Let f: R R be defined as f(x)= ll x^ 3 (1- 2 x)^ 2 _ e ( 1+2 x e^… - Vidyadip

MCQ

Let $f: R \rightarrow R$ be defined as

$f(x)=\left\{\begin{array}{ll}\frac{x^{3}}{(1-\cos 2 x)^{2}} \log _{e}\left(\frac{1+2 x e^{-2 x}}{\left(1-x e^{-x}\right)^{2}}\right), & x \neq 0 \\ \,\alpha & , x=0\end{array}\right.$ If $\mathrm{f}$ is continuous at $\mathrm{x}=0$, then $\alpha$ is equal to :

✓
$1$
B
$0$
C
$3$
D
$2$

✓

Answer

Correct option: A.

$1$

a
For continuity

$\lim _{x \rightarrow 0} \frac{x^{3}}{4 \sin ^{4} x}\left(\ln \left(1+2 e^{-2 x}\right)-2 \ln \left(1-x e^{-x}\right)\right)$

$=\alpha$

$\lim _{x \rightarrow 0} \frac{1}{4 x}\left[2 x e^{-2 x}+2 x e^{-x}\right]=\alpha$

$=\frac{1}{4}(4)=\alpha=1$

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

More "M.C.Q (1 Marks)" from 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Consider the following statements in respect of the differential equation $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}+\cos\Big(\frac{\text{dx}}{\text{dy}}\Big)=0:$
1. The degree of the differential equation is not defined.
2. The order of the differential equation is 2.
Which of the above statements is/are correct ?

1 only
2 only
Both 1 and 2
not defined

A random variable has the following probability distribution:

X = x_i	0	1	2	3	4	5	6	7
P(X = X_i)	0	2p	2p	3p	p²	2p²	7p²	2p

$\frac{1}{10}$
$-1$
$-\frac{1}{10}$
$\frac{1}{5}$

The region formed by the inequalities $x, y \geq 0, y \leq 6, x+y \leq 3$ is $.....$

$\int {\frac{{{{\cot }^{ - 1}}({e^x})}}{{{e^x}}}} $ $dx$ is equal to -

$\left| {\,\begin{array}{*{20}{c}}1&{1 + ac}&{1 + bc}\\1&{1 + ad}&{1 + bd}\\1&{1 + ae}&{1 + be}\end{array}\,} \right| = $

If a spherical balloon has a variable diameter $3x + {9 \over 2}$, then the rate of change of its volume with respect to $x$ is

Let $f, g: R \to R$ be two functions defined by $f(x)\, = \,\left\{ {\begin{array}{*{20}{c}}
{x\,\sin \,\left( {\frac{1}{x}} \right),\,x\, \ne \,0\,\,\,\,\,\,\,\,\,\,}\\
{0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x\, = 0\,\,\,\,\,\,\,\,\,}
\end{array}} \right.,$ and $g(x) =x\,f(x)$

Statement $I:$ $f$ is a continuous function at $x = 0.$
Statement $II:$ $g$ is a differentiable function at $x = 0.$

The vector $\vec{\text{b}}=3\hat{\text{i}}+4\hat{\text{k}}$ is to be written as the sum of a vector $\vec{\alpha}$ parallel to $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}$ and a vector $\vec{\beta}$ perpendicular to $\vec{\text{a}}.$ Then $\vec{\alpha}=$

$\frac{3}{2}\big(\hat{\text{i}}+\hat{\text{j}}\big)$
$\frac{2}{3}\big(\hat{\text{i}}+\hat{\text{j}}\big)$
$\frac{1}{2}\big(\hat{\text{i}}+\hat{\text{j}}\big)$
$\frac{1}{3}\big(\hat{\text{i}}+\hat{\text{j}}\big)$

In transportation models designed in linear programming, points of demand is classified as:

Ordination
Transportation
Destinations
Origins

The value of $x$ for which the angle between the vectors $a = xi - 3j - k,\,\,b = 2xi + xj - k$ is acute and the angle between the vectors $b$ and the axis of ordinate is obtuse, are