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A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar

Chapter 22

Three Dimensional Geometry - all with Video Answers

Educators

Chapter Questions

02:37

Problem 1

The equation of the plane through the points $(2,3,1)$ and $(4,-5,3)$ and parallel to $x$-axis is
(A) $x-z-1=0$
(B) $4 x+y-11=0$
(C) $y+4 z-7=0$
(D) none of these

Varsha Aggarwal
Varsha Aggarwal
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01:39

Problem 2

The edge of a cube is of length ' $a$ ' then the shortest distance between the diagonal of a cube and an edge skew to it is
(A) $a \sqrt{2}$
(B) $a$
(C) $\sqrt{2} / a$
(D) $a / \sqrt{2}$

Varsha Aggarwal
Varsha Aggarwal
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01:39

Problem 3

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) $(\sqrt{3} / 2) a$

Varsha Aggarwal
Varsha Aggarwal
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02:31

Problem 4

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) $2 \mathbf{i}+7 \mathbf{j}+13 \mathbf{k}$

Varsha Aggarwal
Varsha Aggarwal
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03:59

Problem 5

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) $\sqrt{\frac{5}{12}}$

Varsha Aggarwal
Varsha Aggarwal
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03:13

Problem 6

The position vector of the centre of the circle $|\mathbf{r}|=5, \mathbf{r}$ $(\mathbf{i}+\mathbf{j}+\mathbf{k})=3 \sqrt{3}$ is
(A) $\sqrt{3}(\mathbf{i}+\mathbf{j}+\mathbf{k})$
(B) $\mathbf{i}+\mathbf{j}+\mathbf{k}$
(C) $3(\mathbf{i}+\mathbf{i}+\mathbf{k})$
(D) none of the above

Varsha Aggarwal
Varsha Aggarwal
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01:04

Problem 7

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

Varsha Aggarwal
Varsha Aggarwal
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02:22

Problem 8

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:43

Problem 9

The direction ratios of the line
$x-y+z-5=0=x-3 y-6$ are
(A) $3,1,-2$
(B) $2,-4,1$
(C) $\frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{-2}{\sqrt{14}}$
(D) $\frac{2}{\sqrt{41}}, \frac{-4}{\sqrt{41}}, \frac{1}{\sqrt{41}}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:39

Problem 10

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

Varsha Aggarwal
Varsha Aggarwal
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06:02

Problem 11

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:29

Problem 12

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

Varsha Aggarwal
Varsha Aggarwal
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02:33

Problem 13

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

Varsha Aggarwal
Varsha Aggarwal
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01:51

Problem 14

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) $\frac{[\mathbf{a} \mathbf{b} \mathbf{c}]}{|\mathbf{c} \times \mathbf{a}+\mathbf{a} \times \mathbf{b}|}$

Varsha Aggarwal
Varsha Aggarwal
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02:08

Problem 15

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) $\mathbf{r} \cdot(q \mathbf{a}-p \mathbf{b})=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:43

Problem 16

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

Varsha Aggarwal
Varsha Aggarwal
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01:28

Problem 17

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

Varsha Aggarwal
Varsha Aggarwal
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03:16

Problem 18

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) $1: 3$

Varsha Aggarwal
Varsha Aggarwal
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02:34

Problem 19

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) $-b c x+a c y+a b z=0$

Varsha Aggarwal
Varsha Aggarwal
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03:50

Problem 20

A mirror and a source of light are situated at the origin $O$ and at a point on $O X$, respectively. A ray of light from the source strikes the mirror and is reflected. If the $D R s$ of the normal to the plane are $1,-1,1$, then $d$. $c$ 's of the reflected ray are
(A) $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$
(B) $-\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$
(C) $-\frac{1}{3},-\frac{2}{3},-\frac{2}{3}$
(D) $-\frac{1}{3},-\frac{2}{3}, \frac{2}{3}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:34

Problem 21

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) $\left(-\frac{1}{\lambda},-\frac{1}{\lambda},-\frac{1}{\lambda}\right)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:14

Problem 22

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) $x^{2}+y^{2}+z^{2}-6 y-4=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:38

Problem 23

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) $\Sigma x^{2} \pm 2 a x \pm 2 a y \pm 2 a z+2 a^{2}=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:21

Problem 24

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) $\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:35

Problem 25

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) $\frac{x}{1}=\frac{y+1}{-2}=\frac{z+1}{3}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:30

Problem 26

The cartesian equation of the plane $\mathbf{r}=(1+\lambda-\mu) \mathbf{i}+(2+\lambda) \mathbf{j}+(3-2 \lambda+2 \mu) \mathbf{k}$ is
(A) $2 x+y=5$
(B) $2 x-y=5$
(C) $2 x+z=5$
(D) $2 x-z=5$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:54

