Tetrahedron

Q. Let $L_1:\frac{x-1}{2}=\frac{y-2}{-1}=\frac{z-3}{3},$ $L_2:\frac{x-1}{-1}=\frac{y-2}{3}=\frac{3(z-3)}{5}$ and $L_3:\frac{x-1}{-32}=\frac{y-2}{-19}=\frac{z-3}{15}$ be three lines.
A plane is intersecting these lines at $A,B$ and $C$ respectively such that $PA=2,$ $PB=3$ and $PC=6$ where $P\equiv (1,2,3).$ If $V$ is the volume of the tetrahedron $PABC$ and $d$ is the perpendicular distance of the plane from the point $P,$ then

1. $V=18$ cubic units
2. $V=6$ cubic units
3. $d=\frac{6}{\sqrt{14}}$ units
4. $d=7$ units

Q. Find the coordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, −1, 3) and C(2, −3, −1). [NCERT EXEMPLAR]

Q. The position vectors of the vertices $A,B$ and $C$ of a tetrahedron are $(1,1,1),$ $(1,0,0)$ and $(3,0,0)$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line through $A$ of the $\Delta ABC$ at a point $E.$ If the length of side $AD$ is $4$ units and volume of the tetrahedron is $\frac{2\sqrt{2}}{3}$ cubic units, then the CORRECT statement(s) is (are)

1. The length of altitude from the vertex $D$ is $2$.
2. There is exactly one position for the point $E$.
3. There can be two positions for the point $E$.
4. Vector $\hat{j}-\hat{k}$ is normal to the plane $ABC$.

Q. A tetrahedron has vertices $P(1,2,1),Q(2,1,3),R(-1,1,2)$ and $O(0,0,0).$ The angle between the faces $OPQ$ and $PQR$ is:

1. ${\cos }^{-1}\left(\frac{19}{35}\right)$
2. ${\cos }^{-1}\left(\frac{17}{31}\right)$
3. ${\cos }^{-1}\left(\frac{9}{35}\right)$
4. ${\cos }^{-1}\left(\frac{7}{31}\right)$

Q.

A tetrahedron has vertices at $O(0,0,0),A(1,2,1),B(2,1,3)\&C(–1,1,2).$ Then the angle between the faces $OAB\&ABC$ will be?

Q. If volume of regular tetrahedron of edge length $k$ is $V$ and shortest distance between any pair of opposite edges of same regular tetrahedron is $d,$ then the value of $\frac{d^3}{V}$ is

Q. In the tetrahedron $ABCD,A=(1,2,-3)$ and $G(-3,4,5)$ is the centroid of the tetrahedron. If $P$ is the centroid of the $\Delta BCD$, then $AP=$

1. $\frac{4\sqrt{21}}{3}$
2. $\frac{8\sqrt{21}}{3}$
3. $\frac{\sqrt{21}}{3}$
4. $4\sqrt{21}$

Q. The position vectors of the four angular points of a tetrahedron $OABC$ are $(0,0,0),(0,0,2),(0,4,0),(6,0,0)$ respectively. A point $P$ inside the tetrahedron is at the same distance $r$ from the four plane faces of the tetrahedron. Then the value of $9r$ is

Q.

Draw a diagram of a cuboid and label its vertices as A, B, C, D, E, F, G and H. Now, write the names of its faces and edges?

Q.

The point of trisection of the line joining the points (0, 3) and (6, -3) are

1. (2, 0) and (4, -1)

2. (2, -1) and (4, 1)

3. (2, 1) and (4, -1)

4. (3, 1) and (4, -1)

Q. A plane passing through $(1,1,1)$ cuts positive direction of coordinate axes at $A,B$ anc $C,$ then the volume of tetrahedron $OABC$ satisfies

1. $V\le \frac{9}{2}$
2. $V\ge \frac{9}{2}$
3. none of these
4. $V=\frac{9}{2}$

Q. If the volume of parallelepiped formed by vectors $a\times b,b\times c$ and $c\times a$ is $36$ cubic units, then
$\begin{array}{ccc}& \text{List I}& \text{List II}\\ \text{(I)}& \text{Volume of parallelopiped formed by vectors}a,b\text{and}c\text{is}& \text{(P) 0 cubic units}\\ \text{(II)}& \text{Volume of tetrahedron formed by vectors}a,b\text{and}c\text{is}& \text{(Q) 12 cubic units}\\ \text{(III)}& \text{Volume of parallelopiped formed by vectors}a+b,b+c\text{and}c+a\text{is}& \text{(R) 6 cubic units}\\ \text{(IV)}& \text{Volume of parallelopiped formed by vectors}a-b,b-c\text{and}c-a\text{is}& \text{(S) 1 cubic units}\\ & & \text{(T) 5 cubic units}\end{array}$
Which of the following is CORRECT combination?

