2 Mark Question

Question 512 Marks

Find the following product:
$\left(x^3-y^3\right)\left(x^2+y^2\right)$

Answer

$\left(x^3-y^3\right)\left(x^2+y^2\right)$
$=x^3\left(x^2+y^2\right)-y^3\left(x^2+y^2\right)$
$=\left(x^3 \times x^2+x^3 \times y^2\right)-\left(y^3 \times x^2+y^3 \times y^2\right)$
$=x^5+x^3 y^2-x^2 y^3-y^5$

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Question 522 Marks

Simplify the following:
a(b - c) + b(c - a) + c(a - b)

Answer

a(b - c) + b(c - a) + (a - b)
= a × b - a × c + b × c - b × a + c × a - c × b
= ab - ac + bc - ab + ac - bc
= 0

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Question 532 Marks

Find the following product:
$\left(x^2-y^2\right)(x+2 y)$

Answer

$\left(x^2-y^2\right)(x+2 y)$
$=x^2(x+2 y)-y^2(x+2 y)$
$=\left(x^2 \times x+x^2 \times 2 y\right)-\left(y^2 \times x+y^2 \times 2 y\right)$
$=x^3+2 x^2 y-x y^2-2 y^3$

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Question 542 Marks

Find the following product: $\left(x^2-5 x+8\right)\left(x^2+2\right)$

Answer

By column method:
$\text{x}^2-5\text{x}+8\\ \underline{ \ \ \times(\text{x}^2+2)\ \ \ \ \ \ }\\\text{x}^4-5\text{x}^3+8\text{x}^2\\\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\text{x}^2-10\text{x}+16}\\\text{x}^4-5\text{x}^3+10\text{x}^2-10\text{x}+16$
$\therefore(\text{x}^2-5\text{x}+8)(\text{x}^2+2)$
$=\text{x}^4-5\text{x}^3+10\text{x}^2-10\text{x}+16$

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Question 552 Marks

Find the following product:
$(5 y-1)(3 y-8)$

Answer

$(5 y-1)(3 y-8)$
$=5 y(3 y-8)-1(3 y-8)$
$=(5 y \times 3 y-5 y \times 8)-(3 y-8)$
$=15 y^2-40 y-3 y+8$
$=15 y^2-43 y+8$

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Question 562 Marks

Subtract:
(x - y) from (4y - 5x)

Answer

(x - y) from (4y - 5x)
= (4y - 5x) - (x - y)
= 4y - 5x - x + y
= 4y + y - 5x - x
= 5y - 6x

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Question 572 Marks

Find the products:
$\Big(\frac{-18}{5}\text{x}^2\text{z}\Big)\times \Big(\frac{-25}{6}\text{az}^2\text{y}\Big)$

Answer

$\Big(\frac{-18}{5}\text{x}^2\text{z}\Big)\times \Big(\frac{-25}{6}\text{az}^2\text{y}\Big)$
$=\frac{-18}{5}\times \frac{-25}{6}\text{x}^2\text{x}\times \text{z}^1\times\text{z}^2\times \text{y}$
$=[-3\times(-5)]\times \text{x}^{2+1}\times \text{z}^{1+2}\times \text{y}$
$=15\text{x}^3\text{yz}^3$

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Question 582 Marks

Find the products: $\left(2 a^2 b\right) \times\left(-5 a b^2 c\right) \times\left(-6 b c^2\right)$

Answer

$\left(2 a^2 b\right) \times\left(-5 a b^2 c\right) \times\left(-6 b c^2\right)$
$=2 \times(-5) \times(-6) a^2 \times a \times b \times b^2 \times b \times c \times c^2$
$=60 \times a^{2+1} \times b^{1+2+1} \times c^{1+2}$
$=60 a^3 b^4 c^3$

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Question 592 Marks

Find the following product:
$\Big(\frac{3}{4}\text{a}+\frac{2}{3}\text{b}\Big)(4\text{a}+3\text{b})$

Answer

$\Big(\frac{3}{4}\text{a}+\frac{2}{3}\text{b}\Big)(4\text{a}+3\text{b})$
$=\frac{3}{4}\text{a}\times (4\text{a}+3\text{b})+\frac{2}{3}\text{b}\times (4\text{a}+3\text{b})$
$=\Big(\frac{3}{4}\text{a}\times 4\text{a}+\frac{3}{4}\text{a}\times3\text{b}\big)+\Big(\frac{2}{3}\text{b}\times 4\text{a}+\frac{2}{3}\text{b}\times 3\text{b}\Big)$
$=3\text{a}^2+\frac{9}{4}\text{ab}+\frac{8}{3}\text{ba}+2\text{b}^2$
$=3\text{a}^2+\frac{59}{12}\text{ab}+2\text{b}^2$

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Question 602 Marks

Find the following product:
$(9 x+5 y)(4 x+3 y)$

Answer

$(9 x+5 y)(4 x+3 y)$
$=9 x(4 x+3 y)+5 y(4 x+3 y)$
$=(9 x \times 4 x+9 x \times 3 y)+(5 y \times 4 x+5 y \times 3 y)$
$=36 x^2+27 x y+20 x y+15 y^2$
$=36 x^2+47 x y+15 y^2$

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Question 612 Marks

Find the following product:
$\left(2 x^2-5 y^2\right)\left(x^2+3 y^2\right)$

Answer

$\left(2 x^2-5 y^2\right)\left(x^2+3 y^2\right)$
$= 2 x^2\left(x^2+3 y^2\right)-5 y^2\left(x^2+3 y^2\right)$
$= \left(2 x^2 \times x^2+2 x^2 \times 3 y^2\right)-\left(5 y^2 \times x^2+5 y^2 \times 3 y^2\right)$
$= 2 x^4+6 x^2 y^2-5 y^2 x^2-15 y^4$
$= 2 x^4+x^2 y^2-15 y^4$

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Question 622 Marks

Find the following product:
$4 a(3 a+7 b)$

Answer

$-4 a \times 3 a+4 a \times 7 b$
$-4 \times 3 \times a^{(1+1)}+4 \times 7 \times a \times b$
$=12 a^2+28 a b$

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Question 632 Marks

Find the products:
$\Big(\frac{-7}{5}\text{x}^2\text{y}\Big)\times \Big(\frac{3}{2}\text{xy}^2\Big)\times \Big(\frac{-6}{5}\text{x}^3\text{y}^3\Big)$

Answer

$\Big(\frac{-7}{5}\text{x}^2\text{y}\Big)\times \Big(\frac{3}{2}\text{xy}^2\Big)\times \Big(\frac{-6}{5}\text{x}^3\text{y}^3\Big)$
$=\frac{-7}{5}\times\frac{3}{2}\times \frac{-6}{5}\text{x}^2\times \text{x}\times \text{x}^3\times\text{y}\times \text{y}^2\times \text{y}^3$
$=\frac{63}{25}\text{x}^{2+1+3}\times \text{y}^{1+2+3}$
$=\frac{63}{25}\text{x}^6\text{y}^6$

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Question 642 Marks

Find the following product:
$\left(3 p^2+q^2\right)\left(2 p^2-3 q^2\right)$

Answer

$\left(3 p^2+q^2\right)\left(2 p^2-3 q^2\right)$
$=3 p^2\left(2 p^2-3 q^2\right)+q^2\left(2 p^2-3 q^2\right)$
$=\left(3 p^2 \times 2 p^2-3 p^2 \times 3 q^2\right)+\left(q^2 \times 2 p^2-q^2 \times 3 q^2\right)$
$=6 p^4-9 p^2 q^2+2 q^2 p^2-3 q^4$
$=6 p^4-7 p^2 q^2-3 q^4$

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Question 652 Marks

Find the products:
$\Big(\frac{-3}{14}\text{xy}^4\Big)\times \Big(\frac{7}{6}\text{x}^3\text{y}\Big)$

Answer

$\Big(\frac{-3}{14}\text{xy}^4\Big)\times \Big(\frac{7}{6}\text{x}^3\text{y}\Big)$
$=\frac{-3}{14}\times \frac{7}{6}\text{x}\times \text{x}^3\times \text{y}^4\times \text{y}$
$=\frac{-1}{4}\text{x}^{1+3}\times \text{y}^{4+1}$
$=\frac{-1}{4}\text{x}^4\text{y}^5$

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Question 662 Marks

Find the following product:
$\left(x^2-3 x+7\right)(2 x+3)$

Answer

By column method:
$x^2-3 x+7$
$\times(2 x+3)$
$2 x^3-6 x^2+14 x$
$\frac{3 x^2-9 x+21}{2 x^3-3 x^2+5 x+21}$
$\therefore\left(x^2-3 x+7\right)(2 x+3)$
$=2 x^3-3 x^2+5 x+21$

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Question 672 Marks

Find the products:
$\Big(\frac{-3}{4}\text{ab}^3\Big)\times \Big(\frac{-2}{3}\text{a}^2\text{b}^4\Big)$

Answer

$\Big(\frac{-3}{4}\text{ab}^3\Big)\times \Big(\frac{-2}{3}\text{a}^2\text{b}^4\Big)$
$=\frac{-3}{4}\times \frac{-2}{3}\text{a}\times \text{a}^2\times \text{b}^3\times\text{b}^4$
$=\frac{1}{2}\text{a}^{1+2}\times\text{b}^{3+4}$
$=\frac{1}{2}\text{a}^3\text{b}^7$

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Question 682 Marks

Find the following product:
$\Big(\frac{1}{5}\text{x}+2\text{y}\Big)\Big(\frac{2}{3}\text{x}-\text{y}\Big)$

Answer

$\Big(\frac{1}{5}\text{x}+2\text{y}\Big)\Big(\frac{2}{3}\text{x}-\text{y}\Big)$
$=\frac{1}{5}\text{x}\times \Big(\frac{2}{3}\text{x}-\text{y}\Big)+2\text{y}\times \Big(\frac{2}{3}\text{x}-\text{y}\Big)$
$=\Big(\frac{1}{5}\text{x}\times\frac{2}{3}\text{x}-\frac{1}{5}\text{x}\times \text{y}\Big)+\Big(2\text{y}\times \frac{2}{3}\text{x}-2\text{y}\times\text{y} \Big)$
$=\frac{2}{15}\text{x}^2-\frac{1}{5}\text{xy}+\frac{4}{3}\text{yx}-2\text{y}^2$
$=\frac{2}{15}\text{x}^2+\frac{17}{15}\text{xy}-2\text{y}^2$

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Maths 2 Mark Question