Q1. Find whether given trinomials are perfect square or not:
(i) $\text{x}^{2} + 6\text{x} + 9$
(ii) $4\text{x}^{2} - 4\text{x} + 1$
(iii) $25 - 20\text{x} + 4\text{x}^{2}$
(iv) $\text{x}^{2} - \text{x} + \dfrac{1}{4}$
(v) $\text{x}^{2} + 2 + \dfrac{1}{\text{x}^{2}}$
(vi) $16\text{x}^{2} + 20\text{xy} + 25\text{y}^{2}$
(vii) $36\text{x}^{2} - 12\text{x} + 1$
(viii) $\text{a}^{2} + \text{a}^{2}\text{b}^{2} + \text{b}^{4}$
Show Answer
(i) $\text{x}^{2} + 6\text{x} + 9$
$\because \text{a}^{2} + 2\text{ab} + (\text{b})^{2}$
$\therefore \text{x}^{2} + 2 \ (\text{x}) \ (3) + (3)^{2}$
$\Rightarrow \text{x}^{2} + 6\text{x} + 9 = \text{x}^{2} + 6\text{x} + 9$
$\therefore$ Is perfect square Ans
(ii) $4\text{x}^{2} - 4\text{x} + 1$
$\because \text{a}^{2} - 2\text{ab} + (\text{b})^{2}$
$\therefore (2\text{x})^{2} - 2 \ (2\text{x}) \ (1) + (1)^{2}$
$\Rightarrow 4\text{x}^{2} - 4\text{x} + 1 = 4\text{x}^{2} - 4\text{x} + 1$
$\therefore$ Is perfect square Ans
(iii) $25 - 20\text{x} + 4\text{x}^{2}$
$\because \text{a}^{2} - 2\text{ab} + (\text{b})^{2}$
$\therefore (5)^{2} - 2 \ (5) \ (2\text{x}) + (2\text{x})^{2}$
$\Rightarrow 25 - 20\text{x} + 4\text{x}^{2} = 25 - 20\text{x} + 4\text{x}^{2}$
$\therefore$ Is perfect square Ans
(iv) $\text{x}^{2} - \text{x} + \dfrac{1}{4}$
$\because \text{a}^{2} - 2\text{ab} + (\text{b})^{2}$
$\therefore (\text{x})^{2} - 2 \ (\text{x}) \ (\dfrac{1}{2}) + (\dfrac{1}{2})^{2}$
$\Rightarrow \text{x}^{2} - \text{x} + \dfrac{1}{4} = \text{x}^{2} - \text{x} + \dfrac{1}{4}$
$\therefore$ Is perfect square Ans
(v) $\text{x}^{2} + 2 + \dfrac{1}{\text{x}^{2}}$
$\because \text{a}^{2} + 2\text{ab} + (\text{b})^{2}$
$\therefore (\text{x})^{2} + 2 \ (\text{x}) \ (\dfrac{1}{\text{x}}) + (\dfrac{1}{\text{x}})^{2}$
$\Rightarrow \text{x}^{2} + 2 + \dfrac{1}{\text{x}^{2}} = \text{x}^{2} + 2 + \dfrac{1}{\text{x}^{2}}$
$\therefore$ Is perfect square Ans
(vi) $16\text{x}^{2} + 20\text{xy} + 25\text{y}^{2}$
$\because \text{a}^{2} + 2\text{ab} + (\text{b})^{2}$
$\therefore (4\text{x})^{2} + 2 \ (4\text{x}) \ (5\text{y}) + (5\text{y})^{2}$
$\Rightarrow 16\text{x}^{2} + 80\text{xy} + 25\text{y}^{2} \ne 16\text{x}^{2} + 20\text{xy} + 25\text{y}^{2}$
$\therefore$ Is not perfect square Ans
(vii) $36\text{x}^{2} - 12\text{x} + 1$
$\because \text{a}^{2} - 2\text{ab} + (\text{b})^{2}$
$\therefore (6\text{x})^{2} - 2 \ (6\text{x}) \ (1) + (1)^{2}$
$\Rightarrow 36\text{x}^{2} - 12\text{x} + 1 = 36\text{x}^{2} - 12\text{x} + 1$
$\therefore$ Is perfect square Ans
(viii) $\text{a}^{2} + \text{a}^{2}\text{b}^{2} + \text{b}^{4}$
$\because \text{a}^{2} + 2\text{ab} + (\text{b})^{2}$
$\therefore (\text{a})^{2} + 2 \ (\text{a}) \ (\text{b}^{2}) + (\text{b}^{2})^{2}$
$\Rightarrow \text{a}^{2} + 2\text{ab}^{2} + \text{b}^{4} \ne \text{a}^{2} + \text{a}^{2}\text{b}^{2} + \text{b}^{4}$
$\therefore$ Is not perfect square Ans
$\because \text{a}^{2} + 2\text{ab} + (\text{b})^{2}$
