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 ${(1 + x)^{10}} = {C_0} + {C_1}x + {C_2}{x^2} + ... + {C_{10}}{x^{10}}$
(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.
Explore more
Similar questions
If ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y + {\cos ^{ - 1}}z = \pi $, then
→$\mathop {{\rm{lim}}\,}\limits_{n \to \infty } \left[ {\frac{1}{{{n^2}}}{{\sec }^2}\frac{1}{{{n^2}}} + \frac{2}{{{n^2}}}{{\sec }^2}\frac{4}{{{n^2}}} + ..... + \frac{1}{n}{{\sec }^2}1} \right]$ equals
→The general solution of the differential equation $(x + y)dx + xdy = 0$ is
→If $f({x_1}) - f({x_2}) = f\left( {\frac{{{x_1} - {x_2}}}{{1 - {x_1}{x_2}}}} \right)$ for ${x_1},{x_2} \in [ - 1,\,1]$, then $f(x)$ is
→Value of $\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos (1 - \cos x)}}{{x\tan x - {x^2}}}$ IS
→If $P(A) = 2/3$, $P(B) = 1/2$ and ${\rm{ }}P(A \cup B) = 5/6$ then events $A$ and $B$ are
→If $f ( a + b - x )= f ( x ),$ then $\int_{a}^{b} x f(x) d x$ is equal to
→The centre of the circle, which cuts orthogonally each of the three circles ${x^2} + {y^2} + 2x + 17y + 4 = 0,$ ${x^2} + {y^2} + 7x + 6y + 11 = 0,$ ${x^2} + {y^2} - x + 22y + 3 = 0$ is
→How many even numbers of $3$ different digits can be formed from the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ (repetition is not allowed)
→Consider the $6 \times 6$ square in the figure. Let $A_1, \mathrm{~A}_2, \ldots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.
→(image)
($1$) Let $p_i$ be the probability that a randomly chosen point has $i$ many friends, $i=0,1,2,3,4$. Let $X$ be a random variable such that for $i=0,1,2,3,4$, the probability $P(X=i)=p_i$. Then the value of $7 E(X)$ is
($2$) Two distinct points are chosen randomly out of the points $A_1, A_2, \ldots, A_{4 g}$. Let $p$ be the probability that they are friends. Then the value of $7 p$ is