{6^{2}+7^{2}+8^{2}+9^{2}+10^{2}}/\sqrt{7+4 \sqrt{3}-\sqrt{4+2 \sqrt{3}}is equal to

SSC CGL Tier II 25 Oct 2015 Paper 1

Question : 96 of 100

Marks: +1, -0

{6^{2}+7^{2}+8^{2}+9^{2}+10^{2}}/{√{7+4 √{3}}-√{4+2 √{3}}}

is equal to

Solution:

(1)

√{7+4 √{3}}=√{7+2 × 2 ×√{3}}

=√{4+3+2×2 ×√{3}}

=√{(2+√{3})^{2}}=2+√{3}

√{4+2 ×√{3} × 1}=√{3+1+2×√{3}× 1}

=√{(√{3}+1)^{2}}=√{3}+1

∴√{7+4 √{3}}-\√{4+2 √{3}}

=2+√{3}-√{3}-1=1

Again,

1^{2}+2^{2}+...+10^{2}

= {10(10+1)(20+1)}/{6}

= {10 × 11× 21}/{6}=385

1^{2}+2^{2}+...+5^{2}

= {5(5+1)(10+1)}/{6}

= {5 × 6 × 11}/{6}=55

[ {∵ 1^{2}+2^{2}+3^{2}+---n^{2}} {= {n(n+1)(2 n+1)}/{6}} ]

∴ 6^{2}+7^{2}+8^{2}+9^{2}+10^{2}

=385-55=330

∴

Expression

= {330}/{1}=330

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