2 Mark Question

Question 12 Marks

Factorize : $16 a^3-\frac{128}{b^3}$

Answer

$16 a^3-\frac{128}{b^3}$
$=16\left(a^3-\frac{8}{b^3}\right)$
... [Taking out the common factor 16]
$=16\left[a^3-\left(\frac{2}{b}\right)^3\right]$
Here, $\mathrm{A}=\mathrm{a}$ and $\mathrm{B}=\frac{2}{\mathrm{~b}}$
$\begin{aligned}
& \therefore \quad 16 \mathrm{a}^3-\frac{128}{b^3}=16\left(\mathrm{a}-\frac{2}{b}\right)\left[\mathrm{a}^2+a\left(\frac{2}{b}\right)+\left(\frac{2}{b}\right)^2\right] \\
& \cdots\left[\because A^3-B^3=(A-B)\left(A^2+A B+B^2\right)\right] \\
&=16\left(a-\frac{2}{b}\right)\left(a^2+\frac{2 a}{b}+\frac{4}{b^2}\right)
\end{aligned}$

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Question 22 Marks

Factorize : $64 x^3-729 y^3$

Answer

64x³ – 729y³
= (4x)³ – (9y)³
Here, a = 4x and b = 9y
∴ 64x³ – 729y³
= (4x – 9y) [(4x)² + (4x) (9y) + (9y)²]
…[∵ a³ – b³ = (a – b)(a² + ab + b²)]
= (4x – 9y) (16x² + 36xy + 81y²)

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Question 32 Marks

Factorize : $343 a^3-512 b^3$

Answer

343a³ – 512b³
= (7a)³ – (8b)³
Here, A = 7a and B = 8b
∴ 343a³ – 512b³
= (7a – 8b) [(7a)² + (7a)(8b) + (8b)²]
…[∵ A³ – B³ = (A – B)(A² + AB + B²)]
= (7a – 8b) (49a² + 56ab + 64b²)

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Question 42 Marks

Factorize : $8 p^3-\frac{27}{p^3}$

Answer

$\begin{aligned} & \text { } 8 p^3-\frac{27}{p^3} \\ & =(2 p)^3-\left(\frac{3}{p}\right)^3 \\ & \text { Here, } a=2 p \text { and } b=\frac{3}{p} \\ & \therefore \quad 8 p^3-\frac{27}{p^3}=\left(2 p-\frac{3}{p}\right)\left[(2 p)^2+(2 p)\left(\frac{3}{p}\right)+\left(\frac{3}{p}\right)^2\right] \\ & \ldots\left[\because a^3-b^3=(a-b)\left(a^2+a b+b^2\right)\right] \\ & =\left(2 p-\frac{3}{p}\right)\left(4 p^2+6+\frac{9}{p^2}\right) \\ & \end{aligned}$

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Question 52 Marks

Factorize : $125 y^3-1$

Answer

125y³ – 1
= (5y)³ – 1³
Here, a = 5y and b = 1
∴ 125y³ – 1 = (5y – 1) [(5y)² + (5y)(1) + (1)²]
…[∵ a³ – b³ = (a – b)(a² + ab + b²)]
= (5y – 1) (25y² + 5y + 1)

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Question 62 Marks

Factorize : $27 m^3-216 n^3$

Answer

27m³ – 216n³
= 27 (m³ – 8n³)
… [Taking out the common factor 27]
= 27 [m³ – (2n)³]
Here, a = m and b = 2n
∴ 27m³ – 216n³
= 27 {(m – 2n) [m² + m(2n) + (2n)²]}
….[∵ a³ – b³ = (a – b) (a² + ab + b²)]
= 27 (m – 2n)(m² + 2mn + 4n²)

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Question 72 Marks

Factorize : $x^3-64 y^3$

Answer

x³ – 64y³
= x³ – (4y)³
Here, a = x and b = 4y
∴ x³ – 64y³ = (x – 4y)[x² + x(4y) + (4y)²]
…[∵ a³ – b³ = (a – b)(a² + ab + b²)]
= (x – 4y)(x² + 4xy + 16y²)

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Question 82 Marks

Factorize : $y^3-27$

Answer

y³ – 27
= y³ – (3)³
Here, a = y and b = 3
∴ y³ – 27 = (y – 3)[y² + y(3) + (3)2]
…[∵ a³ – b³ = (a – b) (a² + ab + b²)]
= (y – 3)(y² + 3y + 9)

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