Let S be set of all real roots of equation, 3^{x}left(3^{x}-1right)+2=left|3^{x}-1right|+left|3^{x}-

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4-2.Quadratic Equations and Inequations

hard

Let $S$ be the set of all real roots of the equation, $3^{x}\left(3^{x}-1\right)+2=\left|3^{x}-1\right|+\left|3^{x}-2\right| .$ Then $\mathrm{S}$

is an empty set.

contains at least four elements.

contains exactly two elements

is a singleton

(JEE MAIN-2020)

Solution

Let $3^{x}=t ; t>0$

$t(t-1)+2=|t-1|+|t-2|$

$t^{2}-t+2=|t-1|+|t-2|$

Case$-I $: $t<1$

$t^{2}-t+2=1-t+2-t$

$t^{2}+2=3-t$

$t^{2}+t-1=0$

$\mathrm{t}=\frac{-1 \pm \sqrt{5}}{2}$

$\mathrm{t}=\frac{\sqrt{5}-1}{2}$ is only acceptable

Case-II $: 1 \leq t<2$

$t^{2}-t+2=t-1+2-t$

$t^{2}-t+1=0$

$\mathrm{D}<0 $ no real solution

Case-III : $t \geq 2$

$t^{2}-t+2=t-1+t-2$

$\mathrm{t}^{2}-3 \mathrm{t} \quad 5=0 \Rightarrow \mathrm{D}<0$ no real solution

Standard 11

Mathematics

Let $A=\left\{x \in(0, \pi)-\left\{\frac{\pi}{2}\right\}: \log _{(2 / \pi)}|\sin x|+\log _{(2 / \pi)}|\cos x|=2\right\}$ and $B=\{x \geq 0: \sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}$. Then $n(A \cup B)$ is equal to:

normal

(JEE MAIN-2025)

The number of ordered pairs $(x, y)$ of real numbers that satisfy the simultaneous equations $x+y^2=x^2+y=12$ is

normal

(KVPY-2015)

If the product of roots of the equation ${x^2} – 3kx + 2{e^{2\log k}} – 1 = 0$ is $7$, then its roots will real when

medium

(IIT-1984)

Let $\mathrm{S}$ be the set of positive integral values of $a$ for which $\frac{\mathrm{ax}^2+2(\mathrm{a}+1) \mathrm{x}+9 \mathrm{a}+4}{\mathrm{x}^2-8 \mathrm{x}+32}<0, \forall \mathrm{x} \in \mathbb{R}$. Then, the number of elements in $\mathrm{S}$ is :

hard

(JEE MAIN-2024)

Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{x}^{2}-\mathrm{x}-1=0 .$ If $\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,$ then which one of the following statements is not true?

hard

(JEE MAIN-2020)

Let $S$ be the set of all real roots of the equation, $3^{x}\left(3^{x}-1\right)+2=\left|3^{x}-1\right|+\left|3^{x}-2\right| .$ Then $\mathrm{S}$

Solution

Similar Questions

Let $A=\left\{x \in(0, \pi)-\left\{\frac{\pi}{2}\right\}: \log _{(2 / \pi)}|\sin x|+\log _{(2 / \pi)}|\cos x|=2\right\}$ and $B=\{x \geq 0: \sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}$. Then $n(A \cup B)$ is equal to:

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Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{x}^{2}-\mathrm{x}-1=0 .$ If $\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,$ then which one of the following statements is not true?

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