(v) 3x³ + 2x² - 6x + 3, 2x³ - 3x² - x - 4, 1 + 2x - 3x² - 4x³

Arranging the expressions into descending powers of x (x³, x², x, constant):

$\begin{array}{rcccccc} 3x^3 & + & 2x^2 & - & 6x & + & 3 \\ +2x^3 & - & 3x^2 & - & x & - & 4 \\ -4x^3 & - & 3x^2 & + & 2x & + & 1 \\ \hline x^3 & - & 4x^2 & - & 5x & + & 0 \\ \hline \end{array}$

Hence, the answer is x³ - 4x² - 5x

(vi) 3n² + 5mn - 6m², 2m² - 3mn - 4n², 2mn - 3m² - 7n²

Arranging the expressions so that m² is under m², mn is under mn, and n² is under n²:

$\begin{array}{rcccc} -6m^2 & + & 5mn & + & 3n^2 \\ +2m^2 & - & 3mn & - & 4n^2 \\ -3m^2 & + & 2mn & - & 7n^2 \\ \hline -7m^2 & + & 4mn & - & 8n^2 \\ \hline \end{array}$

Hence, the answer is -7m² + 4mn - 8n²

(vii) 3z³ - z² + 5, 1 - 2z + z², 3 + 2z - z³

Arranging the expressions in descending powers of z and use 0 as a placeholder for any missing terms:

$\begin{array}{rcccccc} 3z^3 & - & z^2 & + & 0 & + & 5 \\ +\phantom{3} 0 & + & z^2 & - & 2z & + & 1 \\ -z^3 & + & 0 & + & 2z & + & 3 \\ \hline 2z^3 & + & 0 & + & 0 & + & 9 \\ \hline \end{array}$

Hence, the answer is 2z³ + 9

Simplify:

(i) 5x + 3y - 8z + 2y - 3x + 5z + z - 7y - 2x

(ii) 4x³ - 2x² + 5x - 1 + 8x + x² - 6x³ + 7 - 6x + 3 - 3x² - x³

(iii) 2x² + 3xy - 3y² + x² - xy + y²

(iv) 2 - 3z² + 5yz + 7y² - 8 + z² - 6yz - 9y² + 1 - 2z² - 2yz - y²

(v) 2m - 3n + 5p + 2m + n - 2p - 3m - 4n + p

Answer

(i) 5x + 3y - 8z + 2y - 3x + 5z + z - 7y - 2x

Arranging the like terms together, we have:

5x + 3y - 8z + 2y - 3x + 5z + z - 7y - 2x

= (5x - 3x - 2x) + (3y + 2y - 7y) + (-8z + 5z + z)

Combining coefficients:

= (5 - 3 - 2)x + (3 + 2 - 7)y + (-8 + 5 + 1)z

= (0)x + (-2)y + (-2)z

= -2y - 2z

Hence, the answer is -2y - 2z

(ii) 4x³ - 2x² + 5x - 1 + 8x + x² - 6x³ + 7 - 6x + 3 - 3x² - x³

Arranging the like terms together, we have:

4x³ - 2x² + 5x - 1 + 8x + x² - 6x³ + 7 - 6x + 3 - 3x² - x³

= (4x³ - 6x³ - x³) + (-2x² + x² - 3x²) + (5x + 8x - 6x) + (-1 + 7 + 3)

Combining coefficients:

= (4 - 6 - 1)x³ + (-2 + 1 - 3)x² + (5 + 8 - 6)x + (9)

= (-3)x³ + (-4)x² + (7)x + 9

= -3x³ - 4x² + 7x + 9

Hence, the answer is -3x³ - 4x² + 7x + 9

(iii) 2x² + 3xy - 3y² + x² - xy + y²

Arranging the like terms together, we have:

2x² + 3xy - 3y² + x² - xy + y²

= (2x² + x²) + (3xy - xy) + (-3y² + y²)

Combining coefficients:

= (2 + 1)x² + (3 - 1)xy + (-3 + 1)y²

= 3x² + 2xy - 2y²

Hence, the answer is 3x² + 2xy - 2y²

(iv) 2 - 3z² + 5yz + 7y² - 8 + z² - 6yz - 9y² + 1 - 2z² - 2yz - y²

Arranging the like terms together, we have:

2 - 3z² + 5yz + 7y² - 8 + z² - 6yz - 9y² + 1 - 2z² - 2yz - y²

= (2 - 8 + 1) + (-3z² + z² - 2z²) + (5yz - 6yz - 2yz) + (7y² - 9y² - y²)

Combining coefficients:

= (-5) + (-3 + 1 - 2)z² + (5 - 6 - 2)yz + (7 - 9 - 1)y²

= -5 + (-4)z² + (-3)yz + (-3)y²

= -5 - 4z² - 3yz - 3y²

Hence, the answer is -5 - 4z² - 3yz - 3y²

(v) 2m - 3n + 5p + 2m + n - 2p - 3m - 4n + p

Arranging the like terms together, we have:

2m - 3n + 5p + 2m + n - 2p - 3m - 4n + p

= (2m + 2m - 3m) + (-3n + n - 4n) + (5p - 2p + p)

Combining coefficients:

= (2 + 2 - 3)m + (-3 + 1 - 4)n + (5 - 2 + 1)p

= (1)m + (-6)n + (4)p

= m - 6n + 4p

Hence, the answer is m - 6n + 4p

The two adjacent sides of a rectangle are 3a - b and 6b - a. Find its perimeter.

Answer

Given:

Length = (3a - b)

Breadth = (6b - a)

Perimeter of a rectangle = 2 x (Length + Breadth)

Substituting the values above, we get:

Perimeter of a rectangle = 2 x (3a - b + 6b - a)

= 2 x [(3a - a) + (-b + 6b)] $\quad$[Grouping like terms]

= 2 x (2a + 5b) $\quad$[Distributive property of multiplication]

= 4a + 10b

The perimeter of the rectangle is 4a + 10b.

