Let S be the sample space of all 3 $$ \times $$ 3 matrices with entries from the set {0, 1}. Let the events E₁ and E₂ be given by

E₁ = {A$$ \in $$S : det A = 0} and

E₂ = {A$$ \in $$S : sum of entries of A is 7}.

If a matrix is chosen at random from S, then the conditional probability P(E₁ | E₂) equals ...............

Let the point B be the reflection of the point A(2, 3) with respect to the line $$8x - 6y - 23 = 0$$. Let $$\Gamma_{A} $$ and $$\Gamma_{B} $$ be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles $$\Gamma_{A} $$ and $$\Gamma_{B} $$ such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is .................

Let $$\omega \ne 1$$ be a cube root of unity. Then the minimum of the set $$\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c$$ distinct non-zero integers} equals ..................

If $$I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}} $$, then 27I² equals .................