Algebra - Study Mode
[#381] If a + b = 10 and $$sqrt {frac{a}{b}} - 13 = - sqrt {frac{b}{a}} - 11$$ xa0 xa0 then what is the value of 3ab + 4a 2 + 5b 2 ?
Correct Answer
(B) 300
Explanation
Solution: $$eqalign{
& { ext{Let }}sqrt {frac{a}{b}} = x cr
& herefore ,x - 13 = frac{{ - 1}}{x} - 11 cr
& x + frac{1}{x} = 2 cr
& herefore ,x = 1 cr
& sqrt {frac{a}{b}} = 1{ ext{ and }}a + b = 10 cr
& herefore ,a = b = 5 cr
& 3ab + 4{a^2} + 5{b^2} cr
& = 3{a^2} + 4{a^2} + 5{a^2} cr
& = 12{a^2} cr
& = 12 imes 25 cr
& = 300 cr} $$
[#382] If 16x 2 + 9y 2 + 4z 2 = 24(x - y + z) - 61, then the value of (xy + 2z) is:
Correct Answer
(D) 5
Explanation
Solution: 16x 2 + 9y 2 + 4z 2 = 24(x - y + z) - 61 ⇒ 16x 2 + 9y 2 + 4z 2 = 2 × 4x × 3 - 2 × 3x × 4 + 2 × 2z × 6 ⇒ (4x - 3) 2 + (3y + 4) 2 + (2z - 6) 2 = 0 x = $$frac{3}{4}$$ y = $$ - frac{4}{3}$$ z = 3 ⇒ xy + 2z $$eqalign{
& = frac{3}{4} imes left( { - frac{4}{3}}
ight) + 2 imes 3 cr
& = - 1 + 6 cr
& = 5 cr} $$
[#383] If $$x = frac{a}{b} + frac{b}{a},,y = frac{b}{c} + frac{c}{b}$$ xa0 xa0 and $$z = frac{c}{a} + frac{a}{c},$$ xa0 then what is the value of xyz - x 2 - y 2 - z 2 ?
Correct Answer
(A) -4
Explanation
Solution: Let a = b = c = 1 ∴ x = 2, y = 2, z = 2 xyz - x 2 - y 2 - z 2 = 2 × 2 × 2 - 4 - 4 - 4 = 8 - 12 = -4
[#384] If $${x^2} + frac{1}{{25{x^2}}} = frac{8}{5}$$ xa0 and x > 0, then what is the value of $${x^3} + frac{1}{{125{x^3}}} = ?$$
Correct Answer
(C) $$frac{{7sqrt 2 }}{5}$$
Explanation
Solution: $$eqalign{
& x08ecause ,{x^2} + frac{1}{{25{x^2}}} = frac{8}{5} cr
& Rightarrow {x^2} + frac{1}{{25{x^2}}} + 2x imes frac{1}{{5x}} = frac{8}{5} + frac{2}{5} cr
& Rightarrow {left( {x + frac{1}{{5x}}}
ight)^2} = 2 cr
& Rightarrow x + frac{1}{{5x}} = sqrt 2 cr
& { ext{On cubing both sides}} cr
& Rightarrow {x^3} + frac{1}{{125{x^3}}} + 3 imes frac{1}{5}left( {sqrt 2 }
ight) = {left( {sqrt 2 }
ight)^3} cr
& Rightarrow {x^3} + frac{1}{{125{x^3}}} = 2sqrt 2 - frac{{3sqrt 2 }}{5} cr
& Rightarrow {x^3} + frac{1}{{125{x^3}}} = frac{{7sqrt 2 }}{5} cr} $$
[#385] The graphs of the equations 3x - 20y - 2 = 0 and 11x - 5y + 61 = 0 intersect at P(a, b). What is the value of (a 2 + b 2 - ab)(a 2 - b 2 + ab)?
Correct Answer
(C) $$frac{{31}}{{41}}$$
Explanation
Solution: It intersect at point P(a, b) = (x, y) 3x - 20y = 2 11x - 5y = -61 44x - 20y = -244 3x - 20y = 2 $$overline {41{ ext{x}},,,,,,,,, = - 246,,} $$ x = -6 y = -1 P(a, b) = (-6, -1) $$eqalign{
& frac{{{a^2} + {b^2} - ab}}{{{a^2} - {b^2} + ab}} cr
& = frac{{36 + 1 - 6}}{{36 - 1 + 6}} cr
& = frac{{31}}{{41}} cr} $$