Let $$A=\left(\begin{array}{cc}1& 2\\ -2& -5\end{array}\right)$$. Let $$\alpha ,\beta \in \mathbb{R}$$ be such that $$\alpha A^2+\beta A=2I$$. Then $$\alpha +\beta$$ is equal to

Let $$A=\left(\begin{array}{rr}4& -2\\ \alpha & \beta \end{array}\right)$$. If $${\mathrm{A}}^2+\gamma \mathrm{A}+18\mathrm{I}=\mathrm{O}$$, then $$\det (\mathrm{A})$$ is equal to _____________.

Let the matrix $$A=\left[\begin{array}{lll}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$$ and the matrix $$B_0=A^{49}+2A^{98}$$. If $$B_n=Adj\left(B_{n-1}\right)$$ for all $$n\ge 1$$, then $$\det \left(B_4\right)$$ is equal to :

Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be any two $$3\times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Let A and B be two $$3\times 3$$ non-zero real matrices such that AB is a zero matrix. Then

Which of the following matrices can NOT be obtained from the matrix $$\left[\begin{array}{cc}-1& 2\\ 1& -1\end{array}\right]$$ by a single elementary row operation ?

If the system of equations $$\begin{array}{rl}& x+y+z=6\\ & 2x+5y+\alpha z=\beta \\ & x+2y+3z=14\end{array}$$ has infinitely many solutions, then $$\alpha +\beta$$ is equal to

The probability that a randomly chosen 2 $$\times$$ 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :

The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to :

The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) $$-$$ f(c) = f(d) is :

The probability, that in a randomly selected 3-digit number at least two digits are odd, is :

If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x $$-$$ 6y = 30, then the probability that y

Five numbers $$x_1,x_2,x_3,x_4,x_5$$ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order $$({x_1}

Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(x = 4), then the sum of the mean and the variance of X is :

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is :

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark n is {1 \over n}. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is :

Let E1 and E2 be two events such that the conditional probabilities P({E_1}|{E_2}) = {1 \over 2}, P({E_2}|{E_1}) = {3 \over 4} and P({E_1} \cap {E_2}) = {1 \over 8}. Then :

A random variable X has the following probability distribution : .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top} X 0 1 2 3 4 P(X) k 2k 4k 6k 8k The value of P(1

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is {6 \over {11}}, then n is equal to __________.