Algebra - Study Mode
[#311] The value of [(a 2 - b 2 ) 3 + (b 2 - c 2 ) 3 + (c 2 - a 2 ) 3 ] ÷ [(a - b) 3 + (b - c) 3 + (c - a) 3 ] is equal to: (Given a ≠ b ≠ c).
Correct Answer
(A) (a + b)(b + c)(c + a)
Explanation
Solution: [(a 2 - b 2 ) 3 + (b 2 - c 2 ) 3 + (c 2 - a 2 ) 3 ] ÷ [(a - b) 3 + (b - c) 3 + (c - a) 3 ] put a = 0, b = 1, c = 2 $$frac{{ - 1 - 27 + 64}}{{ - 1 - 1 + 8}} = frac{{36}}{6} = 6$$ (0 + 1)(1 + 2)(2 + 0) = 3 × 2 = 6 Hence option A is right answer.
[#312] What is the value of $$frac{{left( {{a^2} + {b^2}}
ight)left( {a - b}
ight) - left( {{a^3} - {b^3}}
ight)}}{{{a^2}b - a{b^2}}}?$$
Correct Answer
(C) -1
Explanation
Solution: $$eqalign{
& frac{{left( {{a^2} + {b^2}}
ight)left( {a - b}
ight) - left( {{a^3} - {b^3}}
ight)}}{{{a^2}b - a{b^2}}} cr
& = frac{{left( {{a^2} + {b^2}}
ight)left( {a - b}
ight) - left( {{a^3} - {b^3}}
ight)}}{{ableft( {a - b}
ight)}} cr
& = frac{{left( {a - b}
ight)left( {{a^2} + {b^2} - {a^2} - {b^2} - ab}
ight)}}{{ableft( {a - b}
ight)}} cr
& = - 1 cr} $$
[#313] If $${x^4} + frac{1}{{{x^4}}} = 14159,$$ xa0 xa0then the value of $$x + frac{1}{x}$$ xa0is:
Correct Answer
(D) 11
Explanation
Solution: $$eqalign{
& {x^4} + frac{1}{{{x^4}}} = 14159 cr
& {x^2} + frac{1}{{{x^2}}} = sqrt {14159 + 2} cr
& {x^2} + frac{1}{{{x^2}}} = 119 cr
& x + frac{1}{x} = sqrt {119 + 2} cr
& x + frac{1}{x} = 11 cr} $$
[#314] If $$2x + frac{1}{{2x}} = 2,$$ xa0 then what is the value of $$sqrt {2{{left( {frac{1}{x}}
ight)}^4} + {{left( {frac{1}{x}}
ight)}^5}} ?$$
Correct Answer
(D) 8
Explanation
Solution: $$eqalign{
& 2x + frac{1}{{2x}} = 2 cr
& { ext{Now, }}2x = 1 cr
& Rightarrow x = frac{1}{2} cr
& Rightarrow frac{1}{x} = 2 cr
& sqrt {2{{left( {frac{1}{x}}
ight)}^4} + {{left( {frac{1}{x}}
ight)}^5}} cr
& = sqrt {2 imes {2^4} + {2^5}} cr
& = sqrt {32 + 32} cr
& = sqrt {64} cr
& = 8 cr} $$
[#315] If x = 32.5, y = 34.6 and z = 30.9, then the value x 3 + y 3 + z 3 - 3xyz of is 0.98k, where k is equal to:
Correct Answer
(D) 1033
Explanation
Solution: $$eqalign{
& x = 32.5,,y = 34.6{ ext{ and }}z = 30.9 cr
& {x^3} + {y^3} + {z^3} - 3xyz cr
& = left( {x + y + z}
ight)left[ {frac{1}{2}left{ {{{left( {x - y}
ight)}^2} + {{left( {y - z}
ight)}^2} + {{left( {z - x}
ight)}^2}}
ight}}
ight] cr
& = left( {32.5 + 34.6 + 30.9}
ight)left[ {frac{1}{2}left{ {{{left( {32.5 - 34.6}
ight)}^2} + {{left( {34.6 - 30.9}
ight)}^2} + {{left( {30.9 - 32.5}
ight)}^2}}
ight}}
ight] cr
& = 98left[ {frac{1}{2}left{ {{{left( { - 2.1}
ight)}^2} + {{left( {3.7}
ight)}^2} + {{left( { - 1.6}
ight)}^2}}
ight}}
ight] cr
& = 98left[ {frac{1}{2}left{ {4.41 + 13.69 + 2.56}
ight}}
ight] cr
& = 98left[ {frac{1}{2}left{ {20.66}
ight}}
ight] cr
& = 98 imes 10.33 cr
& = 1012.34 cr
& {x^3} + {y^3} + {z^3} - 3xyz = 0.98k cr
& 1012.34 = 0.98k cr
& k = frac{{1012.34}}{{0.98}} = 1033 cr} $$