Algebra - Study Mode
[#361] If x = 997, y = 998 and z = 999 then the value of x 2 + y 2 + z 2 - xy - yz - zx is?
Correct Answer
(D) 3
Explanation
Solution: $$eqalign{
& {x^2} + {y^2} + {z^2} - xy - yz - zx cr
& = frac{1}{2}left[ {{{left( {x - y}
ight)}^2} + {{left( {y - z}
ight)}^2} + {{left( {z - x}
ight)}^2}}
ight] cr} $$ $$ = frac{1}{2}$$ $$left[ {{{left( {997 - 998}
ight)}^2} + {{left( {998 - 999}
ight)}^2} + {{left( {999 - 997}
ight)}^2}}
ight]$$ $$eqalign{
& = frac{1}{2}left( {1 + 1 + 4}
ight) cr
& = 3 cr} $$
[#362] If $$x + frac{1}{x} = 3{ ext{,}}$$ xa0 then the value of $$frac{{3{x^2} - 4x + 3}}{{{x^2} - x + 1}}$$ xa0 is?
Correct Answer
(C) $$frac{5}{2}$$
Explanation
Solution: $$eqalign{
& x + frac{1}{x} = 3 cr
& frac{{3{x^2} - 4x + 3}}{{{x^2} - x + 1}} cr
& = frac{{frac{{3{x^2}}}{x} - frac{{4x}}{x} + frac{3}{x}}}{{frac{{{x^2}}}{x} - frac{x}{x} + frac{1}{x}}} cr
& = frac{{3left( {x + frac{1}{x}}
ight) - 4}}{{left( {x + frac{1}{x}}
ight) - 1}} cr
& = frac{{3 imes 3 - 4}}{{3 - 1}} cr
& = frac{{9 - 4}}{2} cr
& = frac{5}{2} cr} $$
[#363] If $$x = p + frac{1}{p}$$ xa0 and $$y = p - frac{1}{p}$$ xa0 then the value of x 4 - 2x 2 y 2 + y 4 = ?
Correct Answer
(C) 16
Explanation
Solution: $$eqalign{
& x = p + frac{1}{p}{ ext{ }} cr
& y = p - frac{1}{p} cr
& herefore x + y = p + frac{1}{p} + p - frac{1}{p} cr
& Leftrightarrow x + y = 2p cr
& herefore x - y = p + frac{1}{p} - p + frac{1}{p} cr
& Leftrightarrow x - y = frac{2}{p} cr
& herefore {x^4} - 2{x^2}{y^2} + {y^4} cr
& = {x^4} + {y^4} - 2{x^2}{y^2} cr
& = {left( {{x^2} - {y^2}}
ight)^2} cr
& = {left[ {left( {x + y}
ight)left( {x - y}
ight)}
ight]^2} cr
& = {left( {2p imes frac{2}{p}}
ight)^2} cr
& = {left( 4
ight)^2} cr
& = 16 cr} $$
[#364] If a + b + c = 0, then the value of (a + b - c) 2 + (b + c - a) 2 + (c + a - b) 2 is?
Correct Answer
(C) 4(a 2 + b 2 + c 2 )
Explanation
Solution: $${left( {a + b - c}
ight)^2}{ ext{ + }}{left( {b + c - a}
ight)^2}$$ xa0 xa0 $${ ext{ + }}{left( {c + a - b}
ight)^2}$$ $$eqalign{
& Rightarrow a + b + c = 0{ ext{ }}left( {{ ext{ Given}}}
ight) cr
& Rightarrow a + b = - c cr
& Rightarrow b + c = - a cr
& Rightarrow a + c = - b cr} $$ $$ Rightarrow {left( {a + b - c}
ight)^2} + {left( {b + c - a}
ight)^2}$$ xa0 xa0xa0 $$ + {left( {c + a - b}
ight)^2}$$ $$eqalign{
& Rightarrow {left( { - c - c}
ight)^2}{ ext{ + }}{left( { - a - a}
ight)^2}{ ext{ + }}{left( { - b - b}
ight)^2} cr
& Rightarrow {left( { - 2c}
ight)^2}{ ext{ + }}{left( { - 2a}
ight)^2}{ ext{ + }}{left( { - 2b}
ight)^2} cr
& Rightarrow 4{c^2} + 4{a^2} + 4{b^2} cr
& Rightarrow 4left( {{a^2} + {b^2} + {c^2}}
ight) cr} $$
[#365] If x = 2015, y = 2014, z = 2013, then the value of x 2 + y 2 + z 2 - xy - yz - zx is?
Correct Answer
(A) 3
Explanation
Solution: $$eqalign{
& x = 2015 cr
& y = 2014 cr
& z = 2013 cr
& herefore {x^2} + {y^2} + {z^2} - xy - yz - zx cr
& = frac{1}{2}left[ {{{left( {x - y}
ight)}^2} + {{left( {y - z}
ight)}^2} + {{left( {z - x}
ight)}^2}}
ight] cr
& = frac{1}{2}left[ {{{left( {2015 - 2014}
ight)}^2} + {{left( {2014 - 2013}
ight)}^2} + {{left( {2013 - 2015}
ight)}^2}}
ight] cr
& = frac{1}{2}left( {1 + 1 + 4}
ight) cr
& = 3 cr} $$