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Test: Three Dimensional Geometry- 2 - JEE MCQ


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25 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Three Dimensional Geometry- 2

Test: Three Dimensional Geometry- 2 for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The Test: Three Dimensional Geometry- 2 questions and answers have been prepared according to the JEE exam syllabus.The Test: Three Dimensional Geometry- 2 MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Three Dimensional Geometry- 2 below.
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Test: Three Dimensional Geometry- 2 - Question 1

Skew lines are lines in different planes which are

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

By definition : The Skew lines are lines in different planes which are neither parallel nor intersecting .

Test: Three Dimensional Geometry- 2 - Question 2

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

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

If a line has the direction ratios – 18, 12, – 4, then its direction cosines are given by:








Test: Three Dimensional Geometry- 2 - Question 3

Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is.

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 3

In Cartesian co – ordinate system Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is given by : lx + my + nz = d.

Test: Three Dimensional Geometry- 2 - Question 4

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 4

Test: Three Dimensional Geometry- 2 - Question 5

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 – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 5

Test: Three Dimensional Geometry- 2 - Question 6

Angle between skew lines is

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 6

Angle between skew lines is the angle between two intersecting lines drawn from any point 

Test: Three Dimensional Geometry- 2 - Question 7

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 7

The equation of the line which passes through the point (1, 2, 3) and is parallel to the vector  Let vector   and vector 

Test: Three Dimensional Geometry- 2 - Question 8

The equation of a plane through a point whose position vector is  perpendicular to the vector  . is

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 8

In vector form The equation of a plane through a point whose position vector is  perpendicular to the vector  Is given by : 

Test: Three Dimensional Geometry- 2 - Question 9

Find the Cartesian equation of the plane 

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 9

Test: Three Dimensional Geometry- 2 - Question 10

In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 4x + 8y + z – 8 = 0 and y + z – 4 = 0

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 10

We have, 4x + 8y + z – 8 = 0 and y + z – 4 = 0. Let be the angle between the planes, then 

Test: Three Dimensional Geometry- 2 - Question 11

If l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines; and θ is the acute angle between the two lines; then

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 11

If l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines; and θ is the acute angle between the two lines; then the cosine of the angle between these two lines is given by : 

Test: Three Dimensional Geometry- 2 - Question 12

Find the equation of the line in cartesian form that passes through the point with position vector   and is in the direction

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 12

, then , its Cartesian equation is given by :

Test: Three Dimensional Geometry- 2 - Question 13

Equation of a plane passing through three non collinear points (x1, y1, z1),(x2, y2, z2) and (x3, y3, z3) is

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 13

In cartesian co – ordinate system : Equation of a plane passing through three non collinear points (x1, y1, z1),(x2, y2, z2) and (x3, y3, z3) is given  

Test: Three Dimensional Geometry- 2 - Question 14

Find the Cartesian equation of the plane 

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 14

Test: Three Dimensional Geometry- 2 - Question 15

Find the distance of the point (0, 0, 0) from the plane 3x – 4y + 12 z = 3

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 15

As we know that the length of the perpendicular from point 
P(x1,y1,z1) from the plane a1x+b1y+c1z+d1 = 0 is given by: 

Test: Three Dimensional Geometry- 2 - Question 16

If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θ is the acute angle between the two lines; then

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 16

If a1, b1, c1 and a2, b2, c2 are the direction ratios of two lines and θθ is the acute angle between the two lines; then , the cosine of the angle between these two lines is given by :

Test: Three Dimensional Geometry- 2 - Question 17

Find the equation of the line in cartesian form that passes through the point (– 2, 4, – 5) and parallel to the line given by 

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 17

Find the equation of the line in cartesian form that passes through the point (– 2, 4, – 5) and parallel to the line given by

is given by:
 And l = 3 , m = 5 and n = 6 .

Test: Three Dimensional Geometry- 2 - Question 18

Vector equation of a plane that contains three non collinear points having position vectors 

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 18

Vector equation of a plane that contains three non collinear points having position vectors

Test: Three Dimensional Geometry- 2 - Question 19

The vector and cartesian equations of the planes that passes through the point (1, 0, – 2) and the normal to the plane is

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 19

Let 
be the position vector of the point  here,
. Therefore, the required vector equation of the plane is: 


Test: Three Dimensional Geometry- 2 - Question 20

Find the distance of the point (3, – 2, 1) from the plane 2x – y + 2z + 3 = 0

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 20

As we know that the length of the perpendicular from point P(x1,y1,z1) from the plane

Here, P(3, - 2,1) is the point and equation of Plane is 2x - y + 2z+3 = 0

Therefore, the perpendicular distance is :

 

Test: Three Dimensional Geometry- 2 - Question 21

Vector equation of a line that passes through the given point whose position vector is  and parallel to a given vector  is

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 21

 

Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector is given by : 

Test: Three Dimensional Geometry- 2 - Question 22

Find the values of p so that the lines   are at right angles.

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 22

Give lines are :
 and 
The D.R's of the lines are -3, 2p/7, 2 and -3p/7, 1, -5


Test: Three Dimensional Geometry- 2 - Question 23

Vector equation of a plane that passes through the intersection of planes  expressed in terms of a non – zero constant λ is

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 23

In vector form:
Vector equation of a plane that passes through the intersection of planes
 expressed in terms of a non – zero constant λ is given by :

Test: Three Dimensional Geometry- 2 - Question 24

Find the equations of the planes that passes through three points (1, 1, 0), (1, 2, 1), (– 2, 2, – 1)

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 24

In cartesian co-ordinate system :
Equation of a plane passing through three non collinear

Points (x1, y1, z1) , (x2, y2, z2) and (x3, y3, z3) is given by :



Therefore, the equations of the planes that passes through three points (1,1,0), (1,2,1),  (-2,2,-1) is given by :



⇒ (x-1)(-2) - (y-1) (3) + 3z = 0
⇒ 2x+3y - 3z = 5

Test: Three Dimensional Geometry- 2 - Question 25

Find the distance of the point (2, 3, – 5) from the plane x + 2y – 2z = 9

Detailed Solution for Test: Three Dimensional Geometry- 2 - Question 25

As we know that the length of the perpendicular from point P(x1,y1,z1) from the plane


Here, P(2,3,-5) is the point and equation of plane is x+2y - 2z = 9

Therefore, the perpendicular distance is :

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