Algebra - Study Mode
[#286] If $$frac{1}{4} imes $$ $$frac{2}{6} imes $$ $$frac{3}{8} imes $$ $$frac{4}{{10}} imes $$ $$frac{5}{{12}} imes $$ . . . . . $$ imes frac{{31}}{{64}}$$ $$ = frac{1}{{{2^x}}}$$ xa0 then the value of x is?
Correct Answer
(C) 36
Explanation
Solution: $$frac{1}{4} imes $$ $$frac{2}{6} imes $$ $$frac{3}{8} imes $$ $$frac{4}{{10}} imes $$ $$frac{5}{{12}} imes $$ . . . . . $$ imes frac{{31}}{{64}}$$ $$ = frac{1}{{{2^x}}}$$ Denominator of first term is cut by the numerator of next terms, second term to the next one and so on. $$eqalign{
& Rightarrow {left( {frac{1}{2}}
ight)^{30}} imes {left( {frac{1}{2}}
ight)^6} = frac{1}{{{2^x}}} cr
& Rightarrow {left( {frac{1}{2}}
ight)^{30 + 6}} = frac{1}{{{2^x}}} cr
& Rightarrow frac{1}{{{2^{36}}}} = frac{1}{{{2^x}}} cr
& Rightarrow x = 36 cr} $$
[#287] If $$x = 3 + 2sqrt 2 { ext{,}}$$ xa0xa0 then the value of $$left( {sqrt x - frac{1}{{sqrt x }}}
ight){ ext{ is?}}$$
Correct Answer
(B) 2
Explanation
Solution: $$eqalign{
& x = 3 + 2sqrt 2 cr
& Rightarrow x = 2 + 1 + 2sqrt 2 cr
& Rightarrow x = {left( {sqrt 2 + 1}
ight)^2} cr
& Rightarrow sqrt x = sqrt 2 + 1 cr
& Rightarrow frac{1}{{sqrt x }} = frac{1}{{sqrt 2 + 1}} cr
& Rightarrow frac{1}{{sqrt x }} = frac{1}{{sqrt 2 + 1}} imes frac{{sqrt 2 - 1}}{{sqrt 2 - 1}} cr
& Rightarrow frac{1}{{sqrt x }} = sqrt 2 - 1 cr
& herefore sqrt x - frac{1}{{sqrt x }} cr
& = sqrt 2 + 1 - left( {sqrt 2 - 1}
ight) cr
& = sqrt 2 + 1 - sqrt 2 + 1 cr
& = 2 cr} $$
[#288] If p = 999, then the value of $$
oot 3 of {pleft( {{p^2} + 3p + 3}
ight) + 1} { ext{ is?}}$$
Correct Answer
(A) 1000
Explanation
Solution: $$eqalign{
& x08ecause p = 999 cr
&
oot 3 of {pleft( {{p^2} + 3p + 3}
ight) + 1} cr
& =
oot 3 of {{p^3} + 3{p^2} + 3p + 1} cr
& =
oot 3 of {{{left( {p + 1}
ight)}^3}} cr
& =
oot 3 of {{{left( {999 + 1}
ight)}^3}} cr
& =
oot 3 of {{{left( {1000}
ight)}^3}} cr
& = 1000 cr} $$
[#289] If $$frac{a}{b} = frac{7}{9},{ ext{ }}frac{b}{c} = frac{3}{5}{ ext{,}}$$ xa0xa0 then the value of a : b : c is?
Correct Answer
(A) 7 : 9 : 15
Explanation
Solution: $$eqalign{
& { ext{ }}frac{a}{b} = frac{7}{9},{ ext{ }}frac{b}{c} = frac{3}{5} cr
& herefore a,,,:,,,b,,,:,,,c cr
& ,,,,,7,,,:,,,9 cr
& ,,,,,,,,,,,,,,,{3_{ imes 3}}:,,,{5_{ imes 3}} cr
& overline {{ ext{ }},,{ ext{ }}7,,,:,,,,9,,:,,15{ ext{ }}} cr} $$
[#290] If a = 2, b = -3, then the value of 27a 3 - 54a 2 b + 36ab 2 - 8b 3 is?
Correct Answer
(D) 1728
Explanation
Solution: $$eqalign{
& a = 2,{ ext{ }}b = - 3 cr
& { ext{So,}}27{a^3} - 54{a^2}b + 36a{b^2} - 8{b^3} cr} $$ $$ = 27 imes {left( 2
ight)^3} - 54 imes {left( 2
ight)^2} imes left( { - 3}
ight) + $$ xa0 xa0 xa0 $$36 imes $$ $$left( 2
ight) imes $$ $${left( { - 3}
ight)^2}$$ $$ -, 8 imes $$ $${left( { - 3}
ight)^3}$$ $$eqalign{
& = 27 imes 8 - 54 imes 4 imes - 3 + 36 imes 2 imes 9 - 8 imes - 27 cr
& = 216 + 216 + 648 + 648 cr
& = 1728 cr} $$