$\therefore \text{x}^{2} + 2 \ (\text{x}) \ (3) + (3)^{2}$
$\Rightarrow \text{x}^{2} + 6\text{x} + 9 = \text{x}^{2} + 6\text{x} + 9$
$\therefore$ Is perfect square Ans
(ii) $4\text{x}^{2} - 4\text{x} + 1$
$\because \text{a}^{2} - 2\text{ab} + (\text{b})^{2}$
$\therefore (2\text{x})^{2} - 2 \ (2\text{x}) \ (1) + (1)^{2}$
$\Rightarrow 4\text{x}^{2} - 4\text{x} + 1 = 4\text{x}^{2} - 4\text{x} + 1$
$\therefore$ Is perfect square Ans
(iii) $25 - 20\text{x} + 4\text{x}^{2}$
$\because \text{a}^{2} - 2\text{ab} + (\text{b})^{2}$
$\therefore (5)^{2} - 2 \ (5) \ (2\text{x}) + (2\text{x})^{2}$
$\Rightarrow 25 - 20\text{x} + 4\text{x}^{2} = 25 - 20\text{x} + 4\text{x}^{2}$
$\therefore$ Is perfect square Ans
(iv) $\text{x}^{2} - \text{x} + \dfrac{1}{4}$
$\because \text{a}^{2} - 2\text{ab} + (\text{b})^{2}$
$\therefore (\text{x})^{2} - 2 \ (\text{x}) \ (\dfrac{1}{2}) + (\dfrac{1}{2})^{2}$
$\Rightarrow \text{x}^{2} - \text{x} + \dfrac{1}{4} = \text{x}^{2} - \text{x} + \dfrac{1}{4}$
$\therefore$ Is perfect square Ans
(v) $\text{x}^{2} + 2 + \dfrac{1}{\text{x}^{2}}$
$\because \text{a}^{2} + 2\text{ab} + (\text{b})^{2}$
$\therefore (\text{x})^{2} + 2 \ (\text{x}) \ (\dfrac{1}{\text{x}}) + (\dfrac{1}{\text{x}})^{2}$
$\Rightarrow \text{x}^{2} + 2 + \dfrac{1}{\text{x}^{2}} = \text{x}^{2} + 2 + \dfrac{1}{\text{x}^{2}}$
$\therefore$ Is perfect square Ans
(vi) $16\text{x}^{2} + 20\text{xy} + 25\text{y}^{2}$
$\because \text{a}^{2} + 2\text{ab} + (\text{b})^{2}$
$\therefore (4\text{x})^{2} + 2 \ (4\text{x}) \ (5\text{y}) + (5\text{y})^{2}$
$\Rightarrow 16\text{x}^{2} + 80\text{xy} + 25\text{y}^{2} \ne 16\text{x}^{2} + 20\text{xy} + 25\text{y}^{2}$
$\therefore$ Is not perfect square Ans
(vii) $36\text{x}^{2} - 12\text{x} + 1$
$\because \text{a}^{2} - 2\text{ab} + (\text{b})^{2}$
$\therefore (6\text{x})^{2} - 2 \ (6\text{x}) \ (1) + (1)^{2}$
$\Rightarrow 36\text{x}^{2} - 12\text{x} + 1 = 36\text{x}^{2} - 12\text{x} + 1$
$\therefore$ Is perfect square Ans
(viii) $\text{a}^{2} + \text{a}^{2}\text{b}^{2} + \text{b}^{4}$
$\because \text{a}^{2} + 2\text{ab} + (\text{b})^{2}$
$\therefore (\text{a})^{2} + 2 \ (\text{a}) \ (\text{b}^{2}) + (\text{b}^{2})^{2}$
$\Rightarrow \text{a}^{2} + 2\text{ab}^{2} + \text{b}^{4} \ne \text{a}^{2} + \text{a}^{2}\text{b}^{2} + \text{b}^{4}$
$\therefore$ Is not perfect square Ans
Q2. Simplify: $\dfrac{3\text{x} + 3\text{y}}{\text{x}^{2} - \text{y}^{2}} \div \dfrac{3}{\text{x} - \text{y}}$
Show Answer
$\Rightarrow \dfrac{3\text{x} + 3\text{y}}{\text{x}^{2} - \text{y}^{2}} \div \dfrac{3}{\text{x} - \text{y}}$
$\Rightarrow \dfrac{3(\text{x} + \text{y})}{(\text{x} - \text{y})^{2}} \div \dfrac{3}{\text{x} - \text{y}}$
$\Rightarrow \dfrac{3(\text{x} + \text{y})}{(\text{x} + \text{y}) \ (\text{x} - \text{y})} \div \dfrac{3}{\text{x} - \text{y}}$
$\Rightarrow \dfrac{3(\text{x} + \text{y})}{(\text{x} + \text{y}) \ (\text{x} - \text{y})} \times \dfrac{\text{x} - \text{y}}{3}$
