Let $$\alpha \in (0,\infty )$$ and $$A=\left[\begin{array}{lll}1& 2& \alpha \\ 1& 0& 1\\ 0& 1& 2\end{array}\right]$$. If $$\det \left(\mathrm{adj}\left(2A-A^T\right)\cdot \mathrm{adj}\left(A-2A^T\right)\right)=2^8$$, then $$(\det (A){)}^2$$ is equal to:

If the system of equations $$\begin{array}{rl}& x+(\sqrt{2}\sin \alpha )y+(\sqrt{2}\cos \alpha )z=0\\ & x+(\cos \alpha )y+(\sin \alpha )z=0\\ & x+(\sin \alpha )y-(\cos \alpha )z=0\end{array}$$ has a non-trivial solution, then $$\alpha \in \left(0,\frac{\pi }{2}\right)$$ is equal to :

Let $$A=\left[\begin{array}{ll}1& 2\\ 0& 1\end{array}\right]$$ and $$B=I+\mathrm{adj}(A)+(\mathrm{adj}A{)}^2+\dots +(\mathrm{adj}A{)}^{10}$$. Then, the sum of all the elements of the matrix $$B$$ is:

If $$\alpha \ne \mathrm{a},\beta \ne \mathrm{b},\gamma \ne \mathrm{c}$$ and $$\left|\begin{array}{lll}\alpha & \mathrm{b}& \mathrm{c}\\ \mathrm{a}& \beta & \mathrm{c}\\ \mathrm{a}& \mathrm{b}& \gamma \end{array}\right|=0$$, then $$\frac{\mathrm{a}}{\alpha -\mathrm{a}}+\frac{\mathrm{b}}{\beta -\mathrm{b}}+\frac{\gamma }{\gamma -\mathrm{c}}$$ is equal to :

If the system of equations $$x+4y-z=\lambda ,7x+9y+\mu z=-3,5x+y+2z=-1$$ has infinitely many solutions, then $$(2\mu +3\lambda )$$ is equal to :

Let $$A=\left[\begin{array}{lll}2& a& 0\\ 1& 3& 1\\ 0& 5& b\end{array}\right]$$. If $$A^3=4A^2-A-21I$$, where $$I$$ is the identity matrix of order $$3\times 3$$, then $$2a+3b$$ is equal to

Let A and B be two square matrices of order 3 such that $$|A|=3$$ and $$|B|=2$$. Then $$|{\mathrm{A}}^{\mathrm{T}}\mathrm{A}(\mathrm{adj}(2\mathrm{A}){)}^{-1}(\mathrm{adj}(4\mathrm{B}))(\mathrm{adj}(\mathrm{AB}){)}^{-1}{\mathrm{AA}}^{\mathrm{T}}|$$ is equal to :

If the system of equations $$\begin{array}{r}11x+y+\lambda z=-5\\ 2x+3y+5z=3\\ 8x-19y-39z=\mu \end{array}$$ has infinitely many solutions, then $${\lambda }^4-\mu$$ is equal to :

The values of $$m,n$$, for which the system of equations $$\begin{array}{rl}& x+y+z=4,\\ & 2x+5y+5z=17,\\ & x+2y+\mathrm{m}z=\mathrm{n}\end{array}$$ has infinitely many solutions, satisfy the equation :

Let $$\alpha \beta \ne 0$$ and $$A=\left[\begin{array}{rrr}\beta & \alpha & 3\\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{array}\right]$$. If $$B=\left[\begin{array}{rrr}3\alpha & -9& 3\alpha \\ -\alpha & 7& -2\alpha \\ -2\alpha & 5& -2\beta \end{array}\right]$$ is the matrix of cofactors of the elements of $$A$$, then $$\det (AB)$$ is equal to :

If $$A$$ is a square matrix of order 3 such that $$\det (A)=3$$ and $$\det \left(\mathrm{adj}\left(-4\mathrm{adj}\left(-3\mathrm{adj}\left(3\mathrm{adj}\left((2\mathrm{A}{)}^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}}{3}^{\mathrm{n}}$$, then $$\mathrm{m}+2\mathrm{n}$$ is equal to :

For $$\alpha ,\beta \in \mathbb{R}$$ and a natural number $$n$$, let $$A_r=\left|\begin{array}{ccc}r& 1& \frac{n^2}{2}+\alpha \\ 2r& 2& n^2-\beta \\ 3r-2& 3& \frac{n(3n-1)}{2}\end{array}\right|$$. Then $$2A_{10}-A_8$$ is

Let Ajay will not appear in JEE exam with probability $\mathrm{p}=\frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $\mathrm{q}=\frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is :

A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is :

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $$x$$ to be the number of rotten apples in a draw of two apples, the variance of $$x$$ is

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is

An integer is chosen at random from the integers $$1,2,3,\dots ,50$$. The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is