Algebra - Study Mode

[#331] If $$frac{{4left[ {{{left( {17}
ight)}^3} - {{left( 7
ight)}^3}}
ight]}}{{left( {{{17}^2} + {7^2} + p}
ight)}} = 40,$$ xa0 xa0 then what is the value of p?
Correct Answer

(C) 119

Explanation

Solution: $$eqalign{
& frac{{4left[ {{{left( {17}
ight)}^3} - {{left( 7
ight)}^3}}
ight]}}{{left( {{{17}^2} + {7^2} + p}
ight)}} = 40 cr
& { ext{We know,}} cr
& left[ {{a^3} - {b^3} = left( {a - b}
ight)left( {{a^2} + ab + {b^2}}
ight)}
ight] cr
& frac{{4left[ {left( {17 - 7}
ight)left( {{{17}^2} + {7^2} + 17 imes 7}
ight)}
ight]}}{{{{17}^2} + {7^2} + p}} = 40 cr
& {17^2} + {7^2} + 119 = {17^2} + {7^2} + p cr
& p = 119 cr} $$

[#332] If $$frac{{3left( {{x^2} + 1}
ight) - 7x}}{{3x}} = 6,$$ xa0 xa0x ≠ 0 the value $$sqrt x + frac{1}{{sqrt x }}$$ xa0is:
Correct Answer

(B) $$sqrt {frac{{31}}{3}} $$

Explanation

Solution: $$eqalign{
& frac{{3left( {{x^2} + 1}
ight) - 7x}}{{3x}} = 6 cr
& 3{x^2} + 3 - 7x = 18x cr
& 3{x^2} + 3 = 25x cr
& 3left( {{x^2} + 1}
ight) = 25x cr
& { ext{divided by }}'x' cr
& 3left( {x + frac{1}{x}}
ight) = 25 cr
& x + frac{1}{x} = 25 cr
& x + frac{1}{x} + 2 = frac{{25}}{3} cr
& {left( {x + frac{1}{x}}
ight)^2} = frac{{25}}{3} + 2 cr
& {left( {sqrt x + frac{1}{{sqrt x }}}
ight)^2} = frac{{31}}{3} cr
& sqrt x + frac{1}{{sqrt x }} = sqrt {frac{{31}}{3}} cr} $$

[#333] If x = 255, y = 256, z = 257, then find the value of x 3 + y 3 + z 3 - 3xyz.
Correct Answer

(B) 2304

Explanation

Solution: $$eqalign{
& {x^3} + {y^3} + {z^3} - 3xyz cr
& = frac{{left( {x + y + z}
ight)}}{2}left[ {{{left( {x - y}
ight)}^2} + {{left( {y - z}
ight)}^2} + {{left( {z - x}
ight)}^2}}
ight] cr
& = frac{{255 + 256 + 257}}{2}left[ {{1^2} + {1^2} + {2^2}}
ight] cr
& = frac{{768 imes 6}}{2} cr
& = frac{{4608}}{2} cr
& = 2304 cr} $$

[#334] If $$sqrt {left( {1 - {p^2}}
ight)left( {1 - {q^2}}
ight)} = frac{{sqrt 3 }}{2},$$ xa0 xa0 then what is the value of $$sqrt {2{p^2} + 2{q^2} + 2pq} + sqrt {2{p^2} + 2{q^2} - 2pq} ,?$$
Correct Answer

(B) √2

Explanation

Solution: $$eqalign{
& sqrt {left( {1 - {p^2}}
ight)left( {1 - {q^2}}
ight)} = frac{{sqrt 3 }}{2},........,left( { ext{i}}
ight) cr
& { ext{Put value of }}p{ ext{ and }}q cr
& p = 0,,q = frac{1}{2} cr
& { ext{Equation }}left( { ext{i}}
ight){ ext{ is satisfying}} cr
& { ext{Then, }} cr
& sqrt {2{p^2} + 2{q^2} + 2pq} + sqrt {2{p^2} + 2{q^2} - 2pq} cr
& = sqrt {0 + frac{2}{4} + 0} + sqrt {0 + frac{2}{4} - 0} cr
& = frac{1}{{sqrt 2 }} + frac{1}{{sqrt 2 }} cr
& = frac{2}{{sqrt 2 }} cr
& = sqrt 2 cr} $$

[#335] If (4x + 2y) 3 + (4x - 2y) 3 = 16(Ax 3 + Bxy 2 ), then what is the value of $$frac{1}{2}left( {sqrt {{A^2} + {B^2}} }
ight)?$$
Correct Answer

(C) 5

Explanation

Solution: (4x + 2y) 3 + (4x - 2y) 3 = 16(Ax 3 + Bxy 2 ) ⇒ (4x) 3 + (2y) 3 + 3 × 4x × 2y(4x + 2y) + (4x) 3 - (2y) 3 - 3 × 4x × 2y(4x - 2y) = 16(Ax 3 + Bxy 2 ) ⇒ 2(4x) 3 + 96x 2 y + 48xy 2 - 96x 2 y + 48xy 2 = 16(Ax 3 + Bxy 2 ) ⇒ 128x 3 + 96xy 2 = 16(Ax 3 + Bxy 2 ) ⇒ 16(8x 3 + 6xy 2 ) = 16(Ax 3 + Bxy 2 ) On comparing on both side- A = 8 & B = 6 Then, $$eqalign{
& frac{1}{2}left( {sqrt {{A^2} + {B^2}} }
ight) cr
& = frac{1}{2}left( {sqrt {64 + 36} }
ight) cr
& = frac{1}{2} imes 10 cr
& = 5{ ext{ Answer}} cr} $$