class 11

Math

Algebra

Binomial Theorem

If the third term in expansion of $(1+x_{log_{2}x})_{5}$ is $2560$ then $x$ is equal to (a) $22 $ (b) $81 $ (c) $41 $ (d) $42 $

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Using binomial theorem, write down the expansions of the following: $(2x−3y)_{4}$

If $x_{p}$ occurs in the expansion of $(x_{2}+1/x)_{2n}$ , prove that its coefficient is $[31 (4n−p)]![31 (2n+p)]!(2n)! $ .

Let $S_{1}=0≤i<j≤100∑∑ C_{i}C_{j}$, $S_{2}=0≤j<i≤100∑∑ C_{i}C_{j}$ and $S_{3}=0≤i=j≤100∑∑ C_{i}C_{j}$ where $C_{r}$ represents cofficient of $x_{r}$ in the binomial expansion of $(1+x)_{100}$ If $S_{1}+S_{2}+S_{3}=a_{b}$ where $a$, $b∈N$, then the least value of $(a+b)$ is

Find an approximation of $(0.99)_{5}$ using the first three terms of its expansion.

If $N$ is a prime number which divides $S=_{39}P_{19}+_{38}P_{19}+_{37}P_{19}+…+_{20}P_{19}$, then the largest possible value of $N$ among following is

Consider a $G.P.$ with first term $(1+x)_{n}$, $∣x∣<1$, common ratio $21+x $ and number of terms $(n+1)$. Let $_{′}S_{′}$ be sum of all the terms of the $G.P.$, then The coefficient of $x_{n}$ is $_{′}S_{′}$ is

The remainder when $27_{10}+7_{51}$ is divided by $10$

$(_{m}C_{0}+_{m}C_{1}−_{m}C_{2}−_{m}C_{3})+(_{′m}C_{4}+_{m}C_{5}−_{m}C_{6}−_{m}C_{7})+..=0$ if and only if for some positive integer $k$, $m=$