Chapter 22

A square \(A B C D\) of diagonal \(2 a\) is folded along the diagonal \(A C\) so that the planes \(D A C\) and \(B A C\) are at right angle. The shortest distance between \(D C\) and \(A B\) is (A) \(\sqrt{2} a\) (B) \(2 a / \sqrt{3}\) (C) \(2 a / \sqrt{5}\) (D) \((\sqrt{3} / 2) a\)

The line of intersection of the planes \(\mathbf{r} \cdot(3 \mathbf{i}-\mathbf{j}+\mathbf{k})=\) 1 and \(\mathbf{r} \cdot(\mathbf{i}+4 \mathbf{j}-2 \mathbf{k})=2\) is parallel to the vector (A) \(-2 \mathbf{i}+7 \mathbf{j}+13 \mathbf{k}\) (B) \(2 \mathbf{i}+7 \mathbf{j}-13 \mathbf{k}\) (C) \(-2 \mathbf{i}-7 \mathbf{j}+13 \mathbf{k}\) (D) \(2 \mathbf{i}+7 \mathbf{j}+13 \mathbf{k}\)

The smallest radius of the sphere passing through (1, \(0,0),(0,1,0)\) and \((0,0,1)\) is (A) \(\sqrt{\frac{2}{3}}\) (B) \(\sqrt{\frac{3}{8}}\) (C) \(\sqrt{\frac{5}{6}}\) (D) \(\sqrt{\frac{5}{12}}\)

Perpendicular distance of the point \((3,4,5)\) from the \(y\)-axis, is (A) \(\sqrt{34}\) (B) \(\sqrt{41}\) (C) 4 (D) 5

A plane passes through a fixed point \((a, b, c)\). The locus of the foot of the perpendicular to it from the origin is a sphere of radius (A) \(\sqrt{a^{2}+b^{2}+c^{2}}\) (B) \(\frac{1}{2} \sqrt{a^{2}+b^{2}+c^{2}}\) (C) \(a^{2}+b^{2}+c^{2}\) (D) none of these

A straight line \(\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}\) meets the \(p\) lane \(\mathbf{r} \cdot \mathbf{n}=0\) in \(P\). The position vector of \(P\) is (A) \(a+\frac{a \cdot n}{b \cdot n} b\) (B) \(\mathrm{a}-\frac{\mathbf{a} \cdot \mathbf{n}}{\mathbf{b} \cdot \mathbf{n}} \mathbf{b}\) (C) \(\mathrm{a}-\frac{\mathrm{a} \cdot \mathbf{n}}{\mathrm{b} \cdot \mathrm{n}} \mathbf{b}\) (D) none of these

From the point \((1,-2,3)\), lines are drawn to meet the sphere \(x^{2}+y^{2}+z^{2}=4\) and they are divided internally in the ratio \(2: 3\). The locus of the point of division is (A) \(5 x^{2}+5 y^{c}+5 z^{1}-6 x+12 y+2 z=0\) (B) \(5\left(x^{2}+y^{2}+z^{2}\right)=22\) (C) \(5 x^{2}+5 y^{c}+5 z^{1}-2 x y-3 y z-z x-6 x\) \(+12 y+5 z+22=0\) (D) none of these

The length of the perpendicular from the origin to the plane passing though three non-collinear points \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) is (A) \(\frac{[\mathrm{abc}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{c} \times \mathbf{a}+\mathbf{b} \times \mathbf{c}|}\) (B) \(\frac{2[\mathbf{a} \mathbf{b c}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}|}\) (C) \([\mathbf{a} \mathbf{b} \mathbf{c}]\) (D) none of these

The lines \(\mathbf{r}=\mathbf{a}+\lambda(\mathbf{b} \times \mathbf{c})\) and \(\mathbf{r}=\mathbf{b}+\mu(\mathbf{c} \times \mathbf{a})\) will intersect if (A) \(\mathbf{a} \times \mathbf{c}=\mathbf{b} \times \mathbf{c}\) (B) \(\mathbf{a} \cdot \mathbf{c}=\mathbf{b} \cdot \mathbf{c}\) (C) \(\mathbf{b} \times \mathbf{a}=\mathbf{c} \times \mathbf{a}\) (D) none of these

The length of the perpendicular from the origin to the plane passing through the point a and containing the line \(\mathbf{r}=\mathbf{b}+\lambda \mathbf{c}\) is (A) \(\frac{[\mathbf{a b c}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}|}\) (B) \(\frac{[\mathrm{abc}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}|}\) (C) \(\frac{[a \mathbf{b} \mathbf{c}]}{|\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}|}\) (D) \(\frac{[\mathbf{a} \mathbf{b} \mathbf{c}]}{|\mathbf{c} \times \mathbf{a}+\mathbf{a} \times \mathbf{b}|}\)

The equation of the plane which contains the origin and the line of intersection of the planes \(\mathbf{r} \cdot \mathbf{a}=p\) and \(\mathbf{r} \cdot \mathbf{b}=q\) is (A) \(\mathbf{r} \cdot(p \mathbf{a}-q \mathbf{b})=0\) (B) \(\mathbf{r} \cdot(p \mathbf{a}+q \mathbf{b})=0\) (C) \(\mathbf{r} \cdot(q \mathbf{a}+p \mathbf{b})=0\) (D) \(\mathbf{r} \cdot(q \mathbf{a}-p \mathbf{b})=0\)

The vector equation of the line of intersection of the planes \(\mathbf{r} \cdot(\mathbf{i}+2 \mathbf{j}+3 \mathbf{k})=0\) and \(\mathbf{r} \cdot(3 \mathbf{i}+2 \mathbf{j}+\mathbf{k})=0\) is (A) \(\mathbf{r}=\lambda(\mathbf{i}+2 \mathbf{i}+\mathbf{k})\) (B) \(\mathbf{r}=\lambda(\mathbf{i}-2 \mathbf{i}+\mathbf{k})\) (C) \(\mathbf{r}=\lambda(\mathbf{i}+2 \mathbf{i}-3 \mathbf{k})\) (D) none of these

The plane \(x+y+z=5 \sqrt{3}\) and sphere \(x^{2}+y^{2}+z^{2}=5\) (A) touch each other (B) cut in a circle (C) do not meet (D) none of these

If \(P(x, y, z)\) is a point on the line segment joining \(Q(2,\), \(2,4)\) and \(R(3,5,6)\) such that the projection of \(O P\) on the axes are \(\frac{13}{5}, \frac{19}{5}, \frac{26}{5}\) respectively, then \(P\) divides \(Q R\) in the ratio (A) \(1: 2\) (B) \(3: 2\) (C) \(2: 3\) (D) \(1: 3\)

From the point \(P(a, b, c)\) the normals drawn to planes \(y z\) and \(z x\) are \(P A, P B\), then the equation of plane \(O A B\) is (A) \(b c x+a c y+a b z=0\) (B) \(b c x+a c y-a b z=0\) (C) \(b c x-a c y+a b z=0\) (D) \(-b c x+a c y+a b z=0\)

A variable plane moves so that the sum of reciprocals of its intercepts on the three coordinate axes is constant \(\lambda\). It passes through a fixed point, which has coordinates \(\begin{array}{ll}\text { (A) }(\lambda, \lambda, \lambda) & \text { (B) }\left(\frac{1}{\lambda}, \frac{1}{\lambda}, \frac{1}{\lambda}\right)\end{array}\) (C) \((-\lambda,-\lambda,-\lambda)\) (D) \(\left(-\frac{1}{\lambda},-\frac{1}{\lambda},-\frac{1}{\lambda}\right)\)

