Algebra - Study Mode
[#431] If $$x + frac{1}{x} = 3{ ext{,}}$$ xa0 where $$x
e 0{ ext{,}}$$ xa0then the value of $$frac{{{x^4} + 3{x^3} + 5{x^2} + 3x + 1}}{{{x^4} + 1}}$$ xa0 xa0 = ?
Correct Answer
(A) 3
Explanation
Solution: $$eqalign{
& x + frac{1}{x} = 3 cr
& Rightarrow {x^2} + 1 = 3x,.....(i) cr
& Rightarrow {left( {{x^2} + 1}
ight)^2} = {left( {3x}
ight)^2} cr
& Rightarrow {x^4} + 1 + 2{x^2} = 9{x^2} cr
& Rightarrow {x^4} + 1 = 7{x^2},.....(ii) cr
& herefore frac{{{x^4} + 3{x^3} + 5{x^2} + 3x + 1}}{{{x^4} + 1}} cr
& Rightarrow frac{{7{x^2} + 3{x^3} + 5{x^2} + 3x}}{{{x^4} + 1}} cr
& Rightarrow frac{{12{x^2} + 3{x^3} + 3x}}{{7{x^2}}} cr
& { ext{From equation (i)}} cr
& Rightarrow frac{{12x + 3left( {{x^2} + 1}
ight)}}{{7x}} cr
& Rightarrow frac{{12x + 3 imes 3x}}{{7x}} cr
& Rightarrow frac{{21x}}{{7x}} cr
& Rightarrow 3 cr} $$
[#432] If p(x + y) 2 = 5 and q(x - y) 2 = 3, then the simplified value of p 2 (x + y) 2 + 4pqxy - q 2 (x - y) 2 is?
Correct Answer
(A) 2(p + q)
Explanation
Solution: $$eqalign{
& p{left( {x + y}
ight)^2} = 5{ ext{ and }}q{left( {x - y}
ight)^2} = 3 cr
& { ext{Put the value of }}x = 2{ ext{ and }}y = 1 cr
& p{left( {2 + 1}
ight)^2} = 5 cr
& Leftrightarrow p = frac{5}{9} cr
& q{left( {2 - 1}
ight)^2} = 3 cr
& Leftrightarrow q = 3 cr
& {p^2}{left( {x + y}
ight)^2} + 4pqxy - {q^2}{left( {x - y}
ight)^2} cr} $$ $$ = {left( {frac{5}{9}}
ight)^2}{left( {2 + 1}
ight)^2} + 4 imes frac{5}{9} imes 3 imes $$ xa0 xa0 xa0$$2 imes $$ $$1 - $$ $${left( 3
ight)^2}$$ $${left( {2 - 1}
ight)^2}$$ $$eqalign{
& = frac{{25}}{{81}} imes 9 + frac{{40}}{3} - 9 cr
& = frac{{25}}{9} + frac{{40}}{3} - 9 cr
& = frac{{25 + 120 - 81}}{9} cr
& = frac{{64}}{9} cr
& { ext{Put the value of p and q in option (A)}} cr
& { ext{Option 1}} o 2left( {p + q}
ight) cr
& = 2left( {frac{5}{9} + 3}
ight) cr
& = 2 imes frac{{32}}{9} cr
& = frac{{64}}{9} cr
& { ext{Option A is satisfied}} cr
& { ext{So, }}2left( {p + q}
ight){ ext{ is answer}} cr} $$
[#433] If α and β are the roots of equation x 2 + αx + β = 0 then find α 3 + β 3 = ?
Correct Answer
(A) -7
Explanation
Solution: $$eqalign{
& { ext{ }}{x^2}{ ext{ + }}alpha x + x08eta = 0 cr
& { ext{Sum of root}} cr
& alpha + x08eta = frac{{ - alpha }}{1},......(i) cr
& alpha x08eta = x08eta ,......(ii) cr
& { ext{From (i) and (ii)}} cr
& { ext{Then, }}alpha = 1 cr
& { ext{Then, }}x08eta = - 2 cr
& { ext{Then value of }} cr
& Leftrightarrow {alpha ^3} + {x08eta ^3} cr
& = 1 + {left( { - 2}
ight)^3} cr
& = - 7 cr} $$
[#434] If $${x^2} + frac{1}{{{x^2}}} = 1{ ext{,}}$$ xa0 then the value of $${x^{102}}$$ $$ + $$ $${x^{96}}$$ $$ + $$ $${x^{90}}$$ $$ + $$ $${x^{84}}$$ $$ + $$ $${x^{78}}$$ $$ + $$ $${x^{72}}$$ $$ + $$ $$5$$ is?
Correct Answer
(B) 5
Explanation
Solution: $$eqalign{
& {x^2} + frac{1}{{{x^2}}} = 1 cr
& { ext{Then, }}{left( {x + frac{1}{x}}
ight)^2} = 1 + 2 cr
& Rightarrow x + frac{1}{x} = sqrt 3 cr
& Rightarrow {x^3} + frac{1}{{{x^3}}} cr
& = {left( {sqrt 3 }
ight)^3} - 3sqrt 3 cr
& = 3sqrt 3 - 3sqrt 3 cr
& = 0 cr
& { ext{Then,}} cr
& {x^{102}} + {x^{96}} + {x^{90}} + {x^{84}} + {x^{78}} + {x^{72}} + 5 cr} $$ $$ = {x^{96}}left( {{x^6} + 1}
ight) + $$ xa0xa0 $${x^{84}}left( {{x^6} + 1}
ight) + $$ xa0 $${x^{72}}left( {{x^6} + 1}
ight) + $$ xa0 $$5$$ $$ = 5$$
[#435] Find the value of a and b if (x - 1) and (x + 1) are factors of x 4 + ax 3 - 3x 2 + 2x + b = ?
Correct Answer
(C) -2, 2
Explanation
Solution: If (x - 1) and (x + 1) are the factors y equation then, $$eqalign{
& x - 1 = 0 cr
& x = 1 cr
& Rightarrow { ext{Put }}x = 1{ ext{, we get }} cr
& 1 + a - 3 + 2 + b = 0 cr
& a + b = 0,.....(i) cr
& Rightarrow x + 1 = 0 cr
& Rightarrow x = - 1 cr
& { ext{Put }}x = - 1,{ ext{we get }} cr
& 1 - a - 3 - 2 + b = 0 cr
& b - a = 4,.....(ii) cr
& { ext{After solving (i) & (ii),}} cr
& { ext{We get }} cr
& a = - 2,{ ext{ }}b = 2 cr} $$