Algebra - Study Mode
[#171] If a 2 + b 2 + c 2 + $$frac{1}{{{a^2}}} + frac{1}{{{b^2}}} + frac{1}{{{c^2}}} = 6,$$ xa0 xa0then what is the value of a 2 + b 2 + c 2 ?
Correct Answer
(A) 3
Explanation
Solution: a 2 + b 2 + c 2 + $$frac{1}{{{a^2}}} + frac{1}{{{b^2}}} + frac{1}{{{c^2}}} = 6,$$ Put a = b = c = 1 ∴ a 2 + b 2 + c 2 = 1 + 1 + 1 = 3
[#172] The factors of x 2 + 4y 2 + 4y - 4xy - 2x - 8 are:
Correct Answer
(A) (x - 2y - 4)(x - 2y + 2)
Explanation
Solution: x 2 + 4y 2 - 4xy + 4y - 2x - 8 = (x - 2y) 2 - 2(x - 2y) - 8 put y = (x - 2y) = y 2 - 2y - 8 = y 2 - 4y + 2y - 8 = y(y - 4) + 2(y - 4) = (y - 4)(y + 2) = (x - 2y - 4)(x - 2y + 2)
[#173] If $$frac{4}{3}left( {{x^2} + frac{1}{{{x^2}}}}
ight) = 110frac{2}{3},$$ xa0 xa0xa0find $$frac{1}{9}left( {{x^3} - frac{1}{{{x^3}}}}
ight),$$ xa0 where x > 0.
Correct Answer
(A) 84
Explanation
Solution: $$eqalign{
& frac{4}{3}left( {{x^2} + frac{1}{{{x^2}}}}
ight) = 110frac{2}{3} cr
& {x^2} + frac{1}{{{x^2}}} = frac{{332}}{3} imes frac{3}{4} cr
& {x^2} + frac{1}{{{x^2}}} = 83 cr
& {left( {x + frac{1}{x}}
ight)^2} = {x^2} + frac{1}{{{x^2}}} - 2 imes x imes frac{1}{x} cr
& {left( {x + frac{1}{x}}
ight)^2} = 83 - 2 cr
& {left( {x + frac{1}{x}}
ight)^2} = 81 cr
& x + frac{1}{x} = 9 cr
& { ext{Hence,}} cr
& frac{1}{9}left( {{x^3} - frac{1}{{{x^3}}}}
ight) cr
& = frac{1}{9}left[ {{{left( {x - frac{1}{x}}
ight)}^3} + 3 imes left( {x - frac{1}{x}}
ight)}
ight] cr
& = frac{1}{9}left[ {729 + 3 imes 9}
ight] cr
& = 84 cr} $$
[#174] If a + 3b = 12 and ab = 9, then the value of (a - 3b) is:
Correct Answer
(C) 6
Explanation
Solution: If a + 3b = 12 ab = 9 (a - 3b) 2 = a 2 + 9b 2 - 6ab (a - 3b) 2 = (a + 3b) 2 - 12ab (a - 3b) 2 = 144 - 108 (a - 3b) 2 = 36 (a - 3b) = 6
[#175] Determine the value of 'x', if $$x = frac{{{{left( {943 + 864}
ight)}^2} - {{left( {943 - 864}
ight)}^2}}}{{left( {1886 imes 1728}
ight)}}$$
Correct Answer
(A) 1
Explanation
Solution: $$eqalign{
& a = 943,, herefore 2a = 1886 cr
& b = 864,, herefore 2b = 1728 cr
& herefore { ext{Expression}} cr
& = frac{{{{left( {a + b}
ight)}^2} - {{left( {a - b}
ight)}^2}}}{{2a imes 2b}} cr
& = frac{{4ab}}{{4ab}} cr
& = 1 cr} $$