Simplification - Study Mode

[#81] Given that $$sqrt {574.6} $$xa0 = 23.97, $$sqrt {5746} $$xa0 = 75.8 then $$sqrt {0.00005746} $$ xa0 = ?
Correct Answer

(D) 0.00758

Explanation

Solution: $$eqalign{
& { ext{According to question,}} cr
& sqrt {0.00005746} cr
& Rightarrow sqrt {frac{{5746}}{{100000000}}} cr
& Rightarrow frac{{75.8}}{{10000}} cr
& Rightarrow 0.00758 cr} $$

[#82] If $$a = frac{x}{{x + y}}$$ xa0 and $$b = frac{y}{{x - y}}{ ext{,}}$$ xa0 then $$frac{{ab}}{{a + b}}$$ xa0 is equal to = ?
Correct Answer

(A) $$frac{{xy}}{{{x^2} + {y^2}}}$$

Explanation

Solution: $$eqalign{
& frac{{ab}}{{a + b}} = frac{{frac{x}{{x + y}} imes frac{y}{{x - y}}}}{{frac{x}{{x + y}} + frac{y}{{x - y}}}} cr
& frac{{ab}}{{a + b}} = frac{{frac{{xy}}{{{x^2} - {y^2}}}}}{{frac{{xleft( {x - y}
ight) + yleft( {x + y}
ight)}}{{{x^2} - {y^2}}}}} cr
& frac{{ab}}{{a + b}} = frac{{xy}}{{{x^2} + {y^2}}} cr} $$

[#83] If $$frac{m}{n}{ ext{ = }}frac{4}{3}$$ xa0 and $$frac{r}{t}{ ext{ = }}frac{9}{{14}}{ ext{,}}$$ xa0 then the value of $$frac{{3mr - nt}}{{4nt - 7mr}}{ ext{ is}} = { ext{?}}$$
Correct Answer

(B) $$ - frac{{11}}{{14}}$$

Explanation

Solution: $$eqalign{
& frac{m}{n}{ ext{ = }}frac{4}{3}{ ext{ and }}frac{r}{t}{ ext{ = }}frac{9}{{14}} cr
& Rightarrow frac{{mr}}{{nt}} = frac{4}{3} imes frac{9}{{14}} cr
& ,,,,,,,,,,,,,,,,, = frac{6}{7} cr
& herefore frac{{3mr - nt}}{{4nt - 7mr}}{ ext{ }} cr
& = frac{{3frac{{mr}}{{nt}} - 1}}{{4 - 7frac{{mr}}{{nt}}}} cr
& = frac{{3 imes frac{6}{7} - 1}}{{4 - 7 imes frac{6}{7}}} cr
& = frac{{frac{{18}}{7} - 1}}{{4 - 6}} cr
& = frac{{11}}{7} imes left( { - frac{1}{2}}
ight) cr
& = - frac{{11}}{{14}} cr} $$

[#84] $$sqrt {{{left( {0.798}
ight)}^2} + 0.404 imes 0.798 + {{left( {0.202}
ight)}^2}} $$ xa0 xa0 xa0 $$ + 1$$ $$ = ?$$
Correct Answer

(B) 2

Explanation

Solution: According to question, $$sqrt {{{left( {0.798}
ight)}^2} + 0.404 imes 0.798 + {{left( {0.202}
ight)}^2}} + 1,$$ $$ Rightarrow sqrt {{{left( {0.798}
ight)}^2} + {{left( {0.202}
ight)}^2} + 2 imes 0.202 imes 0.798,} + 1,$$ $$ Rightarrow sqrt {{{left( {0.798 + 0.202}
ight)}^2}} + 1,,left[ {x08ecause {{left( {a + b}
ight)}^2} = {a^2} + {b^2} + 2ab}
ight]$$ $$eqalign{
& Rightarrow sqrt {{{left( 1
ight)}^2}} + 1 cr
& Rightarrow 1 + 1 cr
& = 2 cr} $$

[#85] $$frac{2}{{2 + frac{2}{{3 + frac{2}{{3 + frac{2}{3}}}}} imes 0.39}}$$ xa0 xa0 is simplified to = ?
Correct Answer

(D) None of these

Explanation

Solution: $$eqalign{
& { ext{According to question,}} cr
& ,,,,,frac{2}{{2 + frac{2}{{3 + frac{2}{{3 + frac{2}{3}}}}} imes 0.39}} cr
& = frac{2}{{2 + frac{2}{{3 + frac{6}{{11}}}} imes 0.39}} cr
& = frac{2}{{2 + frac{{22}}{{39}} imes 0.39}} cr
& = frac{2}{{2 + frac{{22}}{{100}}}} cr
& = frac{{200}}{{222}} cr
& = frac{{100}}{{111}} cr} $$