The value of p for which the function f(x) = l ( 4^x - 1) ^3 x p … - Vidyadip

MCQ

The value of $p$ for which the function $f(x) = \left\{ \begin{array}{l}\frac{{{{({4^x} - 1)}^3}}}{{\sin \frac{x}{p}\log \left[ {1 + \frac{{{x^2}}}{3}} \right]}},\,x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,12{(\log 4)^3},\,\,x = 0\end{array} \right.$ may be continuous at $x = 0$, is

A
$1$
B
$2$
C
$3$
✓
$4$

✓

Answer

Correct option: D.

$4$

d
(d) For $f(x)$ to be continuous at $x = 0,$ we should have $\mathop {\lim }\limits_{x \to 0} \,\,f(x) = f(0) = 12\,{(\log \,4)^3}$

$\mathop {\lim }\limits_{x \to 0} \,\,f(x) = \mathop {\lim }\limits_{x \to 0} \,{\left( {\frac{{{4^x} - 1}}{x}} \right)^3} \times \frac{{\left( {\frac{x}{p}} \right)}}{{\left( {\sin \frac{x}{p}} \right)}}.\frac{{p{x^2}}}{{\log \,\left( {1 + \frac{1}{3}{x^2}} \right)}}$

$ = {(\log 4)^3}\,.\,1\,.\,p\,.\mathop {\lim }\limits_{x \to 0} \,\left( {\frac{{{x^2}}}{{\frac{1}{3}{x^2} - \frac{1}{{18}}{x^4} + .........}}} \right)$

$ = 3p\,\,{(\log 4)^3}.$ Hence $p = 4.$

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

Choose the correct answer from the given four options.
Family $\text{y}=\text{A}\text{x}+\text{A}^3$ of curves will correspond to a differential equation of order:

3
2
1
Not defined

If the length of a vector be $21$ and direction ratios be $2, -3, 6$ then its direction cosines are

If A = [a_ij] is a square matrix of even order such that a_ij = i² - j², then

A is a skew-symmetric matrix and |A| = 0
A is symmetric matrix and |A| is a square
A is symmetric matrix and |A| = 0
None of these.

Which of the following transformation reduce the differential quation into the form $\frac{\text{du}}{\text{dx}}+\text{P}(\text{x})\text{u}=\text{Q}(\text{x})$ into the from $\frac{\text{dz}}{\text{dx}}+\frac{\text{z}}{\text{x}}\log\text{z}=\frac{\text{z}}{\text{x}^{2}}(\log\text{z})^{2}$

$\text{u}=\log\text{x}$
$\text{u}=\text{e}^{\text{z}}$
$\text{u}=(\log\text{z})^{-1}$
$\text{u}=(\log\text{z})^{2}$

A solid hemisphere is attached to the top of a cylinder, having the same radius as that of the cylinder. If the height of the cylinder were doubled (keeping both radii fixed), the volume of the entire system would have increased by $50\,\%$. By what percentage would the volume have increased if the radii of the hemisphere and the cylinder were doubled (keeping the height fixed)?

The differential equation of all parabolas whose axes are parallel to $y$-axis is

If $a, b, c $ are any three coplanar unit vectors, then

The area bounded by the curves $y = \ln x$, $y = \ln |x|$, $y = \,|\ln x|$ and $y = \,|\ln |x||$ is ......... $sq. \,unit$

In tossing $10$ coins, the probability of getting exactly $5$ heads is

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three units vectors such that $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .$ If $\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} $ and $\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},$ then the ordered pair $(\lambda, {\mathrm{\vec d}})$ is equal to