Tag: maths

Questions Related to maths

If the planes $ 2x-y+ \lambda z- 5=0$ and $x+4y+2z- 7= 0$ are perpendicular, then $\lambda=$

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $1$

  2. $-1$

  3. $2$

  4. $-2$

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Correct Option: A
Explanation:

Since, the planes $2x-y+\lambda z-5=0$ & $x+4y+2z-7=0$ are perpendicular to each other
Therefore, $\left( 2i-j+\lambda k \right) .\left( i+4j+2k \right) =0$
$\Rightarrow 2-4+2\lambda =0$
$\Rightarrow \lambda =1$

Ans: A

If the planes $\vec{r}. (2\widehat{i}- \widehat{j}+ 2\widehat{k})= 4$ and $\vec{r}. (3\widehat{i}+ 2\widehat{j}+\lambda\widehat{k})= 3$ are perpendicular, then $\lambda =$

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $2$

  2. $-2$

  3. $3$

  4. $-3$

Show answer
Correct Option: B
Explanation:

Since, the planes $\vec { r } .(2\widehat { i } -\widehat { j } +2\widehat { k } )=4$ & $\vec{r}. (3\widehat{i}+ 2\widehat{j}+\lambda\widehat{k})= 3\ $ are perpendicular to each other
Therefore, $\left( 2i-j+2 k  \right) .\left( 3i+2j+\lambda k \right) =0$
$\Rightarrow 6-2+2\lambda =0$
$\Rightarrow \lambda =-2$

Ans: B

The angle between the planes, $\vec{r}.(2\widehat{i}- \widehat{j}+\widehat {k})=6$ and $\vec{r}.(\widehat{i}+ \widehat{j}+2\widehat {k})=5$ , is:

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $\dfrac{\pi}{3}$

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

  3. $\dfrac{\pi}{6}$

  4. $\dfrac{5\pi}{6}$

Show answer
Correct Option: A
Explanation:

Angle between $2x-y+z=6$ & $x+y+2z=5$ is
$\theta =\cos ^{ -1

}{ \left[ \dfrac { \left( 2i-j+k \right) .\left( i+j+2k \right)  }{

\sqrt { \left( { 2 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } \right) \left( 1^{ 2

}+{ 1 }^{ 2 }+2^{ 2 } \right)  }  }  \right]  } =\dfrac { \pi  }{ 3 } $

Ans: B

The angle between the planes $ 3x-6y+2z+5=0 $ 7 $ 4x-12y+3z=3 $.Which is bisected by the plane
$ 67x-162y+47z+44 = 0 $is the angle which-

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. contains origin

  2. is acute

  3. is obtuse

  4. is right angle

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Correct Option: A

A plane$ P _{1}$ has the equation $2x-y+z=4$ and the plane $P _{2}$ has the equation $x+ny+2z=11.$ If the angle between $P _{1}$ and $P _{2}$ is $\pi /3$ then the value (s) of '$n$' is (are)

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $7/2$

  2. $17,-1$

  3. $-17,1$

  4. $-7/2$

Show answer
Correct Option: C
Explanation:
A plane$ P _{1}$ has the equation $2x-y+z=4$ and the plane $P _{2}$ has the equation $x+ny+2z=11.$  
The direction vector of the normal of the first plane is $2i-j+k$ and second plane is $i+nj+2k$.
The angle between $P _{1}$ and $P _{2}$ is $\pi /3$
$\cos { \dfrac { \pi  }{ 3 }  } =\dfrac { \left( 2i-j+k \right) .\left( i+nj+2k \right)  }{ \sqrt { \left( { 2 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } \right) \left( { 1 }^{ 2 }+{ n }^{ 2 }+{ 2 }^{ 2 } \right)  }  } $
$\Rightarrow \dfrac { 1 }{ 2 } =\dfrac { 4-n }{ \sqrt { 6\left( 5+{ n }^{ 2 } \right)  }  } $
$\Rightarrow \cos^2(\dfrac{\pi}{3})=\dfrac{(4-n)^2}{6(5+n^2)} $
$\Rightarrow n^2+16n-17=0\Rightarrow (n+17)(n-1)=0$
$\Rightarrow n=-17,1$

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

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $\displaystyle \cos ^{-1}\left ( \frac{1}{5} \sqrt{\frac{2}{5}} \right )$

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

  3. $\displaystyle \frac{\pi }{3}$

  4. $\displaystyle \cos ^{-1}\left ( \frac{2}{5} \sqrt{\frac{2}{3}} \right )$

Show answer
Correct Option: D
Explanation:

The angle between $x+y+z=0$ & $3x-4y+5z=0$ is
$\theta =\cos ^{ -1

}{ \left[ \dfrac { \left(\vec  i+\vec j+\vec k \right) .\left( 3\vec i-4\vec j+5\vec k \right)  }{

\sqrt { \left( { 1 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } \right) \left( 3^{ 2

}+{ 4 }^{ 2 }+5^{ 2 } \right)  }  }  \right]  } $

$= \cos ^{ -1 } \left( \dfrac { 2 }{ 5 } \sqrt { \dfrac { 2 }{ 3 }  }  \right) $

Ans: D

Which of the following planes is equally inclined to the planes $\displaystyle 4x + 3y - 5z = 0$ and $\displaystyle 5x - 12y + 13z = 0$?