Equation of the sphere with centre in the positive octant which passess through the circle \(x^{2}+y^{2}=4, z=\) 0 and is cut by the plane \(x+2 y+2 z=0\) in a circle of radius 3 is (A) \(x^{2}+y^{2}+z^{2}-6 x-4=0\) (B) \(x^{2}+y^{2}+z^{2}-6 z+4=0\) (C) \(x^{2}+y^{2}+z^{2}-6 z-4=0\) (D) \(x^{2}+y^{2}+z^{2}-6 y-4=0\)

The equation of the sphere touching the three coordinate planes is (A) \(\sum x^{2}+2 a(x+y+z)+2 a^{2}=0\) (B) \(\sum x^{2}-2 a(x+y+z)+2 a^{2}=0\) (C) \(\Sigma x^{2} \pm 2 a(x+y+z)+2 a^{2}=0\) (D) \(\Sigma x^{2} \pm 2 a x \pm 2 a y \pm 2 a z+2 a^{2}=0\)

The line \(\mathbf{r}=\mathbf{a}+t \mathbf{b}\) touches the sphere \(\mathbf{r}^{2}-2 \mathbf{r} \cdot \mathbf{c}+\mathbf{h}=\) \(0, c^{2}>h\) at the point with position vector \(a\) if (A) \((\mathrm{a}-\mathbf{b}) \cdot \mathbf{c}=0\) (B) \((\mathbf{a}-\mathbf{c}) \cdot \mathbf{b}=0\) (C) \((\mathbf{b}-\mathbf{c}) \cdot \mathbf{a}=0\) (D) \(\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}=0\)

Equation of the projection of the line \(8 x-y-7 z=8, x\) \(+y+z=1\) on the plane \(5 x-4 y-z=5\) is (A) \(\frac{x-1}{1}=\frac{y}{2}=\frac{z}{-3}\) (B) \(\frac{x}{1}=\frac{y-1}{2}=\frac{z}{-3}\) (C) \(\frac{x}{1}=\frac{y}{2}=\frac{z-1}{-3}\) (D) \(\frac{x}{1}=\frac{y+1}{-2}=\frac{z+1}{3}\)

The angle between the straight lines whose direction cosines are given by \(2 l+2 m-n=0, m n+n l+l m=0\), is (A) \(\frac{\pi}{2}\) (B) \(\frac{\pi}{3}\) (C) \(\frac{\pi}{=}\) (D) none of these

If a variable line in two adjacent positions has direction cosines \(l, m, n\) and \(l+\delta, m+\delta m, n+\delta n\), then the small angle \(\delta \theta\) between the two positions is given by (A) \(\delta \theta^{2}=4\left(\delta^{2}+\delta m^{2}+\delta n^{2}\right)\) (B) \(\delta \theta^{2}=2\left(\delta l^{2}+\delta m^{2}+\delta n^{2}\right)\) (C) \(\delta \theta^{2}=\left(\delta^{2}+\delta m^{2}+\delta n^{2}\right)\) (D) none of these

If \(l_{1}, m_{1}, n_{1}\) and \(l_{2}, m_{2}, n_{2}\) are d.c.'s of the two lines inclined to each other at an angle \(\theta\), then the d.c.'s of the internal bisector of the angle between these lines are (A) \(\frac{l_{1}+l_{2}}{2 \sin \theta / 2}, \frac{m_{1}+m_{2}}{2 \sin \theta / 2}, \frac{n_{1}+n_{2}}{2 \sin \theta / 2}\) (B) \(\frac{l_{1}+l_{2}}{2 \cos \theta / 2}, \frac{m_{1}+m_{2}}{2 \cos \theta / 2}, \frac{n_{1}+n_{2}}{2 \cos \theta / 2}\) (C) \(\frac{l_{1}-l_{2}}{2 \sin \theta / 2}, \frac{m_{1}-m_{2}}{2 \sin \theta / 2}, \frac{n_{1}-n_{2}}{2 \sin \theta / 2}\) (D) \(\frac{l_{1}-l_{2}}{2 \cos \theta / 2}, \frac{m_{1}-m_{2}}{2 \cos \theta / 2}, \frac{n_{1}-n_{2}}{2 \cos \theta / 2}\)

