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Questions Related to maths

The distance between the planes $x + 2y + 3z + 7 = 0$ and $2x + 4y + 6z + 7 = 0$ is

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $\displaystyle \frac{\sqrt{7}}{2 \sqrt{2}}$

  2. $\displaystyle \frac{7}{2}$

  3. $\displaystyle \frac{\sqrt{7}}{2}$

  4. $\displaystyle \frac{7}{2 \sqrt{2}}$

Show answer
Correct Option: A
Explanation:

The equation of second plane can be rearrange as $x+2y+3z+\dfrac{7}{2}=0$. 

The distance between parallel planes $Ax+By+cZ+D _1=0$ and $Ax+By+Cz+D _2$ is given by 
$\dfrac{|D _1-D _2|}{\sqrt{A^2+B^2+C^2}}$.
Hence, for the given problem distance between planes is given by:

 $\dfrac{|7-\dfrac{7}{2}|}{\sqrt{1+4+9}}=\dfrac{\sqrt{7}}{2\sqrt{2}}$.

If the distance between the planes $8x + 12y - 14 z = 2$ and $4x + 6y - 7z = 2$ can be expressed in the form $\displaystyle \frac{1}{\sqrt{N}}$, where N is natural, then the value of $\displaystyle \frac{N(N + 1)}{2}$ is

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $4950$

  2. $5050$

  3. $5150$

  4. $5151$

Show answer
Correct Option: D
Explanation:

Given planes can be written as $4x + 6y - 7z - 1 = 0$ and $4x + 6y - 7z - 2 = 0$
Since both the planes are parallel so distance between them is given by,
$d =\left | \dfrac{(-1-2)}{\sqrt{4^2+6^2+7^2}}\right |=\dfrac{1}{\sqrt{101}}$  


$\therefore N=101$

Hence, $\dfrac{N(N+1)}{2}=101\times  51 = 5151$

If the distance between the planes $8x + 12y - 14z = 2$ and $4x + 6y - 7z = 2$ can be expressed in the form of $ \displaystyle \frac {1}{ \sqrt N} $ where $N$ is a natural number, then the value of $ \displaystyle \frac { N(N+1)}{2} $ is

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $4950$

  2. $5050$

  3. $5150$

  4. $5151$

Show answer
Correct Option: D
Explanation:

Given planes can be written as $4x + 6y - 7z - 1 = 0$ and $4x + 6y - 7z - 2 = 0$
Since both the planes are parallel so distance between them is given by,
$d =\left |  \dfrac{(-1-2)}{\sqrt{4^2+6^2+7^2}}\right |=\dfrac{1}{\sqrt{101}}$ $\therefore N=101$
Hence $\dfrac{N(N+1)}{2}=101 \times 51 = 5151$

If the distance between the planes $8x + 12y - 14z = 2$ and $4x + 6y - 7z = 2$ can be expressed int he form $\dfrac{1}{\sqrt{N}}$ where $N$ is natural, then the value of $\dfrac{N(N+1)}{2}$ is

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $4950$

  2. $5050$

  3. $5150$

  4. $5151$

Show answer
Correct Option: D
Explanation:

Given planes can be written as $4x + 6y - 7z - 1 = 0$ and $4x + 6y - 7z - 2 = 0$
Since both the planes are parallel so distance between them is given by,
$d =\left |\dfrac{(-1-2)}{\sqrt{4^2+6^2+7^2}}\right |=\dfrac{1}{\sqrt{101}}$

$\therefore N=101$
Ergo $\dfrac{N(N+1)}{2}=101\times 51 = 5151$

If ${ p } _{ 1 },{ p } _{ 2 },{ p } _{ 3 }$ denote the distance of the plane $2x-3y+4z+2=0$ from the planes $2x-3y+4z+6=0, 4x-6y+8z+3=0$ and $2x-3y+4z-6=0$ respectively, then 

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. ${ p } _{ 1 }+8{ p } _{ 2 }-{ p } _{ 3 }=0$

  2. ${ { p } _{ 3 } }^{ 2 }=16{ { p } _{ 2 } }^{ 2 }$

  3. $8{ { p } _{ 2 } }^{ 2 }={ { p } _{ 1 } }^{ 2 }$

  4. ${ p } _{ 1 }+2{ p } _{ 2 }+3{ p } _{ 3 }=\sqrt { 29 } $

Show answer
Correct Option: A
Explanation:

Since the planes are all parallel planes,

$\displaystyle { p } _{ 1 }=\dfrac { \left| 2-6 \right|  }{ \sqrt { { 2 }^{ 2 }+{ 3 }^{ 2 }+{ 4 }^{ 2 } }  } =\dfrac { 4 }{ \sqrt { 4+9+16 }  } =\dfrac { 4 }{ \sqrt { 29 }  } $

Equation of the plane $4x-6y+8z+3=0$ can be written as $2x-3y+4z+\displaystyle\dfrac { 3 }{ 2 } =0$

So, $\displaystyle { p } _{ 2 }=\dfrac { \left| 2-\dfrac { 3 }{ 2 }  \right|  }{ \sqrt { { 2 }^{ 2 }+{ 3 }^{ 2 }+{ 4 }^{ 2 } }  } =\dfrac { 1 }{ 2\sqrt { 29 }  } $

and $\displaystyle { p } _{ 3 }=\dfrac { \left| 2+6 \right|  }{ \sqrt { { 2 }^{ 2 }+{ 3 }^{ 2 }+{ 4 }^{ 2 } }  } =\dfrac { 8 }{ \sqrt { 29 }  } $

$\Rightarrow { p } _{ 1 }+8{ p } _{ 2 }-{ p } _{ 3 }=0$

Distance between two parallel planes  $2 x + y + 2 x = 8$  and  $4 x + 2 y + 4 x + 5 = 0$  is

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $\dfrac { 5 } { 2 }$

  2. $\dfrac { 7 } { 2 }$

  3. $\dfrac { 9 } { 2 }$

  4. $\dfrac { 3 } { 2 }$

Show answer
Correct Option: A

The distance between the planes $\displaystyle 4x - 5y + 3z = 5$ and $\displaystyle 4x - 5y + 3z + 2 = 0$ is

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $\displaystyle \frac{7}{2 \sqrt{5}}$

  2. $\displaystyle 7$

  3. $\displaystyle \frac{7}{5 \sqrt{2}}$

  4. $\displaystyle 3$

Show answer
Correct Option: C
Explanation:

Given planes are $4x -5y+ 3z - 5 = 0$ and $4x -5y+ 3z + 2 = 0$
Since both the planes are parallel so distance between them is,
$= \left| \dfrac{(-5-2)}{\sqrt{4^2+5^2+3^2}}\right|=\dfrac{7}{5\sqrt{2}}$

The distance between the planes $\displaystyle 2x + y + 2z = 8$ and $\displaystyle 4x + 2y + 4z + 5 = 0$ is

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $\displaystyle \frac{3}{2}$

  2. $\displaystyle \frac{5}{2}$

  3. $\displaystyle \frac{7}{2}$

  4. $\displaystyle \frac{9}{2}$

Show answer
Correct Option: C
Explanation:
Given planes are $2x + y + 2z = 8$ and $4x + 2y + 4z + 5 = 0$
Multiplying first equation by 2 we get,
$4x+2y+4z=16 , 4x+2y+4z=-5$
Distance betweeb the planes $= \dfrac{21}{\sqrt {4^2+2^2+4^2}}$
$\therefore \dfrac{21}{6}=\dfrac{7}{2}$

The distance between the planes given by $\vec{r}.\left ( i:+:2j:-:2k \right ):+:5= 0$ and $\vec{r}.\left ( i:+:2j:-:2k \right ):-:8= 0$ is

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $1$ unit

  2. $13/3$ units

  3. $13$ units

  4. none of these

Show answer
Correct Option: B
Explanation:
Given planes
$\vec{r}\cdot(\hat{i}+2\hat{j}-2\hat{k})+5=0$---------(1)
$\vec{r}\cdot(\hat{i}+2\hat{j}-2\hat{k})-8=0$---------(2)
from Here $a=1,b=2,c=-2$ and $d _{1}=-5\ and\ d _{2}=8$
$distance=\left | \dfrac{d _{1}-d _{2}}{\sqrt{a^2+b^2+c^2}} \right |$
$distance=\left | \dfrac{-5-8}{\sqrt{1^2+2^2+(-2)^2}} \right |$
$distance=\left | \dfrac{-13}{\sqrt{1+4+4}} \right |$
$distance=\left | \dfrac{-13}{\sqrt{9}} \right |$
$distance=\left | \dfrac{-13}{3} \right |$
$distance=\dfrac{13}{3}$

The distance between the parallel planes given by the equations, $\vec{r}\,. \, (2\, \hat{i}\, -\, 2\, \hat{j}\, +\, \hat{k})\, +\, 3\, =\, 0$ and $\vec{r}\,. \, (4\, \hat{i}\, -\, 4\, \hat{j}\, +\, 2\hat{k})\, +\, 5\, =\, 0$ is:

maths introduction to three-dimensional geometry distance between two parallel planes three dimensional geometry - ii distance formula

  1. $\dfrac{1}{2}$

  2. $\displaystyle \frac{1}{6}$

  3. $\displaystyle \frac{\sqrt{2}}{3}$

  4. $1$

Show answer
Correct Option: B
Explanation:

$\vec { r } .\left( 2\hat { i } -2\hat { j } +\hat { k }  \right) =-3$

Multiplying by $2$
$\vec { r } .\left( 4\hat { i } -4\hat { j } +2\hat { k }  \right) =-6$
Equation of other plane
$\vec { r } .\left( 4\hat { i } -4\hat { j } +2\hat { k }  \right) =-5$
Distance between the two parallel planes
$=\cfrac { { c } _{ 1 }+{ c } _{ 2 } }{ \sqrt { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }  } $
$=\cfrac { \left( -6 \right) -\left( -5 \right)  }{ \sqrt { { 4 }^{ 2 }+{ 4 }^{ 2 }+2^{ 2 } }  } $
$=\cfrac { 1 }{ \sqrt { 16+16+4 }  } $
$=\cfrac { 1 }{ 6 } $