Algebra - Study Mode

[#356] If x - y = 2, xy = 24, then the value of (x 2 + y 2 ) is?
Correct Answer

(D) 52

Explanation

Solution: $$eqalign{
& { ext{Given,}} cr
& x - y = 2{ ext{ and }}xy = 24 cr
& { ext{By squaring}} cr
& Rightarrow {x^2} + {y^2} - 2xy = 4 cr
& Rightarrow {x^2} + {y^2} - 2 imes 24 = 4 cr
& Rightarrow {x^2} + {y^2} = 4 + 48 cr
& Rightarrow {x^2} + {y^2} = 52 cr} $$

[#357] If the expression $$frac{{{x^2}}}{{{y^2}}} + tx + frac{{{y^2}}}{4}$$ xa0 is a perfect square, then the value of t is?
Correct Answer

(A) ±1

Explanation

Solution: $$frac{{{x^2}}}{{{y^2}}} + tx + frac{{{y^2}}}{4}left( {{ ext{ Given}}}
ight)$$ To make it a perfect square it should be in the form $$eqalign{
& {{ ext{A}}^2} pm 2{ ext{AB}} + {{ ext{B}}^2} = {left( {{ ext{A}} pm { ext{B}}}
ight)^2} cr
& = {left( {frac{x}{y}}
ight)^2} pm tx + {left( {frac{y}{2}}
ight)^2} cr
& = {{ ext{A}}^2} pm 2{ ext{AB}} + {{ ext{B}}^2} cr
& { ext{A}} = frac{x}{y}{ ext{, B}} = frac{y}{2},,& ,,{ ext{2AB}} = tx cr
& { ext{So, }}tx = pm 2 imes frac{x}{y} imes frac{y}{2} cr
& Rightarrow tx = pm x cr
& Rightarrow t = pm 1 cr} $$

[#358] The factors of (a 2 + 4b 2 + 4b - 4ab - 2a - 8) are?
Correct Answer

(A) (a - 2b - 4)(a - 2b + 2)

Explanation

Solution: $$eqalign{
& {a^2} + 4{b^2} + 4b - 4ab - 2a - 8 cr
& = {a^2} - 4ab + 4{b^2} - 2a + 4b - 8 cr
& = {left( {a - 2b}
ight)^2} - 2left( {a - 2b}
ight) - 8 cr
& { ext{Put }} t = a - 2b cr
& = {t^2} - 2t - 8 cr
& = {t^2} - 4t + 2t - 8 cr
& = tleft( {t - 4}
ight) + 2left( {t - 4}
ight) cr
& = left( {t + 2}
ight)left( {t - 4}
ight) cr
& = left( {a - 2b - 4}
ight)left( {a - 2b + 2}
ight) cr
& left( {{ ext{Put the value of assume }}t}
ight) cr} $$

[#359] The value of $$frac{1}{{{a^2} + ax + {x^2}}}$$ xa0 $$ - $$ $$frac{1}{{{a^2} - ax + {x^2}}}$$ xa0 $$ + $$ $$frac{2ax}{{{a^4} + {a^2}{x^2} + {x^4}}}$$ xa0xa0 is?
Correct Answer

(D) 0

Explanation

Solution: $$frac{1}{{{a^2} + ax + {x^2}}}$$ xa0 $$ - $$ $$frac{1}{{{a^2} - ax + {x^2}}}$$ xa0 $$ + $$ $$frac{2ax}{{{a^4} + {a^2}{x^2} + {x^4}}}$$ $$ = frac{{{a^2} - ax + {x^2} - {a^2} - ax - {x^2}}}{{left( {{a^2} + {x^2} + ax}
ight)left( {{a^2} + {x^2} - ax}
ight)}} + $$ xa0 xa0 xa0 $$frac{{2ax}}{{{a^4} + {a^2}{x^2} + {x^4}}}$$ $$eqalign{
& = frac{{ - 2ax}}{{{{left( {{a^2} + {x^2}}
ight)}^2} - {{left( {ax}
ight)}^2}}} + frac{{2ax}}{{{a^4} + {x^4} + {a^2}{x^2}}} cr
& = frac{{ - 2ax}}{{{a^4} + {x^4} + 2{a^2}{x^2} - {a^2}{x^2}}} + frac{{2ax}}{{{a^4} + {x^4} + {a^2}{x^2}}} cr
& = frac{{ - 2ax}}{{{a^4} + {x^4} + {a^2}{x^2}}} + frac{{2ax}}{{{a^4} + {x^4} + {a^2}{x^2}}} cr
& = 0 cr} $$

[#360] If x = 11, then the value of x 5 - 12x 4 + 12x 3 - 12x 2 + 12x - 1 is?
Correct Answer

(B) 10

Explanation

Solution: $$x08ecause $$ x = 11 x 5 - 12x 4 + 12x 3 - 12x 2 + 12x - 1 = x 5 - 11x 4 - x 4 + 11x 3 + x 3 - 11x 2 - x 2 + 11x + x - 1 = 11 5 - 11.11 4 - 11 4 + 11.11 3 + 11 3 - 11.11 2 - 11 2 + 11.11 + 11 - 1 = 0 - 0 + 0 + 0 + 11 - 1 = 10