AIEEE 2009 PYQ Mock Test

If $A,B$ and $C$ are three sets such that $A\cap B=A\cap C$ and $A\cup B=A\cup C$, then:

Let $\vec{u},\vec{v},\vec{w}$ be non-coplanar vectors and $p,q$ are real numbers, then the equality $[3\vec{u}p\vec{v}p\vec{w}]-[p\vec{v}\vec{w}q\vec{u}]-[2\vec{w}q\vec{v}q\vec{u}]=0$ holds for:

Let the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies in the plane $x+3y-\alpha z+\beta =0$. Then $(\alpha ,\beta )$ equals:

From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is:

${\int }_0^{\pi }[\cot x]dx$, where $[\cdot ]$ denotes the greatest integer function, is equal to: