Simplification - Study Mode

[#146] $$left( {x + frac{1}{x}}
ight)$$ $$left( {x - frac{1}{x}}
ight)$$ $$left( {{x^2} + frac{1}{{{x^2}}} - 1}
ight)$$xa0 $$left( {{x^2} + frac{1}{{{x^2}}} + 1}
ight)$$ xa0 is equal to ?
Correct Answer

(A) $${x^6} - frac{1}{{{x^6}}}$$

Explanation

Solution: Given expression, $$left[ {left( {x + frac{1}{x}}
ight)left( {{x^2} + frac{1}{{{x^2}}} - x.frac{1}{x}}
ight)}
ight]$$ xa0 xa0 $$left[ {left( {x - frac{1}{x}}
ight)left( {{x^2} + frac{1}{{{x^2}}} + x.frac{1}{x}}
ight)}
ight]$$ $$eqalign{
& = left( {{x^3} + frac{1}{{{x^3}}}}
ight)left( {{x^3} - frac{1}{{{x^3}}}}
ight) cr
& = {x^6} - frac{1}{{{x^6}}} cr} $$

[#147] If $$left( {x + frac{1}{x}}
ight){ ext{ = 2,}}$$ xa0 xa0then $$left( {x - frac{1}{x}}
ight)$$ xa0 is equal to = ?
Correct Answer

(A) 0

Explanation

Solution: $$eqalign{
& left( {x + frac{1}{x}}
ight){ ext{ = 2}} cr
& Rightarrow {left( {x + frac{1}{x}}
ight)^2}{ ext{ = }}{{ ext{2}}^2} cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} + 2 = 4 cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} = 2 cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} - 2.x.frac{1}{x} cr
& ,,,,,,,, = 2 - 2 = 0 cr
& Rightarrow {left( {x - frac{1}{x}}
ight)^2}{ ext{ = 0}} cr
& Rightarrow x - frac{1}{x} = 0 cr} $$

[#148] If 1 2 + 2 2 + 3 2 + . . . . . + p 2 = $$left[ {frac{{{ ext{p}}left( {{ ext{p}} + 1}
ight)left( {2{ ext{p}} + 1}
ight)}}{6}}
ight]{ ext{,}}$$ xa0 xa0 then 1 2 + 3 2 + 5 2 + . . . . . + 17 2 is = ?
Correct Answer

(A) 969

Explanation

Solution: 1 2 + 3 2 + 5 2 + . . . . . + 17 2 = (1 2 + 2 2 + 3 2 + 4 2 + . . . . . + 17 2 ) - (2 2 + 4 2 + . . . . . + 16 2 ) = (1 2 + 2 2 + 3 2 + 4 2 + . . . . . + 17 2 ) - 2 2 (1 2 + 2 2 + . . . . . + 8 2 ) Using given formula $$eqalign{
& left[ {frac{{pleft( {p + 1}
ight)left( {2p + 1}
ight)}}{6}}
ight] cr
& = frac{{17 imes 18 imes 35}}{6} - 4left[ {frac{{8 imes 9 imes 17}}{6}}
ight] cr
& = 1785 - 4 imes 204 cr
& = 1785 - 816 cr
& = 969 cr} $$

[#149] The simplest value of $$left( {frac{1}{{sqrt 9 - sqrt 8 }} - frac{1}{{sqrt 8 - sqrt 7 }} + frac{1}{{sqrt 7 - sqrt 6 }} - frac{1}{{sqrt 6 - sqrt 5 }}}
ight)$$ xa0 xa0 xa0 xa0xa0 is = ?
Correct Answer

(A) $$3 - sqrt 5 $$

Explanation

Solution: $$left( {frac{1}{{sqrt 9 - sqrt 8 }} - frac{1}{{sqrt 8 - sqrt 7 }} + frac{1}{{sqrt 7 - sqrt 6 }} - frac{1}{{sqrt 6 - sqrt 5 }}}
ight)$$ $$eqalign{
& = sqrt 9 + sqrt 8 - sqrt 8 - sqrt 7 + sqrt 7 + sqrt 6 - sqrt 6 - sqrt 5 cr
& = sqrt 9 - sqrt 5 cr
& = 3 - sqrt 5 cr} $$

[#150] The simplified value of
$$frac{{sqrt {32} + sqrt {48} }}{{sqrt 8 + sqrt {12} }}{ ext{ is = ?}}$$
Correct Answer

(B) 2

Explanation

Solution: $$eqalign{
& { ext{According to question,}} cr
& frac{{sqrt {32} + sqrt {48} }}{{sqrt 8 + sqrt {12} }} cr
& = frac{{sqrt {4 imes 4 imes 2} + sqrt {4 imes 4 imes 3} }}{{sqrt {2 imes 2 imes 2} + sqrt {2 imes 2 imes 3} }} cr
& = frac{{4sqrt 2 + 4sqrt 3 }}{{2sqrt 2 + 2sqrt 3 }} cr
& = frac{{4left( {sqrt 2 + sqrt 3 }
ight)}}{{2left( {sqrt 2 + sqrt 3 }
ight)}} cr
& = frac{4}{2} cr
& = 2 cr} $$