Let f be a differentiable function satisfying f ( x )=2 3 _ 0 ^ 3… - Vidyadip

MCQ

Let $f$ be a differentiable function satisfying $f ( x )=\frac{2}{\sqrt{3}} \int_{0}^{\sqrt{3}} f \left(\frac{\lambda^{2} x }{3}\right) d \lambda, x >0$ and $f (1)=\sqrt{3}$. If $y=f(x)$ passes through the point $(\alpha, 6)$, then $\alpha$ is equal to $.........$

A
$6$
✓
$12$
C
$4$
D
$3$

✓

Answer

Correct option: B.

$12$

b
Let, $\frac{\lambda^{2} x }{3}= t$

$\frac{2 \lambda x }{3} d \lambda= dt$

$d \lambda=\frac{3}{2} \cdot \frac{1 \sqrt{ x }}{ x \cdot \sqrt{3} \sqrt{ t }} dt$

$d \lambda=\frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{ x }} \cdot \frac{ dt }{\sqrt{ t }}$

So, $f(x)=\frac{1}{\sqrt{x}} \int_{0}^{x} \frac{f(t)}{\sqrt{t}} d t$

$\sqrt{x} \cdot f^{\prime}(x)+\frac{f(x)}{2 \sqrt{x}}=\frac{f(x)}{\sqrt{x}}$

$\sqrt{ x } \cdot f ^{\prime}( x )=\frac{ f ( x )}{2 \sqrt{ x }}$

$\frac{ dy }{ y }=\frac{ dx }{2 x }$

$\ln y=\frac{1}{2} \ln x+c \Rightarrow f(x)=\sqrt{x}$

$y =\sqrt{3 x } \quad$ $\{as$ $f (1)=\sqrt{3}\}$

So, $f(x)=\sqrt{3 x}$

Now, $f(\alpha)=6 \Rightarrow 36=3 \alpha$

$\alpha=12$

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 STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

If points $(1, 2), (3, 5)$ and $(0, b)$ are collinear the value of $b$ is :

Let $z=\frac{-1+\sqrt{3} i}{2}$, where $i=\sqrt{-1}$, and $r, s \in\{1,2,3\}$. Let$P=\left[\begin{array}{cc}(-z)^r & z^{2 s} \\ z^{2 s} & z^r\end{array}\right]$ and $I$ be the identity matrix of order $2$ . Then the total number of ordered pairs $(r, s)$ for which $P^2=-I$ is

Integrate the following functions with respect to x: $\int\frac{\text{dx}}{4\text{x}+5}$

The radius of a circle is increasing at the rate of $0.4 \ cm/ s.$ The rate of increasing of its circumference is:

If the system of linear equations $x + y + z = 5$ ; $x = 2y + 2z = 6$ ; $x + 3y + \lambda z = u (\lambda \, \mu \in R)$, has infinitely many solutions then the value of $\lambda + \mu $ is

Find the principal value of $\tan ^{-1}(-\sqrt{3})$

Solution of $(xy\cos xy + \sin xy)dx + {x^2}\cos xy\,dy = 0$ is

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x ^2- px +\frac{5}{4} p =0$ are rational. Then the area of the region $\{(x, y): 0 \leq y$ $\left.\leq(x-q)^2, 0 \leq x \leq q\right\}$ is

$\int_{}^{} {{{(\sec x + \tan x)}^2}dx = } $

$\big(\vec{\text{a}}+2\vec{\text{b}}-\vec{\text{c}}\big).\big\{\big(\vec{\text{a}}-\vec{\text{b}}\big)\times\big(\vec{\text{a}}-\vec{\text{b}}-\vec{\text{c}}\big)\big\}$ is equal to: