Let A = {X = (x, y, z)T: PX = 0 and x2 + y2 + z2 = 1} where P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} & 3 & { - 4} \cr 1 & 9 & { - 1} \cr } } \right], then the set A :

If $$\Delta$$ = \left| {\matrix{ {x - 2} & {2x - 3} & {3x - 4} \cr {2x - 3} & {3x - 4} & {4x - 5} \cr {3x - 5} & {5x - 8} & {10x - 17} \cr } } \right| = Ax3 + Bx2 + Cx + D, then B + C is equal to :

Let $$\alpha$$ be a root of the equation x2 + x + 1 = 0 and the matrix A = {1 \over {\sqrt 3 }}\left[ {\matrix{ 1 & 1 & 1 \cr 1 & \alpha & {{\alpha ^2}} \cr 1 & {{\alpha ^2}} & {{\alpha ^4}} \cr } } \right] then the matrix A31 is equal to

If the system of linear equations 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0, where a, b, c $$\in$$ R are non-zero distinct; has a non-zero solution, then:

Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i+j-2)aji, where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is:

For which of the following ordered pairs ($$\mu$$, $$\delta$$), the system of linear equations x + 2y + 3z = 1 3x + 4y + 5z = $$\mu$$ 4x + 4y + 4z = $$\delta$$ is inconsistent ?

The system of linear equations $$\lambda$$x + 2y + 2z = 5 2$$\lambda$$x + 3y + 5z = 8 4x + $$\lambda$$y + 6z = 10 has

If the matrices A = \left[ {\matrix{ 1 & 1 & 2 \cr 1 & 3 & 4 \cr 1 & { - 1} & 3 \cr } } \right], B = adjA and C = 3A, then {{\left| {adjB} \right|} \over {\left| C \right|}} is equal to :

If for some $$\alpha$$ and $$\beta$$ in R, the intersection of the following three places x + 4y – 2z = 1 x + 7y – 5z = b x + 5y + $$\alpha$$z = 5 is a line in R3, then $$\alpha$$ + $$\beta$$ is equal to :

The following system of linear equations 7x + 6y – 2z = 0 3x + 4y + 2z = 0 x – 2y – 6z = 0, has

Let S be the set of all $$\lambda$$ $$\in$$ R for which the system of linear equations 2x – y + 2z = 2 x – 2y + $$\lambda$$z = –4 x + $$\lambda$$y + z = 4 has no solution. Then the set S :

Let A be a 2 $$\times$$ 2 real matrix with entries from {0, 1} and |A| $$\ne$$ 0. Consider the following two statements : (P) If A $$\ne$$ I2 , then |A| = –1 (Q) If |A| = 1, then tr(A) = 2, where I2 denotes 2 $$\times$$ 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :

Let a, b, c $$\in$$ R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A = \left( {\matrix{ a & b & c \cr b & c & a \cr c & a & b \cr } } \right) satisfies ATA = I, then a value of abc can be :

If A = \left( {\matrix{ 2 & 2 \cr 9 & 4 \cr } } \right) and I = \left( {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right) then 10A–1 is equal to :

Let EC denote the complement of an event E. Let E1 , E2 and E3 be any pairwise independent events with P(E1) > 0 and P(E1 $$\cap$$ E2 $$\cap$$ E3) = 0. Then P($$E_2^C\cap E_3^C/E_1$$) is equal to :

The probabilities of three events A, B and C are given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5. If P(A$$\cup$$B) = 0.8, P(A$$\cap$$C) = 0.3, P(A$$\cap$$B$$\cap$$C) = 0.2, P(B$$\cap$$C) = $$\beta$$ and P(A$$\cup$$B$$\cup$$C) = $$\alpha$$, where 0.85 $$\le \alpha \le$$ 0.95, then $$\beta$$ lies in the interval :

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is :

In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws total a of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops as soon as either of the players wins. The probability of A winning the game is :

The probability that a randomly chosen 5-digit number is made from exactly two digits is :

A dice is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is :