If 72^x · 48^y=6^{x y} , where x and y are non-zero rational numbers, then x+y equals

English

Gujarati

Hindi

4-2.Quadratic Equations and Inequations

normal

If $72^x \cdot 48^y=6^{x y}$, where $x$ and $y$ are non-zero rational numbers, then $x+y$ equals

A

$3$

B

$\frac{10}{3}$

C

$-3$

D

$-\frac{10}{3}$

(KVPY-2017)

Solution

(d)

Given, $72^x \cdot 48^y=6^{x y}$

$\left(2^3 \cdot 3^2\right)^x \cdot\left(2^4 \cdot 3\right)^y=2^{x y} \cdot 3^{x y}$

$2^{3 x+4 y} \cdot 3^{2 x+y}=2^{x y} \cdot 3^{x y}$

Equating the exponent of $2$ and $3$ , we get

$3 x+4 y=x y$ and $2 x+y=x y$

On solving these equation, we get

$x=\frac{-15}{3}$ and $y=\frac{5}{3}$

$\therefore \quad x+y=\frac{-15}{3}+\frac{5}{3}=\frac{-10}{3}$

Standard 11

Mathematics

Share

Similar Questions

If $\alpha , \beta , \gamma$ are roots of equation $x^3 + qx -r = 0$ then the equation, whose roots are

$\left( {\beta \gamma + \frac{1}{\alpha }} \right),\,\left( {\gamma \alpha + \frac{1}{\beta }} \right),\,\left( {\alpha \beta + \frac{1}{\gamma }} \right)$

normal

Let $f(x)={{x}^{2}}-x+k-2,k\in R$ then the complete set of values of $k$ for which $y=\left| f\left( \left| x \right| \right) \right|$ is non-derivable at $5$ distinict points is

normal

If $a,b,c$ are distinct real numbers and $a^3 + b^3 + c^3 = 3abc$ , then the equation $ax^2 + bx + c = 0$ has two roots, out of which one root is

normal

If $2 + i$ is a root of the equation ${x^3} – 5{x^2} + 9x – 5 = 0$, then the other roots are

hard

The number of integers $a$ in the interval $[1,2014]$ for which the system of equations $x+y=a$, $\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$ has finitely many solutions is

normal

(KVPY-2014)