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Test: Three Dimensional Geometry- 1 - JEE MCQ


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25 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Three Dimensional Geometry- 1

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

Direction cosines of a line are

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 1

Direction cosines of a line are the cosines of the angles made by the line with the positive direction of the coordinate axis.i.e. x- axis , y-axis and z – axis respectively.

Test: Three Dimensional Geometry- 1 - Question 2

Shortest distance between two skew lines is

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 2

Shortest distance between two skew lines is The line segment perpendicular to both the lines .

Test: Three Dimensional Geometry- 1 - Question 3

Find the shortest distance between the lines 

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 3

On comparing the given equations with :
In the cartesian form two lines


we get ;

x1 = -1, y1 = -1,z1 = -1, ; a1 = 7, b1 = -6, c1 = 1 and 

x2 = 3, y2 = 5, z2 = 7; a2 = 1, b2 = -2, c2 = 1


Now the shortest distance between the lines is given by :








Test: Three Dimensional Geometry- 1 - Question 4

The angle θ between the planes A1x + B1y + C1z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0 is given by

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 4

By definition , The angle θ between the planes A1x + B1y + C1z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0 is given by :

Test: Three Dimensional Geometry- 1 - Question 5

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 5

The equation of the plane through the line of intersection of the planes






Test: Three Dimensional Geometry- 1 - Question 6

If l, m, n are the direction cosines of a line, then

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 6

If l, m , n are the direction cosines of a line then , we know that,  l2+ m2+ n2 = 1.

Test: Three Dimensional Geometry- 1 - Question 7

Shortest distance between 

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 7

Test: Three Dimensional Geometry- 1 - Question 8

Find the shortest distance between the lines :   

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 8

On comparing the given equations with: 
, we get: 





Test: Three Dimensional Geometry- 1 - Question 9

The distance of a point whose position vector is  from the plane

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 9

The distance of a point whose position vector is  from the plane  given by :

Test: Three Dimensional Geometry- 1 - Question 10

Find the angle between the planes whose vector equations are

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 10







Test: Three Dimensional Geometry- 1 - Question 11

is a vector joining two points P(x1, y1, z1) and Q(x2, y2, z2). If  Direction cosines of  are

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 11

is a vector joining two points P(x1, y1, z1) and Q(x2, y2, z2). If  Direction cosines of  are given by : 

Test: Three Dimensional Geometry- 1 - Question 12

Shortest distance between the lines 

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 12

In Cartesian coordinate system Shortest distance between the lines

Test: Three Dimensional Geometry- 1 - Question 13

Find the shortest distance between the lines  and 

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 13

Find the shortest distance between the lines 

On comparing them with :

we get : 






Test: Three Dimensional Geometry- 1 - Question 14

The distance d from a point P(x1, y1, z1) to the plane Ax + By + Cz + D = 0 is

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 14

The distance d from a point P(x1, y1, z1) to the plane Ax + By + Cz + D = 0 is given by :

Test: Three Dimensional Geometry- 1 - Question 15

Determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 15



Test: Three Dimensional Geometry- 1 - Question 16

If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 16

If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are Proportional to the direction cosines of the line.

Test: Three Dimensional Geometry- 1 - Question 17

Distance between 

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 17

In vector form Distance between two parallel lines  given by :

Test: Three Dimensional Geometry- 1 - Question 18

Find the angle between the following pairs of lines:   and 

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 18

If θ is the acute angle between


then cosine of the angle between
these two lines is given by :


Here, 


Then, 



Test: Three Dimensional Geometry- 1 - Question 19

Determine the direction cosines of the normal to the plane and the distance from the origin. Plane z = 2

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 19

We have z = 2 . , it can be written as : 0x+0y+1z = 2. Compare it with lx+my+nz = d , we get ; l = 0 , m = 0 , n = 1 and d = 2 . therefore , D.C.’s of normal to the plane are 0 , 0 , 1 and distance from the origin = 2.

Test: Three Dimensional Geometry- 1 - Question 20

In the following case, determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angles between them. 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 20

We have , 
2x + y + 3z – 2 = 0 and x – 2y + 5 = 0. Let θ be the angle between the planes , then 

Test: Three Dimensional Geometry- 1 - Question 21

If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 21

If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then , the directions cosines of the line are given by :

Test: Three Dimensional Geometry- 1 - Question 22

If a line makes angles 90, 135, 45 with the x, y and z – axes respectively, find its direction cosines.

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 22

If a line makes angles 90, 135, 45 with the x, y and z – axes respectively, then the direction cosines of this line is given by :

Test: Three Dimensional Geometry- 1 - Question 23

In the vector form, equation of a plane which is at a distance d from the origin, and  is the unit vector normal to the plane through the origin is

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 23

In the vector form, equation of a plane which is at a distance d from the origin, and  is the unit vector normal to the plane through the origin is given by : 

Test: Three Dimensional Geometry- 1 - Question 24

Determine the direction cosines of the normal to the plane and the distance from the origin. Plane x + y + z = 1

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 24

Here , D.R’s of normal to the plane are 1, 1 , 1 ,its D.C ‘s are :

On dividing x + y + z = 1 by √3 , we get :
 It is of the form : lx+my+nz = d , therefore , d = 1/√3 .

Test: Three Dimensional Geometry- 1 - Question 25

In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

Detailed Solution for Test: Three Dimensional Geometry- 1 - Question 25

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