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Test: Introduction To 3D Geometry - JEE MCQ


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20 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Introduction To 3D Geometry

Test: Introduction To 3D Geometry for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The Test: Introduction To 3D Geometry questions and answers have been prepared according to the JEE exam syllabus.The Test: Introduction To 3D Geometry MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Introduction To 3D Geometry below.
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Test: Introduction To 3D Geometry - Question 1

Find the direction cosines of a line which makes equal angles with all three the coordinate axes.​

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Test: Introduction To 3D Geometry - Question 2

Three planes, viz the XY Plane, XZ Plane and the YZ Plane divide the space into eight parts. Each part is called an OCTANT. What is the relation between these three planes​

Detailed Solution for Test: Introduction To 3D Geometry - Question 2

The three mutually perpendicular coordinate planes which in turn divide the space into eight parts and each part is known as octant.

Test: Introduction To 3D Geometry - Question 3

The co-ordinates of the vertices of the triangle are A(-2, 3, 6), B(-4, 4, 9) and C(0, 5, 8). The direction cosines of the median BE are:​

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Test: Introduction To 3D Geometry - Question 4

If the direction cosines of a line from the positive X-axis and Y-axis areThe angle of the line through Z-axis is:​

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Test: Introduction To 3D Geometry - Question 5

The direction cosines of the line equally inclined with the axes are:​

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Test: Introduction To 3D Geometry - Question 6

The direction cosines of the line whose direction ratios are 6, – 6, 3 are:

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Test: Introduction To 3D Geometry - Question 7

Find the direction cosines of the x axis.​

Detailed Solution for Test: Introduction To 3D Geometry - Question 7

To find Direction Cosines of X-axis.
Take any two points on X-axis : A(a,0,0) & B(b,0,0)
DR of AB : (b-a,0,0)
DC of AB : ((b-a)/(((b-a)2 + 0 + 0)1/2), 0, 0)
: ((b-a)/(b-a) , 0 , 0)
: (1,0,0)

Test: Introduction To 3D Geometry - Question 8

The direction cosines of the line joining the points (2, -1, 8) and (-4, -3, 5) are:

Detailed Solution for Test: Introduction To 3D Geometry - Question 8

Pt. A(2, -1, 8)
Pt. B(-4, -3, 5)
Direction Ratio DR of AB : ( -4-2 , -3+1 , 5-8 )
: (-6,-2,-3)
Direction cosine of AB : ( -6/(62+22+32)1/2 , -2/(62+22+32)1/2 , -3/(62+22+32)1/2)
: ( -6/7, -2/7, -3/7)
 

Test: Introduction To 3D Geometry - Question 9

What are direction numbers of a line.​

Detailed Solution for Test: Introduction To 3D Geometry - Question 9

The numbers which are proportional to direction cosines of a line are called direction numbers of the line.

Test: Introduction To 3D Geometry - Question 10

What are direction ratios of a line.​

Test: Introduction To 3D Geometry - Question 11

Find the direction cosines of the side AB of the triangle whose vertices are A(3, 5, -4), B(-1, 1, 2) and C(-5, -5, -2)

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Test: Introduction To 3D Geometry - Question 12

If l, m , n are the direction cosines of any line, then sum of the squares of the direction cosines of the line is always​

Test: Introduction To 3D Geometry - Question 13

If a line has the direction ratios -4, 18, -12 then what are its direction cosines?​

Detailed Solution for Test: Introduction To 3D Geometry - Question 13

DR of the line :  (-4, 18 -12)
DC of the line : (-4/k, 18/k, -12/k)
where k = ((42) + (182) + (12)2)1/2
= (16 + 324 + 144)1/2
= (484)1/2
= 22
So, DC : (-4/22, 18/22, -12/22)
: (-2/11 , 9/11 , -6/11)

Test: Introduction To 3D Geometry - Question 14

The direction cosines of the line equally inclined with the axes, are:​

Detailed Solution for Test: Introduction To 3D Geometry - Question 14
Cos^2 alpha+cos ^2 beta +cos^ 2 gamma =1
put alpha =beta. =gamma
we get cosalpha= 1/√3
DC's of given line (1/√3,1/√3,1/3)
Test: Introduction To 3D Geometry - Question 15

If a line makes angles 45°,150°, 135°, with x, y and z-axes respectively, find its direction cosines.

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Test: Introduction To 3D Geometry - Question 16

Find the equation of the set of points which are equidistant from the points (1, 2 , 3) and (3, 2, -1)​

Detailed Solution for Test: Introduction To 3D Geometry - Question 16

Pt. A(1, 2 , 3)
Pt. B(3, 2, -1)
Let P(x,y,z)
So, AP = BP
((x-1)2 + (y-2)2 + (z-3)2)1/2 = ((x-3)2 + (y-2)2 + (z+1)2)1/2
(x-1)2 + (y-2)2 + (z-3)2) = (x-3)2 + (y-2)2 + (z+1)2
x2 +1 -2x + y2 + 4 - 4y + z2 + 9 – 6z = x2 +9 -6x + y2 + 4 - 4y + z2 + 1 + 2z
4x – 8z = 0
x – 2z = 0

Test: Introduction To 3D Geometry - Question 17

If a line in the ZX-plane makes an angle 60o with Z-axis, the direction cosines of this line are:

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Test: Introduction To 3D Geometry - Question 18

A line makes angles α, β, γ with the positive directions of X-axis, Y-axis and Z-axis, respectively, then the directions cosines of the line are:

Detailed Solution for Test: Introduction To 3D Geometry - Question 18

cos α, cos β, cos γ
By the definition of Direction Cosines

Test: Introduction To 3D Geometry - Question 19

The signs of the X,Y and Z coordinates of a point that lies in the octant OXYZ’ is​

Detailed Solution for Test: Introduction To 3D Geometry - Question 19

X,Y,Z imply positive X,Y,Z axis & X’,Y’,Z’ imply negative X,Y,Z axis.
So, OXYZ’ will have a point of signs (+, +, -).

Test: Introduction To 3D Geometry - Question 20

If a line in the ZX-plane makes an angle 30o with Z-axis, the direction cosines of this line are:

Detailed Solution for Test: Introduction To 3D Geometry - Question 20
The line makes 30o with z-axis

Since it is in z-x axis the angle made with y-axis is 90

And angle made by x-axis is 60

Therefore direction cosines are

1/2, 0, √3/2
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