Algebra - Study Mode

[#341] If $$a - frac{1}{{a - 3}} = 5{ ext{,}}$$ xa0 then the value of $${left( {a - 3}
ight)^3}$$ xa0 - $$frac{1}{{{{left( {a - 3}
ight)}^3}}} = ?$$
Correct Answer

(B) 14

Explanation

Solution: $$eqalign{
& a - frac{1}{{a - 3}} = 5 cr
& Rightarrow a - 3 - frac{1}{{a - 3}} = 5 - 3 cr
& Rightarrow left( {a - 3}
ight) - frac{1}{{left( {a - 3}
ight)}} = 2 cr
& { ext{Cubing both sides}} cr
& Rightarrow {left[ {left( {a - 3}
ight) - frac{1}{{left( {a - 3}
ight)}}}
ight]^3} = {left( 2
ight)^3} cr} $$ $$ Rightarrow {left( {a - 3}
ight)^3} - frac{1}{{{{left( {a - 3}
ight)}^3}}} - 3 imes {left( {a - 3}
ight)} imes $$ xa0 xa0 xa0 $$frac{1}{{{{left( {a - 3}
ight)}}}}$$ $$left[ {left( {a - 3}
ight) - frac{1}{{left( {a - 3}
ight)}}}
ight]$$ xa0xa0 $$ = 8$$ $$eqalign{
& Rightarrow {left( {a - 3}
ight)^3} - frac{1}{{{{left( {a - 3}
ight)}^3}}} - 3left( 2
ight) = 8 cr
& Rightarrow {left( {a - 3}
ight)^3} - frac{1}{{{{left( {a - 3}
ight)}^3}}} = 8 + 6 cr
& Rightarrow {left( {a - 3}
ight)^3} - frac{1}{{{{left( {a - 3}
ight)}^3}}} = 14 cr} $$

[#342] (3x - 2y) : (2x + 3y) = 5 : 6, then one of the value of $${left( {frac{{
oot 3 of x +
oot 3 of y }}{{
oot 3 of x -
oot 3 of y }}}
ight)^2}$$ xa0 is?
Correct Answer

(D) 25

Explanation

Solution: $$eqalign{
& frac{{left( {3x - 2y}
ight)}}{{left( {2x + 3y}
ight)}} = frac{5}{6} cr
& Rightarrow 18x - 12y = 10x + 15y cr
& Rightarrow 8x = 27y cr
& Rightarrow frac{x}{y} = frac{{27}}{8} cr
& herefore {left( {frac{{
oot 3 of x +
oot 3 of y }}{{
oot 3 of x -
oot 3 of y }}}
ight)^2} cr
& Rightarrow {left( {frac{{
oot 3 of {27} +
oot 3 of 8 }}{{
oot 3 of {27} -
oot 3 of 8 }}}
ight)^2} cr
& Rightarrow {left( {frac{{3 + 2}}{{3 - 2}}}
ight)^2} cr
& Rightarrow {left( 5
ight)^2} cr
& Rightarrow 25 cr} $$

[#343] If $${ ext{ }}x - sqrt 3 - sqrt 2 = 0$$ xa0xa0 and $$y - sqrt 3 + sqrt 2 { ext{,}}$$ xa0xa0 then the value of $$left( {{x^3} - 20sqrt 2 }
ight) - $$ xa0 $$left( {{y^3} + 2sqrt 2 }
ight)?$$
Correct Answer

(D) 0

Explanation

Solution: $$eqalign{
& { ext{According to the question,}} cr
& x = sqrt 3 + sqrt 2 cr
& y = sqrt 3 - sqrt 2 cr
& left( {{x^3} - 20sqrt 2 }
ight) - left( {{y^3} + 2sqrt 2 }
ight) cr
& = left[ {{{left( {sqrt 3 + sqrt 2 }
ight)}^3} - 20sqrt 2 - {{left( {sqrt 3 - sqrt 2 }
ight)}^3} - 2sqrt 2 }
ight] cr} $$ $$ = 3sqrt 3 + 2sqrt 2 + 9sqrt 2 + 6sqrt 3 , - $$ xa0 xa0 xa0 $$20sqrt 2 , - $$xa0 $$3sqrt 3 ,, + $$xa0 $$2sqrt 2,, + $$xa0 $$9sqrt 2 , - $$xa0 $$6sqrt 3, - $$xa0 $$2sqrt 2 $$ $$eqalign{
& = 9sqrt 3 - 9sqrt 2 - 9sqrt 3 + 9sqrt 2 cr
& = 0 cr} $$

[#344] 3(a 2 + b 2 + c 2 ) = (a + b + c) 2 then the relation between a, b and c is ?
Correct Answer

(D) a = b = c

Explanation

Solution: Always do these types of question with the help of, $$eqalign{
& { ext{Put }}a = b = c = 1 cr
& { ext{3}}left( {{a^2} + {b^2} + {c^2}}
ight) = {left( {a + b + c}
ight)^2} cr
& 3 = 3{ ext{ Satisfied}} cr
& { ext{So, this is answer}} o a = b = c cr} $$

[#345] If $$m = sqrt {5 + sqrt {5 + sqrt {5.....} } } $$ xa0 xa0xa0 and $$n = sqrt {5 - sqrt {5 - sqrt {5.....} } } $$ xa0 xa0xa0 then among the following the relation between m & n holds is?
Correct Answer

(D) m - n - 1 = 0

Explanation

Solution: $$eqalign{
& { ext{Let }}m = sqrt {5 + sqrt {5 + sqrt 5 } } cr
& { ext{Factor}} = left( a
ight) imes left( {a + 1}
ight) cr
& { ext{Here }}m = a + 1 cr
& Rightarrow m - 1 = a,.........(i) cr
& { ext{Let }}n = sqrt {5 - sqrt {5 - sqrt 5 } } cr
& { ext{Factor}} = left( a
ight) imes left( {a + 1}
ight) cr
& { ext{Here }}n = a,.........(ii) cr
& { ext{From (i) & (ii)}} cr
& Leftrightarrow m - 1 = n cr
& Leftrightarrow m - n - 1 = 0 cr} $$