Find the perimeter of a triangle whose sides are 2y + 3z, z - y, 4y - 2z.

Answer

Given:

Side 1 = 2y + 3z

Side 2 = z - y

Side 3 = 4y - 2z

Perimeter of a triangle = Side 1 + Side 2 + Side 3

Substituting the values above, we get:

Perimeter of a triangle = (2y + 3z + z - y + 4y - 2z)

= (2y - y + 4y) + (3z + z - 2z) $\quad$[Grouping like terms]

= (2 - 1 + 4)y + (3 + 1 - 2)z $\quad$[Combining coefficients]

= 5y + 2z

The perimeter of the triangle is 5y + 2z.

Subtract:

(i) 3a - 2b + 4c from 5a - 3b - 5c

(ii) 5x² - 3xy - 7y² from 3x² - xy - 2y²

(iii) 3p³ - 5p²q + 2q² from q² + p²q - 4p³

(iv) ab - bc - ca from 3ab + 2bc - 4ca

(v) 3z³ - 2z² + 7z - 8 from 8 - z - z²

(vi) 2abc - a² - b² from b² + a² - 2abc

Answer

(i) 3a - 2b + 4c from 5a - 3b - 5c

We have:

$\begin{array}{rcccc} 5a & - & 3b & - & 5c \\ +3a & - & 2b & + & 4c \\ -\phantom{3a} & + & & - \\ \hline 2a & - & b & - & 9c \\ \hline \end{array}$

Hence, the answer is 2a - b - 9c

(ii) 5x² - 3xy - 7y² from 3x² - xy - 2y²

We have:

$\begin{array}{rcccc} 3x^2 & - & xy & - & 2y^2 \\ +5x^2 & - & 3xy & - & 7y^2 \\ -\phantom{5x^2} & + & & + \\ \hline -2x^2 & + & 2xy & + & 5y^2 \\ \hline \end{array}$

Hence, the answer is -2x² + 2xy + 5y²

(iii) 3p³ - 5p²q + 2q² from q² + p²q - 4p³

Arranging the terms to match (p³, p²q, q²):

$\begin{array}{rcc} -4p^3 & + & p^2q & + & q^2 \\ +3p^3 & - & 5p^2q & + & 2q^2 \\ -\phantom{3p^3} & + & & - \\ \hline -7p^3 & + & 6p^2q & - & q^2 \\ \hline \end{array}$

Hence, the answer is -7p³ + 6p²q - q²

(iv) ab - bc - ca from 3ab + 2bc - 4ca

Arranging the terms to match(ab, bc, ca):

$\begin{array}{rcccc} 3ab & + & 2bc & - & 4ca \\ +ab & - & bc & - & ca \\ -\phantom{ab} & + & & + \\ \hline 2ab & + & 3bc & - & 3ca \\ \hline \end{array}$

Hence, the answer is 2ab + 3bc - 3ca

(v) 3z³ - 2z² + 7z - 8 from 8 - z - z²

Arranging the expressions in ascending powers of z and use 0 as a placeholder for any missing terms:

$\begin{array}{rcccccc} 8 & - & z & - & z^2 & + & 0 \\ -8 & + & 7z & - & 2z^2 & + & 3z^3 \\ +\phantom{8} & - & & + & & - \\ \hline 16 & - & 8z & + & z^2 & - & 3z^3 \\ \hline \end{array}$

Hence, the answer is 16 - 8z + z² - 3z³

(vi) 2abc - a² - b² from b² + a² - 2abc

Arranging the terms to match (a², b², abc):

$\begin{array}{rcc} a^2 & + & b^2 & - & 2abc \\ -a^2 & - & b^2 & + & 2abc \\ +\phantom{a^2} & + & & - \\ \hline 2a^2 & + & 2b^2 & - & 4abc \\ \hline \end{array}$

Hence, the answer is 2a² + 2b² - 4abc

(i) Subtract 6x³ - 5x² + 4x - 3 from the sum of x + 2x² - 3x³ and 2 - x² + 6x - x³.

(ii) Subtract the sum of a + 2b - 3c and 2c - 3b - 4a from the sum of 5b - 4c + a and 2c - 3b - 4a.

(iii) Subtract the sum of x² - 5xy + 2y² and y² - 2xy - 3x² from the sum of 6x² - 8xy - y² and 2xy - 2y² - x².

Answer

(i) Subtract 6x³ - 5x² + 4x - 3 from the sum of x + 2x² - 3x³ and 2 - x² + 6x - x³.

Let's find the sum of the last two expressions:

We have:

x + 2x² - 3x³ and 2 - x² + 6x - x³

Arranging the terms in descending powers of x and use 0 as a placeholder for any missing term:

$\begin{array}{rcccccc} -3x^3 & + & 2x^2 & + & x & + & 0 \\ -\phantom{3} x^3 & - & x^2 & + & 6x & + & 2 \\ \hline -4x^3 & + & x^2 & + & 7x & + & 2 \\ \hline \end{array}$

The sum is -4x³ + x² + 7x + 2

Now, subtract 6x³ - 5x² + 4x - 3 from the above sum:

$\begin{array}{rcccccc} -4x^3 & + & x^2 & + & 7x & + & 2 \\ +6x^3 & - & 5x^2 & + & 4x & - & 3 \\ -\phantom{6x^3} & + & & - & & + \\ \hline -10x^3 & + & 6x^2 & + & 3x & + & 5 \\ \hline \end{array}$

Hence, the answer is -10x³ + 6x² + 3x + 5

(ii) Subtract the sum of a + 2b - 3c and 2c - 3b - 4a from the sum of 5b - 4c + a and 2c - 3b - 4a.

