JEE (Advanced) 2019 Paper-1

Let S be the sample space of all $$3 \times 3$$ matrices with entries from the set {0,1}. Let the events $$E_1$$ and $$E_2$$ be given by
$$E_1 = \left\{A \in S : \det A = 0\right\}$$ and
$$E_2 =$$ {$$A \in S$$ : sum of entries of A is 7}.
If a matrix is chosen at random from S, then the conditional probability $$P(E_1|E_2)$$ equals _______

Let the point B be the reflection of the point A(2, 3) with respect to the line 8x — 6y — 23 = 0. Let $$\lceil_A$$ and $$\lceil_B$$ be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles $$\lceil_A$$ and $$\lceil_B$$ 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 __________.

If
$$I = \frac{2}{\pi} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac {1}{(1 + e^{\sin x})(2 - \cos 2x)}$$
then 27 $$I^2$$ equals _______ .

Three lines are given by
$$\overrightarrow{r} = \lambda \widehat{i}, \lambda \in R$$
$$\overrightarrow{r} = \mu (\widehat{i} + \widehat{j}), \mu \in R$$ and
$$\overrightarrow{r} = v (\widehat{i} + \widehat{j} + \widehat{k} ), v \in R$$.
Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the area of the triangle ABC is $$\triangle$$ then the value of $$(6\triangle)^2$$ equals__________ .