91Ó°ÊÓ

If the distance between the planes $$ 8 x+12 y-14 z=2 \text { and } 4 x+6 y-7 z=2 $$ can be expressed in the form \(\frac{1}{\sqrt{N}}\), where \(N\) is natural, then the value of \(\frac{N(N+1)}{2}\) is (a) 4950 (b) 5050 (c) 5150 (d) 5151

A plane passes through the points \(P(4,0,0)\) and \(Q(0,0,4)\) and is parallel to the \(Y\)-axis. The distance of the plane from the origin is (a) 2 (b) 4 (c) \(\sqrt{2}\) (d) \(2 \sqrt{2}\)

If from the point \(P(f, g, h)\) perpendiculars \(P L\) and \(P M\) be drawn to \(y z\) and \(z x\)-planes, then the equation to the plane \(O L M\) is (a) \(\frac{x}{f}+\frac{y}{g}-\frac{z}{h}=0\) (b) \(\frac{x}{f}+\frac{y}{g}+\frac{z}{h}=0\) (c) \(\frac{x}{f}-\frac{y}{g}+\frac{z}{h}=0\) (d) \(-\frac{x}{f}+\frac{y}{g}+\frac{z}{h}=0\)

Let \(A B C D\) be a tetrahedron such that the edges \(A B, A C\) and \(A D\) are mutually perpendicular. Let the area of \(\triangle A B C, \triangle A C D\) and \(\triangle A D B\) be 3,4 and 5 sq units, respectively. Then, the area of the \(\triangle B C D\), is (a) \(5 \sqrt{2}\) (b) 5 (c) \(5 / \sqrt{2}\) (d) \(\frac{5}{2}\)

Equation of the line which passes through the point with position vector \((2,1,0)\) and perpendicular to the plane containing the vectors \(\mathbf{i}+\mathbf{j}\) and \(\mathbf{j}+\mathbf{k}\) is (a) \(\mathrm{r}=(2,1,0)+t(1,-1,1)\) (b) \(\mathbf{r}=(2,1,0)+t(-1,1,1)\) (c) \(\mathbf{r}=(2,1,0)+t(1,1,-1)\) (d) \(\mathbf{r}=(2,1,0)+t(1,1,1)\) Where, \(t\) is a parameter.

Which of the following planes are parallel but not identical? $$ \begin{aligned} &P_{1}: 4 x-2 y+6 z=3 \\ &P_{2}: 4 x-2 y-2 z=6 \\ &P_{3}:-6 x+3 y-9 z=5 \\ &P_{4}: 2 x-y-z=3 \end{aligned} $$ (a) \(P_{2}\) and \(P_{3}\) (b) \(P_{2}\) and \(P_{4}\) (c) \(P_{1}\) and \(P_{3}\) (d) \(P_{1}\) and \(P_{4}\)

A parallelopiped is formed by planes drawn through the points \((1,2,3)\) and \((9,8,5)\) parallel to the coordinate planes, then which of the following is not the length of an edge of this rectangular parallelopiped ? (a) 2 (b) 4 (c) 6 (d) 8

A parallelopiped is formed by planes drawn through the points \((1,2,3)\) and \((9,8,5)\) parallel to the coordinate planes, then which of the following is not the length of an edge of this rectangular parallelopiped ? (a) 2 (b) 4 (c) 6 (d) 8

The vector equations of the two lines \(L_{1}\) and \(L_{2}\) are given by \(L_{1}: \mathbf{r}=(2 \mathbf{i}+9 \mathbf{j}+13 \mathbf{k})+\lambda(\mathbf{i}+2 \mathbf{j}+3 \mathbf{k})\) and \(L_{2}: \mathbf{r}=(-3 \mathbf{i}+7 \mathbf{j}+p \mathbf{k})+\mu(-\mathbf{i}+2 \mathbf{j}-3 \mathbf{k})\). Then, the lines \(L_{1}\) and \(L_{2}\) are (a) skew lines for all \(p \in R\) (b) intersecting for all \(p \in R\) and the point of intersection is \((-1,3,4)\) (c) intersecting lines for \(p=-2\) (d) intersecting for all real \(p \in R\)

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 (a) \(-5\) (b) \(-2\) (c) 5 (d) \(-3\)