Let f: R R be a function defined by f(x)=4^x 4^x+2 and M= _ f(a) … - Vidyadip

Question

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and $M=\int_{f(a)}^{f(1-a)} x \sin ^4(x(1-x)) d x,$ $N=\int_{f(a)}^{f(1-a)} \sin ^4(x(1-x)) d x ; a \neq \frac{1}{2} . \text { If }$ $\alpha \mathrm{M}=\beta \mathrm{N}, \alpha, \beta \in \mathbb{N}$, then the least value of $\alpha^2+\beta^2$ is equal to $ . . . . .$

✓

Answer

b
$f(a)+f(1-a)=1 .$

$M=\int_{f(a)}^{f(1-a)}(1-x) \cdot \sin ^4 x(1-x) d x$

$M=N-M \quad 2 M=N$

$\alpha=2 ; \beta=1 ;$

Ans. $5$

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

Let the functions $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{ll}x+2, & x<0 \\ x^{2}, & x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{lr}x^{3}, & x<1 \\ 3 x-2, & x \geq 1\end{array}\right.$

Then, the number of points in $R$ where $(fog)( x )$ is $NOT$ differentiable is equal to

Let $A$ denote the set of all $2-$digit numbers in base $10$ that are equal to four times the sum of the factorial of their digits. The sum of the numbers in $A$ is

If $A$ and $B$ are two events such that $A \subseteq B,$ then $P\,\left( {\frac{B}{A}} \right) = $

A stone thrown vertically upwards rises $ ‘s’$ metre in $t $ seconds, where $s = 80t - 16{t^2}$, then the velocity after $2 $ seconds is .......... $m\, per\, sec$

If the area of the region $\left\{( x , y ): x ^{\frac{2}{3}}+ y ^{\frac{2}{3}} \leq 1 x + y \geq 0, y \geq 0\right\}$ is $A$, then $\frac{256 A }{\pi}$

If the coefficients of the three consecutive terms in the expansion of $(1+ x )^{ n }$ are in the ratio $1: 5: 20$, then the coefficient of the fourth term is $............$.

The distance of the point $ - \hat i + 2\hat j + 6\hat k$ from the straight line that passes through the point $2\hat i + 3\hat j - 4\hat k$ and is parallel to the vector $6\hat i + 3\hat j - 4\hat k$ is

Two tangents are drawn from the point $\mathrm{P}(-1,1)$ to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}-6 \mathrm{y}+6=0$. If these tangents touch the circle at points $A$ and $B$, and if $D$ is a point on the circle such that length of the segments $A B$ and $A D$ are equal, then the area of the triangle $A B D$ is eqaul to:

Let $P$ be a non-zero polynomial such that $P(1+x)=P(1-x)$ for all real $x$ and $P(1)=0$. Let $m$ be the largest integer such that $(x-1)^m$ divides $P(x)$ for all such $P(x)$. Then, $m$ equals

Let $n \in N$ and $[x]$ denote the greatest integer less than or equal to $x$. If the sum of $(n+1)$ terms ${ }^{n} C_{0}, 3 .{ }^{n} C_{1}, 5 .{ }^{n} C_{2}, 7 .{ }^{n} C_{3}, \ldots$ is equal to $2^{100} \cdot 101$, then $2\left[\frac{n-1}{2}\right]$ is equal to $....$