The number of symmetric matrices of order 3, with all the entries from the set $$\{0,1,2,3,4,5,6,7,8,9\}$$ is :

Let $$A=\left[\begin{array}{cc}1& \frac{1}{51}\\ 0& 1\end{array}\right]$$. If $$\mathrm{B}=\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right]A\left[\begin{array}{cc}-1& -2\\ 1& 1\end{array}\right]$$, then the sum of all the elements of the matrix \sum_\limits{n=1}^{50} B^{n} is equal to

If the system of linear equations $$\begin{array}{rl}& 7x+11y+\alpha z=13\\ & \\ & 5x+4y+7z=\beta \\ & \\ & 175x+194y+57z=361\end{array}$$ has infinitely many solutions, then $$\alpha +\beta +2$$ is equal to :

$$\left|\begin{array}{ccc}x+1& x& x\\ x& x+\lambda & x\\ x& x& x+{\lambda }^2\end{array}\right|=\frac{9}{8}(103x+81)$$, then $$\lambda ,\frac{\lambda }{3}$$ are the roots of the equation :

Let $$\mathrm{A}$$ be a $$2\times 2$$ matrix with real entries such that $${\mathrm{A}}^{\prime }=\alpha \mathrm{A}+\mathrm{I}$$, where $$\alpha \in \mathbb{R}-\{-1,1\}$$. If $$\det \left(A^2-A\right)=4$$, then the sum of all possible values of $$\alpha$$ is equal to :

If $$\mathrm{A}=\frac{1}{5!6!7!}\left[\begin{array}{ccc}5!& 6!& 7!\\ 6!& 7!& 8!\\ 7!& 8!& 9!\end{array}\right]$$, then $$|\mathrm{adj}(\mathrm{adj}(2\mathrm{A}))|$$ is equal to :

If A is a 3 $$\times$$ 3 matrix and $$|A|=2$$, then $$|3adj(|3A|A^2)|$$ is equal to :

For the system of linear equations $$2x-y+3z=5$$ $$3x+2y-z=7$$ $$4x+5y+\alpha z=\beta$$, which of the following is NOT correct?

If $$A=\left[\begin{array}{cc}1& 5\\ \lambda & 10\end{array}\right],{\mathrm{A}}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$$ and $$\alpha +\beta =-2$$, then $$4{\alpha }^2+{\beta }^2+{\lambda }^2$$ is equal to :

Let S be the set of all values of $$\theta \in [-\pi ,\pi ]$$ for which the system of linear equations $$x+y+\sqrt{3}z=0$$ $$-x+(\tan \theta )y+\sqrt{7}z=0$$ $$x+y+(\tan \theta )z=0$$ has non-trivial solution. Then \frac{120}{\pi} \sum_\limits{\theta \in \mathrm{s}} \theta is equal to :

Let $$A=\left[\begin{array}{ccc}2& 1& 0\\ 1& 2& -1\\ 0& -1& 2\end{array}\right]$$. If $$|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}2A))|=(16{)}^n$$, then $$n$$ is equal to :

Let $$P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right],A=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]$$ and $$Q=PA{P}^T$$. If $$P^T{Q}^{2007}P=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$$, then $$2a+b-3c-4d$$ equal to :

If the system of equations $$x+y+az=b$$ $$2x+5y+2z=6$$ $$x+2y+3z=3$$ has infinitely many solutions, then $$2a+3b$$ is equal to :

Let $$\mathrm{A}={\left[{\mathrm{a}}_{\mathrm{ij}}\right]}_{2\times 2}$$, where $${\mathrm{a}}_{\mathrm{ij}}\ne 0$$ for all $$\mathrm{i},\mathrm{j}$$ and $${\mathrm{A}}^2=\mathrm{I}$$. Let a be the sum of all diagonal elements of $$\mathrm{A}$$ and $$\mathrm{b}=|\mathrm{A}|$$. Then $$3a^2+4b^2$$ is equal to :

Let $$P$$ be a square matrix such that $$P^2=I-P$$. For $$\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$$, if $$P^{\alpha }+P^{\beta }=\gamma I-29P$$ and $$P^{\alpha }-P^{\beta }=\delta I-13P$$, then $$\alpha +\beta +\gamma -\delta$$ is equal to :

For the system of equations $$x+y+z=6$$ $$x+2y+\alpha z=10$$ $$x+3y+5z=\beta$$, which one of the following is NOT true?

Two dice are thrown independently. Let $$\mathrm{A}$$ be the event that the number appeared on the $$1^{\text{st }}$$ die is less than the number appeared on the $$2^{\text{nd }}$$ die, $$\mathrm{B}$$ be the event that the number appeared on the $$1^{\text{st }}$$ die is even and that on the second die is odd, and $$\mathrm{C}$$ be the event that the number appeared on the $$1^{\text{st }}$$ die is odd and that on the $$2^{\text{nd }}$$ is even. Then :

In a binomial distribution $$B(n,p)$$, the sum and the product of the mean and the variance are 5 and 6 respectively, then $$6(n+p-q)$$ is equal to :

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is :

If an unbiased die, marked with $$-2,-1,0,1,2,3$$ on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is :