Let's find the sum of first pair:

We have:

a + 2b - 3c and 2c - 3b - 4a

Arranging the terms to match (a, b, c):

$\begin{array}{rcccc} a & + & 2b & - & 3c \\ -4a & - & 3b & + & 2c \\ \hline -3a & - & b & - & c \\ \hline \end{array}$

Sum 1 = -3a - b - c

Now, let's find the sum of second pair:

We have:

5b - 4c + a and 2c - 3b - 4a

Arranging the terms to match (a, b, c):

$\begin{array}{rcccc} a & + & 5b & - & 4c \\ -4a & - & 3b & + & 2c \\ \hline -3a & + & 2b & - & 2c \\ \hline \end{array}$

Sum 2 = -3a + 2b - 2c

Let's subtract sum 1 from sum 2:

$\begin{array}{rcccc} -3a & + & 2b & - & 2c \\ -3a & - & b & - & c \\ +\phantom{3a} & + & & + \\ \hline 0 & + & 3b & - & c \\ \hline \end{array}$

Hence, the answer is 3b - c

(iii) Subtract the sum of x² - 5xy + 2y² and y² - 2xy - 3x² from the sum of 6x² - 8xy - y² and 2xy - 2y² - x².

Let's find the sum of first pair:

We have:

x² - 5xy + 2y² and y² - 2xy - 3x²

Arranging the terms to match (x², xy, y²):

$\begin{array}{rcccc} x^2 & - & 5xy & + & 2y^2 \\ -3x^2 & - & 2xy & + & y^2 \\ \hline -2x^2 & - & 7xy & + & 3y^2 \\ \hline \end{array}$

Sum 1 = -2x² - 7xy + 3y²

Now, let's find the sum of second pair:

We have:

6x² - 8xy - y² and 2xy - 2y² - x²

Arranging the terms to match (x², xy, y²):

$\begin{array}{rcccc} 6x^2 & - & 8xy & - & y^2 \\ -x^2 & + & 2xy & - & 2y^2 \\ \hline 5x^2 & - & 6xy & - & 3y^2 \\ \hline \end{array}$

Sum 2 = 5x² - 6xy - 3y²

Let's subtract sum 1 from sum 2:

$\begin{array}{rcccc} 5x^2 & - & 6xy & - & 3y^2 \\ -2x^2 & - & 7xy & + & 3y^2 \\ +\phantom{2x^2} & + & & - \\ \hline 7x^2 & + & xy & - & 6y^2 \\ \hline \end{array}$

Hence, the answer is 7x² + xy - 6y²

(i) What should be subtracted from x + 2y - 3z to get 3x - 2y + z?

(ii) What should be subtracted from 2x² - y² + 4z² to get x² + y² - z²?

(iii) What should be subtracted from 1 + x - x² to get 2x + x²?

Answer

(i) What should be subtracted from x + 2y - 3z to get 3x - 2y + z?

To find what must be subtracted from A to get B, we simply calculate A - B.

∴ Subtract 3x - 2y + z from x + 2y - 3z

$\begin{array}{rcccc} x & + & 2y & - & 3z \\ +3x & - & 2y & + & z \\ -\phantom{3x} & + & & - \\ \hline -2x & + & 4y & - & 4z \\ \hline \end{array}$

∴ The required expression is -2x + 4y - 4z

(ii) What should be subtracted from 2x² - y² + 4z² to get x² + y² - z²?

To find what must be subtracted from A to get B, we simply calculate A - B.

∴ Subtract x² + y² - z² from 2x² - y² + 4z²

$\begin{array}{rcccc} 2x^2 & - & y^2 & + & 4z^2 \\ +x^2 & + & y^2 & - & z^2 \\ -\phantom{x^2} & - & & + \\ \hline x^2 & - & 2y^2 & + & 5z^2 \\ \hline \end{array}$

∴ The required expression is x² - 2y² + 5z²

(iii) What should be subtracted from 1 + x - x² to get 2x + x²?

To find what must be subtracted from A to get B, we simply calculate A - B.

∴ Subtract 2x + x² from 1 + x - x²

Arranging the expressions in ascending powers (starting with the constant) and use 0 as a placeholder for missing terms.

$\begin{array}{rcccc} 1 & + & x & - & x^2 \\ +0 & + & 2x & + & x^2 \\ -\phantom{0} & - & & - \\ \hline 1 & - & x & - & 2x^2 \\ \hline \end{array}$

∴ The required expression is 1 - x - 2x²

(i) What should be added to 7a - 9b + 13c to get 9a + b - c?

(ii) What should be added to 1 + 2x - 3x² to get x² - x - 1?

(iii) What should be added to m² - 2mn + 5n² to get n² + mn - m²?

Answer

(i) What should be added to 7a - 9b + 13c to get 9a + b - c?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract 7a - 9b + 13c from 9a + b - c:

$\begin{array}{rcccc} 9a & + & b & - & c \\ +7a & - & 9b & + & 13c \\ -\phantom{7a} & + & & - \\ \hline 2a & + & 10b & - & 14c \\ \hline \end{array}$

∴ The required expression is 2a + 10b - 14c

(ii) What should be added to 1 + 2x - 3x² to get x² - x - 1?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract 1 + 2x - 3x² from x² - x - 1:

Arranging the expressions in descending powers of x (x², x, constant).