$\Rightarrow \dfrac{\text{x} + \text{y}}{\text{x} + \text{y}} = 1$ Ans
$\Rightarrow \dfrac{3(\text{x} + \text{y})}{(\text{x} - \text{y})^{2}} \div \dfrac{3}{\text{x} - \text{y}}$
$\Rightarrow \dfrac{3(\text{x} + \text{y})}{(\text{x} + \text{y}) \ (\text{x} - \text{y})} \div \dfrac{3}{\text{x} - \text{y}}$
$\Rightarrow \dfrac{3(\text{x} + \text{y})}{(\text{x} + \text{y}) \ (\text{x} - \text{y})} \times \dfrac{\text{x} - \text{y}}{3}$
$\Rightarrow \dfrac{\text{x} + \text{y}}{\text{x} + \text{y}} = 1$ Ans
Q3. Simplify: $\dfrac{(\text{c} - \text{d})^{2} + (\text{d} - \text{c})}{\text{c} - \text{d} - 1}$
Show Answer
$\Rightarrow \dfrac{(\text{c} - \text{d})^{2} + (\text{d} - \text{c})}{\text{c} - \text{d} - 1}$
Upon rearranging terms of numerator, we have:
$\Rightarrow \dfrac{(\text{c} - \text{d})^{2} - (\text{c} + \text{d})}{\text{c} - \text{d} - 1}$
$\Rightarrow \dfrac{(\text{c} - \text{d}) \ (\text{c} - \text{d}) - 1(\text{c} - \text{d})}{\text{c} - \text{d} - 1}$
$\Rightarrow \dfrac{(\text{c} - \text{d}) \ (\text{c} - \text{d} - 1)}{\text{c} - \text{d} - 1} = \text{c} - \text{d}$ Ans
Upon rearranging terms of numerator, we have:
$\Rightarrow \dfrac{(\text{c} - \text{d})^{2} - (\text{c} + \text{d})}{\text{c} - \text{d} - 1}$
$\Rightarrow \dfrac{(\text{c} - \text{d}) \ (\text{c} - \text{d}) - 1(\text{c} - \text{d})}{\text{c} - \text{d} - 1}$
$\Rightarrow \dfrac{(\text{c} - \text{d}) \ (\text{c} - \text{d} - 1)}{\text{c} - \text{d} - 1} = \text{c} - \text{d}$ Ans
Q4. Simplify: $\dfrac{\text{a}(\text{a} - \text{b}) - \text{a} + \text{b}}{\text{a} - \text{b}}$
Show Answer
$\Rightarrow \dfrac{\text{a}(\text{a} - \text{b}) - \text{a} + \text{b}}{\text{a} - \text{b}}$
$\Rightarrow \dfrac{\text{a}(\text{a} - \text{b}) - 1(\text{a} - \text{b})}{\text{a} - \text{b}}$
$\Rightarrow \dfrac{(\text{a} - \text{b}) \ (\text{a} - 1)}{\text{a} - \text{b}} = \text{a} - 1$ Ans
$\Rightarrow \dfrac{\text{a}(\text{a} - \text{b}) - 1(\text{a} - \text{b})}{\text{a} - \text{b}}$
$\Rightarrow \dfrac{(\text{a} - \text{b}) \ (\text{a} - 1)}{\text{a} - \text{b}} = \text{a} - 1$ Ans
Q5. Simplify: $\dfrac{\text{x} - 5}{\text{x} + 6} - \dfrac{\text{x} + 5}{\text{x} - 6}$
Show Answer
$\Rightarrow \dfrac{\text{x} - 5}{\text{x} + 6} - \dfrac{\text{x} + 5}{\text{x} - 6}$
$\Rightarrow \dfrac{(\text{x} - 6) \ (\text{x} - 5) - (\text{x} + 6) \ (\text{x} + 5)}{(\text{x} + 6) \ (\text{x} - 6)}$
$\Rightarrow \dfrac{(\text{x}^{2} - 5\text{x} - 6\text{x} + 30) - (\text{x}^{2} + 5\text{x} + 6\text{x} + 30)}{\text{x}^{2} - 6\text{x} + 6\text{x} - 36}$
$\Rightarrow \dfrac{(\text{x}^{2} - 11\text{x} + 30) - (\text{x}^{2} + 11\text{x} + 30)}{\text{x}^{2} - 36}$
$\Rightarrow \dfrac{\text{x}^{2} - 11\text{x} + 30 - \text{x}^{2} - 11\text{x} - 30}{\text{x}^{2} - 36}$
$\Rightarrow \dfrac{- \ 22\text{x}}{\text{x}^{2} - 36} = \dfrac{- \ 22\text{x}}{(\text{x} - 6)^{2}}$ Ans
