Algebra - Study Mode

[#136] If $$frac{1}{p} + frac{1}{q}$$xa0 = $$frac{1}{{p + q}}{ ext{,}}$$ xa0 then the value of p 3 - q 3 is?
Correct Answer

(D) 0

Explanation

Solution: $$eqalign{
& frac{1}{p} + frac{1}{q} = frac{1}{{p + q}} cr
& Rightarrow frac{{p + q}}{{pq}} = frac{1}{{p + q}} cr
& Rightarrow {left( {p + q}
ight)^2} = pq cr
& Rightarrow left( {{p^2} + {q^2} + 2pq - pq}
ight) = 0 cr
& Rightarrow left( {{p^2} + {q^2} + pq}
ight) = 0 cr
& { ext{Multiply by }}left( {p - q}
ight){ ext{ both side}} cr
& Rightarrow left( {p - q}
ight)left( {{p^2} + {q^2} + pq}
ight) = left( {p - q}
ight) imes 0 cr
& Rightarrow {p^3} - {q^3} = 0 cr} $$

[#137] If ab = 21 and $$frac{{{{left( {a + b}
ight)}^2}}}{{{{left( {a - b}
ight)}^2}}}$$ xa0 = $$frac{{25}}{4}{ ext{,}}$$ xa0then the value of a 2 + b 2 + 3ab is?
Correct Answer

(B) 121

Explanation

Solution: $$eqalign{
& frac{{{{left( {a + b}
ight)}^2}}}{{{{left( {a - b}
ight)}^2}}} = frac{{25}}{4} cr
& Rightarrow frac{{a + b}}{{a - b}} = frac{5}{2} cr
& Rightarrow { ext{By Componendo & Dividendo}} cr
& Rightarrow frac{{a + b + a - b}}{{a + b - a + b}} = frac{{5 + 2}}{{5 - 2}} cr
& Rightarrow frac{{2a}}{{2b}} = frac{7}{3} cr
& Rightarrow frac{a}{b} = frac{7}{3} cr
& { ext{Now, the value of}} cr
& Rightarrow {a^2} + {b^2} + 3ab cr
& Rightarrow {7^2} + {3^2} + 3.7.3 cr
& Rightarrow 49 + 9 + 63 cr
& Rightarrow 121 cr} $$

[#138] Given a - b = 2, a 3 - b 3 = 26, then (a + b) 2 is?
Correct Answer

(C) 16

Explanation

Solution: $$eqalign{
& a - b = 2{ ext{ }} cr
& {a^3} - {b^3} = 26 cr
& Rightarrow {a^3} - {b^3} = left( {a - b}
ight)left( {{a^2} + ab + {b^2}}
ight) cr
& Rightarrow 26 = left( 2
ight)left( {{a^2} + ab + {b^2}}
ight) cr
& Rightarrow 13 = left( {{a^2} + ab + {b^2}}
ight),....(i) cr
& Rightarrow 4 = 13 + ab cr
& Rightarrow {left( {a - b}
ight)^2} = {a^2} + {b^2} - 2ab cr
& Rightarrow {left( 2
ight)^2} = {a^2} + {b^2} + ab - 3ab cr
& Rightarrow 3ab = 9 cr
& Rightarrow ab = 3 cr
& herefore {left( {a + b}
ight)^2} cr
& = {left( {a - b}
ight)^2} + 4ab cr
& = 4 + 4 imes 3 cr
& = 16 cr} $$

[#139] If x + y + z = 9, then the value of (x - 4) 3 + (y - 2) 3 + (z - 3) 3 - 3(x - 4)(y - 2)(z - 3) is?
Correct Answer

(C) 0

Explanation

Solution: $$eqalign{
& mathop {mathop {{{left( {x - 4}
ight)}^3}}limits_ Downarrow }limits_{{a^3}} + mathop {mathop {{{left( {y - 2}
ight)}^3}}limits_ Downarrow }limits_{{b^3}} + mathop {mathop {{{left( {z - 3}
ight)}^3}}limits_ Downarrow }limits_{{c^3} - 3abc} - 3left( {x - 4}
ight)left( {y - 2}
ight)left( {z - 3}
ight) cr
& Rightarrow a + b + c = x - 4 + y - 2 + z - 3 cr
& Rightarrow a + b + c = x + y + z - 9 cr
& Rightarrow a + b + c = 9 - 9 cr
& Rightarrow a + b + c = 0 cr} $$ $${ ext{So, }}{left( {x - 4}
ight)^3} + {left( {y - 2}
ight)^3} + $$ xa0 xa0 $${left( {z - 3}
ight)^3} - $$ xa0 $$3left( {x - 4}
ight)$$xa0 $$left( {y - 2}
ight)$$ $$left( {z - 3}
ight)$$ $$ Rightarrow 0$$

[#140] If $$x + frac{1}{{9x}} = 4{ ext{,}}$$ xa0 then $${ ext{9}}{x^2} + frac{1}{{9{x^2}}}$$ xa0 is?
Correct Answer

(B) 142

Explanation

Solution: $$eqalign{
& { ext{ }}x + frac{1}{{9x}} = 4 cr
& { ext{Multiply by 3 both side}} cr
& Rightarrow { ext{3}}x + frac{1}{{3x}} = 12 cr
& { ext{Squaring both sides}} cr
& Rightarrow { ext{9}}{x^2} + frac{1}{{9{x^2}}} + 2 imes 3x imes frac{1}{{3x}} = 144 cr
& Rightarrow { ext{9}}{x^2} + frac{1}{{9{x^2}}} + 2 = 144 cr
& Rightarrow { ext{9}}{x^2} + frac{1}{{9{x^2}}} = 142 cr} $$