$\begin{array}{rcccc} x^2 & - & x & - & 1 \\ -3x^2 & + & 2x & + & 1 \\ +\phantom{3x^2} & - & & - \\ \hline 4x^2 & - & 3x & - & 2 \\ \hline \end{array}$

∴ The required expression is 4x² - 3x - 2

(iii) What should be added to m² - 2mn + 5n² to get n² + mn - m²?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract m² - 2mn + 5n² from n² + mn - m²

Arranging the terms to match (m², mn, n²):

$\begin{array}{rcccc} -m^2 & + & mn & + & n^2 \\ +m^2 & - & 2mn & + & 5n^2 \\ -\phantom{m^2} & + & & - \\ \hline -2m^2 & + & 3mn & - & 4n^2 \\ \hline \end{array}$

∴ The required expression is -2m² + 3mn - 4n²

Find the excess of 4p² - 2pq + 3q² over 2p² - pq + 4q².

Answer

"Excess" means how much larger the first expression is than the second. To find it, we subtract the second from the first (A - B).

∴ Subtract 2p² - pq + 4q² from 4p² - 2pq + 3q².

$\begin{array}{rcccc} 4p^2 & - & 2pq & + & 3q^2 \\ +2p^2 & - & pq & + & 4q^2 \\ -\phantom{2p^2} & + & & - \\ \hline 2p^2 & - & pq & - & q^2 \\ \hline \end{array}$

Hence, the answer is 2p² - pq - q²

By how much does 3x³ - 5x² + 2x - 3 exceed 2x³ - 3x² + x + 1?

Answer

"Exceed" is just another way of asking for the difference. Subtract the second expression from the first.

∴ Subtract 2x³ - 3x² + x + 1 from 3x³ - 5x² + 2x - 3

$\begin{array}{rcccccc} 3x^3 & - & 5x^2 & + & 2x & - & 3 \\ +2x^3 & - & 3x^2 & + & x & + & 1 \\ -\phantom{2x^3} & + & - & - \\ \hline x^3 & - & 2x^2 & + & x & - & 4 \\ \hline \end{array}$

Hence, the answer is x³ - 2x² + x - 4

How much is -x² + 7y² - 3xy less than 2x² - y² + xy?

Answer

Here, "how much is A less than B," means B is the larger value. Therefore, we subtract A from B (B - A).

∴ Subtract -x² + 7y² - 3xy from 2x² - y² + xy

$\begin{array}{rcc} 2x^2 & - & y^2 & + & xy \\ -x^2 & + & 7y^2 & - & 3xy \\ +\phantom{x^2} & - & & + \\ \hline 3x^2 & - & 8y^2 & + & 4xy \\ \hline \end{array}$

Hence, the answer is 3x² - 8y² + 4xy

How much is x³ - 3x² + 5x - 2 less than 3 - 2x + x² - x³?

Answer

Here it asks "how much is A less than B," it means B is the larger value. Therefore, we subtract A from B (B - A).

∴ Subtract (x³ - 3x² + 5x - 2) from (3 - 2x + x² - x³)

Arranging the expressions in ascending powers of x (constant, x, x², x³).

$\begin{array}{rcccccc} 3 & - & 2x & + & x^2 & - & x^3 \\ -2 & + & 5x & - & 3x^2 & + & x^3 \\ +\phantom{2} & - & & + & & - \\ \hline 5 & - & 7x & + & 4x^2 & - & 2x^3 \\ \hline \end{array}$

Hence, the answer is 5 - 7x + 4x² - 2x³

If x = 2a² + 3b² - 5ab, y = b² - 3a² + 7ab and z = 6a² - b² + ab, find :

(i) x + y - z

(ii) x - y + z

Answer

Given:

x = 2a² + 3b² - 5ab

y = b² - 3a² + 7ab

z = 6a² - b² + ab

(i) x + y - z

First, let's calculate x + y:

Arranging the terms to match (ab, b², a²):

$\begin{array}{rcccc} -5ab & + & 3b^2 & + & 2a^2 \\ +7ab & + & b^2 & - & 3a^2 \\ \hline 2ab & + & 4b^2 & - & a^2 \\ \hline \end{array}$

The sum is 2ab + 4b² - a² .

Now, subtract z from the above sum:

$\begin{array}{rcccc} 2ab & + & 4b^2 & - & a^2 \\ +ab & - & b^2 & + & 6a^2 \\ -\phantom{ab} & + & & - \\ \hline ab & + & 5b^2 & - & 7a^2 \\ \hline \end{array}$

∴ x + y - z = ab + 5b² - 7a²

(ii) x - y + z

First, let's calculate x - y:

Arranging the terms to match (a², b², ab):

$\begin{array}{rcccc} 2a^2 & + & 3b^2 & - & 5ab \\ -3a^2 & + & b^2 & + & 7ab \\ +\phantom{3a^2} & - & & - \\ \hline 5a^2 & + & 2b^2 & - & 12ab \\ \hline \end{array}$

The result is 5a² + 2b² - 12ab.

Now, add z to the above result:

$\begin{array}{rcccc} 5a^2 & + & 2b^2 & - & 12ab \\ +6a^2 & - & b^2 & + & ab \\ \hline 11a^2 & + & b^2 & - & 11ab \\ \hline \end{array}$

∴ x - y + z = 11a² + b² - 11ab

The perimeter of a triangle is 8 + 13a + 7a² and two of its sides are 2a² + 3a + 2 and 3a² - 4a - 1. Find the third side of the triangle.

Answer

Given:

Perimeter of a triangle = 8 + 13a + 7a²

Side 1 (S1) = 2a² + 3a + 2

Side 2 (S2) = 3a² - 4a - 1

Side 3 (S3) = ?

