If A = \left[ {\matrix{ {\cos \theta } & { - \sin \theta } \cr {\sin \theta } & {\cos \theta } \cr } } \right], then the matrix A–50 when $$\theta$$ = $$\pi \frac{1}{2}$$, is equal to :

The system of linear equations x + y + z = 2 2x + 3y + 2z = 5 2x + 3y + (a2 – 1) z = a + 1 then

If the system of linear equations x $$-$$ 4y + 7z = g 3y $$-$$ 5z = h $$-$$2x + 5y $$-$$ 9z = k is consistent, then :

If A = \left[ {\matrix{ {{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr {{e^t}} & { - {e^{ - t}}\cos t - {e^{ - t}}\sin t} & { - {e^{ - t}}\sin t + {e^{ - t}}co{\mathop{\rm s}\nolimits} t} \cr {{e^t}} & {2{e^{ - t}}\sin t} & { - 2{e^{ - t}}\cos t} \cr } } \right] then A is :

If the system of equations x + y + z = 5 x + 2y + 3z = 9 x + 3y + az = $$\beta$$ has infinitely many solutions, then $$\beta$$ $$-$$ $$\alpha$$ equals -

Let d $$\in$$ R, and A = \left[ {\matrix{ { - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr 1 & {\left( {\sin \theta } \right) + 2} & d \cr 5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \right) + 2 + 2d} \cr } } \right], $$\theta \in \left[0,2\pi \right]$$ If the minimum value of det(A) is 8, then a value of d is -

Let A = \left[ {\matrix{ 2 & b & 1 \cr b & {{b^2} + 1} & b \cr 1 & b & 2 \cr } } \right] where b > 0. Then the minimum value of {{\det \left( A \right)} \over b} is -

The number of values of $$\theta$$ $$\in$$ (0, $$\pi$$) for which the system of linear equations x + 3y + 7z = 0 $$-$$ x + 4y + 7z = 0 (sin3$$\theta$$)x + (cos2$$\theta$$)y + 2z = 0. has a non-trival solution, is -

If the system of linear equations 2x + 2y + 3z = a 3x – y + 5z = b x – 3y + 2z = c where a, b, c are non zero real numbers, has more one solution, then :

Let A = \left( {\matrix{ 0 & {2q} & r \cr p & q & { - r} \cr p & { - q} & r \cr } } \right). If AAT = I3, then $$\left|p\right|$$ is :

The set of all values of $$\lambda$$ for which the system of linear equations x – 2y – 2z = $$\lambda$$x x + 2y + z = $$\lambda$$y – x – y = $$\lambda$$z has a non-trivial solutions :

Let A and B be two invertible matrices of order 3 $$\times$$ 3. If det(ABAT) = 8 and det(AB–1) = 8, then det (BA–1 BT) is equal to :

An ordered pair ($$\alpha$$, $$\beta$$) for which the system of linear equations (1 + $$\alpha$$) x + $$\beta$$y + z = 2 $$\alpha$$x + (1 + $$\beta$$)y + z = 3 $$\alpha$$x + $$\beta$$y + 2z = 2 has a unique solution, is :

Let P = \left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 9 & 3 & 1 \cr } } \right] and Q = [qij] be two 3 $$\times$$ 3 matrices such that Q – P5 = I3. Then {{{q_{21}} + {q_{31}}} \over {{q_{32}}}} is equal to :

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is :

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate solve any problem is {4 \over 5} , then the probability that he is unable to solve less than two problems is :

A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is :

Let a random variable X have a binomial distribution with mean 8 and variance 4. If P\left( {X \le 2} \right) = {k \over {{2^{16}}}}, then k is equal to :

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :