If the system of equations $$\begin{array}{rl}& x+2y-3z=2\\ & 2x+\lambda y+5z=5\\ & 14x+3y+\mu z=33\end{array}$$ has infinitely many solutions, then $\lambda +\mu$ is equal to :

Let α be a solution of $x^2+x+1=0$, and for some a and b in $R,\left[\begin{array}{ccc}4& a& b\end{array}\right]\left[\begin{array}{ccc}1& 16& 13\\ -1& -1& 2\\ -2& -14& -8\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$. If $\frac{4}{{\alpha }^4}+\frac{m}{{\alpha }^a}+\frac{n}{{\alpha }^b}=3$, then m + n is equal to _______

Let $A=\left[\begin{array}{ccc}2& 2+p& 2+p+q\\ 4& 6+2p& 8+3p+2q\\ 6& 12+3p& 20+6p+3q\end{array}\right]$. If $\det (\text{adj}(\text{adj}(3A)))=2^m\cdot 3^n$, $m,n\in \mathbb{N}$, then $ m + n $ is equal to

Let $\mathrm{A}=\left[\begin{array}{cc}\alpha & -1\\ 6& \beta \end{array}\right],\alpha >0$, such that $\det (\mathrm{A})=0$ and $\alpha +\beta =1$. If I denotes $2\times 2$ identity matrix, then the matrix $(I+A{)}^8$ is :

Let $a\in R$ and $A$ be a matrix of order $3\times 3$ such that $\det (A)=-4$ and $A+I=\left[\begin{array}{lll}1& a& 1\\ 2& 1& 0\\ a& 1& 2\end{array}\right]$, where $I$ is the identity matrix of order $3\times 3$. If $\det ((a+1)\mathrm{adj}((a-1)A))$ is $2^{\mathrm{m}}{3}^{\mathrm{n}},\mathrm{m}$, $\mathrm{n}\in \{0,1,2,\dots ,20\}$, then $\mathrm{m}+\mathrm{n}$ is equal to :

If the system of linear equations $$\begin{array}{rl}& 3x+y+\beta z=3\\ & 2x+\alpha y-z=-3\\ & x+2y+z=4\end{array}$$ has infinitely many solutions, then the value of $22\beta -9\alpha$ is :

Let the system of equations x + 5y - z = 1 4x + 3y - 3z = 7 24x + y + λz = μ λ, μ ∈ ℝ, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy 7 ≤ x + y + z ≤ 77, is :

Let $A$ be a $3\times 3$ matrix such that $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}\mathrm{A}))|=81$. If $S=\left\{n\in \mathbb{Z}:(|\mathrm{adj}(\mathrm{adj}A)|{)}^{\frac{(n-1{)}^2}{2}}=|A{|}^{\left(3n^2-5n-4\right)}\right\}$, then \sum_\limits{n \in S}\left|A^{\left(n^2+n\right)}\right| is equal to :

Let the system of equations : $$\begin{array}{rl}& 2x+3y+5z=9\\ & 7x+3y-2z=8\\ & 12x+3y-(4+\lambda )z=16-\mu \end{array}$$ have infinitely many solutions. Then the radius of the circle centred at $(\lambda ,\mu )$ and touching the line $4 x=3 y$ is :

If the system of equations $$\begin{array}{rl}& 2x+\lambda y+3z=5\\ & 3x+2y-z=7\\ & 4x+5y+\mu z=9\end{array}$$ has infinitely many solutions, then $\left({\lambda }^2+{\mu }^2\right)$ is equal to :

Let $A$ be a $3\times 3$ real matrix such that $A^2(A-2I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha ,\beta$, and $\gamma$ are real constants, then $\alpha +\beta +\gamma$ is equal to :

Let $A$ be a matrix of order $3\times 3$ and $|A|=5$. If $|2\mathrm{adj}(3A\mathrm{adj}(2A))|=2^{\alpha }\cdot 3^{\beta }\cdot 5^{\gamma },\alpha ,\beta ,\gamma \in N$, then $\alpha +\beta +\gamma$ is equal to

Let the matrix $A=\left[\begin{array}{lll}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A^2-I$ for $n⩾3$. Then the sum of all the elements of ${\mathrm{A}}^{50}$ is :

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is $\frac{29}{45}$, then n is equal to:

A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and ${\sigma }^2$ denote the mean and variance of $X$, then the value of $64\left(\mu +{\sigma }^2\right)$ is:

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd (m,n)=1$, then $m+n$ is equal to :

If $A$ and $B$ are two events such that $P(A\cap B)=0.1$, and $P(A\mid B)$ and $P(B\mid A)$ are the roots of the equation $12x^2-7x+1=0$, then the value of $\frac{P(\bar{A}\cup \bar{B})}{P(\bar{A}\cap \bar{B})}$ is :

Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag $B_2$ is:

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is