We know the formula,

Perimeter of a triangle = S1 + S2 + S3

⟹ S3 = Perimeter of a triangle - (S1 + S2)

Substituting the values above, we get:

S3 = (8 + 13a + 7a²) - ((2a² + 3a + 2) + (3a² - 4a - 1))

First, let's find S1 + S2:

$\begin{array}{rcccc} 2 & + & 3a & + & 2a^2 \\ -1 & - & 4a & + & 3a^2 \\ \hline 1 & - & a & + & 5a^2 \\ \hline \end{array}$

Sum (S1 + S2) = 1 - a + 5a²

Now, we have S3 = (8 + 13a + 7a²) - (1 - a + 5a²)

Subtract Sum (S1 + S2) from perimeter of a triangle:

$\begin{array}{rcccc} 8 & + & 13a & + & 7a^2 \\ +1 & - & a & + & 5a^2 \\ -\phantom{1} & + & & - \\ \hline 7 & + & 14a & + & 2a^2 \\ \hline \end{array}$

S3 = 7 + 14a + 2a²

∴ The third side of the triangle is 7 + 14a + 2a².

The perimeter of a rectangle is 16x³ - 6x² + 12x + 4. If one of its sides is 8x² + 3x, find the other side.

Answer

Given:

Perimeter of a rectangle = 16x³ - 6x² + 12x + 4

Length (Known side) = 8x² + 3x

Breadth = ?

We know the formula,

Perimeter of a rectangle = 2(Length + Breadth)

$\Rightarrow \text {Breadth} = \dfrac{\text{Perimeter - 2(Length)}}{2}$

Multiply Length by 2:

2(Length) = 2(8x² + 3x) = 16x² + 6x

Now, subtract 2(Length) from Perimeter:

$\begin{array}{rcccccc} 16x^3 & - & 6x^2 & + & 12x & + & 4 \\ +0 & + & 16x^2 & + & 6x & + & 0 \\ -\phantom{0} & - & & - & & - \\ \hline 16x^3 & - & 22x^2 & + & 6x & + & 4 \\ \hline \end{array}$

Now we have:

Breadth = $\dfrac{16x^3 - 22x^2 + 6x + 4}{2}$ = 8x³ - 11x² + 3x + 2

∴ The other side of the rectangle is 8x³ - 11x² + 3x + 2.

Which of the following is a literal?

1. 0.5
2. 0
3. s
4. $\dfrac{3}{7}$

Answer

A literal is a letter (like x, y, s) used to represent a number.

Here 's' is the literal and remaining are all fixed numerical constants.

Hence, option 3 is the correct option.

The coefficient of x in - 6 xyz² is

1. -6
2. -6x
3. yz²
4. -6 yz²

Answer

The coefficient is the part of the term other than the chosen variable.

In - 6 xyz², if we remove the x, we are left with -6yz².

Hence, option 4 is the correct option.

Which of the following is a pair of like terms?

1. ab, xy

2. 2xz², -7x²z

3. 4a²b², -$\dfrac{1}{3}$a²bc

4. 0.9 xy, - $\dfrac{2}{11}$ xy

Answer

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

Only option 4 has the same variable part (xy) in both terms.

Hence, option 4 is the correct option.

The degree of the polynomial ma²b³ + 6ab² - m is

1. 6
2. 5
3. 3
4. 2

Answer

The degree of a term is the sum of the powers of its variables. The degree of the polynomial is the highest degree among its terms.

Term 1 (ma²b³): 1 + 2 + 3 = 6

Term 2 (6ab²): 1 + 2 = 3

Term 3 (m): 1

Here, term 1 has the highest degree i.e., 6

Hence, option 1 is the correct option.

The expression -3h⁴ + 7(h - 1) is a

1. monomial
2. binomial
3. trinomial
4. none of these

Answer

Given:

-3h⁴ + 7(h - 1)

Let's simplify:

-3h⁴ + 7(h - 1) = -3h⁴ + 7h - 7

It has three terms: -3h⁴, 7h and -7. So, it is a trinomial.

Hence, option 3 is the correct option.

Which of the following is a binomial?

1. a² - 6b² + 1000
2. -7x²y + xy² + 3x²y
3. ma²b - (m + n)a²b² + nab²
4. st² - st + 1

Answer

A binomial must have exactly two terms after simplification.

Analysis:

1. a² - 6b² + 1000

There are no like terms to combine. It has 3 terms.

2. -7x²y + xy² + 3x²y

Let's simplify:

-7x²y + 3x²y + xy² $\quad$[Grouping like terms]

= (-7 + 3)x²y + xy²

= -4x²y + xy²

It has 2 terms. It is a binomial.

3. ma²b - (m + n)a²b² + nab²

There are no like terms to combine. It has 3 terms.

4. st² - st + 1

There are no like terms to combine. It has 3 terms.

Hence, option 2 is the correct option.

Which of the following is a polynomial?

1. $x - \dfrac{1}{x}$

2. $x + \sqrt{x}$

3. $\dfrac{x}{\sqrt{2}} - \dfrac{1}{\sqrt[3]{2}}$

4. $x^{\dfrac{1}{2}} + x^{\dfrac{-1}{2}}$

Answer

In a polynomial, the variables must have whole number exponents (0, 1, 2...). They cannot be in the denominator or under a root.

Analysis:

1. $1/x$ is x^-1 (negative). It is not a polynomial.

2. $\sqrt{x}$ is x^1/2 (fraction). It is not a polynomial.

3. $\dfrac{x}{\sqrt{2}} - \dfrac{1}{\sqrt[3]{2}}$ has a variable with power 1. The root is only on the constant. So, it is a polynomial.

4. $x^\dfrac{1}{2} + x^\dfrac{-1}{2}$ are fractions. It is not a polynomial.

Hence, option 3 is the correct option.

Which of the following has the highest degree?

1. $x^5 + 2x^5$

2. $2^8$

3. $-\dfrac{3a^3b^2c}{4}$

4. $\dfrac{7a^6b^5}{-5a^3b^4}$

Answer

The highest degree of a polynomial is the greatest sum of the exponents of the variables found in any single term of that expression.

Analysis:

1. $x^5 + 2x^5 = 3x^5$ → Degree = 5

2. $2^8$ → Constant, Degree = 0

3. $a^3b^2c$ → Degree = 3 + 2 + 1 = 6

4. $\dfrac{a^6b^5}{a^3b^4} = a^3b^1$ → Degree = 3 + 1 = 4

Among these, $-\dfrac{3a^3b^2c}{4}$ has the highest degree i.e., 6.

Hence, option 3 is the correct option.

The value of the expression 2m² + 3n²m + $\dfrac{7}{2}n^2 + 8$, when m = -2 and n = 2 is

1. 1
2. -4
3. 6
4. -14

Answer

Given expression = 2m² + 3n²m + $\dfrac{7}{2}n^2 + 8$

m = -2, n = 2

Substituting the given values in the expression, we get:

= 2(-2)² + 3(2)²(-2) + $\dfrac{7}{2}(2)^2 + 8$

= 2(4) + 3(4)(-2) + $\dfrac{7}{2}(4) + 8$

= 8 + (-24) + $\dfrac{28}{2} + 8$

= 8 - 24 + 14 + 8

= 6

Hence, option 3 is the correct option.

The sum of the polynomials bn - 8am - 8cp, 5cp - am + 2bn and 6am + 5bn - 3cp is equal to

1. -2am + 2bn - 4cp
2. -2am + 8bn - 2cp
3. -3am + 2bn - 4cp
4. -3am + 8bn - 6cp

Answer

Given polynomials = bn - 8am - 8cp, 5cp - am + 2bn and 6am + 5bn - 3cp

Arranging the expressions so that am is under am, bn is under bn and cp is under cp.

$\begin{array}{rccc} -8am & + & bn & - & 8cp \\ -\phantom{8} am & + & 2bn & + & 5cp \\ +6am & + & 5bn & - & 3cp \\ \hline -3am & + & 8bn & - & 6cp \\ \hline \end{array}$

Sum is -3am + 8bn - 6cp

Hence, option 4 is the correct option.

Fill in the blanks :

(i) The highest power of the ............... in a polynomial is called its degree.

(ii) The degree of the polynomial 7⁵ is ............... .

(iii) Several parts of an algebraic expression separated by + or - signs are called the ............... of the expression.

(iv) Any number is a polynomial of degree ............... .

(v) Terms having the same ............... are called like terms.

(vi) The algebraic expression for the statement ‘the number of times b is contained in x' is ............... .

(vii) 1 + x + y + xy is a polynomial having ............... terms and degree .............. .

(viii) The length of a side of a square having perimeter 8x² - 2y + 16xy is ............... .

(ix) In a polynomial, the exponents of the variables are always ............... .

(x) If the length of a rectangle having perimeter (6 m² - 2 mn + 2 m²n - 4n²) units, is (3m² + m²n - 2 mn) units, then its breadth is equal to ............... units.

Answer

(i) The highest power of the variable in a polynomial is called its degree.

(ii) The degree of the polynomial 7⁵ is 0.

(iii) Several parts of an algebraic expression separated by + or - signs are called the terms of the expression.

(iv) Any number is a polynomial of degree 0.

(v) Terms having the same literal coefficients are called like terms.

(vi) The algebraic expression for the statement ‘the number of times b is contained in x' is $\dfrac{x}{b}$.

(vii) 1 + x + y + xy is a polynomial having 4 terms and degree 2.

(viii) The length of a side of a square having perimeter 8x² - 2y + 16xy is $2x^2 - \dfrac{1}{2}y + 4xy$.

(ix) In a polynomial, the exponents of the variables are always non-negative integers.

(x) If the length of a rectangle having perimeter (6 m² - 2 mn + 2 m²n - 4n²) units, is (3m² + m²n - 2 mn) units, then its breadth is equal to mn - 2n² units.

Explanation

(i) In algebra, the degree represents the maximum power of the variable present. For example, in x³ + x, the degree is 3.

(ii) The degree refers to the power of the variable. Since 7⁵ is just a constant with no variable, its degree is 0.

(iii) Terms are the individual building blocks of an expression. In 3x + 5y, "3x" and "5y" are the terms.

(iv) Any constant number k can be written as k.x⁰. Since the variable power is 0, the degree is 0.

(v) Like terms must have the exact same variables raised to the exact same powers, such as 5ab² and -2ab².

(vi) To find how many times one number is "contained" in another, we use division. For example, 2 is contained in 10 five times (10 ÷ 2).

(vii) Given polynomial: 1 + x + y + xy

Terms: 1, x, y, xy = 4 terms

Degree: The term xy has degree 1 + 1 = 2.

(viii)

Given:

Perimeter of square = 8x² - 2y + 16xy

Side = ?

We have the formula:

Perimeter of a square = 4 x (Side)

$\Rightarrow \text {Side} = \dfrac{\text{Perimeter}}{4}$

Substituting the values above, we get:

$\text {Side} = \dfrac{8x^2 - 2y + 16xy}{4} = 2x^2 - \dfrac{1}{2}y + 4xy$

(ix) Polynomials are defined by having non-negative integer exponents. They cannot have variables with negative powers (x^-1) or roots ($\sqrt{x}$).

(x)

Given:

Length = (3m² + m²n - 2 mn)

Perimeter of a rectangle = (6 m² - 2 mn + 2 m²n - 4n²)

Breadth = ?

We know the formula:

Perimeter of a rectangle = 2(Length + Breadth)

$\Rightarrow \text {Breadth} = \dfrac{\text{Perimeter - 2(Length)}}{2}$

First, find 2 x Length:

2(Length) = 2(3m² + m²n - 2 mn) = 6m² + 2m²n - 4 mn

Now, find Perimeter - 2(Length):

$\begin{array}{rcccccc} 6m^2 & + & 2m^2n & - & 2mn & - & 4n^2 \\ -6m^2 & + & 2m^2n & - & 4mn & + & 0 \\ -\phantom{6m^2} & - & & + & & - \\ \hline 0 & & 0 & & 2mn & - & 4n^2 \\ \hline \end{array}$

Perimeter - 2(Length) = 2mn - 4n²

Now, we have:

Breadth = $\dfrac{2mn - 4n^2}{2}$

Breadth = mn - 2n²

Write true (T) or false (F) :

(i) A literal can take on various numerical values.

