2 C_0 + 2^2 2 C_1 + 2^3 3 C_2 + .... + 2^ 11 11 C_ 10 = . . . - Vidyadip

MCQ

$2{C_0} + \frac{{{2^2}}}{2}{C_1} + \frac{{{2^3}}}{3}{C_2} + .... + \frac{{{2^{11}}}}{{11}}{C_{10}}$ = . . .

✓
$\frac{{{3^{11}} - 1}}{{11}}$
B
$\frac{{{2^{11}} - 1}}{{11}}$
C
$\frac{{{{11}^3} - 1}}{{11}}$
D
$\frac{{{{11}^2} - 1}}{{11}}$

✓

Answer

Correct option: A.

$\frac{{{3^{11}} - 1}}{{11}}$

a
(a) We have ${(1 + x)^{10}} = {C_0} + {C_1}x + {C_2}{x^2} + ... + {C_{10}}{x^{10}}$

Integrating both sides from $0$ to $2$, we get

$\frac{{{3^{11}} - 1}}{{11}} = 2{C_0} + \frac{{{2^2}}}{2}{C_1} + \frac{{{2^3}}}{3}{C_2} + .... + \frac{{{2^{11}}}}{{11}}{C_{10}}$.

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 11 - 7. binomial theoram→STD 11 - 7. binomial theoram — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

The roots of the equation $a({x^2} + 1) - ({a^2} + 1)x = 0$ are

The cube roots of unity when represented on the Argand plane form the vertices of an [IIT 1988; Pb. CET 2004]

The number of solutions of the equation $\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0, x\,>\,0$, is $....$

The equation of latus rectum of a parabola is $x + y = 8$ and the equation of the tangent at the vertex is $x + y = 12$, then length of the latus rectum is

If the chord $y = mx + 1$ of the circle ${x^2} + {y^2} = 1$ subtends an angle of measure ${45^o}$ at the major segment of the circle then value of $m$ is

The sum of $100$ observations and the sum of their squares are $400$ and $2475$, respectively. Later on, three observations, $3, 4$ and $5$, were found to be incorrect . If the incorrect observations are omitted, then the variance of the remaining observations is

If $\alpha $ and $\beta $ be the roots of the equation $x^2 - 2x + 2 = 0$ , then the least value of $n$ for ${\left( {\frac{\alpha }{\beta }} \right)^n} = 1$ is

Suppose $\theta $ and $\phi (\ne 0)$ are such that $sec\,(\theta + \phi ),$ $sec\,\theta $ and $sec\,(\theta - \phi )$ are in $A.P.$ If $cos\,\theta = k\,cos\,( \frac {\phi }{2})$ for some $k,$ then $k$ is equal to

If one of the diameters of the circle $x^2+y^2-10 x+$ $4 y+13=0$ is a chord of another circle $C,$ whose center is the point of intersection of the lines $2 x+$ $3 y=12$ and $3 x-2 y=5$, then the radius of the circle $\mathrm{C}$ is

For an integer $n$ let $S_n=\{n+1, n+2, \ldots \ldots, n+18\}$. Which of the following is true for all $n \geq 10$ ?