Three Dimensional Geometry - Algebra - Mathematics Previous Year Questions

If \((3,4,-1)\) and \((-1,2,3)\) be end points of the diameter of a sphere, then the radius of the sphere is

The position vector of a point \(R\) which divides the line joining \(P(6,3,-2)\) and \(Q(3,1,-4)\) in the ratio \(2 : 1\) externally is

The following lines are

$$\begin{aligned} \[\mathbf{r} & =(\hat{\mathbf{i}}+\hat{\mathbf{j}})+\lambda^{\prime}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}), \\\] \[\text { and } \quad \mathbf{r} & =(\hat{\mathbf{i}}+\hat{\mathbf{j}})+\mu(-\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}})\] \end{aligned}$$

The angle between the lines \(\frac{x-5}{-3}=\frac{y+3}{-4}=\frac{z-7}{0}, \frac{x}{1}=\frac{y-1}{-2}=\frac{z-6}{2}\) is

The image of the point \((2,-3,4)\) with respect to the plane \(4 x+2 y-4 z+3=0\), is

If a variable plane cuts the coordinate axes in \(A, B, C\) and is at constant distance $p$ from the origin, then the locus of the centroid of the tetrahedron \(A B C\) is equal to

A force of magnitude \(\sqrt{6}\) acting along, the line joining points \(A(2,-1,1)\) and \(B(3,1,2)\) displaces a particle from \(A\) to \(B\). The work done by the force is

If \(\theta_1, \theta_2\) and \(\theta_3\) are the angles made by a line with the positive direction of \(X, Y\) and \(Z\)-axes, then \(\cos 2 \theta_1+\cos 2 \theta_2+\cos 2 \theta_3\) is equal to

The plane is perpendicular to the planes \(x-y+2 z-4=0\) and \(2 x-2 y+z=0\) and passes through \((1,-2,1)\) is

The distance between the point \((7,2,4)\) and the plane determined by the points \((2,5,-3),(-2,-3,5)\) and \((5,3,-3)\) is

The image of the point \((2,3,7)\) in the plane \(2 x+5 y-3 z-19=0\), is

If the distance between the planes $8 x+12 y-14 z=2$ and $4 x+6 y-7 z=2$ can be expressed as $\frac{1}{\sqrt{N}}$, then the value of $\frac{N(N+1)}{2}$ is

The value of ' $a$ ' for which the lines $\frac{x-2}{1}=\frac{y-9}{2}=\frac{z-13}{3}$ and $\frac{x-a}{-1}=\frac{y-7}{2}$ $=\frac{z+2}{-3}$ intersect, is