The plane \(l x+m y=0\) is rotated about its line of intersection with the plane \(z=0\) through an angle \(\alpha\). The equation of the plane in its new position is (A) \(l x+m y \pm z \sqrt{l^{2}+m^{2}} \sin \alpha=0\) (B) \(l x+m y \pm z \sqrt{l^{2}+m^{2}} \tan \alpha=0\) (C) \(l x+m y \pm z \sqrt{l^{2}+m^{2}} \cot \alpha=0\) (D) none of these

The planes \(3 x-y+z+1=0,5 x+y+3 z=0\) intersect in the line \(P Q\). The equation of the plane through the point \((2,1,4)\) and perpendicular to \(P Q\) is (A) \(x+y-2 z=5\) (B) \(x+y-2 z=-5\) (C) \(x+y+2 z=5\) (D) \(x+y+2 z=-5\).

The equation of the plane containing the lines \(\mathbf{r}=\mathbf{a}_{1}+\) \(\lambda \mathbf{b}\) and \(\mathbf{r}=\mathbf{a}_{2}+\mu \mathbf{b}\) is (A) \(\mathbf{r} \cdot\left(\mathbf{a}_{1}-\mathbf{a}_{2}\right) \times \mathbf{b}=\left[\mathbf{a}_{1} \mathbf{a}_{2} \mathbf{b}\right]\) (B) \(\mathbf{r} \cdot\left(\mathbf{a}_{2}-\mathbf{a}_{1}\right) \times \mathbf{b}=\left[\mathbf{a}_{1} \mathbf{a}_{2} \mathbf{b}\right]\) (C) \(\mathbf{r} \cdot\left(\mathbf{a}_{1}+\mathbf{a}_{2}\right) \times \mathbf{b}=\left[\mathbf{a}_{2} \mathbf{a}_{1} \mathbf{b}\right]\) (D) none of these

The equation of the sphere inscribed in a tetrahedron, whose faces are \(x=0, y=0, z=0\) and \(x+2 y+2 z=1\) is (A) \(32\left(x^{2}+y^{2}+z^{2}\right)+8(x+y+z)+1=0\) (B) \(32\left(x^{2}+y^{2}+z^{2}\right)-8(x+y+z)-1=0\) (C) \(32\left(x^{2}+y^{2}+z^{2}\right)-8(x+y+z)+1=0\) (D) none of these

The perpendicular distance of a corner of a unit cube from a diagonal not passing through it is (A) \(\frac{1}{\sqrt{3}}\) (B) \(\frac{2}{\sqrt{3}}\) (C) \(\sqrt{\frac{2}{3}}\) (D) none of these

If \(P\) be any point on the plane \(l x+m y+n z=p\) and \(Q\) be a point on the line \(O P\) such that \(O P . O Q=p^{2}\). The locus of the point \(Q\) is (A) \(l x+m y+n z=x^{2}+y^{2}+z^{2}\) (B) \(b x+m y+n z=p\left(x^{2}+y^{2}+z^{2}\right)\) (C) \(p(b x+m y+n z)=x^{2}+y^{2}+z^{2}\) (D) none of these

Through a point \(P(h, k, l)\) a plane is drawn at right angles to \(O P\) to meet the coordinate axes in \(A, B\) and C. If \(O P=p\), then the area of \(\Delta A B C\) is (A) \(\frac{p^{5}}{2 h k l}\) (B) \(\frac{p^{5}}{h k l}\) (C) \(\frac{p^{5}}{4 h k l}\) (D) none of these

A variable plane passes through a fixed point \((a, b, c)\) and meets the coordinate axes in \(A, B, C\). The locus of the point common to the planes through \(A, B, C\) parallel to coordinate planes is (A) \(a y z+b z x+c x y=x y z\) (B) \(a y z+b z x+c x y=2 x y z\) (C) \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1\) (D) none of these

