Simplification - Study Mode
[#1] If x = y = 2z and xyz = 256, then x = ?
Correct Answer
(C) 8
Explanation
Solution: xyz = 256 ⇒ (2z) (2z) z = 256 ⇒ 4z 3 = 256 ⇒ z 3 = 64 ⇒ z = 4 ∴ x = 2z = (2 × 4) = 8
[#2] The value of $$left( {1 - frac{1}{{{3^2}}}}
ight)$$ $$left( {1 - frac{1}{{{4^2}}}}
ight)$$ $$left( {1 - frac{1}{{{5^2}}}}
ight)$$ xa0 . . . . . $$left( {1 - frac{1}{{{{11}^2}}}}
ight)$$ $$left( {1 - frac{1}{{{{12}^2}}}}
ight)$$ xa0 $$ = ?$$
Correct Answer
(C) $$frac{{13}}{{18}}$$
Explanation
Solution: $$left( {1 - frac{1}{{{3^2}}}}
ight)$$ $$left( {1 - frac{1}{{{4^2}}}}
ight)$$ $$left( {1 - frac{1}{{{5^2}}}}
ight)$$ xa0 . . . . . $$left( {1 - frac{1}{{{{11}^2}}}}
ight)$$ $$left( {1 - frac{1}{{{{12}^2}}}}
ight)$$ $$ = left( {frac{{{3^2} - 1}}{{{3^2}}}}
ight)left( {frac{{{4^2} - 1}}{{{4^2}}}}
ight)left( {frac{{{5^2} - 1}}{{{5^2}}}}
ight)....$$ xa0 xa0 xa0 $$left( {frac{{{{11}^2} - 1}}{{{{11}^2}}}}
ight)$$ $$left( {frac{{{{12}^2} - 1}}{{{{12}^2}}}}
ight)$$ $$ = left[ {frac{{left( {3 + 1}
ight)left( {3 - 1}
ight)}}{{left( {2 + 1}
ight)left( {4 - 1}
ight)}}}
ight]$$ xa0 $$left[ {frac{{left( {4 + 1}
ight)left( {4 - 1}
ight)}}{{left( {3 + 1}
ight)left( {5 - 1}
ight)}}}
ight]$$ xa0 $$left[ {frac{{left( {5 + 1}
ight)left( {5 - 1}
ight)}}{{left( {4 + 1}
ight)left( {6 - 1}
ight)}}}
ight]$$ xa0 . . . . . $$left[ {frac{{left( {11 + 1}
ight)left( {11 - 1}
ight)}}{{left( {10 + 1}
ight)left( {12 - 1}
ight)}}}
ight]$$ xa0 $$left[ {frac{{left( {12 + 1}
ight)left( {12 - 1}
ight)}}{{left( {11 + 1}
ight)left( {13 - 1}
ight)}}}
ight]$$ $$eqalign{
& = frac{{left( {3 - 1}
ight)}}{{left( {2 + 1}
ight)}} imes frac{{left( {12 + 1}
ight)}}{{left( {13 - 1}
ight)}} cr
& = frac{2}{3} imes frac{{13}}{{12}} cr
& = frac{{13}}{{18}} cr} $$
[#3] $$frac{{{{left( {7.5}
ight)}^3} + 1}}{{{{left( {7.5}
ight)}^2} - 6.5}}$$ xa0 xa0is equal to = ?
Correct Answer
(D) 8.5
Explanation
Solution: $$eqalign{
& { ext{According to question,}} cr
& Rightarrow frac{{{{left( {7.5}
ight)}^3} + 1}}{{{{left( {7.5}
ight)}^2} - 6.5}},,,,,,, cr
& ,,,,,,,,,,,left[ { herefore {a^3} + {b^3} = left( {a + b}
ight)left( {{a^2} + {b^2} - ab}
ight)}
ight] cr
& Rightarrow frac{{left( {7.5 + 1}
ight)left[ {{{left( {7.5}
ight)}^2} + 1 - 7.5 imes 1}
ight]}}{{{{left( {7.5}
ight)}^2} - 7.5 imes 1 + {1^2}}} cr
& Rightarrow 8.5 cr} $$
[#4] Given that $$sqrt {13} $$ = 3.6 and $$sqrt {130} $$xa0 = 11.4, then the value of $$sqrt {13} $$ + $$sqrt {1300} $$xa0 + $$sqrt {0.013} $$ xa0 is equal to = ?
Correct Answer
(C) 39.714
Explanation
Solution: $$eqalign{
& { ext{According to question}} cr
& sqrt {13} { ext{ + }}sqrt {1300} { ext{ + }}sqrt {0.013} cr
& = 3.6 + 10sqrt {13} + sqrt {frac{{130}}{{10000}}} cr
& = 3.6 + 10 imes 3.6 + frac{{11.4}}{{100}} cr
& = 39.714 cr} $$
[#5] Find the sum : $$frac{1}{2} + $$ $$frac{1}{6} + $$ $$frac{1}{{12}} + $$ $$frac{1}{{20}} + $$ $$frac{1}{{30}} + $$ $$frac{1}{{42}} + $$ $$frac{1}{{56}} + $$ $$frac{1}{{72}} + $$ $$frac{1}{{90}} + $$ $$frac{1}{{110}} + $$ $$frac{1}{{132}}$$ $$ = ?$$
Correct Answer
(B) $$frac{{11}}{{12}}$$
Explanation
Solution: Given expression, $${ ext{ = }}left( {1 - frac{1}{2}}
ight)$$ xa0 $$ + $$ $$left( {frac{1}{2} - frac{1}{3}}
ight)$$ xa0 $$ + $$ $$left( {frac{1}{3} - frac{1}{4}}
ight)$$ xa0 $$ + $$ $$left( {frac{1}{4} - frac{1}{5}}
ight)$$ xa0 $$ + $$ . . . . . $$ + $$ $$left( {frac{1}{{11}} - frac{1}{{12}}}
ight)$$ $$eqalign{
& = left( {1 - frac{1}{{12}}}
ight) cr
& = frac{{11}}{{12}} cr} $$