Q. Find the number of ways of arranging the letters of the word, $CORRUPTION$.

Q. If $-1,2$ and $\alpha$ are the roots of the equation $2x^3+x^2-7x-6=0$, then find $\alpha$.

1. $-\frac{3}{2}$
2. $\frac{3}{2}$
3. $-\frac{1}{2}$
4. $\frac{1}{2}$

Q. If $A=\left[\begin{array}{ccc}1& 2& 1\\ -1& 0& 2\\ 0& 1& -1\end{array}\right]$, find the rank of $A$.

Q. Find the probability that a non-leap year contain $53$ Sundays.

Q. If $A=\left[\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}3& 2& 1\\ 1& 2& 3\end{array}\right]$, find $3B-2A$.

Q. If ${}^n{P}_5:{}^{n-1}{P}_5=3:2$, find $n$.

Q. If the letters of the word $EAMCET$ are permitted in all possible ways and if the words thus formed are arranged in the dictionary order, find the rank of the word $EAMCET$.

Q. Simplify: ${{}^{34}C}_5+{\sum }_{r=0}^4{{}^{(38-r)}C}_4$.

1. ${}^{38}{C}_4$
2. ${}^{39}{C}_5$
3. ${}^{39}{C}_4$
4. ${}^{38}{C}_5$

Q. If $X$ is a Poisson's variate such that $P(X=1)=3P(X=2)$, then find the variance of $X$.

1. $\frac{3}{8}$
2. $\frac{1}{3}$
3. $\frac{5}{4}$
4. $\frac{2}{3}$

Q. Solve the equation $x^4-10x^3+26x^2-10x+1=0$.

Q. Let $A$ and $B$ be independent events with $P\left(A\right)=0.2,P\left(B\right)=0.5$. Then find:
(i) $P\left(A|B\right)$
(ii) $P\left(B|A\right)$
(iii) $P(A\cap B)$
(iv) $P(A\cup B)$

Q. Write the middle terms in the expansion of ${(\frac{3x}{7}-2y)}^{10}$.

Q. Write the coefficient of $x^n$ in the expansion of $e^{a-bx}$.

Q. Find the maximum or minimum value of the quadratic expression $2x-7-5x^2$.