question_answer 1)

                Use a suitable identity to get each of the following products: (i) \[(x+3)(x-3)\] (ii) \[(2y+5)(2y+5)\] (iii) \[(2a-7)(2a-7)\]  (iv) \[\left( 3a-\frac{1}{2} \right)\,\left( 3a-\frac{1}{2} \right)\] (v) \[(1.1\,m\,-\,0.4)\,(1.1\,m\,+\,0.4)\] (vi) \[({{a}^{2}}+{{b}^{2}})\,(-{{a}^{2}}+{{b}^{2}})\] (vii) \[(6x-7)\,(6x+7)\] (viii) \[(-a+c)\,(-a+c)\] (ix) \[\left( \frac{x}{2}+\frac{3y}{4} \right)\,\left( \frac{x}{2}\,+\frac{3y}{4} \right)\] (x) \[(7a-9b)\,(7a-9b)\].

Answer:

                (i) \[(x+3)\,(x+3)\]                 \[(x+3)\,(x+3)={{(x+3)}^{2}}\]                 \[{{(x)}^{2}}+2(x)\,(3)+{{(3)}^{2}}\]                         |Using Identity I \[={{x}^{2}}+6x+9\] (ii) \[(2y+5)\,(2y+5)\] \[(2y+5)\,(2y+5)\]\[={{(2y+5)}^{2}}\] \[{{(=2y)}^{2}}+2(2y)(5)+{{(5)}^{2}}\]     |Using Identity I \[=4{{y}^{2}}+20y+25\]  (iii) \[(2a-7)\,(2a-7)\] \[(2a-7)\,(2a-7)\] \[={{(2a-7)}^{2}}\] \[={{(2a)}^{2}}-2(2a)\,(7)+{{(7)}^{2}}\]   |Using Identity II \[=4{{a}^{2}}-28a+49\] (iv) \[\left( 3a-\frac{1}{2} \right)\,\left( 3a-\frac{1}{2} \right)\] \[\left( 3a-\frac{1}{2} \right)\,\left( 3a-\frac{1}{2} \right)\]\[={{\left( 3a-\frac{1}{2} \right)}^{2}}\] \[={{(3a)}^{2}}-2(3a)\,\left( \frac{1}{2} \right)+{{\left( \frac{1}{2} \right)}^{2}}\] |Using Identity II \[=9{{a}^{2}}-3a+\frac{1}{4}\] (v) \[(1.1m-0.4)\,(1.1\,m+0.4)\] \[(1.1m-0.4)\,(1.1\,m+0.4)\] \[={{(1.1\,m)}^{2}}-0.16\]                            |Using Identity III \[=1.21\,{{m}^{2}}-0.16\] (vi) \[({{a}^{2}}+{{b}^{2}})\,(-{{a}^{2}}+{{b}^{2}})\] \[({{a}^{2}}+{{b}^{2}})\,(-{{a}^{2}}+{{b}^{2}})\]\[=({{b}^{2}}+{{a}^{2}})\,({{b}^{2}}-{{a}^{2}})\] \[={{({{b}^{2}})}^{2}}\,-{{({{a}^{2}})}^{2}}\]                                         |Using Identity III \[={{b}^{4}}-{{a}^{4}}\] (vii) \[(6x-7)\,(6x+7)\] \[(6x-7)\,(6x+7)\]\[={{(6x)}^{2}}-{{(7)}^{2}}\]     |Using  Identity III \[=36{{x}^{2}}-49\] (viii) \[(-a+c)\,(-a+c)\] \[(-a+c)\,(-a+c)\]\[={{(-a+c)}^{2}}\] \[={{(c-a)}^{2}}\] \[={{c}^{2}}-2ca+{{a}^{2}}\]                                         |Using Identity II (ix) \[\left( \frac{x}{2}+\frac{3y}{4} \right)\,\left( \frac{x}{2}+\frac{3y}{4} \right)\] \[\left( \frac{x}{2}+\frac{3y}{4} \right)\,\left( \frac{x}{2}+\frac{3y}{4} \right)\] \[={{\left( \frac{x}{2} \right)}^{2}}+2\left( \frac{x}{2} \right)\,\left( \frac{3y}{4} \right)\,+{{\left( \frac{3y}{4} \right)}^{2}}\]    |Using Identity I \[=\frac{{{x}^{2}}}{4}+\frac{3xy}{4}\,+\frac{9{{y}^{2}}}{16}\] (x) \[(7a-9b)\,(7a-9b)\] \[(7a-9b)\,(7a-9b)\]\[={{(7a-9b)}^{2}}\] \[={{(7a)}^{2}}-2(7a)(9b)\,+{{(9b)}^{2}}\]             |Using Identity II \[=49{{a}^{2}}-126ab+81{{b}^{2}}\].