Algebraic Expressions and Identities — Multiple Choice Questions

The literal coefficient of $-9xyz^2$ is

1. -9

2. $xy$

3. $xyz^2$

4. $-9xy$

Answer

Any factor of a term is called the coefficient of the remaining factor of the term. The constant part is the numerical coefficient and the remaining part is the literal coefficient.

In $-9xyz^2$, the numerical coefficient is -9 and the remaining part is $xyz^2$.

Hence, option 3 is the correct option.

Which of the following algebraic expression is not a polynomial?

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

2. $\dfrac{5x^3}{2x} + 3x^2 + 8$

3. $3x + \dfrac{2}{x} + 7$

4. $\sqrt{2}x^2 + \sqrt{3}x + \sqrt{6}$

Answer

An algebraic expression is a polynomial if the powers of the variables in each term are non-negative integers.

Option 1: In $3x^2 - 2x + 7$ the powers of $x$ are 2, 1 and 0. So, it is a polynomial.

Option 2: $\dfrac{5x^3}{2x} + 3x^2 + 8 = \dfrac{5}{2}x^2 + 3x^2 + 8$, in which the powers of $x$ are 2 and 0. So, it is a polynomial.

Option 3: In $3x + \dfrac{2}{x} + 7$, the term $\dfrac{2}{x} = 2x^{-1}$, in which the power of $x$ is -1, a negative integer. So, it is not a polynomial.

Option 4: In $\sqrt{2}x^2 + \sqrt{3}x + \sqrt{6}$ the powers of $x$ are 2, 1 and 0. So, it is a polynomial.

Hence, option 3 is the correct option.

Which of the following algebraic expression is not a monomial?

1. $3x \times y \times z$

2. $-5pq$

3. $8m^2 \times n \div 3l$

4. $7x + y - z$

Answer

An algebraic expression having only one term is called a monomial. Multiplication and division do not separate the terms of an algebraic expression.

Option 1: $3x \times y \times z$ has only one term, so it is a monomial.

Option 2: $-5pq$ has only one term, so it is a monomial.

Option 3: $8m^2 \times n \div 3l$ has only one term, so it is a monomial.

Option 4: $7x + y - z$ has three terms, so it is not a monomial. It is a trinomial.

Hence, option 4 is the correct option.

Degree of the polynomial $7x^2yz^2 + 6x^3y^2z^2 - 5x + 8y$ is

1. 4

2. 5

3. 6

4. 7

Answer

In a polynomial in two or more variables, the degree is the greatest sum of the powers of the variables in a term.

The given polynomial is $7x^2yz^2 + 6x^3y^2z^2 - 5x + 8y$

Sum of powers of $7x^2yz^2 = 2 + 1 + 2 = 5$

Sum of powers of $6x^3y^2z^2 = 3 + 2 + 2 = 7$

Sum of powers of $-5x = 1$

Sum of powers of $8y = 1$

The greatest sum is 7.

Hence, option 4 is the correct option.

$a(b - c) + b(c - a) + c(a - b)$ is equal to

1. $ab + bc + ca$

2. 0

3. $2(ab + bc + ca)$

4. none of these

Answer

Solving,

$\Rightarrow a(b - c) + b(c - a) + c(a - b)\\[1em] \Rightarrow ab - ac + bc - ab + ac - bc\\[1em] \Rightarrow (ab - ab) + (-ac + ac) + (bc - bc)\\[1em] \Rightarrow 0$

Hence, option 2 is the correct option.

$\dfrac{7}{5}xy - \dfrac{2}{3}xy + \dfrac{8}{9}xy$ is equal to

1. $\dfrac{73}{45}xy$

2. $-\dfrac{73}{45}xy$

3. $xy$

4. none of these

Answer

Solving,

$\Rightarrow \dfrac{7}{5}xy - \dfrac{2}{3}xy + \dfrac{8}{9}xy\\[1em] \Rightarrow \Big(\dfrac{7}{5} - \dfrac{2}{3} + \dfrac{8}{9}\Big)xy\\[1em] \Rightarrow \dfrac{63 - 30 + 40}{45}xy\\[1em] \Rightarrow \dfrac{73}{45}xy$

Hence, option 1 is the correct option.

$(3p^2qr^3) \times (-4p^3q^2r^2) \times (7pq^2r)$ is equal to

1. $84p^6q^6r^6$

2. $-84p^6q^6r^6$

3. $84p^6q^5r^6$

4. $-84p^6q^5r^6$

Answer

Solving,

$\Rightarrow (3p^2qr^3) \times (-4p^3q^2r^2) \times (7pq^2r)\\[1em] \Rightarrow (3 \times (-4) \times 7) \times (p^2 \times p^3 \times p) \times (q \times q^2 \times q^2) \times (r^3 \times r^2 \times r)\\[1em] \Rightarrow -84p^{(2+3+1)}q^{(1+2+2)}r^{(3+2+1)}\\[1em] \Rightarrow -84p^6q^5r^6$

Hence, option 4 is the correct option.

$3m \times (2m^2 - 5mn + 4n^2)$ is equal to

1. $6m^3 + 15m^2n - 12mn^2$

2. $6m^3 - 15m^2n + 12mn^2$

3. $6m^3 - 15m^2n - 12mn^2$

4. $6m^3 + 15m^2n + 12mn^2$

Answer

Solving,

$\Rightarrow 3m \times (2m^2 - 5mn + 4n^2)\\[1em] \Rightarrow 3m \times 2m^2 - 3m \times 5mn + 3m \times 4n^2\\[1em] \Rightarrow 6m^3 - 15m^2n + 12mn^2$

Hence, option 2 is the correct option.

Volume of a rectangular box whose adjacent edges are $3x^2y, 4y^2z$ and $5z^2x$ respectively is

1. $60xyz$

2. $60x^2y^2z^2$

3. $60x^3y^3z^3$

4. none of these

Answer

Volume of a rectangular box = length × breadth × height

$\Rightarrow 3x^2y \times 4y^2z \times 5z^2x\\[1em] \Rightarrow (3 \times 4 \times 5) \times (x^2 \times x) \times (y \times y^2) \times (z \times z^2)\\[1em] \Rightarrow 60x^3y^3z^3$

Hence, option 3 is the correct option.

$(x + 1)(x + 2)$ is equal to

1. $2x + 3$

2. $x^2 + 2x + 2$

3. $x^2 + 3x + 2$

4. $x^2 + 2x + 3$

Answer

$(x + 1)(x + 2)$

Using the identity $(x + a)(x + b) = x^2 + (a + b)x + ab$,

$\Rightarrow x^2 + (1 + 2)x + 1 \times 2\\[1em] \Rightarrow x^2 + 3x + 2$

Hence, option 3 is the correct option.

If $x + \dfrac{1}{x} = 2$, then $x^2 + \dfrac{1}{x^2}$ is equal to

1. 4

2. 2

3. 0

4. none of these

Answer

Given, $x + \dfrac{1}{x} = 2$

Squaring both sides,

$\Rightarrow \Big(x + \dfrac{1}{x}\Big)^2 = 2^2\\[1em] \Rightarrow x^2 + 2 \times x \times \dfrac{1}{x} + \dfrac{1}{x^2} = 4\\[1em] \Rightarrow x^2 + 2 + \dfrac{1}{x^2} = 4\\[1em] \Rightarrow x^2 + \dfrac{1}{x^2} = 4 - 2\\[1em] \Rightarrow x^2 + \dfrac{1}{x^2} = 2$

Hence, option 2 is the correct option.

If $x^2 + y^2 = 9$ and $xy = 8$, then $x + y$ is equal to

1. 25

2. 5

3. -5

4. $\pm5$

Answer

Given,

$x^2 + y^2 = 9$ and $xy = 8$

We know that,

$(x + y)^2 = x^2 + y^2 + 2xy$

Substituting the values,

$\Rightarrow (x + y)^2 = 9 + 2 \times 8\\[1em] \Rightarrow (x + y)^2 = 9 + 16\\[1em] \Rightarrow (x + y)^2 = 25\\[1em] \Rightarrow x + y = \pm\sqrt{25}\\[1em] \Rightarrow x + y = \pm 5$

Hence, option 4 is the correct option.

(102)² - (98)² is equal to

1. 200

2. 400

3. 600

4. 800

Answer

Using the identity,

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

⇒ (102)² - (98)²

⇒ (102 + 98)(102 - 98)

⇒ 200 × 4

⇒ 800

Hence, option 4 is the correct option.

$-50x^3y^2z^2$ divided by $-5xyz$ is equal to

1. $10xyz$

2. $10x^2yz$

3. $-10xyz$

4. $-10x^2yz$

Answer

Solving,

$\Rightarrow -50x^3y^2z^2 \div (-5xyz)\\[1em] \Rightarrow \dfrac{-50x^3y^2z^2}{-5xyz}\\[1em] \Rightarrow \dfrac{50}{5} \times x^{3-1}y^{2-1}z^{2-1}\\[1em] \Rightarrow 10x^2yz$

Hence, option 2 is the correct option.

$96 \times 104$ is equal to

1. 9984

2. 9974

3. 9964

4. none of these

Answer

$96 \times 104 = (100 - 4)(100 + 4)$

Using the identity $(a - b)(a + b) = a^2 - b^2$,

$\Rightarrow (100)^2 - (4)^2\\[1em] \Rightarrow 10000 - 16\\[1em] \Rightarrow 9984$

Hence, option 1 is the correct option.

If the area of a rectangle is $24(x^2yz + xy^2z + xyz^2)$ and its length is $8xyz$, then its breadth is

1. $3(x + y + z)$

2. $3xyz$

3. $3(x + y - z)$

4. none of these

Answer

Area of rectangle = length × breadth

⇒ Breadth = Area of rectangle ÷ length

$\Rightarrow \dfrac{24(x^2yz + xy^2z + xyz^2)}{8xyz}\\[1em] \Rightarrow \dfrac{24xyz(x + y + z)}{8xyz}\\[1em] \Rightarrow 3(x + y + z)$

Hence, option 1 is the correct option.