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Introduction To 3D Geometry - Free MCQ Practice Test with solutions, JEE


MCQ Practice Test & Solutions: Test: Introduction To 3D Geometry (20 Questions)

You can prepare effectively for JEE Chapter-wise Tests for JEE Main & Advanced with this dedicated MCQ Practice Test (available with solutions) on the important topic of "Test: Introduction To 3D Geometry". These 20 questions have been designed by the experts with the latest curriculum of JEE 2026, to help you master the concept.

Test Highlights:

  • - Format: Multiple Choice Questions (MCQ)
  • - Duration: 20 minutes
  • - Number of Questions: 20

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Test: Introduction To 3D Geometry - Question 1
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Find the direction cosines of a line which makes equal angles with all three the coordinate axes.​

Detailed Solution: Question 1

Test: Introduction To 3D Geometry - Question 2
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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: Question 2

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

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Test: Introduction To 3D Geometry - Question 3
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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:​

Detailed Solution: Question 3


Test: Introduction To 3D Geometry - Question 4
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If the direction cosines of a line from the positive X-axis and Y-axis areThe angle of the line through Z-axis is:​

Detailed Solution: Question 4



Test: Introduction To 3D Geometry - Question 5
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The direction cosines of the line equally inclined with the axes are:​

Detailed Solution: Question 5

Test: Introduction To 3D Geometry - Question 6
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The direction cosines of the line whose direction ratios are 6, – 6, 3 are:

Detailed Solution: Question 6

Test: Introduction To 3D Geometry - Question 7
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Find the direction cosines of the x axis.​

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

Detailed Solution: 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
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What are direction numbers of a line.​

Detailed Solution: 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
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What are direction ratios of a line.​

Test: Introduction To 3D Geometry - Question 11
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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)

Detailed Solution: Question 11

Test: Introduction To 3D Geometry - Question 12
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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​

Detailed Solution: Question 12

Let OP be any line through the origin O which has direction cosines l, m, n.

Let P be the point having coordinates (x, y, z) and OP = r.

Then OP² = x² + y² + z² = r² ... (1)

From P draw PA, PB, PC perpendicular on the coordinate axes, so that
OA = x, OB = y, OC = z.

Also, ∠POA = α, ∠POB = β, ∠POC = γ.

From triangle AOP, l = cos α = x / r ⇒ x = lr.
Similarly, y = mr, z = nr.

Adding all we get x² + y² + z² = r²(l² + m² + n²).

⇒ r² = r²(l² + m² + n²)

⇒ l² + m² + n² = 1.

Test: Introduction To 3D Geometry - Question 13
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If a line has the direction ratios -4, 18, -12 then what are its direction cosines?​

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

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

Detailed Solution: Question 15

Test: Introduction To 3D Geometry - Question 16
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Find the equation of the set of points which are equidistant from the points (1, 2 , 3) and (3, 2, -1)​

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

Detailed Solution: Question 17

Test: Introduction To 3D Geometry - Question 18
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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: Question 18

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

Test: Introduction To 3D Geometry - Question 19
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The signs of the X,Y and Z coordinates of a point that lies in the octant OXYZ’ is​

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

Detailed Solution: 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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This test contains 20 MCQs on Introduction To 3D Geometry with detailed solutions for each question. It is part of the Chapter-wise Tests for JEE Main & Advanced course on EduRev, designed to align with the current JEE syllabus. Each solution explains not just the correct answer but also gives you an understanding of the topic — not just to attempt the question but also help you prepare for the exam with practice.

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