Algebra - Study Mode

[#506] If a 3 = 117 + b 3 and a = 3 + b, then the value of a + b is?
Correct Answer

(A) 7

Explanation

Solution: $$eqalign{
& {a^3} = 117 + {b^3}{ ext{ , }}a = 3 + b cr
& {a^3} - {b^3}{ ext{ = 117}},......{ ext{(i)}} cr
& a - b = 3,........(ii) cr
& { ext{Put }}a = 5,{ ext{ }}b = 2 cr
& { ext{Both equation satisfy}} cr
& { ext{Now, }}a + b = 5 + 2 = 7 cr} $$

[#507] If $${x^2} + frac{1}{{{x^2}}} = 98{ ext{,}}$$ xa0 $$left( {x > 0}
ight){ ext{,}}$$ xa0 then the value of $$x^3 + frac{1}{{{x^3}}}$$ xa0is?
Correct Answer

(A) 970

Explanation

Solution: $$eqalign{
& {x^2} + frac{1}{{{x^2}}} = 98 cr
& Rightarrow {x^2} + frac{1}{{{x^2}}} + 2 = 100 cr
& Rightarrow {left( {x + frac{1}{x}}
ight)^2} = 100 cr
& Rightarrow x + frac{1}{x} = 10 cr
& { ext{Cubing}},{ ext{both}},{ ext{sides}} cr
& Rightarrow { ext{ }}{x^3} + frac{1}{{{x^3}}} = {10^3} - 3 imes 10 cr
& Rightarrow {x^3} + frac{1}{{{x^3}}} = 1000 - 3 imes 10 cr
& Rightarrow {x^3} + frac{1}{{{x^3}}} = 970 cr} $$

[#508] If x = y + z then x 3 - y 3 - z 3 is?
Correct Answer

(B) 3xyz

Explanation

Solution: $$eqalign{
& { ext{Given , }}x = y + z cr
& herefore x - y - z = 0 cr
& { ext{Then }}{x^3} - {y^3} - {z^3} = 3xyz cr} $$

[#509] If a + b + c + d = 4, then the value of $$frac{1}{{left( {1 - a}
ight)left( {1 - b}
ight)left( {1 - c}
ight)}}$$ xa0 xa0 + $$frac{1}{{left( {1 - b}
ight)left( {1 - c}
ight)left( {1 - d}
ight)}}$$ xa0 xa0 + $$frac{1}{{left( {1 - c}
ight)left( {1 - d}
ight)left( {1 - a}
ight)}}$$ xa0 xa0 + $$frac{1}{{left( {1 - d}
ight)left( {1 - a}
ight)left( {1 - b}
ight)}}$$ xa0 xa0 is?
Correct Answer

(A) 0

Explanation

Solution: $$eqalign{
& frac{1}{{left( {1 - a}
ight)left( {1 - b}
ight)left( {1 - c}
ight)}} + frac{1}{{left( {1 - b}
ight)left( {1 - c}
ight)left( {1 - d}
ight)}} + frac{1}{{left( {1 - c}
ight)left( {1 - d}
ight)left( {1 - a}
ight)}} + frac{1}{{left( {1 - d}
ight)left( {1 - a}
ight)left( {1 - b}
ight)}} cr
& = frac{{1 - d + 1 - a + 1 - b + 1 - c}}{{left( {1 - a}
ight)left( {1 - b}
ight)left( {1 - c}
ight)left( {1 - d}
ight)}} cr
& = frac{{4 - left( {a + b + c + d}
ight)}}{{left( {1 - a}
ight)left( {1 - b}
ight)left( {1 - c}
ight)left( {1 - d}
ight)}} cr
& = frac{{4 - 4}}{{left( {1 - a}
ight)left( {1 - b}
ight)left( {1 - c}
ight)left( {1 - d}
ight)}} cr
& = 0 cr} $$

[#510] If x - 11, then the value of x 5 - 12x 4 + 12x 3 - 12x 2 + 12x - 1 is?
Correct Answer

(B) 10

Explanation

Solution: $${x^5} - 12{x^4} + 12{x^3} - 12{x^2} + 12x - 1$$ $$ = {x^5} - 11{x^4} - {x^4} + 11{x^3} + {x^3} - 11{x^2} - $$ xa0 xa0 xa0 $$,{x^2} + $$ $$11x ,+, $$ $$x - $$ $$1$$ $${ ext{Put }}x = 11$$ $$ = {11^5} - {11.11^4} - {11^4} + {11.11^3} + {11^3} - $$ xa0 xa0 xa0 $${11.11^2} -, $$ $$,{11^2} ,+, $$ $$11.11 +, $$ $$11 - $$ $$1$$ $$eqalign{
& = 11 - 1 cr
& = 10 cr} $$