Embibe Experts Solutions for Exercise 2: Assignment

If the orthocentre and centroid of a triangle are $\left(-3,5,1\right)$ and $\left(3,3,-1\right)$ respectively, then its circumcentre is

The circumcentre of triangle $ABC$ with vertices $\left(0,7,10\right),\left(-1,6,6\right)$ and $\left(-4,9,6\right)$ is $\left(\alpha ,\beta ,\gamma \right)$ then $\left|\alpha ·\beta ·\gamma \right|$ is equal to

If $A,B,C,D$ are points $\left(4,5,3\right),\left(6,3,4\right),\left(2,4,-1\right),\left(0,5,1\right)$ respectively, then length of projection of $CD$ on $AB$ is equal to $\lambda$ then $6\lambda$ is

The points $\left(8,-5,6\right),\left(11,1,8\right),\left(9,4,2\right)$ and $\left(6,-2,0\right)$ are vertices of a cyclic quadrilateral and its area is $k$ sq. units, then the value of $k$ is

There area of the triangle whose vertices are the points $\left(2,1,1\right),\left(3,1,2\right),\left(-4,0,1\right)$ is $\lambda$ square units then ${\left(2\lambda \right)}^2$ is equal to

If $\alpha ,\beta ,\gamma$ are the angle made by line joining points $A\left(3,5,-2\right)$ and $B\left(-1,3,-4\right)$ then the value of ${\left(\cos 2\alpha +\cos 2\beta +\cos 2\gamma \right)}^2+3\left({\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma \right)$ is equal to

The sum of values of $\lambda$, for which the triangle with vertices $\left(6,10,10\right),\left(1,0,-5\right)$ and $\left(6,-10,\lambda \right)$ will be right-angled is $k$, then $3k$ is equal to

The distance of point $P\left(5,-7,-6\right)$, from the point $Q$ which divides line segment joining points $A\left(-2,3,5\right)$ and $B\left(7,0,-1\right)$ in ratio $1:2$ internally is $\lambda$ then ${\lambda }^2$ is equal to