There are five students S₁, S₂, S₃, S₄ and S₅ in a music class and for them there are five seats R₁, R₂, R₃, R₄ and R₅ arranged in a row, where initially the seat R_i is allotted to the student S_i, i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

(There are two questions based on Paragraph "A", the question given below is one of them)

The probability that, on the examination day, the student S₁ gets the previously allotted seat R₁, and NONE of the remaining students gets the seat previously allotted to him/her is

There are five students S₁, S₂, S₃, S₄ and S₅ in a music class and for them there are five seats R₁, R₂, R₃, R₄ and R₅ arranged in a row, where initially the seat R_i is allotted to the student S_i, i = 1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

(There are two questions based on Paragraph "A", the question given below is one of them)

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

Two vectors $$\overrightarrow A $$ and $$\overrightarrow B $$ are defined as $$\overrightarrow A $$ $$=$$ $$a\widehat i$$ and $$\overrightarrow B = a$$ $$\left( {\cos \,\omega T\widehat i + \sin \,\omega t\,\widehat j} \right),$$ where $$a$$ is a constant and $$\omega = \pi /6\,\,rad{s^{ - 1}}.$$ If $$\left| {\overrightarrow A + \overrightarrow B } \right| = \sqrt 3 \left| {\overrightarrow A - \overrightarrow B } \right|$$ at time $$t = \tau $$ for the first time, the value of $$\tau ,$$ in second, is ______________.

Consider a body of mass $$1.0$$ $$kg$$ at rest at the origin at time $$t=0.$$ A force $$\overrightarrow F = \left( {\alpha t \widehat i + \beta \widehat j} \right)$$ is applied on the body, where $$\alpha = 1.0N{s^{ - 1}}$$ and $$\beta = 1.0\,N.$$ The torque acting on the body about the origin at time $$t=1.0s$$ is $$\overrightarrow \tau .$$ Which of the following statements is (are) true?