91Ó°ÊÓ

\(A(3,2,0), B(5,3,2)\) and \(C(-9,6,-3)\) are the vertices of a triangle \(A B C\). If the bisector of \(\angle A B C\) meets \(\mathrm{BC}\) at \(\mathrm{D}\), then coordinates of \(D\) are (a) \(\left(\frac{19}{8}, \frac{57}{16}, \frac{17}{16}\right)\) (b) \(\left(-\frac{19}{8}, \frac{57}{16}, \frac{17}{16}\right)\) (c) \(\left(\frac{19}{8},-\frac{57}{16}, \frac{17}{16}\right)\) (d) None of these

The equation of the plane perpendicular to the line \(\frac{x-1}{1}, \frac{y-2}{-1}, \frac{z+1}{2}\) and passing through the point \((2,3,1)\), is \(\begin{array}{ll}\text { (a) } \mathbf{r} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})=1 & \text { (b) } \mathbf{r} .(\hat{\mathbf{i}}-\mathbf{j}+2 \hat{\mathbf{k}})=1\end{array}\) (c) \(\mathbf{r} \cdot(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})=7\) (d) None of these

The number of planes that are equidistant from four non-coplanar points is (a) 3 (b) 4 (c) 7 (d) 9

The shortest distance between the two lines \(L_{1}: x=k_{1}\); \(y=k_{2}\) and \(L_{2}: x=k_{3}: y=k_{4}\) is equal to (a) \(\left|\sqrt{k_{1}^{2}+k_{2}^{2}}-\sqrt{k_{3}^{2}+k_{4}^{2}}\right|\) (b) \(\sqrt{k_{1} k_{3}+k_{2} k_{4}}\) (c) \(\sqrt{\left(k_{1}+k_{3}\right)^{2}+\left(k_{2}+k_{4}\right)^{2}}\) (d) \(\sqrt{\left(k_{1}-k_{3}\right)^{2}+\left(k_{2}-k_{4}\right)^{2}}\)

In a regular tetrahedron, if the distance between the mid-points of opposite edges is unity, its volume is (a) \(\frac{1}{3}\) (b) \(\frac{1}{2}\) (c) \(\frac{1}{\sqrt{2}}\) (d) \(\frac{1}{6 \sqrt{2}}\)

Let \(\mathbf{A}\) be vector parallel to line of intersection of planes \(P_{1}\) and \(P_{2} .\) Plane \(P_{1}\) is parallel to the vectors \(2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}\) and that \(P_{2}\) is parallel to \(\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\), then the angle between vector \(\mathbf{A}\) and a given vector \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}\) is (a) \(\frac{\pi}{2}\) (b) \(\frac{\pi}{4}\) (c) \(\frac{\pi}{6}\) (d) \(\frac{3 \pi}{4}\)

A rod of length 2 units whose one end is \((1,0,-1)\) and other end touches the plane \(x-2 y+2 z+4=0\), then (a) The rod sweeps the figure whose volume is \(\pi\) cubic units. (b) The area of the region which the rod traces on the plane is \(2 \pi\) (c) The length of projection of the rod on the plane is \(\sqrt{3}\) units. (d) The centre of the region which the rod traces on the plane is \(\left(\frac{2}{3}, \frac{2}{3}, \frac{-5}{3}\right)\)

If the perpendicular distance of the point \((6,5,8)\) from the \(Y\)-axis is \(5 \lambda\) units, then \(\lambda\) is equal to .........

If the circumcentre of the triangle whose vertices are \((3,2,-5),(-3,8,-5)\) and \((-3,2,1)\) is \((-1, \lambda,-3)\) the integer \(\lambda\) must be equal to ..........

Find the angle between the lines whose direction cosines has the relation \(l+m+n=0\) and \(2 l^{2}+2 m^{2}-n^{2}=0 .\)