(ii) 2a³b - a²b - 3a²b² + 7ba² - ba³ is a trinomial.

(iii) 3mn is a factor of -9mn².

(iv) If we add a monomial and a trinomial, the answer can be a monomial.

(v) The coefficient of a²b in - 9a²b²c is -9bc.

(vi) The degree of the monomial 3³ is 3.

Answer

(i) True
Reason — A literal is a letter (like x, a, or b) used in algebra to represent a variable quantity. Unlike a constant (like 5), which has a fixed value, a literal can represent various numerical values depending on the problem or context.

(ii) True
Reason —

Given expression:

2a³b - a²b - 3a²b² + 7ba² - ba³

= (2a³b - ba³) + (- a²b + 7ba²) - 3a²b² $\quad$[Arranging like terms together]

= (2-1)a³b + (-1 + 7)a²b - 3a²b² $\quad$[Combining coefficients]

= a³b + 6a²b - 3a²b²

Since the simplified expression has exactly 3 terms, it is a trinomial.

(iii) True
Reason — A term is a factor if it divides the other term completely without leaving a remainder.

$\dfrac{-9mn^2}{3mn} = \dfrac{-9}{3} \times \dfrac{m}{m} \times \dfrac{n^2}{n} = -3n$

Since 3mn x (-3n) = -9mn², it is a factor.

(iv) False
Reason — Usually, when we add a monomial (1 term) to a trinomial (3 terms), the maximum number of terms we can get is 4, and the minimum is 2 (if the monomial is a like term that combines with one of the trinomial's terms).

Example:

Add monomial (2x) and trinomial (x² + 3x + 5):

2x + (x² + 3x + 5) = x² + 5x + 5

The result is a trinomial, not a monomial.

(v) True
Reason — To find the coefficient of a²b in the term -9a²b²c, we remove a²b from the term:

$\dfrac{-9a^2b^2c}{a^2b} = -9bc$

(vi) False
Reason — The degree of a polynomial is determined by the power of the variables. Since 3³ is a constant number and has no variable attached to it, its degree is 0.

Saket wrote four different algebraic expressions in his notebook. These are $f(x) = 2x^3 - \dfrac{1}{3}x,\ g(x) = x - \dfrac{1}{x} + 2,\ h(x) = x^2 - 2\sqrt{x}\ \text{and}\ k(y) = 2y - 1$.

(1) Which of these expressions is/are binomials ?

1. f(x), g(x), h(x) only
2. f(x), g(x), k(y) only
3. f(x), h(x), k(y) only
4. f(x), h(x) only

(2) Which of these expressions is/are not polynomials ?

1. f(x), g(x), h(x) only
2. f(x), h(x) only
3. g(x), h(x) only
4. h(x) only

(3) Which of these expressions is/are polynomials in two variables ?

1. f(x) only
2. k(y) only
3. f(x), k(y) only
4. None of these

(4) The degree of the polynomial f (x) is :

1. 0
2. 1
3. 2
4. 3

Answer

Given:

$f(x) = 2x^3 - \dfrac{1}{3}x$

$g(x) = x - \dfrac{1}{x} + 2$

$h(x) = x^2 - 2\sqrt{x}$

k(y) = 2y - 1

(1)

f(x): 2 terms (2x³ and $-\dfrac{1}{3}x$) → Binomial.

g(x): 3 terms (x, $-\dfrac{1}{x}$, and $2$) → Trinomial.

h(x): 2 terms (x² and $-2\sqrt{x}$) → Binomial.

k(y): 2 terms (2y and -1) → Binomial.

Binomial expressions are f(x), h(x), k(y).

Hence, option 3 is the correct option.

(2) A polynomial cannot have variables in the denominator or under a root (fractional powers).

Analysis:

$f(x) = 2x^3 - \dfrac{1}{3}x$

The powers of x are 3 and 1. Both are whole numbers. It is a polynomial.

$g(x) = x - \dfrac{1}{x} + 2$

The term $-\dfrac{1}{x}$ can be written as -x^-1, the exponent -1 which is a negative integer not a whole number. So, it is not a polynomial.

$h(x) = x^2 - 2\sqrt{x}$

The term $-2\sqrt{x}$ can be written as -2x^1/2, the exponent $\dfrac{1}{2}$ is a fraction, not a whole number. So, it is not a polynomial.

k(y) = 2y - 1

The power of y is 1, which is a whole number. So, it is a polynomial.

So, g(x) and h(x) are not polynomials.

Hence, option 3 is the correct option.

(3) A "polynomial in two variables" must contain exactly two different letters (like x and y, or a and b) throughout its terms.

Analysis:

$f(x) = 2x^3 - \dfrac{1}{3}x$

Only x appears in both terms. This is a polynomial in one variable (x).

$g(x) = x - \dfrac{1}{x} + 2$

Only x appears. This is an algebraic expression in one variable (x). It is not a polynomial.

$h(x) = x^2 - 2\sqrt{x}$

Only x appears. This is an algebraic expression in one variable (x). It is not a polynomial.

k(y) = 2y - 1

Only y appears. This is a polynomial in one variable (y).

No expression is a polynomial in two variables.

Hence, option 4 is the correct option.

(4) $f(x) = 2x^3 - \dfrac{1}{3}x$

The highest power of x in $2x^3 - \dfrac{1}{3}x$ is 3.

Hence, option 4 is the correct option.

