Algebra - Study Mode
[#226] If x 2 + y 2 + 2x +1 = 0, then the value of x 31 + y 35 is?
Correct Answer
(A) -1
Explanation
Solution: $$eqalign{
& {x^2} + {y^2} + 2x + 1 = 0 cr
& Rightarrow {left( {x + 1}
ight)^2} + {y^2} = 0 cr
& { ext{Let }}{left( {x + 1}
ight)^2} = 0 cr
& {y^2} = 0 cr
& x = - 1,y = 0 cr
& { ext{Now, }}{x^{31}} + {y^{35}} cr
& = {left( { - 1}
ight)^{31}} + {left( 0
ight)^{35}} cr
& = - 1 cr} $$
[#227] If $$x = frac{{sqrt 5 + 1}}{{sqrt 5 - 1}}$$ xa0 and $${ ext{y}} = frac{{sqrt 5 - 1}}{{sqrt 5 + 1}}{ ext{,}}$$ xa0 then the value of $$frac{{{x^2} + xy + {y^2}}}{{{x^2} - xy + {y^2}}}$$ xa0 is?
Correct Answer
(C) $$frac{4}{3}$$
Explanation
Solution: $$eqalign{
& x = frac{{sqrt 5 + 1}}{{sqrt 5 - 1}}{ ext{ and y}} = frac{{sqrt 5 - 1}}{{sqrt 5 + 1}} cr
& herefore x = frac{1}{y} cr
& Leftrightarrow xy = 1 cr
& x + y = frac{{sqrt 5 + 1}}{{sqrt 5 - 1}}{ ext{ + }}frac{{sqrt 5 - 1}}{{sqrt 5 + 1}} cr
& Rightarrow x + y = frac{{5 + 1 + 2sqrt 5 + 5 + 1 - 2sqrt 5 }}{{5 - 1}} cr
& Rightarrow x + y = frac{{12}}{4} cr
& Rightarrow x + y = 3 cr
& Rightarrow x + frac{1}{x} = 3 cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} = {left( 3
ight)^2} - 2 cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} = 7 cr
& { ext{Now,}}frac{{{x^2} + xy + {y^2}}}{{{x^2} - xy + {y^2}}} cr
& = frac{{{x^2} + {y^2} + xy}}{{{x^2} + {y^2} - xy}} cr
& = frac{{7 + 1}}{{7 - 1}} cr
& = frac{8}{6} cr
& = frac{4}{3} cr} $$
[#228] If x 4 + 2x 3 + ax 2 + bx + 9 is a perfect square where a and b are positive real numbers, then the value of a and b is?
Correct Answer
(B) a = 6, b = 7
Explanation
Solution: $$eqalign{
& {x^4} + 2{x^3} + a{x^2} + bx + 9 cr
& { ext{Put }}x = 1 cr
& = 1 + 2 imes 1 + a + b + 9 cr
& = 1 + 2 + a + b + 9 cr
& = 13 + a + b cr} $$ To make a perfect square numbers value of a + b must be either 3 or 13 Now, option (B) a = 6, b = 7 $$eqalign{
& herefore a + b = 13 cr
& { ext{make perfect square}} cr
& left( {25 = {5^2}}
ight) cr} $$
[#229] If a 2 + b 2 + c 2 = 16, x 2 + y 2 + z 2 = 25 and ax + by + cz = 20 then the value of $$frac{{a + b + c}}{{x + y + z}}$$ xa0 = ?
Correct Answer
(B) $$frac{4}{5}$$
Explanation
Solution: $$eqalign{
& {a^2} + {b^2} + {c^2} = 16,{ ext{ }}{x^2} + {y^2} + {z^2} = 25{ ext{ }} cr
& { ext{But }}b = c = 0,{ ext{ But }}y = z = 0 cr
& { ext{}}a = 4{ ext{ }},,,,,,,,,,,,,,,,x = 5 cr
& { ext{Now, }} cr
& ax + by + cz = 20 cr
& 4 imes 5 + 0 + 0 = 20 cr
& 20 = 20{ ext{ }}left( {{ ext{Satisfy}}}
ight) cr
& { ext{Now, }}frac{{a + b + c}}{{x + y + z}} cr
& = frac{{4 + 0 + 0}}{{5 + 0 + 0}} cr
& = frac{4}{5} cr} $$
[#230] If 3x + 4y - 2z + 9 = 17, 7x + 2y + 11z + 8 = 23 and 5x + 9y + 6z - 4 = 18, then what is the value of x + y + z - 34?
Correct Answer
(C) -31
Explanation
Solution: 3x + 4y - 2z + 9 = 17 . . . . . . . . (1) 7x + 2y + 11z + 8 = 23 . . . . . . . (2) 5x + 9y + 6z - 4 = 18 . . . . . . . . (3) ____________________________ 15x + 15y + 15z + 13 = 58 (By adding equation (1), (2) & (3)) 15(x + y + z) = 45 x + y + z = 3 x + y + z - 34 = 3 - 34 = -31