Algebra - Study Mode
[#471] If a + b + c = 8 and ab + bc + ca = 12, then a 3 + b 3 + c 3 - 3abc is equal to:
Correct Answer
(B) 224
Explanation
Solution: a + b + c = 8 and ab + bc + ca = 12 a 3 + b 3 + c 3 - 3abc = ? Let c = 0 a + b = 8, ab = 12 a 3 + b 3 = (a + b)(a 2 + b 2 - ab) a 3 + b 3 = (a + b)[(a + b) 2 - 3ab] a 3 + b 3 = 8[8 2 - 3 × 12] a 3 + b 3 = 224
[#472] If $$a - frac{1}{a} = b,,b - frac{1}{b} = c$$ xa0 xa0 and $$c - frac{1}{c} = a,$$ xa0 then what is the value $$frac{1}{{ab}} + frac{1}{{bc}} + frac{1}{{ca}} = ?$$
Correct Answer
(A) -3
Explanation
Solution: $$eqalign{
& a - frac{1}{a} = b,........left( { ext{i}}
ight) cr
& b - frac{1}{b} = c,........left( {{ ext{ii}}}
ight) cr
& c - frac{1}{c} = a,........left( {{ ext{iii}}}
ight) cr
& { ext{Add Equation }}left( { ext{i}}
ight),,left( {{ ext{ii}}}
ight){ ext{ and}}left( {{ ext{iii}}}
ight) cr
& a + b + c - left[ {frac{1}{a} + frac{1}{b} + frac{1}{c}}
ight] = a + b + c cr
& frac{1}{a} + frac{1}{b} + frac{1}{c} = 0,........left( {{ ext{iv}}}
ight) cr
& Rightarrow a - frac{1}{a} = b cr
& { ext{Squaring both sides}} cr
& {a^2} + frac{1}{{{a^2}}} - 2 = {b^2},........left( { ext{v}}
ight) cr
& Rightarrow b - frac{1}{b} = c cr
& {b^2} + frac{1}{{{b^2}}} - 2 = {c^2},........left( {{ ext{vi}}}
ight) cr
& Rightarrow c + frac{1}{c} = a cr
& {c^2} + frac{1}{{{c^2}}} - 2 = {a^2},........left( {{ ext{vii}}}
ight) cr
& { ext{Add Equation }}left( { ext{v}}
ight),,left( {{ ext{vi}}}
ight){ ext{ and}}left( {{ ext{vii}}}
ight) cr
& {a^2} + {b^2} + {c^2} + frac{1}{{{a^2}}} + frac{1}{{{b^2}}} + frac{1}{{{c^2}}} - 2 - 2 - 2 = {a^2} + {b^2} + {c^2} cr
& frac{1}{{{a^2}}} + frac{1}{{{b^2}}} + frac{1}{{{c^2}}} = 6,........left( {{ ext{viii}}}
ight) cr
& Rightarrow left( {frac{1}{a} + frac{1}{b} + frac{1}{c}}
ight) cr
& = frac{1}{{{a^2}}} + frac{1}{{{b^2}}} + frac{1}{{{c^2}}} + frac{2}{{ab}} + frac{2}{{bc}} + frac{2}{{ca}} cr
& {0^2} = 6 + 2left[ {frac{1}{{ab}} + frac{1}{{bc}} + frac{1}{{ca}}}
ight] cr
& frac{1}{{ab}} + frac{1}{{bc}} + frac{1}{{ca}} = frac{{ - 6}}{2} = - 3 cr} $$
[#473] If x + y + z = 22 and xy + yz + zx = 35, then what is the value of (x - y) 2 + (y - z) 2 + (z - x) 2 ?
Correct Answer
(C) 758
Explanation
Solution: x + y + z = 22 xy + yz + zx = 35 (x + y + z) 2 = x 2 + y 2 + z 2 + 2(xy + yz + zx) (22) 2 = x 2 + y 2 + z 2 + 2 × 35 484 - 70 = x 2 + y 2 + z 2 x 2 + y 2 + z 2 = 414 (x - y) 2 + (y - z) 2 + (z - x) 2 = 2(x 2 + y 2 + z 2 - xy - yz - zx) = 2(414 - 35) = 2 × 379 = 758
[#474] If $${x^2} + frac{1}{{{x^2}}} = frac{{31}}{9}$$ xa0 and x > 0, then what is the value of $${x^3} + frac{1}{{{x^3}}} = ?$$
Correct Answer
(B) $$frac{{154}}{{27}}$$
Explanation
Solution: $$eqalign{
& {x^2} + frac{1}{{{x^2}}} = frac{{31}}{9} cr
& x + frac{1}{x} = sqrt {frac{{31}}{9} + 2} cr
& x + frac{1}{x} = frac{7}{3} cr
& {x^3} + frac{1}{{{x^3}}} = {n^3} - 3n cr
& {x^3} + frac{1}{{{x^3}}} = {left( {frac{7}{3}}
ight)^3} - 3 imes frac{7}{3} cr
& {x^3} + frac{1}{{{x^3}}} = frac{{343}}{{27}} - 7 cr
& {x^3} + frac{1}{{{x^3}}} = frac{{343 - 7 imes 27}}{{27}} cr
& {x^3} + frac{1}{{{x^3}}} = frac{{154}}{{27}} cr} $$
[#475] If x 4 + x -4 = 194, x > 0, then the value of (x - 2) 2 is:
Correct Answer
(D) 3
Explanation
Solution: $$eqalign{
& {x^4} + frac{1}{{{x^4}}} = 194, cr
& {x^4} + frac{1}{{{x^4}}} + 2 = 194 + 2 cr
& {x^2} + frac{1}{{{x^2}}} + 2 = 16 cr
& x + frac{1}{x} = 4 cr
& {x^2} + 1 = 4x cr
& {x^2} - 4x = - 1 cr
& {x^2} - 4x + 1 = 0 cr
& {x^2} - 4x + 4 - 3 = 0 cr
& {x^2} - 4x + 4 = 3 cr
& {left( {x - 2}
ight)^2} = 3 cr} $$