Tag: maths

Questions Related to maths

lf a plane meets the coordinate axes at $A,B,C$ , then equation of plane is such that centroid of triangle $ABC$ is $\left (\displaystyle \dfrac{1}{3}\dfrac{2} {3},\dfrac{4}{3}\right)$

maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

  1. $4x+2y+z=4$

  2. $4x+2y+z=3$

  3. $x+y+z=3$

  4. $x+y+z=9$

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Correct Option: A
Explanation:

Assume equation of plane is, $ax+by+cz=d$
Now this plane intersect axes at $A,B$ and $C$,
$\Rightarrow A = \left (\dfrac{d}{a}, 0, 0\right), B =  \left (0, \dfrac{d}{b}, 0\right)$ and $ C =\left (0, 0, \dfrac{d}{c}\right)$
So the centroid of triangle $ABC$ is, $\left (\dfrac{d}{3a}, \dfrac{d}{3b}, \dfrac{d}{3c}\right)$
Comparing this with given value $ a=d, b = 2d$ and $ c = 4d$
Hence equation of the plane is 

$4x+2y+z = 4$

If from a point $P(a,b,c)$ perpendicular $PA$ and $PB$ are drawn to $yz$ and $zx$ planes, find the equation of the plane $OAB$:

maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

  1. $\displaystyle \dfrac { x }{ a } +\dfrac { y }{ b } -\dfrac { z }{ c } =0$

  2. $\displaystyle \dfrac { x }{ a } +\dfrac { y }{ b } +\dfrac { z }{ c } =0$

  3. xa−yb−z

  4. None of these

Show answer
Correct Option: A
Explanation:

The coordinates of $A$ and $B$ are $(0,b,c)$ and $(a,0,c)$ respectively.

The equation of the plane passing through $O(0,0,0),A(0,b,0)$ and $B(a,0,c)$ is given by 
$\displaystyle \begin{vmatrix} x-0 & y-0 & z-0 \ 0-0 & b-0 & c-0 \ a-0 & 0-0 & c-0 \end{vmatrix}=0\Rightarrow bcx+acy-abz=0$
$\displaystyle \Rightarrow \frac { x }{ a } +\frac { y }{ b } -\frac { z }{ c } =0$

The equation of the plane which is parallel to y-axis and cuts off intercepts of length 2 and 3 from x-axis and z-axis is :

maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

  1. $ 3x + 2z = 1$

  2. $ 3x+ 2z = 6 $

  3. $ 2x+ 3z = 6 $

  4. $ 3x+ 2z = 0 $

Show answer
Correct Option: B
Explanation:

The equation of plane parallel to $y-axis$ is, $ax+bz+1=0$      ----- ( 1 )


Here,  $x=2$ and $z=3$


Substituting $x=2$ and $z=0$ in equation ( 1 ),

$\Rightarrow$  $2a+0+1=0$

$\Rightarrow$  $x=\dfrac{-1}{2}$

Substituting $x=0$ and $z=3$ in equation ( 1 ),

$\Rightarrow$  $0+3b+1=0$

$\Rightarrow$  $b=\dfrac{-1}{3}$

Substituting value of $a$ and $b$ equation (  1 ) we get,

$\Rightarrow$  $\dfrac{-1}{2}x-\dfrac{1}{3}z+1=0$

$\Rightarrow$  $3x+2z=6$

The expression of $x+y+z=1$ in form of $x\cos { \alpha  } +y\cos { \beta  } +z\cos { \gamma  } =p$ is _______.

maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

  1. $x+y+z=1$

  2. $\cfrac { x }{ 2\sqrt { 3 } } +\cfrac { y }{ 2\sqrt { 3 } } +\cfrac { z }{ 2\sqrt { 3 } } =\cfrac { 1 }{ \sqrt { 3 } } $

  3. $\cfrac { x }{ \sqrt { 3 } } +\cfrac { y }{ \sqrt { 3 } } +\cfrac { z }{ \sqrt { 3 } } =1$

  4. $\cfrac { x }{ \sqrt { 3 } } +\cfrac { y }{ \sqrt { 3 } } +\cfrac { z }{ \sqrt { 3 } } =\cfrac { 1 }{ \sqrt { 3 } } $

Show answer
Correct Option: D
Explanation:

$\cfrac { x }{ \sqrt { 3 }  } +\cfrac { y }{ \sqrt { 3 }  } +\cfrac { z }{ \sqrt { 3 }  } =1$
$\quad \rightarrow P=\cfrac { \left| -1 \right|  }{ \sqrt { 1+1+1 }  } =\cfrac { 1 }{ \sqrt { 3 }  } $
$\therefore x+y+z=1$
$\therefore \cfrac { x }{ \sqrt { 3 }  } +\cfrac { y }{ \sqrt { 3 }  } +\cfrac { z }{ \sqrt { 3 }  } =\cfrac { 1 }{ \sqrt { 3 }  } $

The sum of Y and Z intercepts of the plane $3x+4y-6z=12$ is ___________.

maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

  1. $10$

  2. $4$

  3. $1$

  4. $5$

Show answer
Correct Option: C
Explanation:

$3x+4y-6z=12$
$\therefore\dfrac{3x}{12}+\dfrac{4y}{12}+\dfrac{(-6)z}{12}=1$
$\therefore \dfrac{x}{4}+\dfrac{y}{3}+\dfrac{z}{(-2)}=1$
$\therefore$ y intercept $b=3$ and z intercept $c=-2$
$\therefore$ y intercept $+$ z intercept $=3+(-2)=1$

