Surds And Indices - Study Mode

[#111] $${ ext{If}},{kern 1pt} {left( {frac{a}{b}}
ight)^{x - 1}} = {left( {frac{b}{a}}
ight)^{x - 3}},$$ xa0 xa0 then the value of x is
Correct Answer

(C) 2

Explanation

Solution: $$eqalign{
& { ext{Given}},{left( {frac{a}{b}}
ight)^{x - 1}} = {left( {frac{b}{a}}
ight)^{x - 3}} cr
& Rightarrow {left( {frac{a}{b}}
ight)^{x - 1}} = {left( {frac{a}{b}}
ight)^{ - left( {x - 3}
ight)}} = {left( {frac{a}{b}}
ight)^{left( {3 - x}
ight)}} cr
& Rightarrow x - 1 = 3 - x cr
& Rightarrow 2x = 4 cr
& Rightarrow x = 2 cr} $$

[#112] Given that 10 0.48 = x , 10 0.70 = y and x z = y 2 , then the value of z is close to:
Correct Answer

(C) 2.9

Explanation

Solution: $$eqalign{
& {x^z} = {y^2} Leftrightarrow {10^{left( {0.48z}
ight)}} = {10^{2 imes 0.70}} = {10^{1.40}} cr
& Rightarrow 0.48z = 1.40 cr
& Rightarrow z = frac{{140}}{{48}} = frac{{35}}{{12}} = 2.9({ ext{approx}}) cr} $$

[#113] If 5 a = 3125, then the value of 5 ( a - 3) is:
Correct Answer

(A) 25

Explanation

Solution: 5 a = 3125 xa0 xa0 ⇔ xa0 xa0 5 a = 5 5 ⇒ a = 5. ∴ 5 ( a - 3) = 5 (5 - 3) = 5 2 = 25

[#114] If 3 ( x - y ) = 27 and 3 ( x + y ) = 243, then x is equal to:
Correct Answer

(C) 4

Explanation

Solution: 3 x - y = 27 = 3 3 ⇔ x - y = 3 ....(i) 3 x + y = 243 = 3 5 ⇔ x + y = 5 ....(ii) On solving (i) and (ii), we get x = 4.

[#115] (256) 0.16 × (256) 0.09 = ?
Correct Answer

(A) 4

Explanation

Solution: (256) 0.16 × (256) 0.09 = (256) (0.16 + 0.09) $$eqalign{
& = {left( {256}
ight)^{0.25}} cr
& = {left( {256}
ight)^{frac{{25}}{{100}}}} cr
& = {left( {256}
ight)^{frac{1}{4}}} cr
& = {left( {{4^4}}
ight)^{frac{1}{4}}} cr
& = {4^1} cr
& = 4 cr} $$