AIEEE 2003 (Maths) Mock Test

The two lines $x=ay+b$, $z=cy+d$ and $x=a^{\prime }y+b^{\prime }$, $z=c^{\prime }y+d^{\prime }$ will be perpendicular, if and only if:

The shortest distance from the plane $12x+4y+3z=327$ to the sphere $x^2+y^2+z^2+4x-2y-6z=155$ is:

Two systems of rectangular axes have the same origin. If a plane cuts them at distances $a,b,c$ and $a^{\prime },b^{\prime },c^{\prime }$ from the origin, then:

$\vec{a},\vec{b},\vec{c}$ are 3 vectors, such that $\vec{a}+\vec{b}+\vec{c}=0$, $|\vec{a}|=1,|\vec{b}|=2,|\vec{c}|=3$, then $\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ is equal to:

If $\vec{u},\vec{v}$ and $\vec{w}$ are three non-coplanar vectors, then $(\vec{u}+\vec{v}-\vec{w})\cdot [(\vec{u}-\vec{v})\times (\vec{v}-\vec{w})]$ equals: