Tag: maths

Questions Related to maths

If the system of equation $x+2y-3z=1$, $(p+2)z=3$, $(2p+1)y+z=2$ has infinite number of solutions, then the value of p is not equal to.

maths set concepts finite and infinite sets types of sets set language

  1. $-2$

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

  3. $0$

  4. $2$

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Correct Option: A
Explanation:
$x+2y-3z=1$
$(p+2)z=3$
$(2p+1)y+z=2$
let $p=5$, $s\in R/ \left\{ -2, 1/2\right\}$
$\therefore z=\dfrac{3}{s+2}$
$\Rightarrow y=\left(2-\dfrac{3}{s+2}\right)\dfrac{1}{(2s+1)}\Rightarrow \dfrac{2s+1}{(s+2)(2s+1)}-\dfrac{1}{s+2}$
$[As\ 2s+1\neq 0]$
$\therefore x=3z+1-2y$
$=\dfrac{9}{s+2}+1-\dfrac{2}{s+2}=\dfrac{7}{s+2}+1$
$\therefore$ solutions $(x, y, z)=\left(\dfrac{7}{s+2}+1, \dfrac{1}{s+2}, \dfrac{3}{s+2}\right)$
is an infinite set,
$\therefore p$ cannot be equal to $-2$ or $1/2$

The coordinates of any point, which lies on $x$ axis are

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. $(0,x,0)$

  2. $(x,0,0)$

  3. $(x,x,0)$

  4. $(x,x,x)$

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Correct Option: B
Explanation:

In 3-dimensional plane, the point which lies on $x$-axis does not have any part in y and z axes.

At that point, the value of $y$ and $z$ will be $0$.
Hence, coordinate of any point which lies on $x$ axis are $(x,0,0)$.

A point at which all the three perpendicular coordinate axes meets is known as 

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. Meeting point

  2. Origin

  3. Triple point

  4. None of these

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Correct Option: B
Explanation:

The three perpendicular coordinate axes meets at one of the point which divides the plane into eight quadrant.

The $1^{st}$ quadrant has all positive points, $2^{nd}$ has $x$ -ve and remaining $2$ +ve points and so on.
Only $(0,0,0)$ is not included in any quadrant and is the intersection point.
Thus, the three axes meet at $(0,0,0)$ from where the eight quadrants originate.
Hence, the point is known as origin.

If point $p$ lies in first octant, then the sign of $x-$ coordinate will always be 

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. $+$

  2. $-$

  3. $x$ coordinate is always $0$

  4. $x$ coordinate can be $+$ or $-$

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Correct Option: A
Explanation:

In the first octant, the values of x,y and z axes are positive.

Any point which lies in first octant has all their coordinate values as positive.
Since point $p$ lies in first octant, so, p will have all its coordinates as positive.
Hence, sign of $x$-coordinate will always be $+$.

 Which one of the following 3D shapes does not have a vertex?  

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. Sphere

  2. Pyramid

  3. Prism

  4. Cone

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Correct Option: A

Examine if the following is true statement.
The cube can cast a shadow in the shape of a rectangle.

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. True

  2. False

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Correct Option: A
Explanation:

The formation of shadow is equivalent to cutting of cube . We know that cutting cube always gives a square or a rectangle . so, it can cast a shadow of a rectangle.

Examine if the following is true statement.
The cube can cast a shadow in the shape of a hexagon.

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. True

  2. False

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Correct Option: B
Explanation:
False
The cube cannot cast shadow in shape of hexaxon because of following reason:
no matter whatever the position of light source focussing on cube it always cast a shadow of either square or rectangle .

State the following statement is True or False
Two distinct lines intersecting each other in a plane can have two points in common

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. True

  2. False

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Correct Option: B
Explanation:

Answer is option B (False)
Two distinct lines intersecting each other  in a plane cannot have more than one point in common

Name three undefined terms.

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. Point

  2. Line

  3. Plane

  4. All of the above

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Correct Option: D
Explanation:

The basic undefined term is point.
Line is formed from points and plane is formed from many lines.
Undefined terms are point, line and plane.

State the following statement is True or False
If two distinct lines are intersecting each other in a plane then they cannot have more than one point in common.

maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

  1. True

  2. False

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Correct Option: A
Explanation:
suppose lines $l1$ and $l2$ intersect at two disticnt points say P and Q.Then $l1$ contains points P and Q.
Also, $l2$ contains points P and Q.
So two lines $l1$ and $l2$ pass through two distinct points P and Q.
But only one line can pass through two different points.  (axiom 3)
Hence, two lines cannot have more than one point in common.
$\therefore$ Option A  is correct.