The plane $ax+by+cz=1$ meets the coordinate axes in $A, B$ and $C$. The centroid of the triangle is:

maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

  1. $(3a, 3b, 3c)$

  2. $\left( \dfrac { a }{ 3 } ,\dfrac { b }{ 3 } ,\dfrac { c }{ 3 } \right)$

  3. $\left( \dfrac { 3 }{ a } ,\dfrac { 3 }{ b }, \dfrac { 3 }{ c } \right)$

  4. $\left( \dfrac { 1 }{ 3a } ,\dfrac { 1 }{ 3b } ,\dfrac { 1 }{ 3c } \right)$

Show answer
Correct Option: D
Explanation:

The plane $ax + by + cz = 1$ meets the coordinate axis in $A,B,C$, then the coordinates will be,

$A\left( {a,0,0} \right)$, $B\left( {0,b,0} \right)$ and $C\left( {0,0,c} \right)$

The equation of the plane in intercept form is,

$\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{c}{z} = 1$

The intercepts that the plane make on the axis is $\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}$

Let C denotes the centroid, then,

$C = \left( {\dfrac{{\dfrac{1}{a} + 0 + 0}}{3},\dfrac{{0 + \dfrac{1}{b} + 0}}{3},\dfrac{{0 + 0 + \dfrac{1}{c}}}{3}} \right)$

Therefore, the coordinates of the centroid will be $\left( {\dfrac{1}{{3a}},\dfrac{1}{{3b}},\dfrac{1}{{3c}}} \right)$.

If a plane passes through a fixed point $\left ( 2, 3, 4 \right )$ and meets the axes of reference in $A$, $B$ and $C$, the point of intersection of the planes through $A$, $B$, $C$ parallel to the coordinate planes can be

maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

  1. $\left ( 6, 9, 12 \right )$

  2. $\left ( 4, 12, 16 \right )$

  3. $\left ( 1, 1, -1 \right )$

  4. $\left ( 2, 3, -4 \right )$

Show answer
Correct Option: A,B,C,D
Explanation:

Let us say a plane P $ax+by+cz=k$ passes through $\left( 2,3,4 \right) $ so $2a+3b+4c=k \quad -(1)$

$A\left( \dfrac { k }{ a } ,0,0 \right) ,\quad B\left( 0,\dfrac { k }{ b } ,0 \right) ,\quad C\left( 0,0,\dfrac { k }{ c }  \right) $

Points of intersection will be $\left< \dfrac { k }{ a } ,\dfrac { k }{ b } ,\dfrac { k }{ c }  \right> $

Let $\dfrac { k }{ a } =x\quad \dfrac { k }{ b } =y\quad \dfrac { k }{ c } =z$ so in $(1)$

$\dfrac { 2k }{ x } +\dfrac { 3k }{ y } +\dfrac { 4k }{ z } =k$

$\dfrac { 2 }{ x } +\dfrac { 3 }{ y } +\dfrac { 4 }{ z } =1\quad -(1)$

$(a)$ if $(x,y,z) = (6,9,12)$

$\dfrac { 2 }{ 6 } +\dfrac { 3 }{ 9 } +\dfrac { 4 }{ 12 } =\dfrac { 1 }{ 3 } +\dfrac { 1 }{ 3 } +\dfrac { 1 }{ 3 } =1$ Hence true.

$(b)$ $\left< 4,12,16 \right> $

$\dfrac { 2 }{ 4 } +\dfrac { 3 }{ 12 } +\dfrac { 4 }{ 16 } =\dfrac { 1 }{ 2 } +\dfrac { 1 }{ 4 } +\dfrac { 1 }{ 4 } =1$ Hence correct

$(c)$ $\left< 1,1,-1 \right> $

$\dfrac { 2 }{ 1 } +\dfrac { 3 }{ 1 } +\dfrac { 4 }{ -1 } =1$ Hence this is also correct.

$(d)$ $\left< 2,3,-4 \right> $

$\dfrac { 2 }{ 2 } +\dfrac { 3 }{ 3 } +\dfrac { 4 }{ -4 } =2-1=1$ This is also correct.

Two polygons of different number of side ...... be similar.

maths properties of parallel lines and their transversal introduction to shapes similarity of triangles introduction to similar triangles

  1. Cannot

  2. Can

  3. Both A and B

  4. None

Show answer
Correct Option: A
Explanation:

Ex: A triangle and a square are not similar figures.
Therefore, A  is the correct answer.

If the corresponding angles are equal then the two figures having samenumber of sides are said to be

maths properties of parallel lines and their transversal introduction to shapes similarity of triangles introduction to similar triangles

  1. Figures

  2. Not similar

  3. None of the above

  4. Similar

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Correct Option: D
Explanation:

The angles are of the same measure means the figures are similar. 
Therefore, D is the correct answer.

If the same photograph is printed in different sizes , we say it is

maths properties of parallel lines and their transversal introduction to shapes similarity of triangles introduction to similar triangles

  1. Not similar

  2. Similar

  3. Common

  4. None

Show answer
Correct Option: B
Explanation:

Photograph being same it is printed in the same shape but in passport size, card size, and mini size. They are said to be similar.
Therefore, B is the correct answer.