If \(O A B C\) is a tetrahedron such that \(O A^{2}+B C^{2}=O B^{2}+\) \(C A^{2}=O C^{2}+A B^{2}\), then (A) \(A B\) is perpendicular to \(O C\) (B) \(B C\) is perpendicular to \(O A\) (C) \(C A\) is perpendicular to \(O B\) (D) \(A B\) is perpendicular to \(C A\)

If the median through \(A\) of a \(\Delta A B C\) having vertices \(A\) \(\equiv(2,3,5), B \equiv(-1,3,2)\) and \(C \equiv(\lambda, 5, \mu)\) is equally inclined to the axes, then (A) \(\lambda=7\) (B) \(\mu=10\) (C) \(\lambda=10\) (D) \(\mu=7\)

Column-I Column-II I. The centre of the sphere hav- (A) \((-1,4,-2)\) \(=\) ing the circle \(x^{2}+y^{2}+z^{2}-3 x+4 y-2 z-5\) \(=0,5 x-2 y+4 z+7=0\) as the great circle is II. The plane \(2 x-2 y+z+12=0\) ( (B) \((2,-3,1)\) touches the sphere \(x^{2}+y^{2}+z^{2}-\) \(2 x-4 y+2 z-3=0\) at the point (C) \((-1,-1,-1)\) III. \(A(3,2,0), B(5,3,2), C(-9,6,\), \(-3\) ) are three points forming a triangle. If \(A D\), the bisector of \(\angle B A C\) meets \(B C\) in \(D\), then coordinates of \(D\) are IV. The point in which the line (D) \(\left(\frac{19}{8}, \frac{57}{16}, \frac{17}{16}\right)\) \(\frac{x+1}{-1}=\frac{y-12}{5}=\frac{z-7}{-2} \quad\) cuts the surface \(11 x^{2}-5 y^{2}-z^{2}=0\) is

A plane which passes through die point \((3,2,0)\) and the line \(\frac{x-4}{1}=\frac{y-7}{5}=\frac{z-4}{4}\) is: \(\quad\) (A) \(x-y+z=1\) (B) \(x+y+z=5\) (C) \(x+2 y-z=1\) (D) \(2 x-y+z=5\)

A parallelopiped is formed by planes drawn through the points \((2,3,5)\) and \((5,9,7)\), parallel to the coordi- nate planes. The length of a diagonal of the parallelopiped is: \([\mathbf{2 0 0 2}]\) (A) 7 unit (B) \(\sqrt{38}\) unit (C) \(\sqrt{155}\) unit (D) none of these

A tetrahedron has vertices at \(O(0,0,0), A(1,2,1)\), \(B(2,1,3)\) and \(C(-1,1,2)\). Then the angle between the faces \(O A B\) and \(A B C\) will be (A) \(\cos ^{-1}\left(\frac{19}{35}\right)\) (B) \(\cos ^{-1}\left(\frac{17}{31}\right)\) (C) \(30^{\circ}\) (D) \(90^{\circ}\)

A line makes the same angle \(\theta\), with each of the \(x\) and \(z\) axis. If the angle \(\beta\), which it makes with \(y\)-axis, is such that \(\sin ^{2} \beta=3 \sin ^{2} \theta\), then \(\cos ^{2} \theta\) equals (A) \(\frac{2}{3}\) (B) \(\frac{1}{5}\) (C) \(\frac{3}{5}\) (D) \(\frac{2}{5}\)

Distance between two parallel planes \(2 x+y+2 z=8\) and \(4 x+2 y+4 z+5=0\) is (A) \(\frac{3}{2}\) (B) \(\frac{5}{2}\) (C) \(\frac{7}{2}\) (D) \(\frac{9}{2}\)

A line with direction cosines proportional to \(2,1,2\) meets each of the lines \(x=y+a=z\) and \(x+a=2 y=2 z\). The co-ordinates of each of the point of intersection are given by (A) \((3 a, 3 a, 3 a),(a, a, a)\) (B) \((3 a, 2 a, 3 a),(a, a, a)\) (C) \((3 a, 2 a, 3 a),(a, a, 2 a)\) (D) \((2 a, 3 a, 3 a),(2 a, a, a)\)

