Three Dimensional Geometry Test

Result

Three Dimensional Geometry Test - 3

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
2 / -0.83

The distance of the point (3, 4, 5) from X- axis is

A
√41

B
3

C
5

D
√34

Solution

x is 3 units to the right from (0, 4, 5)
y is 4 units upwards from (3, 0, 5)
z is 5 units inwards from (3, 4, 0)
Note that the x-axis has the equation ‘y=0, z=0 ’, because the axis is horizontal and always has a y and z values of 0 for all x.
Therefore, the distance of point P(3,4,5) from X-axis is the hypotenuse of the triangle formed by the upward and inward movement along y and z-axis, which can be found using Pythagoras ’Theorem:
sqrt(5² + 4² ) = sqrt(41)
The distance is the square root of 41.
Question 2
2 / -0.83

The direction cosines of X -axis are

A
< 0 , 1 , 0 >

B
< 1 , 0 , 0 >

C
< 0 , 0 , 1 >

D
< 0 , 1 , 1 >

Solution

The x-axis makes angles 0 °, 90 °and 90 °with x, y and z-axis. Again y-axis makes angles 0 °, 90 °, 90 °with x, y and z-axis.
∴direction cosines of x-axes are cos 0 °, cos 90 °, cos 90 °i.e. 1, 0, 0.
Question 3
2 / -0.83

The medians of a triangle are concurrent at the point called

A
orthocentre

B
incentre

C
centroid

D
circumcentre
Question 4
2 / -0.83

The direction cosines of any normal to the XY plane are

A
< 1 , 1 , 0 >

B
< 0 , 0 , 1 >

C
< 1 , 0 , 0 >

D
< 0 , 1 , 0 >

Solution

Any normal of x −y plane is along z −axis.
normal unit vector to x −y plane,
n →= k ˆ= 0i ˆ+0j ˆ+1k ˆ
Direction cosines are <0, 0, 1 >or
n →= k =0i ˆ+0j ˆ−1k ˆ
Direction cosines are <0, 0, −1 >
Question 5
2 / -0.83

How many lines through the origin make equal angles with the coordinate axes?

A
1

B
4

C
8

D
2

Solution

There are two different lines that make equal angles with the coordinate axes.
The are y = x line and y = - x line.
The y = x line divides the 1st and 3rd quadrant equally and makes a 45 °angle with the positive direction of the x-axis and the y-axis and negative direction of the x-axis and the y-axis.
Again, the line y = - x divides the 2nd and 4th quadrant equally and makes a 45 °angle with the positive direction of the x-axis and negative direction of the y-axis &negative direction of the x-axis and positive direction of the y-axis.
Question 6
2 / -0.83

The numbers 3, 4 , 5 can be

A
direction cosines of a line in space

B
coordinates of a point on the line y = 4 , z = 0

C
direction numbers of a line in space

D
coordinates of a point in the plane x + y –z = 0

Solution

Correct Answer : c
Explanation : l² + m² + n² = 1
(a) 9 + 16 + 25 not equal to 1
(b) y = 4z
4 is not equal to 20
(d) x + y - z = 0
3 + 4 - 5 = 0
=>2 is not equal to 0
There option c satisfies the equation.
Question 7
2 / -0.83

If the direction cosines of a straight line are < k , k , k > , then

A
0 < k < 1

B

C
k > 0

D
k = 1

Solution

Given that direction cosine of the line(k,k,k)
The value of k = +-1/(3)^½
We know the sum of the squares of the direction cosine is one.
k² + k² + k² = 1
3k² = 1
k² = +-1/(3)^½
Question 8
2 / -0.83

Volume of a tetrahedron is k X area of one face X length of perpendicular from the opposite vertex upon it, where k is

A
it is 1/6

B
it is 1/4

C
it is 1/3

D
it is 1/2

Solution

Correct Answer : C
Explanation : Volume of tetrahedron = k(Area of one face) * (length of perpendicular from the opposite vertex upon it)
1/3 *(Area of one face) * (length of perpendicular from the opposite vertex upon it)
k = 1/3
Question 9
2 / -0.83

The radius of the sphere through the points (4 ,3 , 0) , (0 , 4 , 3) ,(0 , 5 , 0) and (4 , 0 , 3) is

A
5/7

B
7/5

C
7

D
5
Question 10
2 / -0.83

The points (4,7,8) ,(2, 3,4),(- 1, -2 , 1) and (1, 2, 5) are the vertices of a

A
rectangle

B
square

C
rhombus

D
parallelogram

Solution

ABCD is a parallelogram for this we 'll show that both the diagonal meet at 0 i.e., 0 is the midpoint of AC and BD
AC = [(4-1)/2, (7-2)/2, (8+1)/2]
= [3/2, 5/2, 9/2]
BD = [(2+1)/2, (3+2)/2, (4+5)/2]
= [3/2, 5/2, 9/2]
Question 11
2 / -0.83