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $\displaystyle 11x - 3y = 0$

  2. $\displaystyle 3x + 11y = 0$

  3. $\displaystyle 3x + 11y = 65z$

  4. none of these

Show answer
Correct Option: A
Explanation:

The bisector of planes $4x+3y-5z=0$ & $5x-12y+13z=0$ can be given by

$\dfrac { 4x+3y-5z }{ \sqrt { { 4 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 } }  } \pm \dfrac { 5x-12y+13z }{ \sqrt { { 5 }^{ 2 }+12^{ 2 }+13^{ 2 } }  } =0$

$\Rightarrow 52x+39y-65z\pm \left( 25x-60y+65z \right) =0$

$\Rightarrow 11x-3y=0$ and $27x+99y-130z=0$

Ans: A

The equation of the plane bisecting the acute angle between the planes $\displaystyle x - y + z - 1 = 0$ and $\displaystyle x + y + z = 2$ is

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $\displaystyle x + z = \frac{3}{2}$

  2. $\displaystyle 2y = 1$

  3. $\displaystyle x - y - z = 3$

  4. None of these

Show answer
Correct Option: A
Explanation:

Given planes are  $ x-y+z-1=0.....(1)$ and $x+y+z-2=0.........(2)$
Therefore equation of plane bisecting these planes are
$\dfrac{x-y+z-1}{\sqrt{3}}=\pm\dfrac{x+y+z-2}{\sqrt{3}}$
$\Rightarrow x+z = \dfrac{3}{2}.......(3)$ and $y = \dfrac{1}{2}.......(4)$
If $\theta$ is the angle between (2) and (4) then,
$  \cos\theta = \dfrac{1/2}{(1/2).(\sqrt{3})}=\dfrac{1}{\sqrt{3}}$
$\Rightarrow \theta > 45^\circ$
Hence plane (4) bisects the obtuse angle between the given planes.
Therefore equation of plane bisecting acute angle  between given plane is
$x+z = \dfrac{3}{2}$

An angle between the plane, x+y+z=5 and the line of intersection of the planes, 3x+4y+z-1=0 and 5x+8y+2z+14=0, is

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $

    \sin ^ { - 1 } ( 3 / \sqrt { 17 } )

    $

  2. $

    \cos ^ { - 1 } ( \sqrt { 3 / 17 } )

    $

  3. $

    \sin ^ { - 1 } ( \sqrt { 3 / 17 } )

    $

  4. $

    \cos ^ { - 1 } ( 3 / \sqrt { 17 } )

    $

Show answer
Correct Option: A

The angle between the planes $\bar { r } \cdot \bar { n _{ 1 } } =\left| \bar { { d } _{ 1 } }  \right| $ and $\bar { r } \cdot \bar { n _{ 2 } } =\left| \bar { { d } _{ 2 } }  \right| $

maths the plane angle between planes angle between two planes problems involving equation of plane

  1. $\cos^{-1}\left(\displaystyle \frac{\bar{n _{1} }\cdot\bar{d} _{1}}{\left | \bar{d} _{1}\times \bar{d} _{2} \right |}\right)$

  2. $\cos^{-1}\left(\displaystyle \frac{\bar{n} _{1}.\bar{n} _{2}}{\left |\bar{n} _{1} \right |\left | \bar{n} _{2} \right |}\right)$

  3. $\cos^{-1}\left(\displaystyle \frac{\bar{n} _{1}\bar{n} _{2}}{\bar{n} _{1}\times \bar{n} _{2} }\right)$

  4. $\cos^{-1}\left(\displaystyle \frac{\bar{n} _{1}\cdot \left | \bar{d} _{2} \right |}{\left | \bar{n} _{1} \right |\left | \bar{n} _{2} \right |}\right)$

Show answer
Correct Option: B
Explanation:

Given planes are $\bar { r } \cdot \bar { n _{ 1 } } =\left| \bar { { d } _{ 1 } }  \right| $ and $\bar { r } \cdot \bar { n _{ 2 } } =\left| \bar { { d } _{ 2 } }  \right| $ 

Angle between the planes is same as the angle between the normal vectors.
Hence the angle  $\theta=\cos^{-1}\left(\dfrac{\bar{n} _1.\bar{n} _2}{|\bar{n} _1||\bar{n} _2|}\right)$