Algebra - Study Mode
[#426] If 2, 0 is a solution of the linear equation 2x + 3y = k, then the value of k is?
Correct Answer
(A) 4
Explanation
Solution: $$eqalign{
& 2x + 3y = kleft( {2 = x,{ ext{ }}0 = y}
ight) cr
& herefore 2 imes 2 + 3 imes 0 = k cr
& Leftrightarrow k = 4 cr} $$
[#427] The graph of linear equation y = x passes throughout the point ?
Correct Answer
(C) $$left( {1,1}
ight)$$
Explanation
Solution: Correct option is C.( becasue x = y)
[#428] If $${left( {a + frac{1}{a}}
ight)^2} = 3{ ext{,}}$$ xa0xa0 then find the value of $${a^{30}}$$ + $${a^{24}}$$ + $${a^{18}}$$ + $${a^{12}}$$ + $${a^6}$$ + $$1$$ = ?
Correct Answer
(A) 0
Explanation
Solution: $$eqalign{
& {left( {a + frac{1}{a}}
ight)^2} = 3 cr
& Rightarrow a + frac{1}{a} = sqrt 3 cr
& { ext{Cube both sides}} cr
& Rightarrow {a^3} + frac{1}{{{a^3}}} + 3 imes a imes frac{1}{a}left( {a + frac{1}{a}}
ight) = {left( {sqrt 3 }
ight)^3} cr
& Rightarrow {a^3} + frac{1}{{{a^3}}} + 3left( {sqrt 3 }
ight) = 3sqrt 3 cr
& Rightarrow {a^3} + frac{1}{{{a^3}}} = 3sqrt 3 - 3sqrt 3 cr
& Rightarrow {a^3} + frac{1}{{{a^3}}} = 0 cr
& Rightarrow {a^6} + 1 = 0 cr
& herefore { ext{ }}{a^{30}} + {a^{24}} + {a^{18}} + {a^{12}} + {a^6} + 1 = ? cr
& Rightarrow { ext{ }}{a^{24}}left( {{a^6} + 1}
ight) + {a^{12}}left( {{a^6} + 1}
ight) + {a^6} + 1 cr
& Rightarrow {a^{24}}left( 0
ight) + {a^{12}}left( 0
ight) + 0 cr
& Rightarrow 0 cr} $$
[#429] If x : y = 3 : 5 and x - y = -2, then the value of x + y is?
Correct Answer
(A) 8
Explanation
Solution: $$eqalign{
& x:y = 3:5{ ext{ }} cr
& x - y = - 2 cr
& frac{x}{y} = frac{3}{5} cr
& x - y = 3 - 5 cr
& Leftrightarrow - 2 = - 2 cr
& x = 3,{ ext{ }}y = 5 cr
& x + y = 3 + 5 cr
& Leftrightarrow x + y = 8 cr} $$
[#430] If x = 1 + $$sqrt 2 $$xa0 + $$sqrt 3 $$xa0 and y = 1 + $$sqrt 2 $$xa0 - $$sqrt 3 { ext{,}}$$xa0 then the value of $$frac{{{x^2} + 4xy + {y^2}}}{{x + y}}$$ xa0 is?
Correct Answer
(D) 6
Explanation
Solution: $$eqalign{
& { ext{According to the question,}} cr
& x = 1 + sqrt 2 + sqrt 3 ,.....(i) cr
& y = 1 + sqrt 2 - sqrt 3 ,.....(ii) cr
& Rightarrow frac{{{x^2} + 4xy + {y^2}}}{{x + y}} cr
& Rightarrow frac{{{{left( {x + y}
ight)}^2} + 2xy}}{{x + y}} cr
& { ext{From equation (i)}} + { ext{(ii)}} cr
& Rightarrow x + y = 2 + 2sqrt 2 cr
& xy = {left( {1 + sqrt 2 }
ight)^2} - {left( {sqrt 3 }
ight)^2} cr
& Rightarrow xy = 3 + 2sqrt 2 - 3 cr
& Rightarrow xy = 2sqrt 2 cr
& { ext{So, }}frac{{{{left( {x + y}
ight)}^2} + 2xy}}{{x + y}} cr
& = frac{{{{left( {2 + 2sqrt 2 }
ight)}^2} + 2 imes 2sqrt 2 }}{{2 + 2sqrt 2 }} cr
& = frac{{4 + 8 + 8sqrt 2 + 4sqrt 2 }}{{2 + 2sqrt 2 }} cr
& = frac{{12 + 12sqrt 2 }}{{2 + 2sqrt 2 }} cr
& = frac{{12left( {1 + sqrt 2 }
ight)}}{{2left( {1 + sqrt 2 }
ight)}} cr
& = 6 cr} $$