Problem 27

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:44

Problem 28

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:55

Problem 29

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) $\frac{l_{1}-l_{2}}{2 \cos \theta / 2}, \frac{m_{1}-m_{2}}{2 \cos \theta / 2}, \frac{n_{1}-n_{2}}{2 \cos \theta / 2}$

NW
Nida Wasiq
Numerade Educator
04:11

Problem 30

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:08

Problem 31

$P$ is any point on the plane $l x+m y+n z=p ;$ a point $Q$ is taken on the line $O P$ such that $O P \cdot O Q=p^{2}$, then the locus of $Q$ is
(A) $b x+m y+n z=p\left(x^{2}+y^{2}+z^{2}\right)$
(B) $p(l x+m y+n z)=x^{2}+y^{2}+z^{2}$
(C) $p(x+y+z)=l x^{2}+m y^{2}+n z^{2}$
(D) none of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:39

Problem 32

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) $x+y+2 z=-5$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:33

Problem 33

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:25

Problem 34

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:17

Problem 35

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

Ahmad Reda
Ahmad Reda
Numerade Educator
02:52

Problem 36

The centre of the sphere which touches the lines $y=x$, $z=c$ and $y=-x, z=-c$ lies on
(A) $x y=2 c z$
(B) $x y=-2 c z$
(C) $y z=2 c x$
(D) $y z=-2 c x$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:06

Problem 37

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:22

Problem 38

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

Linh Vu
Linh Vu
Numerade Educator
03:11

Problem 39

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:49

Problem 40

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 $C A$

Akshaya Rs
Akshaya Rs
Numerade Educator
01:58

Problem 41

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) $\mu=7$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:14

Problem 42

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:16

Problem 43

I. The radius of the circle $x^{2}+y^{2}+z^{2}=$ 49
(A) $\frac{10}{3 \sqrt{3}}$
$2 x+3 y-z-5 \sqrt{14}=0$ is
II. The locus of the foot of the perpen-
(B) $\sqrt{14}$
dicular from the origin on the variable plane through the fixed point $(2,-4,6)$ is a sphere of radius
III. The distance of the line $L$ whose vector (C) 3 equation is $\mathbf{r}=2 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k}+\lambda(\mathbf{i}-\mathbf{j}$
$+4 \mathbf{k}$ ) from the plane $\pi$ whose vector equation is $\mathbf{r} \cdot(\mathbf{i}+5 \mathbf{j}+\mathbf{k})=5$, is
IV. The equation of the plane which meets
(D) $2 \sqrt{6}$
the axes in $A, B$ and $C$, given that the centroid of the triangle $A B C$ is the point $(\alpha, \beta, \gamma)$, is $\frac{x}{\alpha}+\frac{y}{\beta}+\frac{z}{\gamma}=k$, where
$k=$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:13

Problem 44

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) $2 x-y+z=5$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:48

Problem 45

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:56

Problem 46

The equation of the plane containing the line $\frac{x-x_{1}}{l}=\frac{y-y_{1}}{m}=\frac{z-z_{1}}{n}$ is $a\left(x-x_{1}\right)+b\left(y-y_{1}\right)+c(z-$
$\left.z_{1}\right)=0$, where:
(A) $a x_{1}+b y_{1}+c z_{1}=0$
(B) $a l+b m+c n=0$
(C) $\frac{a}{l}=\frac{b}{m}=\frac{c}{n}$
(D) $l x_{1}+m y_{1}+n z_{1}=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:31

Problem 47

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) $90^{\circ}$

Ahmad Reda
Ahmad Reda
Numerade Educator
01:42

Problem 48

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) $\frac{2}{5}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:22

Problem 49

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) $\frac{9}{2}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:58

Problem 50

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) $(2 a, 3 a, 3 a),(2 a, a, a)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:54

Problem 51

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:16

Problem 52

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) $2 x-y-z=1$

Lucas Finney
Lucas Finney
Numerade Educator
01:37

Problem 53

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) $\frac{-4}{3}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:28

Problem 54

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:04

Problem 55

The two lines $x=a y+b, z=c y+d ;$ and $x=a^{\prime} y+b^{\prime}, z$ $=c^{\prime} y+d^{\prime}$ are perpendicular to each other if $[\mathbf{2 0 0 6}]$
(A) $a a^{\prime}+c c^{\prime}=-1$
(B) $a a^{\prime}+c c^{\prime}=1$
(C) $\frac{a}{a^{\prime}}+\frac{c}{c^{\prime}}=-1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:23

