JEE (Advanced) 2018 Paper-1

Let $$E_{1} E_{2}$$ and $$F_{1} F_{2}$$ be the chords of 𝑆 passing through the point $$P_{0}$$ (1, 1) and parallel to the x-axis and the y-axis, respectively. Let $$G_{1}G_{2}$$ be the chord of S passing through $$P_{0}$$ and having slope −1. Let the tangents to 𝑆 at $$E_{1}$$ and $$E_{2}$$ meet at $$E_{3}$$, the tangents to S at $$F_{1}$$ and $$F_{2}$$ meet at $$F_{3}$$, and the tangents to S at $$G_{1}$$ and $$G_{2}$$ meet at $$G_{3}$$. Then, the points $$E_{3}$$, $$F_{3}$$ and $$G_{3}$$ lie on the curve

Let 𝑃 be a point on the circle 𝑆 with both coordinates being positive. Let the tangent to 𝑆 at 𝑃 intersect the coordinate axes at the points 𝑀 and 𝑁. Then, the mid-point of the line segment 𝑀𝑁 must lie on the curve

The probability that, on the examination day, the student $$S_{1}$$ gets the previously allotted seat $$𝑅_{1}$$, and NONE of the remaining students gets the seat previously allotted to him/her is

For 𝑖 = 1, 2, 3, 4, let $$𝑇_{𝑖}$$ denote the event that the students $$𝑆_{𝑖}$$ and $$𝑆_{𝑖}$$+1 do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $$𝑇_{1} \cap 𝑇_{2} \cap 𝑇_{3} \cap 𝑇_{4}$$ is