$\Rightarrow \dfrac{(\text{x} - 6) \ (\text{x} - 5) - (\text{x} + 6) \ (\text{x} + 5)}{(\text{x} + 6) \ (\text{x} - 6)}$
$\Rightarrow \dfrac{(\text{x}^{2} - 5\text{x} - 6\text{x} + 30) - (\text{x}^{2} + 5\text{x} + 6\text{x} + 30)}{\text{x}^{2} - 6\text{x} + 6\text{x} - 36}$
$\Rightarrow \dfrac{(\text{x}^{2} - 11\text{x} + 30) - (\text{x}^{2} + 11\text{x} + 30)}{\text{x}^{2} - 36}$
$\Rightarrow \dfrac{\text{x}^{2} - 11\text{x} + 30 - \text{x}^{2} - 11\text{x} - 30}{\text{x}^{2} - 36}$
$\Rightarrow \dfrac{- \ 22\text{x}}{\text{x}^{2} - 36} = \dfrac{- \ 22\text{x}}{(\text{x} - 6)^{2}}$ Ans
Q6. Simplify: $\dfrac{6}{\text{a} + 2} - \dfrac{4}{\text{a} - 2} - \dfrac{3}{\text{a}^{2} - 4}$
Show Answer
$\Rightarrow \dfrac{6}{\text{a} + 2} - \dfrac{4}{\text{a} - 2} - \dfrac{3}{\text{a}^{2} - 4}$
First we take:
$\Rightarrow \dfrac{6}{\text{a} + 2} - \dfrac{4}{\text{a} - 2}$
$\Rightarrow \dfrac{6(\text{a} - 2) - 4(\text{a} + 2)}{(\text{a} + 2) \ (\text{a} - 2)}$
$\Rightarrow \dfrac{6\text{a} - 12 - 4\text{a} - 8}{\text{a}^{2} - 2\text{a} + 2\text{a} - 4} = \dfrac{2\text{a} - 20}{\text{a}^{2} - 4}$
Now we take:
$\Rightarrow \dfrac{2\text{a} - 20}{\text{a}^{2} - 4} - \dfrac{3}{\text{a}^{2} - 4}$
$\Rightarrow \dfrac{2\text{a} - 20 - 3}{\text{a}^{2} - 4}$
$\Rightarrow \dfrac{2\text{a} - 23}{\text{a}^{2} - 4}$ Ans
First we take:
$\Rightarrow \dfrac{6}{\text{a} + 2} - \dfrac{4}{\text{a} - 2}$
$\Rightarrow \dfrac{6(\text{a} - 2) - 4(\text{a} + 2)}{(\text{a} + 2) \ (\text{a} - 2)}$
$\Rightarrow \dfrac{6\text{a} - 12 - 4\text{a} - 8}{\text{a}^{2} - 2\text{a} + 2\text{a} - 4} = \dfrac{2\text{a} - 20}{\text{a}^{2} - 4}$
Now we take:
$\Rightarrow \dfrac{2\text{a} - 20}{\text{a}^{2} - 4} - \dfrac{3}{\text{a}^{2} - 4}$
$\Rightarrow \dfrac{2\text{a} - 20 - 3}{\text{a}^{2} - 4}$
$\Rightarrow \dfrac{2\text{a} - 23}{\text{a}^{2} - 4}$ Ans
Q7. Simplify: $\dfrac{1}{2(\text{x} - 3)^{2}} - \dfrac{1}{\text{x}(\text{x} - 3)}$
Show Answer
$\Rightarrow \dfrac{1}{2(\text{x} - 3)^{2}} - \dfrac{1}{\text{x}(\text{x} - 3)}$
$\Rightarrow$ L.C.M. = $2(\text{x} - 3)^{2}$
$\Rightarrow \dfrac{\text{x} - 2(\text{x} - 3)}{2(\text{x} - 3)^{2}}$
$\Rightarrow \dfrac{\text{x} - 2\text{x} + 6}{2(\text{x} - 3)^{2}}$
$\Rightarrow \dfrac{- \ \text{x} + 6}{2(\text{x} - 3)^{2}} = \dfrac{6 - \text{x}}{2(\text{x} - 3)^{2}}$ Ans
$\Rightarrow$ L.C.M. = $2(\text{x} - 3)^{2}$
$\Rightarrow \dfrac{\text{x} - 2(\text{x} - 3)}{2(\text{x} - 3)^{2}}$
$\Rightarrow \dfrac{\text{x} - 2\text{x} + 6}{2(\text{x} - 3)^{2}}$
$\Rightarrow \dfrac{- \ \text{x} + 6}{2(\text{x} - 3)^{2}} = \dfrac{6 - \text{x}}{2(\text{x} - 3)^{2}}$ Ans
Q8. Simplify: $\dfrac{(\text{x} + \text{y})^{2} - (\text{x} - \text{y})^{2}}{\text{x}^{2}\text{y} + \text{xy}^{2}}$
Show Answer
$\Rightarrow \dfrac{(\text{x} + \text{y})^{2} - (\text{x} - \text{y})^{2}}{\text{x}^{2}\text{y} + \text{xy}^{2}}$