The equation xy = 0 in three dimensional space represents

A
a pair of parallel lines

B
a pair of planes at right angles

C
a plane

D
a pair of straight lines
Question 12
2 / -0.83

The graph of the equation x² +y² = 0 in the three dimensional space is

A
Y - axis

B
X - axis

C
XY - plane

D
Z - axis
Question 13
2 / -0.83

In three dimensional space, locus of the equation x² +z² = 0x² +z² = 0 is

A
y - axis

B
a degenerate sphere

C
a degenerate circle

D
empty set
Question 14
2 / -0.83

The distance of the point (x , y , z) from the XY –plane is

A
y

B
x

C
z

D
I z I
Question 15
2 / -0.83

The angle between the lines x = 1 , y = 2 and y = - 1 , z = 0 is

A
0⁰

B
60⁰

C
30⁰

D
90⁰

Solution

Given lines are
(x-1)/0 = (y-2)/0 = z/1
and x/1=(y+1)0=z/0
∴cos θ=0 ⋅1+0 ⋅0+1 ⋅0=0
⇒θ=90^∘
Question 16
2 / -0.83

The line x = 1 , y = 2 is

A
parallel to Y –axs

B
lies in a plane parallel to XY –plane

C
parallel to Z –axs

D
parallel to X –axs
Question 17
2 / -0.83

The equation (x –1) (x –2) = 0 represents, in three dimensional space, a

A
pair of parallel planes

B
an empty set

C
pair of straight lines

D
sphere
Question 18
2 / -0.83

A point (x , y , z) moves parallel to X- axis. Which of the three variables x , y , z remain fixed ?

A
y and z

B
x and y

C
x

D
z and x
Question 19
2 / -0.83

A point (x , y , z) moves parallel to XY - plane. Which of the three variables x, y, z remain fixed ?

A
y

B
z

C
x

D
x and y
Question 20
2 / -0.83

The plane x=0 divides the joinning of (−2,3,4) and (1,−2,3) in the ratio :

A
it is 1: 2

B
it is - 4 : 3

C
it is 2 : 1

D
it is 3 : 2

Solution

Given place : x=0 and two points →(−2,3,4) and (1,−2,3)
let say a point (x,y,z) in x=0 place
So, x=(m+n(−2))/m+n
0=(m −2n)/m+n
0 = (m −2n)/m+n ⇒m=2n
So, m/n = 2/1
⇒2:1
Question 21
2 / -0.83

The direction cosines of the line joining (1 , - 1 , 1) , and (-1 , 1 , 1) are

A
< 1 , - 1 , 1 >

B
< 1 , - 1 , 0 >

C
< 1/√2, - 1/√2, 0 >

D
< 2 , - 2 , 0 >
Question 22
2 / -0.83

The line

A
at right angles to plane YOZ

B
is parallel to Y- axis

C
at right angles to x –axis

D
none of these
Question 23
2 / -0.83

The line x = x₁ , y = y₁ is

A
parallel to y axis

B
parallel to z axis

C
parallel to x axis

D
none of these
Question 24
2 / -0.83

The locus of the equation xy + yz = 0 is

A
a pair of perpendicular planes

B
a pair of parallel planes

C
a pair of straight lines

D
none of these
Question 25
2 / -0.83

The number of spheres of a given radius r and touching the coordinate axes is

A
4

B
8

C
6

D
none of these

User

Question Analysis

Correct -
Wrong -
Skipped -

My Perfomance

Score
-
out of -
Rank
-
out of -

Re-Attempt Weekly Quiz Competition

Three Dimensional Geometry Test - 3

Result

Three Dimensional Geometry Test - 3

Score

-

Rank

-

SHARING IS CARING

Question 1

√41

3

5

√34

Solution

Question 2

< 0 , 1 , 0 >

< 1 , 0 , 0 >

< 0 , 0 , 1 >

< 0 , 1 , 1 >

Solution

Question 3

orthocentre

incentre

centroid

circumcentre

Question 4

< 1 , 1 , 0 >

< 0 , 0 , 1 >

< 1 , 0 , 0 >

< 0 , 1 , 0 >

Solution

Question 5

1

4

8

2

Solution

Question 6

direction cosines of a line in space

coordinates of a point on the line y = 4 , z = 0

direction numbers of a line in space

coordinates of a point in the plane x + y –z = 0

Solution

Question 7

0 < k < 1

k > 0

k = 1

Solution

Question 8

it is 1/6

it is 1/4

it is 1/3

it is 1/2

Solution

Question 9

5/7

7/5

7

5

Question 10

rectangle

square

rhombus

parallelogram

Solution

Question 11

a pair of parallel lines

a pair of planes at right angles

a plane

a pair of straight lines

Question 12

Y - axis

X - axis

XY - plane

Z - axis

Question 13

y - axis

a degenerate sphere

a degenerate circle

empty set

0⁰

60⁰

30⁰

90⁰