1. $\text{(I)}\to \left(Q\right);\text{(II)}\to \left(P\right);\text{(III)}\to (R,T);\text{(IV)}\to \left(S\right)$
2. $\text{(I)}\to \left(P\right);\text{(II)}\to (Q,T);\text{(III)}\to \left(R\right);\text{(IV)}\to \left(S\right)$
3. $\text{(I)}\to \left(Q\right);\text{(II)}\to \left(P\right);\text{(III)}\to (S,T);\text{(IV)}\to \left(R\right)$
4. $\text{(I)}\to \left(R\right);\text{(II)}\to \left(S\right);\text{(III)}\to \left(Q\right);\text{(IV)}\to \left(P\right)$

Q. a = Number of vertices of a tetrahedron
b = Number of edges of a tetrahedron
c = Number of faces of a tetrahedron
Find the value of a+b+c

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Q. In the tetrahedron $ABCD,A=(1,2,-3)$ and $G(-3,4,5)$ is the centroid of the tetrahedron. If $P$ is the centroid of the $\Delta BCD$, then $AP=$

1. $\frac{8\sqrt{21}}{3}$
2. $\frac{4\sqrt{21}}{3}$
3. $\frac{\sqrt{21}}{3}$
4. $4\sqrt{21}$

Q. If
a = Number of vertices of a tetrahedron
b = Number of edges of a tetrahedron
c = Number of faces of a tetrahedron
Then the value of a+b+c is

1. 14
2. 16
3. 10
4. 12

Q. A variable plane is at a constant distance $p$ from the origin and meets the axes in $A,B$ and $C$. Then locus of the centroid of the tetrahedron $OABC$ is

1. $x^{-2}+y^{-2}+z^{-2}=16p^{-1}$
2. $x^{-2}+y^{-2}+z^{-2}=16p^{-2}$
3. $x^2+y^{-2}+z^{-2}=16$
4. None of these

Q. a = Number of vertices of a tetrahedron
b = Number of edges of a tetrahedron
c = Number of faces of a tetrahedron
Find the value of a+b+c

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Q. Let $A(2\hat{i}+3\hat{j}+5\hat{k}),B(-\hat{i}+3\hat{j}+2\hat{k})$ , and $C(\lambda \hat{i}+5\hat{j}+\mu \hat{k})$ are the vertices of a $\Delta ABC$ and its median through $A$ is equally inclined to the positive directions of axes. Then find the value $2\lambda -\mu$.

Q. If the plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=\frac{1}{2}$ intersect x, y and z-axis at A, B and C respectively then volume of the tetrahedron OABC where 'O' is the origin is

1. 6
2. 4
3. $\frac{1}{2}$
4. 3

Q. The co-ordinates of the vertices of the given tetrahedron are (1, 2, 3), (5, 8, 6), (14, 15, 18) and (0, 0, 0). Find the co-ordinates of the centroid of the tetrahedron?

1. (5, 6.25, 6.75)
2. (5, 7, 6)
3. (7, 5, 6)
4. (6.67, 8.34, 9)

Q. A plane passing through $(1,1,1)$ cuts positive direction of coordinate axes at $A,B$ anc $C,$ then the volume of tetrahedron $OABC$ satisfies

1. $V\le \frac{9}{2}$
2. $V\ge \frac{9}{2}$
3. $V=\frac{9}{2}$
4. none of these

Q. Find the angle between the following pair of lines:
(i) $\overrightarrow{r}=2\hat{i}-5\hat{j}+\hat{k}+\lambda (3\hat{i}-2\hat{j}+6\hat{k})$ and $\overrightarrow{r}=7\hat{i}-6\hat{k}+\mu (\hat{i}+2\hat{j}+2\hat{k})$

Q. Find the foot of the perpendicular drawn from the point A (1, 0, 3) to the joint of the points B (4, 7, 1) and C (3, 5, 3).

Q. A variable plane which remains at a constant distance $p$ from the origin cuts the coordinate axes in $A,B,C$. The locus of the centorid of the tetrahedron $OABC$ is $y^2{z}^2+z^2{x}^2+x^2{y}^2=k{x}^2{y}^2{z}^2$, where $k$ is equal to

1. $9p^2$
2. $\frac{9}{p^2}$
3. $\frac{7}{p^2}$
4. $\frac{16}{p^2}$

Q. The position vectors of the vertices $A,B$ and $C$ of a tetrahedron are $(1,1,1),$ $(1,0,0)$ and $(3,0,0)$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line through $A$ of the $\Delta ABC$ at a point $E.$ If the length of side $AD$ is $4$ units and volume of the tetrahedron is $\frac{2\sqrt{2}}{3}$ cubic units, then the CORRECT statement(s) is (are)

1. The length of altitude from the vertex $D$ is $2$.
2. There is exactly one position for the point $E$.
3. There can be two positions for the point $E$.
4. Vector $\hat{j}-\hat{k}$ is normal to the plane $ABC$.

Q. A vector of magnitude $2$ along a bisector of the angle between the two vectors $2\overrightarrow{i}-2\overrightarrow{j}+\overrightarrow{k}$ and $\overrightarrow{i}+2\overrightarrow{j}-2\overrightarrow{k}$ is

1. $\frac{1}{\sqrt{26}}(\overrightarrow{i}-4\overrightarrow{j}+3\overrightarrow{k})$
2. None of these
3. $\frac{2}{\sqrt{26}}(\overrightarrow{i}-4\overrightarrow{j}+3\overrightarrow{k})$
4. $\frac{2}{\sqrt{10}}(3\overrightarrow{i}-\overrightarrow{k})$

Q. Find the foot of the perpendicular from (1, 2, −3) to the line $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}.$

Q. Using vectors, find the area of the triangle with vertices: $A(1,2,3),B(2,-1,4)$ and $C(4,5,-1)$

Q. a = Number of vertices of a tetrahedron
b = Number of edges of a tetrahedron
c = Number of faces of a tetrahedron
Find the value of a+b+c

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Q. a = Number of vertices of a tetrahedron
b = Number of edges of a tetrahedron
c = Number of faces of a tetrahedron
Find the value of a+b+c

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