If the straight lines \(x=1+s, y=-3-\lambda s, z=1+\lambda s\) and \(x=\frac{t}{2}, y=1+t, z=2-t\) with parameters \(s\) and \(t\) respectively, are co-planar then \(\lambda\). Equals (A) \(-2\) (B) \(-1\) (C) \(-\frac{1}{2}\) (D) 0

The intersection of the spheres \(x^{2}+y^{2}+z^{2}+7 x-2 y\) \(-z=13\) and \(x^{2}+y^{2}+z^{2}-3 x+3 y+4 z=8\) is the same as the intersection of one of the sphere and the plane (A) \(x-y-z=1\) (B) \(x-2 y-z=1\) (C) \(x--2 z=1\) (D) \(2 x-y-z=1\)

If the angle \(Q\) between the line \(\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}\) and the plane \(2 x-y+\sqrt{\lambda z}+4=0\) is such that \(\sin \theta=\frac{1}{3}\) the value of \(\lambda\) is \(\quad\) (A) \(\frac{5}{3}\) (B) \(\frac{-3}{5}\) (C) \(\frac{3}{4}\) (D) \(\frac{-4}{3}\)

If the plane \(2 a x-3 a y+4 a z+6=0\) passes through the midpoint of the line joining the centres of the spheres \([2005]\) \(x^{2}+y^{2}+z^{2}+6 x-8 y-2 z=13\) and \(x^{2}+y^{2}+z^{2}-10 x+4 y-2 z=8\), then a equals (A) \(-1\) (B) 1 (C) \(-2\) (D) 2

Let \(L\) be the line of intersection of the planes \(2 x+3 y+\) \(z=1\) and \(x+3 y+2 z=2\). If \(L\) makes an angles \(\alpha\) with the positive \(x\)-axis, then \(\cos \alpha\) equals [2007] (A) \(\frac{1}{\sqrt{3}}\) (B) \(\frac{1}{2}\) (C) \(\underline{1}\) (D) \(\frac{1}{\sqrt{2}}\)

If \((2,3,5)\) is one end of a diameter of the sphere \(x^{2}+y^{2}\) \(+z^{2}-6 x-12 y-2 z+20=0\), then the coordinates of the other end of the diameter are \([2007]\) (A) \((4,9,-3)\) (B) \((4,-3,3)\) (C) \((4,3,5)\) (D) \((4,3,-3)\)

The line passing through the points \((5,1, a)\) and \((3, b,\), 1) crosses the \(y z\)-plane at the point \(\left(0, \frac{17}{2}, \frac{-13}{2}\right)\) then (A) \(a=2, b=8\) (B) \(a=4, b=6\) (C) \(a=6, b=4\) (D) \(a=8, b=2\)

If the straight lines \(\frac{x-1}{k}=\frac{y-2}{2}=\frac{z-3}{3}\) and \(\frac{x-2}{3}=\frac{y-3}{k}=\frac{z-1}{2}\) intersect at a point, then the integer \(k\) is equal to (A) \(-5\) (B) 5 (C) 2 (D) \(-2\)

Let the line \(\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}\) lies in the plane \(x+\) \(3 y-\alpha z+\beta=0 .\) Then \((\alpha, \beta)\) equals (A) \((6,-17)\) (B) \((-6,7)\) (C) \((5,-15)\) (D) \((-5,15)\)

A line \(\mathrm{AB}\) in 3 -dimensional space makes angles \(45^{\circ}\) and \(120^{\circ}\) with the positive \(x\)-axis and the positive \(y\)-axis respectively. If \(A B\) makes an acute angle \(\theta\) with the positive \(z\)-axis, then \(\theta\) equals [2010] (A) \(45^{\circ}\) (B) \(60^{\circ}\) (C) \(75^{\circ}\) (D) \(30^{\circ}\)