Vasu has a rectangular farmland A. The length of this farmland is x²y - 2xy + 3x² and its perimeter is 2x²y + 6x² - 2xy - 4y². Today he purchased the adjacent farmland B and combined the two farmlands into one. The length of his farmland now increased by x² + xy, while the breadth remained the same.

(1) Find the breadth of farmland A :

1. x²y + 3x² - 4y²
2. xy - 2y²
3. 12x² - 4xy
4. 6x² + xy - 2y²

(2) Find the length of the combined farmland owned by Vasu :

1. x² + 2xy - 2y²
2. 4x² - 3xy + x²y
3. 4x² - xy + x²y
4. 2x² - 3xy + x²y

(3) The perimeter of Vasu's combined farmland is :

1. 8x² + 2x²y - 4y²
2. 4y² + 4xy + 2x²y + 8x²
3. 4x² - 2x²y + 4y²
4. 4x² + 4xy - 2x²y - 8x²

(4) The change in the perimeter after combining farmlands A and B is :

1. 4x² + 2xy + 4x²y - y²
2. 2x² - 2xy + y²
3. 4x² + 2xy - y²
4. 2x² + 2xy

Answer

Given for Farmland A:

Length (L_A) = x²y - 2xy + 3x²

Perimeter (P_A) = 2x²y + 6x² - 2xy - 4y²

(1)

We know the formula:

Perimeter of a rectangle = 2(Length + Breadth)

$\Rightarrow \text {Breadth} = \dfrac{\text{Perimeter - 2(Length)}}{2}$

First, find 2(Length):

2(Length) = 2(x²y - 2xy + 3x²) = 2x²y - 4xy + 6x²

Now, calculate Perimeter - 2(Length):

$\begin{array}{rcccccc} 2x^2y & + & 6x^2 & - & 2xy & - & 4y^2 \\ +2x^2y & + & 6x^2 & - & 4xy & + & 0 \\ -\phantom{2x^2y} & - & & + & & - \\ \hline 0 & & 0 & & 2xy & - & 4y^2 \\ \hline \end{array}$

Perimeter - 2(Length) = 2xy - 4y²

Now we have:

Breadth = $\dfrac{2xy - 4y^2}{2}$

Breadth = xy - 2y²

Hence, option 2 is the correct option.

(2)

Original Length (L_A) = x²y - 2xy + 3x²

Increase = x² + xy

Combined Length = Original Length + Increase

Substituting the values above, we get:

Combined Length = (x²y - 2xy + 3x²) + (x² + xy)

= (3x² + x²) + (- 2xy + xy) + x²y $\quad$[Arranging like terms together]

= (3 + 1)x² + (-1)xy + x²y

= 4x² - xy + x²y

Hence, option 3 is the correct option.

(3)

Combined Length = 4x² - xy + x²y $\quad$[From step 2]

Breadth = xy - 2y² $\quad$[From step 1]

Perimeter of combined farmland = ?

Let's apply the perimeter of a rectangle formula:

Perimeter of combined farmland = 2(Length + Breadth)

Let's first calculate Length + Breadth.

We have:

Length + Breadth = (4x² - xy + x²y) + (xy - 2y²)

= 4x² + (- xy + xy) + x²y + (- 2y²) $\quad$[Arranging like terms together]

= 4x² + 0xy + x²y - 2y²

= 4x² + x²y - 2y²

Now we have:

Perimeter of combined farmland = 2 x (4x² + x²y - 2y²) = 8x² + 2x²y - 4y²

Hence, option 1 is the correct option.

(4)

Original Perimeter = 2x²y + 6x² - 2xy - 4y²

Combined Perimeter = 8x² + 2x²y - 4y²

Change in perimeter = Combined Perimeter - Original Perimeter

Substituting the values above, we get:

Change in perimeter = (8x² + 2x²y - 4y²) - (2x²y + 6x² - 2xy - 4y²)

= 8x² + 2x²y - 4y² - 2x²y - 6x² + 2xy + 4y² $\quad$[Simplifying brackets]

= (8x² - 6x²) + (2x²y - 2x²y) + 2xy + (- 4y² + 4y²) $\quad$[Arranging like terms together]

= 2x² + 0x²y + 2xy + 0y²

= 2x² + 2xy

Hence, option 4 is the correct option.

Assertion: The expression x + y - 2x is a trinomial.

Reason: An algebraic expression containing three terms is called a trinomial.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is false but Reason (R) is true.

Explanation

Given expression:

x + y - 2x

Let's simplify it:

∴ x + y - 2x = (x - 2x) + y $\quad$[Arranging like terms together]

= -x + y

= y - x

∴ Simplified expression is y - x, which has only two terms. Therefore, it is a binomial, not a trinomial. So, Assertion is incorrect.

Reason is true because it is the correct definition of a trinomial.

Hence, option 4 is the correct option.

Assertion: If we subtract 15x² - 9x + 1 from 1, then we get -15x² + 9x.

Reason: The degree of 15x² - 9x + 1 is 3 as it has three terms and degree of -15x² + 9x is 2 as it has two terms.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is true but Reason (R) is false.

Explanation

Assertion:

Subtracting 15x² - 9x + 1 from 1:

1 - (15x² - 9x + 1)

= 1 - 15x² + 9x - 1 $\quad$[Simplifying brackets]

= (1 - 1) - 15x² + 9x

= 0 - 15x² + 9x

= - 15x² + 9x

The result matches the statement exactly.

So, Assertion is true.

Reason is false because, the degree of a polynomial is determined by the highest power of the variable, not by the number of terms.

Degree of 15x² - 9x + 1 = 2 (because the highest power of x is 2).

Degree of -15x² + 9x = 2 (because the highest power of x is 2).

The Reason incorrectly claims the degree is 3 based on the number of terms.

Hence, option 3 is the correct option.