Number of ordered pairs (x, y) of positive integers satisfying 2^x+3^y=5^{x y} is

English

Gujarati

Hindi

4-2.Quadratic Equations and Inequations

hard

The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x+3^y=5^{x y}$ is

$1$

$2$

$5$

Infinite

(KVPY-2020)

Solution

(a)

$\because 2^x+3^y=5^{x y}$

When $x=y=1$, then $2+3=5$ and $\left(\frac{2}{5}\right)^x \cdot \frac{1}{5^y}+\left(\frac{3}{5}\right)^y \cdot \frac{1}{5^x}=1$ here for any $x, y \in Z^{+}$LHS is not equal to 1 as both are less than $\frac{1}{2}$.

$\therefore$ Only one ordered pair $(1,1)$ is possible.

Standard 11

Mathematics

Suppose $a, b, c$ are three distinct real numbers, let $P(x)=\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}+\frac{(x-a)(x-b)}{(c-a)(c-b)}$ When simplified, $P(x)$ becomes

normal

(KVPY-2011)

If $\alpha , \beta , \gamma $ are roots of equation ${x^3} + a{x^2} + bx + c = 0$, then ${\alpha ^{ – 1}} + {\beta ^{ – 1}} + {\gamma ^{ – 1}} = $

easy

In the real number system, the equation $\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1$ has

hard

(KVPY-2012)

The set of all $a \in R$ for which the equation $x | x -1|+| x +2|+a=0$ has exactly one real root is:

hard

(JEE MAIN-2023)

The solutions of the quadratic equation ${(3|x| – 3)^2} = |x| + 7$ which belongs to the domain of definition of the function $y = \sqrt {x(x – 3)} $ are given by

hard

The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x+3^y=5^{x y}$ is

Solution

Similar Questions

Suppose $a, b, c$ are three distinct real numbers, let $P(x)=\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}+\frac{(x-a)(x-b)}{(c-a)(c-b)}$ When simplified, $P(x)$ becomes

If $\alpha , \beta , \gamma $ are roots of equation ${x^3} + a{x^2} + bx + c = 0$, then ${\alpha ^{ – 1}} + {\beta ^{ – 1}} + {\gamma ^{ – 1}} = $

In the real number system, the equation $\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1$ has

The set of all $a \in R$ for which the equation $x | x -1|+| x +2|+a=0$ has exactly one real root is:

The solutions of the quadratic equation ${(3|x| – 3)^2} = |x| + 7$ which belongs to the domain of definition of the function $y = \sqrt {x(x – 3)} $ are given by

Start a Free Trial Now