Problem 56

The image of the point $(-1,3,4)$ in the plane $x-2 y=$ 0 is $[\mathbf{2 0 0 6}]$
(A) $\left(-\frac{17}{3},-\frac{19}{3}, 4\right)$
(B) $(15,11,4)$
(C) $\left(-\frac{17}{3},-\frac{19}{3}, 1\right)$
(D) $(8,4,4)$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:31

Problem 57

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) $\frac{1}{\sqrt{2}}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:19

Problem 58

If a line makes an angle of $\frac{\pi}{4}$ with the positive directions of each of $x$-axis and $y$-axis, then the angle that the line makes with the positive direction of the $z$-axis is [2007]
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{3}$
(C) $\frac{\pi}{4}$
(D) $\frac{\pi}{2}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:01

Problem 59

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) $(4,3,-3)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:49

Problem 60

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) $a=8, b=2$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:30

Problem 61

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) $-2$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:07

Problem 62

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) $(-5,15)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:13

Problem 63

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) $30^{\circ}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:37

Problem 64

If the angle between the line $x=\frac{y-1}{2}=\frac{z-3}{\lambda}$ and the plane $x+2 y+3 z=$ is $\cos ^{-4}\left(\sqrt{\frac{5}{14}}\right)$, then $\lambda$ equals [2011]
(A) $\frac{3}{2}$
(B) $\frac{2}{5}$
(C) $\frac{5}{3}$
(D) $\frac{2}{3}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:17

Problem 65

Statement - 1 : The point $A(1,0,7)$ is the mirror image of the point $B(1,6,3)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. [2011] Statement $-2:$ The line: $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ bisects the line segment joining $A(1,0,7)$ and $B(1,6,3)$.
(A) Statement- 1 is true, Statement- 2 is true; Statement2 is not a correct explanation for Statement - 1
(B) Statement- 1 is true, Statement $-2$ is false.
(C) Statement- 1 is false, Statement- 2 is true.
(D) Statement- 1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement- 1

Km Neeraj
Km Neeraj
Numerade Educator
01:58

Problem 66

An equation of a plane parallel to the plane $x-2 y+2 z$ $=5$ and at a unit distance from the origin is $[\mathbf{2 0 1 2}]$
(A) $x-2 y+2 z-3=0$
(B) $x-2 y+2 z+1=0$
(C) $x-2 y+2 z-1=0$
(D) $x-2 y+2 z+5=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:30

Problem 67

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$
intersect, then the value of $k$ is equal to
(A) $-1$
(B) $\frac{2}{9}$
(C) $\frac{9}{2}$
(D) 0

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:36

Problem 68

If the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}$
$=\frac{z-5}{1}$ are coplanar, then $k$ can have $[2013]$
(A) exactly one value
(B) exactly two values
(C) exactly three values
(D) any value

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:27

Problem 69

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

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:40

Problem 70

The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ on the plane $2 x-y+z+3=0$ is the line [2014]
(A) $\frac{x+3}{3}=\frac{y-5}{1}=\frac{z-2}{-5}$
(B) $\frac{x+3}{-3}=\frac{y-5}{-1}=\frac{z+2}{5}$
(C) $\frac{x-3}{3}=\frac{y+5}{1}=\frac{z-2}{-5}$
(D) $\frac{x-3}{-3}=\frac{y+5}{-1}=\frac{z-2}{5}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:21

Problem 71

The angle between the two lines whose direction cosines satisfy the equations $l+m+n=0$ and $f=m^{2}$ $+n^{2}$ is
(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{6}$
(D) $\frac{\pi}{2}$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:06

Problem 72

The distance of the point $(1,0,2)$ from the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane $x-y+z=16$, is:
(A) 8
(B) $3 \sqrt{21}$
(C) 13
(D) $2 \sqrt{14}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:40

Problem 73

The equation of the plane containing the lines $2 x-5 y$ $+z=3$ and $x+y+4 z=5$, and parallel to the plane, $x+$ $3 y+6 z=1$, is:
(A) $x+3 y+6 z=-7$
(B) $x+3 y+6 z=7$
(C) $2 x+6 y+12 z=-13$
(D) $2 x+6 y+12 z=13$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:32

Problem 74

The distance of the point $(1,-5,9)$ from the plane $x-y$ $+z=5$ measured along the line $x=y=z$ is: $\quad[\mathbf{2 0 1 6}]$
(A) $\frac{20}{3}$
(B) $3 \sqrt{10}$
(C) $10 \sqrt{3}$
(D) $\frac{10}{\sqrt{3}}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:52

Problem 75

If the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane, $l x+$ $m y-z=9$, then $R+m^{2}$ is equal to:
(A) 2
(B) 26
(C) 18
(D) 5

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator