Algebra - Study Mode
[#66] If x + y + z = 19, x 2 + y 2 + z 2 = 133 and xz = y 2 , then the difference between z and x is:
Correct Answer
(A) 5
Explanation
Solution: Given: x + y + z = 19, x 2 + y 2 + z 2 = 133 and xz = y 2 , Formula used: (x + y + z) 2 = x 2 + y 2 + z 2 + 2(xy + yz + zx) Calculation: (x + y + z) 2 = x 2 + y 2 + z 2 + 2(xy + yz + zx) ⇒ (19) 2 = 133 + 2(xy + yz + y 2 ) ⇒ 133 + 2[y(x + y + z)] = 361 ⇒ 2y(19) = 361 - 133 ⇒ y = 6 x + y + z = 19 ⇒ x + z = 13 The possible value of x and z is 9 and 4 x - 4 ⇒ 9 - 4 ⇒ 5 ∴ The value is 5
[#67] If $$a - frac{1}{{a - 5}} = 10$$ xa0 , then the value of $${left( {a - 5}
ight)^3} - frac{1}{{{{left( {a - 5}
ight)}^3}}}$$ xa0 xa0is:
Correct Answer
(A) 140
Explanation
Solution: $$a - frac{1}{{a - 5}} = 10$$ We can write it is as (subtracting 5 from both sides) $$eqalign{
& a - 5 - frac{1}{{a - 5}} = 10 - 5 cr
& { ext{Now, }}left( {a - 5}
ight) - frac{1}{{left( {a - 5}
ight)}} = 5 cr} $$ So, take the cube of this equation, $$eqalign{
& left[ {{{left( {a - 5}
ight)}^3} - frac{1}{{{{left( {a - 5}
ight)}^3}}}}
ight] = 125 + 3 imes 5 cr
& left[ {{{left( {a - 5}
ight)}^3} - frac{1}{{{{left( {a - 5}
ight)}^3}}}}
ight] = 140 cr} $$
[#68] If x 2 - 16x + 59 = 0, then what is the value of $${left( {x - 6}
ight)^2} + frac{1}{{{{left( {x - 6}
ight)}^2}}}?$$
Correct Answer
(B) 18
Explanation
Solution: $$eqalign{
& {x^2} - 16x + 59 = 0 cr
& {left( {x - 6}
ight)^2} + frac{1}{{{{left( {x - 6}
ight)}^2}}} = ? cr
& { ext{Let }}x - 6 = t cr
& x = t + 6 cr
& {t^2} + frac{1}{{{t^2}}} = ? cr
& {left( {t + 6}
ight)^2} - 16left( {t + 6}
ight) + 59 = 0 cr
& {t^2} + 12t - 36 - 16t - 96 + 59 = 0 cr
& {t^2} - 4t - 1 = 0 cr
& { ext{Dividing by }}t,{ ext{ we get}} cr
& t - 4 - frac{1}{t} = 0 cr
& t - frac{1}{t} = 4 cr
& { ext{By squaring, we get}} cr
& {t^2} + frac{1}{{{t^2}}} - 2 = 16 cr
& {t^2} + frac{1}{{{t^2}}} = 18 cr} $$
[#69] If x = √5 + 1 and y = √5 - 1, then what is the value of $$frac{{{x^2}}}{{{y^2}}} + frac{{{y^2}}}{{{x^2}}} + 4left[ {frac{x}{y} + frac{y}{x}}
ight] + 6?$$
Correct Answer
(D) 25
Explanation
Solution: $$eqalign{
& x = sqrt 5 + 1,,y = sqrt 5 - 1 cr
& frac{{{x^2}}}{{{y^2}}} + frac{{{y^2}}}{{{x^2}}} + 4left[ {frac{x}{y} + frac{y}{x}}
ight] + 6 cr
& = {left[ {frac{x}{y} + frac{y}{x}}
ight]^2} - 2 + 4left[ {frac{x}{y} + frac{y}{x}}
ight] + 6 cr
& = {left[ {frac{{sqrt 5 + 1}}{{sqrt 5 - 1}} + frac{{sqrt 5 - 1}}{{sqrt 5 + 1}}}
ight]^2} + 4left[ {frac{{sqrt 5 + 1}}{{sqrt 5 - 1}} + frac{{sqrt 5 - 1}}{{sqrt 5 + 1}}}
ight] + 4 cr
& = {left[ {frac{{5 + 1 + 2sqrt 5 + 5 + 1 - 2sqrt 5 }}{{5 - 1}}}
ight]^2} + 4left[ {frac{{5 + 1 + 2sqrt 5 + 5 + 1 - 2sqrt 5 }}{{5 - 1}}}
ight] + 4 cr
& = {left[ {frac{{12}}{4}}
ight]^2} + 4left[ {frac{{12}}{4}}
ight] + 4 cr
& = {left( 3
ight)^2} + 4left( 3
ight) + 4 cr
& = 9 + 12 + 4 cr
& = 25 cr} $$
[#70] If a 3 + 3a 2 + 9a = 1, then what is the value of $${a^3} + frac{3}{a}?$$
Correct Answer
(C) 28
Explanation
Solution: $$eqalign{
& {a^3} + 3{a^2} + 9a = 1.......left( { ext{i}}
ight) cr
& {a^3} + frac{3}{a} = ? cr
& { ext{Multiply by '}}a{ ext{' and '3' in equation}}left( { ext{i}}
ight) cr
& left( {{a^3} + 3{a^2} + 9a = 1}
ight) imes a cr
& left( {{a^3} + 3{a^2} + 9a = 1}
ight) imes 3 cr
& {a^4} + 3{a^3} + 9{a^2} = 1 imes a cr
& underline {3{a^3} + 9{a^2} + 27a = 1 imes 3} o left( {{ ext{Subtracting}}}
ight) cr
& {a^4} - 27a = a - 3 cr
& {a^4} + 3 = 28a cr
& { ext{On dividing by }}a cr
& {a^3} + frac{3}{a} = 28 cr} $$