$\Rightarrow \dfrac{(\text{x} + \text{y}) \ (\text{x} + \text{y}) - (\text{x} - \text{y}) \ (\text{x} - \text{y})}{\text{xy}(\text{x} + \text{y})}$
$\Rightarrow \dfrac{(\text{x}^{2} + 2\text{xy} + \text{y}^{2}) - (\text{x}^{2} - 2\text{xy} + \text{y}^{2})}{\text{xy}(\text{x} + \text{y})}$
$\Rightarrow \dfrac{\text{x}^{2} + 2\text{xy} + \text{y}^{2} - \text{x}^{2} + 2\text{xy} - \text{y}^{2}}{\text{xy}(\text{x} + \text{y})}$
$\Rightarrow \dfrac{4\text{xy}}{\text{xy}(\text{x} + \text{y})} = \dfrac{4}{\text{x} + \text{y}}$ Ans
$\Rightarrow \dfrac{(\text{x} + \text{y}) \ (\text{x} + \text{y}) - (\text{x} - \text{y}) \ (\text{x} - \text{y})}{\text{xy}(\text{x} + \text{y})}$
$\Rightarrow \dfrac{(\text{x}^{2} + 2\text{xy} + \text{y}^{2}) - (\text{x}^{2} - 2\text{xy} + \text{y}^{2})}{\text{xy}(\text{x} + \text{y})}$
$\Rightarrow \dfrac{\text{x}^{2} + 2\text{xy} + \text{y}^{2} - \text{x}^{2} + 2\text{xy} - \text{y}^{2}}{\text{xy}(\text{x} + \text{y})}$
$\Rightarrow \dfrac{4\text{xy}}{\text{xy}(\text{x} + \text{y})} = \dfrac{4}{\text{x} + \text{y}}$ Ans
Q9. Simplify: $\dfrac{1}{\text{x} - 3} \div \dfrac{3}{\text{x}^{2} - 9}$
Show Answer
$\Rightarrow \dfrac{1}{\text{x} - 3} \div \dfrac{3}{\text{x}^{2} - 9}$
$\Rightarrow \dfrac{1}{\text{x} - 3} \div \dfrac{3}{\text{x}^{2} - 3^{2}}$
$\Rightarrow \dfrac{1}{\text{x} - 3} \div \dfrac{3}{(\text{x} - 3)^{2}}$
$\Rightarrow \dfrac{1}{\text{x} - 3} \div \dfrac{3}{(\text{x} - 3) \ (\text{x} + 3)}$
$\Rightarrow \dfrac{1}{\text{x} - 3} \times \dfrac{(\text{x} - 3) \ (\text{x} + 3)}{3}$
$\Rightarrow \dfrac{\text{x} + 3}{3}$ Ans
$\Rightarrow \dfrac{1}{\text{x} - 3} \div \dfrac{3}{\text{x}^{2} - 3^{2}}$
$\Rightarrow \dfrac{1}{\text{x} - 3} \div \dfrac{3}{(\text{x} - 3)^{2}}$
$\Rightarrow \dfrac{1}{\text{x} - 3} \div \dfrac{3}{(\text{x} - 3) \ (\text{x} + 3)}$
$\Rightarrow \dfrac{1}{\text{x} - 3} \times \dfrac{(\text{x} - 3) \ (\text{x} + 3)}{3}$
$\Rightarrow \dfrac{\text{x} + 3}{3}$ Ans
Q10. Simplify: $\dfrac{\text{x} - 2}{\text{x} - 1} - \dfrac{3 - 3\text{x}}{\text{x}^{2} - 2\text{x} + 1}$
Show Answer
$\because \text{x}^{2} - 2\text{x} + 1 = (\text{x} - 1) \ (\text{x} - 1)$
$\therefore$ L.C.M. = $(\text{x} - 1) \ (\text{x} - 1) = (\text{x} - 1)^{2}$
$\therefore \dfrac{\text{x} - 2}{\text{x} - 1} - \dfrac{3 - 3\text{x}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{(\text{x} - 1) \ (\text{x} - 1) - 3 - 3\text{x}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{(\text{x}^{2} - 2\text{x} - \text{x} + 2) - 3 - 3\text{x}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{\text{x}^{2} - 3\text{x} + 2 - 3 - 3\text{x}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{\text{x}^{2} - 1}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{\text{x}^{2} - 1^{2}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{(\text{x} + 1) \ (\text{x} - 1)}{(\text{x} - 1) \ ((\text{x} - 1))} = \dfrac{\text{x} + 1}{\text{x} - 1}$ Ans
