Embibe Experts Solutions for Chapter: Differential Equations, Exercise 1: JEE Advanced Paper 2 - 2020

For any real numbers $\alpha$ and $\beta$, let $y_{\alpha ,\beta }\left(x\right),x\in R$, be the solution of the differential equation

$\frac{dy}{dx}+\alpha y=x{e}^{\beta x},y\left(1\right)=1$

Let $S=\left\{y_{\alpha ,\beta }\left(x\right) :\alpha ,\beta \in R\right\}$. Then which of the following functions belong(s) to the set $S$?

Let $b$ be a nonzero real number. Suppose $f:\mathrm{ℝ}\to \mathrm{ℝ}$ is a differentiable function such that $f\left(0\right)=1$. If the derivative $f^{\text{'}}$ of $f$ satisfies the equation $f^{\text{'}}\left(x\right)=\frac{f\left(x\right)}{b^2+x^2}$ for all $x\in \mathrm{ℝ},$ then which of the following statements is/are TRUE?

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\mathrm{ℝ}\to \mathrm{ℝ}$ be functions satisfying $f\left(x+y\right)=f\left(x\right)+f\left(y\right)+f\left(x\right)f\left(y\right)\text{ and }f\left(x\right)=xg\left(x\right)$ for all $x,y\in \mathrm{ℝ}.$ If $\underset{x\to 0}{\lim }g\left(x\right)=1,$ then which of the following statements is/are TRUE?

Let $f:\left(0, \pi \right)\to \mathbb{R}$ be a twice differentiable function such that $\underset{t\to x}{\lim }\left\{\frac{f\left(x\right)\sin t-f\left(t\right)\sin x}{t-x}\right\}={\sin }^{2}x$ for all $x\in \left(0,\pi \right)$. If $f\left(\frac{\pi }{6}\right)=-\frac{\pi }{12}$, then which of the following statement(s) is (are) TRUE ?

Let $f: R\to R$ be a differentiable function with $f\left(0\right)=0$. If $y=f\left(x\right)$ satisfies the differential equation $\frac{dy}{dx}=\left(2+5y\right)\left(5y-2\right).$ If the value of  $\underset{x\to -\infty }{\lim }f\left(x\right)=\lambda$, then $10\lambda$ is equal to

Let $f:R\to R$ be a differentiable function with $f\left(0\right)=1$ and satisfying the equation $f\left(x+y\right)=f\left(x\right)f^{\text{'}}\left(y\right)+f^{\text{'}}\left(x\right)f\left(y\right)$ for all $x, y\in R$. Then, the value of ${\log }_e\left(f\left(4\right)\right)$ is 

If $y=y\left(x\right)$ satisfies the differential equation $8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy={\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)}^{-1}dx$ and $x>0, y\left(0\right)=\sqrt{7},$ then $y\left(256\right)=$

Let $f:\mathbb{R}\to \mathbb{R}\mathbb{ }\mathrm{a}\mathrm{n}\mathrm{d} g:\mathbb{R}\to \mathbb{R}$ be two non-constant differentiable functions. If $f^{\prime }\left(x\right)=\left(e^{\left(f\left(x\right)\right)-g\left(x\right)}\right)g^{\prime }\left(x\right) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} x\in \mathbb{R}$, and $f\left(1\right)=g\left(2\right)=1$ then which of the following statement(s) is (are) TRUE?