Algebra - Study Mode

[#131] If (x - 4)(x 2 + 4x + 16) = x 3 - p, then p is equal to?
Correct Answer

(C) 64

Explanation

Solution: $$eqalign{
& { ext{We know that }} cr
& {a^3} - {b^3} = left( {a - b}
ight)left( {{a^2} + ab + {b^2}}
ight) cr
& {x^3} - p = left( {x - 4}
ight)left( {{x^2} + 4x + 16}
ight) cr
& Rightarrow {x^3} - p = left( {{x^3} - {4^3}}
ight) cr
& Rightarrow p = {4^3}{ ext{ }}left( {{ ext{By comparison}}}
ight) cr
& { ext{So, }}p = 64 cr} $$

[#132] If $$4x + frac{1}{x} = 5,$$ xa0 $$x
e 0{ ext{,}}$$ xa0 then the value of $$frac{{5x}}{{4{x^2} + 10x + 1}}$$ xa0xa0 is?
Correct Answer

(B) $$frac{1}{3}$$

Explanation

Solution: $$eqalign{
& frac{{5x}}{{4{x^2} + 10x + 1}} cr
& = frac{5x}{{xleft( {4x + 10 + frac{1}{x}}
ight)}} cr
& = frac{5}{{4x + frac{1}{x} + 10}} cr
& = frac{5}{{5 + 10}} cr
& = frac{5}{{15}} cr
& = frac{1}{3} cr} $$

[#133] If $$c + frac{1}{c} = sqrt 3 { ext{,}}$$ xa0 then the value of $${c^3} + frac{1}{{{c^3}}}$$ xa0 is equal to?
Correct Answer

(A) 0

Explanation

Solution: $$eqalign{
& c + frac{1}{c} = sqrt 3 cr
& { ext{On cubing both side}} cr
& Rightarrow {left( {c + frac{1}{c}}
ight)^3} = 3sqrt 3 cr
& Rightarrow {c^3} + frac{1}{{{c^3}}} + 3.c.frac{1}{c}left( {c + frac{1}{c}}
ight) = 3sqrt 3 cr
& Rightarrow {c^3} + frac{1}{{{c^3}}} + 3sqrt 3 = 3sqrt 3 cr
& Rightarrow {c^3} + frac{1}{{{c^3}}} = 3sqrt 3 - 3sqrt 3 cr
& Rightarrow {c^3} + frac{1}{{{c^3}}} = 0 cr} $$

[#134] If x = 222, y = 223, z = 225, then the value of x 3 + y 3 + z 3 - 3xyz is?
Correct Answer

(B) 4690

Explanation

Solution: $${x^3} + {y^3} + {z^3} - 3xyz$$ $$ = frac{1}{2}left( {x + y + z}
ight)$$xa0 $$left[ {{{left( {x - y}
ight)}^2} + {{left( {y - z}
ight)}^2} + {{left( {z - x}
ight)}^2}}
ight]$$ $$ = frac{1}{2}left( {222 + 223 + 225}
ight)$$ xa0xa0 $$left[ {{{left( {222 - 223}
ight)}^2} + {{left( {223 - 225}
ight)}^2} + {{left( {225 - 222}
ight)}^2}}
ight]$$ $$eqalign{
& = frac{1}{2}left( {670}
ight)left( {1 + 4 + 9}
ight) cr
& = frac{1}{2} imes 670 imes 14 cr
& = 4690 cr} $$

[#135] If a + b + c = 0, then the value of a 3 + b 3 + c 3 is?
Correct Answer

(C) 3abc

Explanation

Solution: $$eqalign{
& a + b + c = 0 cr
& { ext{Let, }}{a^3} + {b^3} + {c^3} = T cr} $$ $$ Rightarrow {a^3} + {b^3} + {c^3} - 3abc = frac{1}{2}left( {a + b + c}
ight)$$ xa0 xa0 xa0$$left[ {{{left( {a - b}
ight)}^2} + {{left( {b - c}
ight)}^2} + {{left( {c - a}
ight)}^2}}
ight]$$ $$ Rightarrow {a^3} + {b^3} + {c^3} - 3abc = left( 0
ight)$$ xa0 xa0xa0$$left[ {{{left( {a - b}
ight)}^2} + {{left( {b - c}
ight)}^2} + {{left( {c - a}
ight)}^2}}
ight]$$ $$eqalign{
& Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 0 cr
& Rightarrow {a^3} + {b^3} + {c^3} = 3abc cr} $$