Question

If p=1/2√3 and q= 1/√3+2 then the value of 7p^2+11 pq-7p2 equals?

Answer 1

/* There is a mistake in the question. It may be like this */

$i) p = \frac{1}{(2-\sqrt{3})}$

$\implies p = \frac{(2+\sqrt{3})}{(2-\sqrt{3})(2-\sqrt{3})}$

$\implies p = \frac{(2+\sqrt{3})}{2^{2}-(\sqrt{3})^{2}}$

$\implies p = \frac{(2+\sqrt{3})}{4-3}$

$\implies p = 2 + \sqrt{3} \: --(1)$

$ii) q = \frac{1}{(2+\sqrt{3})}$

$\implies q = \frac{(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}$

$\implies q = \frac{(2-\sqrt{3})}{2^{2}-(\sqrt{3})^{2}}$

$\implies q = \frac{(2-\sqrt{3})}{4-3}$

$\implies q = 2 -\sqrt{3} \: --(2)$

$iii ) p+q$

$= 2 + \sqrt{3} + 2 -\sqrt{3}$

$= 4 \: --(3)$

$iv ) p-q$

$= 2 + \sqrt{3} -( 2 -\sqrt{3})$

$= 2 + \sqrt{3} - 2 +\sqrt{3}$

$= 2\sqrt{3} \: --(4)$

$v) pq = ( 2 + \sqrt{3})( 2 - \sqrt{3})$

$= 2^{2} - (\sqrt{3})^{2}$

$= 4 - 3$

$= 1\: ---(5)$

$\red{ Value \: of \: 7p^{2} + 11pq - 7q^{2} }$

$= 7(p^{2} - q^{2}) + 11pq$

$= 7(p+q)(p-q) + 11pq$

$= 7 \times 4 \times 2\sqrt{3} + 11 \times 1$

$= 56\sqrt{3} + 11$

Therefore.,

$\red{ Value \: of \: 7p^{2} + 11pq - 7q^{2} }$

$\green {= 56\sqrt{3} + 11 }$

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