| (i) Factorise: \[{{x}^{2}}+\frac{1}{{{x}^{2}}}-3\] |
| (ii) Find the greatest common factors of \[14{{x}^{2}}{{y}^{3}},21{{x}^{3}}{{y}^{2}}\] and \[35{{x}^{4}}{{y}^{5}}z\]. |
| (iii) Divide \[z(5{{z}^{2}}-80)\]by \[5z(z+4)\]. |
A)
(i) (ii) (iii) \[\left( x-\frac{1}{x} \right)\left( x-\frac{1}{x}-2 \right)\] \[7x{{y}^{2}}\] \[z-4\]
B)
(i) (ii) (iii) \[\left( x+\frac{1}{x} \right)\left( x+\frac{1}{x}+2 \right)\] \[7{{x}^{2}}y\] \[z-4\]
C)
(i) (ii) (iii) \[\left( x-\frac{1}{x}+1 \right)\left( x-\frac{1}{x}-1 \right)\] \[7{{x}^{2}}{{y}^{2}}\] \[z-4\]
D)
(i) (ii) (iii) \[\left( x-\frac{1}{x}-1 \right)\left( x+\frac{1}{x}+1 \right)\] \[7{{x}^{2}}{{y}^{2}}\] \[z-2\]
Correct Answer: C
Solution :
(i) We have, \[{{x}^{2}}+\frac{1}{{{x}^{2}}}-3\] \[={{x}^{2}}+\frac{1}{{{x}^{2}}}-2+2-3={{\left( x-\frac{1}{x} \right)}^{2}}-{{(1)}^{2}}\] \[=\left( x-\frac{1}{x}+1 \right)\left( x-\frac{1}{x}-1 \right)\] (ii) The greatest common factor of \[14{{x}^{2}}{{y}^{3}},21{{x}^{3}}{{y}^{2}}\] and \[35{{x}^{4}}{{y}^{5}}z\] is \[7{{x}^{2}}{{y}^{2}}\] (iii) \[z(5{{z}^{2}}-80)=5z({{z}^{2}}+16)=5z(z+4)(z-4)\] Thus, \[z(5{{z}^{2}}-80)\div 5z(z+4)=\frac{z(5{{z}^{2}}-80)}{5z(z+4)}\] \[=\frac{5z(z+4)(z-4)}{5z(z+4)}=z-4\]
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