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

MCQ

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 $ . . . . .$

A
$4$
✓
$5$
C
$6$
D
$7$

✓

Answer

Correct option: B.

$5$

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

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

A bag contains six red four green and eight white balls If a ball is picked at random the probability that it is not white is:

A particle starts at the origin and moves along the $x$-axis in such a way that its velocity at the point $(x, 0)$ is given by the formula $\frac{{dx}}{{dt}} = {\cos ^2}\pi x.$ Then the particle never reaches the point on

Let $A=\left[a_{i j}\right]$ be a square matrix of order $3$ such that $a_{i j}=2^{j-i}$, for all $i, j=1,2,3$. Then, the matrix $A ^{2}+ A ^{3}+\ldots+ A ^{10}$ is equal to

If $A$ and $B $ be symmetric matrices of the same order, then $AB - BA$ will be a

The positive value of the determinant of the matrix $A$, whose $A d j(A d j(A))=\left(\begin{array}{ccc}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{array}\right)$, is

Let $f\left( x \right) = \frac{x}{{\sqrt {{a^2} + {x^2}} }} - \frac{{d - x}}{{\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},x \in R\,$, where $a, b$ and $d$ are non -zero real constant. Then

Let $\text{A}=\begin{bmatrix}\text{a}&0&0\\0&\text{a}&0\\0&0&\text{a}\end{bmatrix},$ then $A^n$ is equal to:

Choose the correct answer from the given four options.The general solution of $\frac{\text{dy}}{\text{dx}}=2\text{x}\ \text{e}^{\text{x}^{2}-\text{y}}$ is:

For every point P(x, y, z) on the xy-plane,

If $\int\frac{1}{5+4\sin\text{x}}\text{ dx}=\text{A}\tan^{-1}\Big(\text{B}\tan\frac{\pi}{2}+\frac{4}{3}\Big)+\text{C},$ then: