Algebraic Expressions and Identities — Statement I-II Type Questions

Statement I: The volume of a cube having edge of $(x + y)$ is $x^3 + y^3 + 3xy(x + y)$

Statement II: The volume of a cube with edge $a$ is $a^3$

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Volume of a cube = (edge)³

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

So, Statement I is true.

Volume of a cube = (edge)³ = $a^3$, where $a$ is the edge of the cube.

So, Statement II is true.

Hence, option 3 is the correct option.

Statement I: If the area of a rectangle is $x^2 + y^2 + 2xy$ and one side is $(x + y)$ then it is a square.

Statement II: $(x - y)^2 = x^2 + y^2 + 2xy$

1. Statement I is true but statement II is false.

2. Statement I is false but statement II is true.

3. Both Statement I and statement II are true.

4. Both Statement I and statement II are false.

Answer

Area of the rectangle = $x^2 + y^2 + 2xy$

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

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

As one side is $(x + y)$, the adjacent side = $(x + y)$.

As both the adjacent sides are equal, the rectangle is a square.

So, Statement I is true.

We know that $(x - y)^2 = x^2 + y^2 - 2xy$ and not $x^2 + y^2 + 2xy$.

So, Statement II is false.

Hence, option 1 is the correct option.

Statement I: If $2x^4 + 5x^3 - 18x^2 + 6x + 8$ is divided by $x^2 + 5x + 2$, then the quotient is $2x^2 + 5x + 3$ and the remainder is $x + 2$

Statement II: Divisor = $\dfrac{\text{Dividend} - \text{Remainder}}{\text{Quotient}}$

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Dividing $2x^4 + 5x^3 - 18x^2 + 6x + 8$ by $x^2 + 5x + 2$

$\begin{array}{l} \phantom{x^2 + 5x + 2\big)}{2x^2 - 5x + 3} \\ x^2 + 5x + 2\overline{\smash{\big)}2x^4 + 5x^3 - 18x^2 + 6x + 8} \\ \phantom{+++++}\underline{\underset{-}{}2x^4 \underset{-}{+} 10x^3 \underset{-}{+} 4x^2 \phantom{+ 6x} \phantom{+ 8}} \\ \phantom{+++++++}-5x^3 - 22x^2 + 6x + 8 \\ \phantom{x^2 + 5x + 2\big)2x^4;}\underline{\underset{+}{-}5x^3 \underset{+}{-} 25x^2 \underset{+}{-} 10x \phantom{+ 8}} \\ \phantom{+++++++++++}3x^2 + 16x + 8 \\ \phantom{x^2 + 5x + 2\big)2x^4 + 5x^3;}\underline{\underset{-}{}3x^2 \underset{-}{+} 15x \underset{-}{+} 6} \\ \phantom{++++++++++++++}x + 2 \end{array}$

Here, quotient = $2x^2 - 5x + 3$ and remainder = $x + 2$

But Statement I says the quotient is $2x^2 + 5x + 3$.

So, Statement I is false.

We know that Dividend = Divisor × Quotient + Remainder

⇒ Dividend - Remainder = Divisor × Quotient

⇒ Divisor = $\dfrac{\text{Dividend} - \text{Remainder}}{\text{Quotient}}$

So, Statement II is true.

Hence, option 2 is the correct option.

Statement I: If $x - \dfrac{1}{x} = 1$, then $x^2 - \dfrac{1}{x^2} = x + \dfrac{1}{x}$

Statement II: $(a - b)^2 = a^2 - b^2 - 2ab$.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Given, $x - \dfrac{1}{x} = 1$

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

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

So, Statement I is true.

We know that $(a - b)^2 = a^2 + b^2 - 2ab$ and not $a^2 - b^2 - 2ab$.

So, Statement II is false.

Hence, option 1 is the correct option.

Statement I: $(x + 3)(x - 3)(x + 1)(x - 1)$ is a polynomial of degree 1.

Statement II: A polynomial is of degree $n$ if the greatest sum of the powers of variables in the terms is $n$.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Solving,

$\Rightarrow (x + 3)(x - 3)(x + 1)(x - 1)\\[1em] \Rightarrow [(x + 3)(x - 3)][(x + 1)(x - 1)]$

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

$\Rightarrow (x^2 - 9)(x^2 - 1)\\[1em] \Rightarrow x^4 - x^2 - 9x^2 + 9\\[1em] \Rightarrow x^4 - 10x^2 + 9$

The greatest power of $x$ is 4, so it is a polynomial of degree 4 and not of degree 1.

So, Statement I is false.

A polynomial is of degree $n$ if the greatest sum of the powers of variables in the terms is $n$.

So, Statement II is true.

Hence, option 2 is the correct option.