Centroid of a Tetrahedron

Q.

If a variable plane forms a tetrahedron of constant volume $64k^3$ with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

1. $xyz=k^3$

2. $xyz=2k^3$

3. $xyz=12k^3$

4. $xyz=6k^3$

Q. $ABCD$ is a tetrahedron where $A(2,0,0),$ $B(0,4,0)$ and $CD=\sqrt{14}.$ The edge $CD$ lies on the line $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}.$ If locus of centroid of tetrahedron is $\frac{x-\frac{3}{2}}{1}=\frac{y-y_1}{a}=\frac{z-z_1}{b},$ then

1. $a+b=5$
2. $y_1+z_1=6$
3. $y_1-z_1=1$
4. $a+b+y_1=8$

Q.

If centroid of the tetrahedron OABC, where coordinates of $A,B,C$ are $(a,2,3),(1,b,3)$ and $(2,1,c)$ respectively be $(1,2,3),$ then find the distance of a point $(a,b,c)$ from the origin, where O is the origin.

1. 

2. 

3. 

4. 7

Q. If centroid of the tetrahedron , whose vertices are given by (0, 0, 0), (a, 2, 3), (1, b, 2) and (2, 1, c) be (1, 2, –1), then distance of P(a, b, c) from origin is equal to

1. $\sqrt{107}$
2. $\frac{\sqrt{107}}{\sqrt{14}}$
3. $10$
4. $\sqrt{14}$

Q.

If centroid of the tetrahedron OABC, where A, B, C are given by (a, 2, 3), (1, b, 2) and (2, 1, c) respectively be (1, 2, -1), then distance of P(a, b, c) from origin is equal to

1. None of these

2. 
   

3. 
   

4. 
   

Q. If centroid of the tetrahedron , whose vertices are given by (0, 0, 0), (a, 2, 3), (1, b, 2) and (2, 1, c) be (1, 2, –1), then distance of P(a, b, c) from origin is equal to

1. $\sqrt{107}$
2. $\sqrt{14}$
3. $\frac{\sqrt{107}}{\sqrt{14}}$
4. $10$

Q. If a variable plane forms a tetrahedron of constant volume $64k^3$ with the coordinate planes, find the locus of the centroid of the tetrahedron.

1. $xyz=32k^3$
2. $xyz=6k^3$
3. $xyz=12k^3$
4. None of these

Q. Assertion :Three concurrent lines with direction cosines $l_1,m_1,n_1;l_2,m_2,n_2;l_3,m_3,n_3$ are coplanar if $\left|\begin{array}{ccc}{l}_1& m_1& n_1\\ l_2& m_2& n_2\\ l_3& m_3& n_3\end{array}\right|=0$ Reason: Lines $\frac{x-x_1}{l_1}=\frac{y-y_1}{m_1}=\frac{z-z_1}{n_1},\frac{x-x_2}{l_2}=\frac{y-y_2}{m_2}=\frac{z-z_2}{n_2}$ are coplanar if $\left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ l_1& m_1& n_1\\ l_2& m_2& n_2\end{array}\right|=0$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q.

If a variable plane forms a tetrahedron of constant volume $64k^3$ with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

1. $xyz=k^3$

2. $xyz=2k^3$

3. $xyz=12k^3$

4. $xyz=6k^3$

Q.

If centroid of the tetrahedron OABC, where A, B, C are given by (a, 2, 3), (1, b, 2) and (2, 1, c) respectively be (1, 2, -1), then distance of P(a, b, c) from origin is equal to

1. $\sqrt{107}$

2. $\sqrt{14}$

3. $\sqrt{\frac{107}{14}}$

4. $Noneofthese$

Q. Centroid of the tetrahedron OABC, where coordinates of A, B, C are (a, 2, 3), (1, b, 3) and (2, 1, c) respectively, is (1, 2, 3). Find the distance of the point (a, b, c) from the origin O

1. 
2. 
3. 
4. 

Q. If centroid of the tetrahedron , whose vertices are given by (0, 0, 0), (a, 2, 3), (1, b, 2) and (2, 1, c) be (1, 2, –1), then distance of P(a, b, c) from origin is equal to

1. $\sqrt{107}$
2. $\sqrt{14}$
3. $10$
4. $\frac{\sqrt{107}}{\sqrt{14}}$

Q.

If a variable plane forms a tetrahedron of constant volume $64k^3$ with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

1. $xyz=k^3$

2. $xyz=2k^3$

3. $xyz=12k^3$

4. $xyz=6k^3$

Q.

If centroid of the tetrahedron OABC, where A, B, C are given by (a, 2, 3), (1, b, 2) and (2, 1, c) respectively be (1, 2, -1), then distance of P(a, b, c) from origin is equal to

1. 
   

2. None of these

3. 
   

4. 
   

Q. Two points $(a,3)$ and $(5,b)$ are the opposite vertices of a rectangle. If the coordinates $(x,y)$ of the other two vertices satisfy the relation $y=2x+c$ where $c^2+2a-b=0$ then the value $c$ can be

1. $2\sqrt{2}+1$
2. $1-2\sqrt{2}$
3. $-1-2\sqrt{2}$
4. $2\sqrt{2}-1$

Q.

If a variable plane forms a tetrahedron of constant volume $64k^3$ with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

1. $xyz=k^3$

2. $xyz=2k^3$

3. $xyz=12k^3$

4. $xyz=6k^3$

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

Q. If the lines $\{\overline{r}-(2\hat{i}-3\hat{j}+4\hat{k}\left)\right\}.\left\{\right(1-p)\hat{i}+3\hat{j}-4\hat{k}\}=0$ and $\{\overline{r}-(3\hat{i}-4\hat{j}+5\hat{k}\left)\right\}.\{2\hat{i}-4\hat{j}+5\hat{k}\}=0$ are coplanar, then the value of $p$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q. The volume of tetrahedron included between the plane $3x+4y–5z–60=0$ and co-ordinate planes is

1. $600\text{cubic unit}$
2. $2400\text{cubic unit}$
3. $1200\text{cubic unit}$
4. $3600\text{cubic unit}$

Q. If $(4,2,p)$ is the centroid of the tetrahedron formed by the points $(k,2,-1),(4,1,1),(6,2,5)$ and
$(3,3,3)$ then $k+p=$

1. $\frac{17}{3}$
2. $\frac{5}{3}$
3. $1$
4. $5$