Algebra - Study Mode
[#436] Find the minimum value of x which the expression x 3 - 7x 2 + 11x - 5 ≥ 0.
Correct Answer
(C) 1
Explanation
Solution: $$eqalign{
& {x^3} - 7{x^2} + 11x - 5 geqslant 0 cr
& Rightarrow {x^3} - 5{x^2} - 2{x^2} + 10x + x - 5 geqslant 0 cr
& Rightarrow {x^2}left( {x - 5}
ight) - 2xleft( {x - 5}
ight) + 1left( {x - 5}
ight) geqslant 0 cr
& Rightarrow left( {x - 5}
ight)left( {{x^2} - 2x + 1}
ight) geqslant 0 cr
& Rightarrow left( {x - 5}
ight){left( {x - 1}
ight)^2} geqslant 0 cr
& Rightarrow left( {x - 5}
ight)left( {x - 1}
ight)left( {x - 1}
ight) geqslant 0 cr
& { ext{So, }}x = 1& 5 cr} $$ Equation satisfies at both the values, but the minimum value of these two x = 1
[#437] If a + b + c = 26 and ab + bc + ca = 109, find the value of a 2 + b 2 + c 2 = ?
Correct Answer
(A) 458
Explanation
Solution: $$eqalign{
& {left( {a + b + c}
ight)^2} = { ext{ }}{a^2} + {b^2} + {c^2} + 2left( {ab + bc + ca}
ight) cr
& Rightarrow {left( {26}
ight)^2} = { ext{ }}{a^2} + {b^2} + {c^2} + 2left( {109}
ight) cr
& Rightarrow {a^2} + {b^2} + {c^2} = 676 - 218 cr
& Rightarrow {a^2} + {b^2} + {c^2} = 458 cr} $$
[#438] If $$x = 3 + 2sqrt 2 { ext{,}}$$ xa0 then the value of $${x^2}{ ext{ + }}frac{1}{{{x^2}}}{ ext{ is?}}$$
Correct Answer
(D) 34
Explanation
Solution: $$eqalign{
& { ext{ }}x = 3 + 2sqrt 2 cr
& Rightarrow {x^2} = {left( {3 + 2sqrt 2 }
ight)^2} cr
& left( {{ ext{Squaring both sides}}}
ight) cr
& Rightarrow {x^2} = 9 + 8 + 12sqrt 2 cr
& Rightarrow {x^2} = 17 + 12sqrt 2 cr
& Rightarrow frac{1}{{{x^2}}} = frac{1}{{17 + 12sqrt 2 }} imes frac{{17 - 12sqrt 2 }}{{17 - 12sqrt 2 }} cr
& Rightarrow frac{1}{{{x^2}}} = 17 - 12sqrt 2 cr
& herefore { ext{ }}{x^2}{ ext{ + }}frac{1}{{{x^2}}} cr
& = 17 + 12sqrt 2 + 17 - 12sqrt 2 cr
& = 34 cr} $$
[#439] If $$xleft( {3 - frac{2}{x}}
ight) = frac{3}{x}{ ext{,}}$$ xa0xa0 then the value of $${x^2}{ ext{ + }}frac{1}{{{x^2}}}$$ xa0 is?
Correct Answer
(B) $${ ext{2}}frac{4}{9}$$
Explanation
Solution: $$eqalign{
& xleft( {3 - frac{2}{x}}
ight) = frac{3}{x} cr
& Rightarrow 3x - 2 = frac{3}{x} cr
& Rightarrow 3x - frac{3}{x} = 2 cr
& Rightarrow 3left( {x - frac{1}{x}}
ight) = 2 cr
& Rightarrow x - frac{1}{x} = frac{2}{3} cr
& left( {{ ext{Squaring both sides}}}
ight) cr
& Rightarrow {x^2}{ ext{ + }}frac{1}{{{x^2}}} - 2 = frac{4}{9} cr
& Rightarrow {x^2}{ ext{ + }}frac{1}{{{x^2}}} = frac{4}{9} + 2 cr
& Rightarrow {x^2}{ ext{ + }}frac{1}{{{x^2}}} = 2frac{4}{9} cr} $$
[#440] If x 2 - 3x + 1 = 0, then the value of $${x^2} + x + frac{1}{x} + frac{1}{{{x^2}}}$$ xa0xa0 is?
Correct Answer
(A) 10
Explanation
Solution: $$eqalign{
& {x^2} - 3x + 1 = 0 cr
& Rightarrow {x^2} + 1 = 3x cr
& Rightarrow x + frac{1}{x} = 3 cr
& { ext{Squaring both sides}} cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} + 2 = 9 cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} = 7 cr
& herefore {x^2} + x + frac{1}{x} + frac{1}{{{x^2}}} cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} + x + frac{1}{x} cr
& Rightarrow 7 + 3 cr
& Rightarrow 10 cr} $$