Let be a non-zero real number. Suppose f: R R is a differentiable… - Vidyadip

MCQ

Let $\alpha$ be a non-zero real number. Suppose $f: \mathrm{R} \rightarrow$ $\mathrm{R}$ is a differentiable function such that $f(0)=2$ and $\lim _{\mathrm{x} \rightarrow-\infty} \mathrm{f}(\mathrm{x})=1$. If $f^{\prime}(\mathrm{x})=\alpha f(x)+3$, for all $\mathrm{x} \in \mathrm{R}$, then $f\left(-\log _e 2\right)$ is equal to . . . . . . . . .

✓
$3$
B
$5$
C
$9$
D
$7$

✓

Answer

Correct option: A.

$3$

a
$ f(0)=2, \lim _{x \rightarrow-\infty} f(x)=1 $
$ f^{\prime}(x)-\alpha \cdot f(x)=3 $
$ \text { I.F }=e^{-\alpha x} $
$ y\left(e^{-\alpha x}\right)=\int 3 \cdot e^{-\alpha x} d x $
$ f(x) \cdot\left(e^{-\alpha x}\right)=\frac{3 e^{-\alpha x}}{-\alpha}+c $
$ x=0 \Rightarrow 2=\frac{-3}{\alpha}+c \Rightarrow \frac{3}{\alpha}=c-2 $ $....(1)$
$ f(x)=\frac{-3}{\alpha}+c \cdot e^{\alpha x}$
Case-I $\alpha > 0$
$\mathrm{x} \rightarrow-\infty \Rightarrow 1=\frac{-3}{\alpha}+\mathrm{c}(0)$
$\alpha=-3 \quad$ (rejected)
Case-II $\alpha < 0$
as $\lim _{x \rightarrow-\infty} \mathrm{f}(\mathrm{x})=1 \Rightarrow \mathrm{c}=0$ and $\frac{-3}{\alpha}=1 \Rightarrow \alpha=-3$
$\Rightarrow \mathrm{f}(\mathrm{x})=1 \quad \text { (rejected) }$
as $f(0)=2$
$\Rightarrow$ data is inconsistent

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\frac{d y}{d x}+\frac{2^{x-y}\left(2^{y}-1\right)}{2^{x}-1}=0, x, y>0, y(1)=1$, then $y (2)$ is equal to

Number of divisors of $n = 38808$ (except $1$ and $n$) is

If line $x + \alpha y + \beta = 0$ touches curve $4x^3 + 4y^3 = xy(xy + 16)$ at points $(x_1, y_1)$ and $\left( {{x_2},{y_2}} \right),{x_1} \ne {x_2},$ then value of $\alpha + \beta $ is $\left( {\alpha ,\beta \in R} \right)$

If two straight lines whose direction cosines are given by the relations $l+m-n=0,3l^{2}+m^{2}+c n l =0$ are parallel, then the positive value of $c$ is

If $x = \sqrt {{2^{\cos e{c^{ - 1}}t}}} $ and $y = \sqrt {{2^{se{c^{ - 1}}t}}} (\left| t \right|\,\, \ge \,1\,),$ then $\frac{{dy}}{{dx}}$ is equal to.

If $G(x) = - \sqrt {25 - {x^2}} $, then $\mathop {\lim }\limits_{x \to 1} \frac{{G(x) - G(1)}}{{x - 1}} = $

If the function $f(x)=\left\{\begin{array}{ll}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ a \log _e 2 \log _c 3 & , x=0\end{array}\right.$ is continuous at $x = 0,$ then the value of $a ^2$ is equal to

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

If a circle cuts a rectangular hyperbola $xy = {c^2}$ in $A, B, C, D$ and the parameters of these four points be ${t_1},\;{t_2},\;{t_3}$ and ${t_4}$ respectively. Then

The median of a set of $9$ distinct observations is $20.5$. If each of the largest $4$ observation of the set is increased by $2$, then the median of the new set