Question 1 :
<b>Binomial theorem<br/><br/></b> If $(x+a)^{n}=x^{n}+^{n}C_{1}x^{n-1}a+^{n}C_{2}x^{n-2}a^{2}+...+a^{n}$, where n is a positive integer. (r + 1)th term is called the general term and is usually denoted by $u_{r+1\cdot }$<br/><b><br/></b>$\therefore u_{r+1}=^{n}C_{r}x^{n-r}a^{r}$ ,<br/><b><br/></b> Then the expansion contains (n+1) terms.

Question 3 :
If $n$ is a positive integer, then the number of terms in the expansion of $ (x+a)^n $ is<br/>

Question 4 :
Let $P(n)$ be the statement $&#34;3^n&gt;n&#34;$. If $P(n)$ is true, $P(n+1)$ is true.

Question 5 :
Let $P(n)$ be the statement $2^{n}&lt;n!$ where $n$ is a natural number, then $P(n)$ is true for:

Question 6 :
The number of terms in the expansion of $ (1+x)^{21} $ is <br/>

Question 8 :
<div>A bag contains $3$ red and $2$ black balls. One ball is drawn from it at random. Find the probability of drawing red ball is $\dfrac 35$</div>

Question 10 :
Let $P(n)$ be a statement and $P(n)=P(n+1) \forall n\in N$, then $P(n)$ is true for what values of $n$?<br>

Question 11 :
State whether the following statement is true or false.<br/>cos x + cos 2x + .... + cos nx =<br/>$\dfrac{cos\left (\dfrac{n \, + \, 1 }{2} \right )x sin \dfrac{nx}{2}}{sin\dfrac{x}{2}}$

Question 12 :
If n is a natural number then $\left( \displaystyle \frac{n + 1}{2} \right)^{n}\, \geq n!$ is true when.

Question 13 :
In the expansion of $\left (x + \dfrac {1}{x}\right )^{n}$, then the coefficient of the term indepenent of x is

Question 15 :
The number of terms with integral coefficient in the expansion of $ \displaystyle \left ( \left ( 27 \right )^{\dfrac 16}+\sqrt[10]{32} x\right )^{600} $ is