 ## How to Register to Vote in India (2023 & 2024)

The worlds largest democracy prevails in India. The heart of democracy lies in the utilizing and acknowledging the right to vote by the citizens. These rights bring the power and responsibility of citizens to shape the future of the country resulting in defining a better future for the country. Citizens choose the leaders as their representatives to run the country for making the decisions for the development, and growth of the country. There is a whole process through which people are registere...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: If the plane $2 x-y+z=0$ is parallel to the line $\frac{2 x-1}{2}=\frac{2-y}{2}=\frac{z+1}{a}$, then the value of $a$ is A. $-4$ B. $-2$ C. 4 D. 2 Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The equation of the plane passing through the points $A(0,-1,0), B(2,1,-1)$ and $C(1,1,1)$ is given by A. $4 x+3 y-2 z-3=0$ B. $4 x-3 y+2 z+3=0$ C. $4 x-3 y+2 z-3=0$ D. None of these Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The equation of the plane passing through the intersection of the planes $3 x-y+2 z-4=0$ and $x+y+z-2=0$ and passing through the point $A(2,2,1)$ is given by A. $7 x+5 y-4 z-8=0$ B. $7 x-5 y+4 z-8=0$ C. $5 x-7 y+4 z-8=0$ D. $5 x+7 y-4 z+8=0$ Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The equation of the plane passing through the points $\mathrm{A}(2,2,1)$ and $\mathrm{B}(9,3,6)$ and perpendicular to the plane $2 x+6 y+6 z=1$, is A. $x+2 y-3 z+5=0$ B. $2 x-3 y+4 z-6=0$ C. $4 x+5 y-6 z+3=0$ D. $3 x+4 y-5 z-9=0$ Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The line $\frac{x-1}{2}=\frac{y-2}{4}=\frac{z-3}{-3}$ meets the plane $2 x+3 y-z=14$ in the point A. $(2,5,7)$ B. $(3,5,7)$ C. $(5,7,3)$ D. $(6,5,3)$ Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The equation of a plane through the point $A(1,0,-1)$ and perpendicular to the line $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+7}{-3}$ is A. $2 x+4 y-3 z=3$ B. $2 x-4 y+3 z=5$ C. $2 x+4 y-3 z=5$ D. $x+3 y+7 z=-6$ Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: If a plane meets the coordinate axes in $A, B$ and $C$ such that the centroid of $\triangle A B C$ is $(1,2,4)$, then the equation of the plane is A. $x+2 y+4 z=6$ B. $4 x+2 y+z=12$ C. $x+2 y+4 z=7$ D. $4 x+2 y+z=7$ Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The plane $2 x+3 y+4 z=12$ meets the coordinate axes in $A, B$ and $C$. The centroid of $\triangle A B C$ is A. $(2,3,4)$ B. $(6,4,3)$ C. $\left(2, \frac{4}{3}, 1\right)$ D. None of these Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: If the line $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}$ lies in the plane $2 x-4 y+z=7$, then the value of $k$ is A. $-7$ B. 7 C. 4 D. $-4$ Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: If $O$ is the origin and $P(1,2,-3)$ is a given point, then the equation of the plane through $P$ and perpendicular to OP is A. $x+2 y-3 z=14$ B. $x-2 y+3 z=12$ C. $x-2 y-3 z=14$ D. None of these Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: If the line $\frac{x+1}{3}=\frac{y-2}{4}=\frac{z+6}{5}$ is parallel to the plane $2 x-3 y+k z=0$, then the value of $k$ is A. $\frac{5}{6}$ B. $\frac{6}{5}$ C. $\frac{3}{4}$ D. $\frac{4}{5}$ Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: A plane cuts off intercepts $3,-4,6$ on the coordinate axes. The length of perpendicular from the origin to this plane is A. $\frac{5}{\sqrt{29}}$ units B. $\frac{8}{\sqrt{29}}$ units C. $\frac{6}{\sqrt{29}}$ units D. $\frac{12}{\sqrt{29}}$ units Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The equation of a plane passing through the point $A(2,-3,7)$ and making equal intercepts on the axes, is A. $x+y+z=3$ B. $x+y+z=6$ C. $x+y+z=9$ D. $x+y+z=4$ Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The length of perpendicular from the origin to the plane $\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-12 \hat{\mathrm{k}})+39=0$ is A. 3 units B. $\frac{13}{5}$ units C. $\frac{5}{3}$ units D. None of these Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The direction cosines of the normal to the plane $5 y+4=0$ are A. $0, \frac{-4}{5}, 0$ B. $0,1,0$ C. $0,-1,0$ D. None of these Solution:...

## Mark against the correct answer in each of the following:

Question: Mark against the correct answer in each of the following: The direction cosines of the perpendicular from the origin to the plane $\overrightarrow{\mathrm{r}} \cdot(6 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})+1=0$ are A. $\frac{6}{7}, \frac{3}{7}, \frac{-2}{7}$ B. $\frac{6}{7}, \frac{-3}{7}, \frac{2}{7}$ C. $\frac{-6}{7}, \frac{3}{7}, \frac{2}{7}$ D. None of these Solution:...

## Write the equation of a plane passing through the point

Question: Write the equation of a plane passing through the point $(2,-1,1)$ and parallel to the plane $3 x+2 y-z=7$. Solution:...

## Write the angle between the line

Question: Write the angle between the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z+3}{-2}$ and the plane $x+y+4=0$ Solution:...

## Solve this following

Question: Find the value of $\lambda$ for which the line $\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-1}{2}$ is parallel to the plane $\bar{r} \cdot(2 \hat{\imath}+3 \hat{\jmath}+4 \hat{k})=4$ Solution:...

## Find the length of perpendicular from the origin to the plane

Question: Find the length of perpendicular from the origin to the plane $\bar{r} \cdot(2 \hat{j}-3 \hat{j}+6 \hat{k})+14-0$. Solution:...

## Solve this following

Question: Show that the line $\vec{r}-(4 \hat{i}-7 \hat{k})+\lambda(4 \hat{i}-2 \hat{j}+3 \hat{k})$ is parallel to the plane $\vec{r} \cdot(5 \hat{i}+4 \hat{j}-4 \hat{k})-7$. Solution: Hence, the given line is parallel to the given plane....

## Find the direction cosines of the perpendicular from the origin to the plane

Question: Find the direction cosines of the perpendicular from the origin to the plane $\bar{r} \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})+1-0$. Solution:...

## Find the length of perpendicular drawn from the origin to the plane

Question: Find the length of perpendicular drawn from the origin to the plane $2 x-3 y+6 z+21=0$. Solution:...

## Write the equation of the plane passing through the point

Question: Write the equation of the plane passing through the point $(\mathrm{a}, \mathrm{b}, \mathrm{c})$ and parallel to the plane $\bar{r} \cdot(\hat{i}+\hat{j}+\hat{k})=2$. Solution:...