$\therefore$ L.C.M. = $(\text{x} - 1) \ (\text{x} - 1) = (\text{x} - 1)^{2}$
$\therefore \dfrac{\text{x} - 2}{\text{x} - 1} - \dfrac{3 - 3\text{x}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{(\text{x} - 1) \ (\text{x} - 1) - 3 - 3\text{x}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{(\text{x}^{2} - 2\text{x} - \text{x} + 2) - 3 - 3\text{x}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{\text{x}^{2} - 3\text{x} + 2 - 3 - 3\text{x}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{\text{x}^{2} - 1}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{\text{x}^{2} - 1^{2}}{(\text{x} - 1)^{2}}$
$\Rightarrow \dfrac{(\text{x} + 1) \ (\text{x} - 1)}{(\text{x} - 1) \ ((\text{x} - 1))} = \dfrac{\text{x} + 1}{\text{x} - 1}$ Ans
Q11. Simplify: $\dfrac{1}{\text{x} - 1} + \dfrac{2\text{x}}{1 - \text{x}}$
Show Answer
$\Rightarrow \dfrac{1}{\text{x} - 1} + \dfrac{2\text{x}}{1 - \text{x}}$
L.C.M. = $(\text{x} - 1) \ (1 - \text{x})$
$\Rightarrow \dfrac{1 - \text{x} + 2\text{x}(\text{x} - 1)}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1 - \text{x} + 2\text{x}^{2} - 2\text{x}}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1 - 3\text{x} + 2\text{x}^{2}}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1 - \text{x} - 2\text{x} + 2\text{x}^{2}}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1(1 - \text{x}) - 2\text{x}(1 - \text{x})}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{(1 - \text{x}) \ (1 - 2\text{x})}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1 - 2\text{x}}{\text{x} - 1}$ Ans
L.C.M. = $(\text{x} - 1) \ (1 - \text{x})$
$\Rightarrow \dfrac{1 - \text{x} + 2\text{x}(\text{x} - 1)}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1 - \text{x} + 2\text{x}^{2} - 2\text{x}}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1 - 3\text{x} + 2\text{x}^{2}}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1 - \text{x} - 2\text{x} + 2\text{x}^{2}}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1(1 - \text{x}) - 2\text{x}(1 - \text{x})}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{(1 - \text{x}) \ (1 - 2\text{x})}{(\text{x} - 1) \ (1 - \text{x})}$
$\Rightarrow \dfrac{1 - 2\text{x}}{\text{x} - 1}$ Ans
Q12. Simplify: $(1 + \dfrac{3}{\text{x}}) \ (\dfrac{9}{\text{x}^{2} - 9} + 1)$
Show Answer
$\Rightarrow (\dfrac{\text{x} + 3}{\text{x}}) \ (\dfrac{9 + \text{x}^{2} - 9}{\text{x}^{2} - 9})$
$\Rightarrow \dfrac{\text{x} + 3}{\text{x}} \times \dfrac{\text{x}^{2}}{\text{x}^{2} - 9}$
$\Rightarrow \dfrac{\text{x}^{2}(\text{x} + 3)}{\text{x}(\text{x}^{2} - 9)}$
$\Rightarrow \dfrac{\text{x}(\text{x} + 3)}{\text{x}^{2} - 9}$
$\Rightarrow \dfrac{\text{x}(\text{x} + 3)}{\text{x}^{2} - 3^{2}}$
$\Rightarrow \dfrac{\text{x}(\text{x} + 3)}{(\text{x} + 3) \ (\text{x} - 3)}$
$\Rightarrow \dfrac{\text{x}}{\text{x} - 3}$ Ans
$\Rightarrow \dfrac{\text{x} + 3}{\text{x}} \times \dfrac{\text{x}^{2}}{\text{x}^{2} - 9}$
$\Rightarrow \dfrac{\text{x}^{2}(\text{x} + 3)}{\text{x}(\text{x}^{2} - 9)}$
$\Rightarrow \dfrac{\text{x}(\text{x} + 3)}{\text{x}^{2} - 9}$
$\Rightarrow \dfrac{\text{x}(\text{x} + 3)}{\text{x}^{2} - 3^{2}}$
$\Rightarrow \dfrac{\text{x}(\text{x} + 3)}{(\text{x} + 3) \ (\text{x} - 3)}$
$\Rightarrow \dfrac{\text{x}}{\text{x} - 3}$ Ans
Q13. Simplify: $\dfrac{\text{a}^{2} + \text{a} + 1}{\text{a} + 1} + \dfrac{\text{a}^{2} - \text{a} + 1}{\text{a} - 1}$
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$\Rightarrow \dfrac{\text{a}^{2} + \text{a} + 1}{\text{a} + 1} + \dfrac{\text{a}^{2} - \text{a} + 1}{\text{a} - 1}$
$\because (\text{a} + 1) \ (\text{a} - 1) = \text{a}^{2} - \text{a} + \text{a} - 1 = \text{a}^{2} - 1$
$\therefore$ L.C.M. = $\text{a}^{2} - 1$
$\Rightarrow \dfrac{(\text{a} - 1) \ (\text{a}^{2} + \text{a} + 1) + (\text{a} + 1) \ (\text{a}^{2} - \text{a} + 1)}{\text{a}^{2} - 1}$
$\Rightarrow \dfrac{(\text{a}^{3} + \text{a}^{2} + \text{a} - \text{a}^{2} - \text{a} - 1) + (\text{a}^{3} - \text{a}^{2} + \text{a} + \text{a}^{2} - \text{a} + 1)}{\text{a}^{2} - 1}$
$\Rightarrow \dfrac{\text{a}^{3} - 1 + \text{a}^{3} + 1}{\text{a}^{2} - 1}$
$\Rightarrow \dfrac{2\text{a}^{3}}{\text{a}^{2} - 1}$ Ans
$\because (\text{a} + 1) \ (\text{a} - 1) = \text{a}^{2} - \text{a} + \text{a} - 1 = \text{a}^{2} - 1$
$\therefore$ L.C.M. = $\text{a}^{2} - 1$
$\Rightarrow \dfrac{(\text{a} - 1) \ (\text{a}^{2} + \text{a} + 1) + (\text{a} + 1) \ (\text{a}^{2} - \text{a} + 1)}{\text{a}^{2} - 1}$
$\Rightarrow \dfrac{(\text{a}^{3} + \text{a}^{2} + \text{a} - \text{a}^{2} - \text{a} - 1) + (\text{a}^{3} - \text{a}^{2} + \text{a} + \text{a}^{2} - \text{a} + 1)}{\text{a}^{2} - 1}$
$\Rightarrow \dfrac{\text{a}^{3} - 1 + \text{a}^{3} + 1}{\text{a}^{2} - 1}$
$\Rightarrow \dfrac{2\text{a}^{3}}{\text{a}^{2} - 1}$ Ans
Q14. Simplify: $\dfrac{1}{\text{a} + 2} - \dfrac{1}{\text{a} - 2} - \dfrac{4}{4 - \text{a}^{2}}$
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$\Rightarrow$ First we take: $\dfrac{1}{\text{a} + 2} - \dfrac{1}{\text{a} - 2}$
$\Rightarrow$ L.C.M. = $(\text{a} + 2) \ (\text{a} - 2) = (\text{a}^{2} - 4)$
$\Rightarrow \dfrac{\text{a} - 2 - (\text{a} + 2)}{\text{a}^{2} - 4}$
$\Rightarrow \dfrac{\text{a} - 2 - \text{a} - 2}{\text{a}^{2} - 4} = \dfrac{- \ 4}{\text{a}^{2} - 4}$
$\Rightarrow$ Now we take: $\dfrac{- \ 4}{\text{a}^{2} - 4} - \dfrac{4}{4 - \text{a}^{2}}$
$\Rightarrow \dfrac{- \ 4(4 - \text{a}^{2}) - 4(\text{a}^{2} - 4)}{(\text{a}^{2} - 4) \ (4 - \text{a}^{2})}$
$\Rightarrow \dfrac{- \ 16 + 4\text{a}^{2} - 4\text{a}^{2} + 16}{(\text{a}^{2} - 4) \ (4 - \text{a}^{2})}$
$\Rightarrow \dfrac{0}{(\text{a}^{2} - 4) \ (4 - \text{a}^{2})}= 0$ Ans
$\Rightarrow$ L.C.M. = $(\text{a} + 2) \ (\text{a} - 2) = (\text{a}^{2} - 4)$
$\Rightarrow \dfrac{\text{a} - 2 - (\text{a} + 2)}{\text{a}^{2} - 4}$
$\Rightarrow \dfrac{\text{a} - 2 - \text{a} - 2}{\text{a}^{2} - 4} = \dfrac{- \ 4}{\text{a}^{2} - 4}$
$\Rightarrow$ Now we take: $\dfrac{- \ 4}{\text{a}^{2} - 4} - \dfrac{4}{4 - \text{a}^{2}}$
$\Rightarrow \dfrac{- \ 4(4 - \text{a}^{2}) - 4(\text{a}^{2} - 4)}{(\text{a}^{2} - 4) \ (4 - \text{a}^{2})}$
$\Rightarrow \dfrac{- \ 16 + 4\text{a}^{2} - 4\text{a}^{2} + 16}{(\text{a}^{2} - 4) \ (4 - \text{a}^{2})}$
$\Rightarrow \dfrac{0}{(\text{a}^{2} - 4) \ (4 - \text{a}^{2})}= 0$ Ans
Q15. Simplify: $\left(\dfrac{\text{a}}{1 + \text{a}} + \dfrac{1 - \text{a}}{\text{a}}\right) \div \left(\dfrac{\text{a}}{1 - \text{a}} - \dfrac{1 + \text{a}}{\text{a}}\right)$
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$\Rightarrow \left(\dfrac{\text{a}}{1 + \text{a}} + \dfrac{1 - \text{a}}{\text{a}}\right) \div \left(\dfrac{\text{a}}{1 - \text{a}} - \dfrac{1 + \text{a}}{\text{a}}\right)$
$\Rightarrow \left(\dfrac{\text{a}(\text{a}) + (1 + \text{a}) \ (1 - \text{a})}{\text{a}(1 + \text{a})}\right) \div \left(\dfrac{\text{a}(\text{a}) - (1 - \text{a}) \ (1 + \text{a})}{\text{a}(1 - \text{a})}\right)$
$\Rightarrow \left(\dfrac{\text{a}^{2} + (1 - \text{a} + \text{a} - \text{a}^{2})}{\text{a}(1 + \text{a})}\right) \div \left(\dfrac{\text{a}^{2} - (1 + \text{a} - \text{a} - \text{a}^{2})}{\text{a}(1 - \text{a})}\right)$
$\Rightarrow \left(\dfrac{\text{a}^{2} + (1 - \text{a}^{2})}{\text{a}(1 + \text{a})}\right) \div \left(\dfrac{\text{a}^{2} - (1 - \text{a}^{2})}{\text{a}(1 - \text{a})}\right)$
$\Rightarrow \dfrac{\text{a}^{2} + 1 - \text{a}^{2}}{\text{a}(1 + \text{a})} \div \dfrac{\text{a}^{2} - 1 + \text{a}^{2}}{\text{a}(1 - \text{a})}$
$\Rightarrow \dfrac{1}{\text{a}(1 + \text{a})} \div \dfrac{2\text{a}^{2} - 1}{\text{a}(1 - \text{a})}$
$\Rightarrow \dfrac{1}{\text{a}(1 + \text{a})} \times \dfrac{\text{a}(1 - \text{a})}{2\text{a}^{2} - 1}$
$\Rightarrow \dfrac{1 - \text{a}}{(1 + \text{a}) \ (2\text{a}^{2} - 1)}$ Ans
$\Rightarrow \left(\dfrac{\text{a}(\text{a}) + (1 + \text{a}) \ (1 - \text{a})}{\text{a}(1 + \text{a})}\right) \div \left(\dfrac{\text{a}(\text{a}) - (1 - \text{a}) \ (1 + \text{a})}{\text{a}(1 - \text{a})}\right)$
$\Rightarrow \left(\dfrac{\text{a}^{2} + (1 - \text{a} + \text{a} - \text{a}^{2})}{\text{a}(1 + \text{a})}\right) \div \left(\dfrac{\text{a}^{2} - (1 + \text{a} - \text{a} - \text{a}^{2})}{\text{a}(1 - \text{a})}\right)$
$\Rightarrow \left(\dfrac{\text{a}^{2} + (1 - \text{a}^{2})}{\text{a}(1 + \text{a})}\right) \div \left(\dfrac{\text{a}^{2} - (1 - \text{a}^{2})}{\text{a}(1 - \text{a})}\right)$
$\Rightarrow \dfrac{\text{a}^{2} + 1 - \text{a}^{2}}{\text{a}(1 + \text{a})} \div \dfrac{\text{a}^{2} - 1 + \text{a}^{2}}{\text{a}(1 - \text{a})}$
$\Rightarrow \dfrac{1}{\text{a}(1 + \text{a})} \div \dfrac{2\text{a}^{2} - 1}{\text{a}(1 - \text{a})}$
$\Rightarrow \dfrac{1}{\text{a}(1 + \text{a})} \times \dfrac{\text{a}(1 - \text{a})}{2\text{a}^{2} - 1}$
$\Rightarrow \dfrac{1 - \text{a}}{(1 + \text{a}) \ (2\text{a}^{2} - 1)}$ Ans