Consider the function f: R R defined by f(x)=2 x 1+9 x^2 . If the… - Vidyadip

MCQ

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $f, \underbrace{(f \circ f \circ f \circ \ldots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to....................

A
$1044$
B
$1075$
C
$1056$
✓
$1024$

✓

Answer

Correct option: D.

$1024$

d
$ \mathrm{f}(\mathrm{f}(\mathrm{x}))=\frac{2 \mathrm{f}(\mathrm{x})}{\sqrt{1+9 \mathrm{f}^2(\mathrm{x})}}=\frac{4 \mathrm{x}}{\sqrt{1+9 \mathrm{x}^2+9.2^2 \mathrm{x}^2}} $

$ \mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x})))=\frac{2^3 \mathrm{x} / \sqrt{1+9 \mathrm{x}^2}}{\sqrt{1+9\left(1+2^2\right) \frac{2^2 \mathrm{x}^2}{1+9 \mathrm{x}^2}}}=\frac{2^3 \mathrm{x}}{\sqrt{1+9 \mathrm{x}^2\left(1+2^2+2^4\right)}} $

$ \therefore \text { By observation } $

$ \alpha=1+2^2+2^4+\ldots+2^{18}=1\left(\frac{\left(2^2\right)^{10}-1}{2^2-1}\right)=\frac{2^{20}-1}{3} $

$ 3 \alpha+1=2^{20} \rightarrow \sqrt{3 \alpha+1}=2^{10}=1024$

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 - 1. relation and function→STD 12 - 1. relation and function — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

$\int_{0}^{1}\frac{(\tan^{-1}\text{x})^2}{1+\text{x}^2}\text{dx}=$

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $ = \sqrt 3 $ is

$\left| {\,\begin{array}{*{20}{c}}{a + b}&{a + 2b}&{a + 3b}\\{a + 2b}&{a + 3b}&{a + 4b}\\{a + 4b}&{a + 5b}&{a + 6b}\end{array}\,} \right| = $

The shortest distance between the lines $x+1=2 y=-$ $12 z$ and $x=y+2=6 z-6$ is

The equation of the plane parallel to the lines $x - 1 = 2y - 5 = 2z$ and $3x = 4y - 11 = 3z -4$ and passing through the point $(2, 3, 3)$ is :

A company has two plants $\mathrm{A}$ and $\mathrm{B}$ to manufacture motorcycles. $60 \%$ motorcycles are manufactured at plant $\mathrm{A}$ and the remaining are manufactured at plant B. $80 \%$ of the motorcycles manufactured at plant $\mathrm{A}$ are rated of the standard quality, while $90 \%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $\mathrm{B}$, then $126 \mathrm{p}$ is

Matrix $\text{A} = [\text{a}_\text{ij}]_{\text{m} \times \text{n}}$ is a square matrix if:

An ordered pair $(\alpha , \beta )$ for which the system of linear equations

$\left( {1 + \alpha } \right)x + \beta y + z = 2$ ; $\alpha x + \left( {1 + \beta } \right)y + z = 3$ ; $\alpha x + \beta y + 2z = 2$ has a unique solution, is

The principal solution of $\sin ^{-1}\left(\sin \left(\frac{5 \pi}{3}\right)\right)$ is

Let $f$ and $g$ be two functions defined by

$f(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\|x-1|, & x \geq 0\end{array} \text { and } g(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\1, & x \geq 0\end{array}\right. \text {. }\right.$

Then (gof) (x) is