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Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry


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82 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry

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Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 1

A plane which passes through the point (3, 2, 0) and the line

 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 1

As the point (3, 2, 0)  lies on the given line

∴ There can be infinite many planes passing through this line. But here out of the four options only first option is satisfied by the coordinates of both the points (3, 2, 0) and (4, 7, 4)
∴ x – y + z = 1 is the required plane.

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 2

 and the angle between   then   is equal to

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 2

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 3

 are vectors such that   then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 3



Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 4

 are vectors show that   and 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 4


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 5

thus what will be the value of   given that 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 5


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 6

If the vectors  are such that   form a right handed system then 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 6

Since   form a right handed system,

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 7

 and   are two vectors and   a vector such that

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 7

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 8

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 8

Above three conditions will be satisfied for non-zero vectors if and only if

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 9

The d.r. of normal to the plane through (1, 0, 0), (0, 1, 0) which makes an angle π /4 with plane  x + y = 3 are

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 9

Equation of plane through (1, 0, 0) is a (x – 1) + by + cz = 0 ...(i)
(i) passes through (0, 1, 0). –a + b = 0 ⇒ b = a; Also, cos 45°

So d.r of normal are a, a 

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 10

Let  is a unit vector such that  is equal to

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 10

since  perpendicular  

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 11

A particle acted on by constant forces   and   is displaced from the point   to the point  The total work done by the forces is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 11

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 12

The vectors  are the sides of a triangle ABC. The length of the median through A is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 12


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 13

The shortest distance from the plane 12x + 4 y + 3z = 327 to the sphere x2 + y2 + z2 + 4x - 2y - 6z = 155 is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 13

Shortest distance = perpendicular distance between the plane and sphere = distance of plane from centre of sphere – radius

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 14

The two lines x = ay + b , z = cy + d and x = a`y + b`, z = c`y + d` will be perpendicular, if and only if

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 14

For perpendicularity of lines  aa'+1 + cc' = 0

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 15

The lines  arecoplanar if

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 15


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 16

 are 3 vectors, such that  ,  then   is equal to

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 16

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 17

The radius of the circle in which the sphere

x2 + y2 + z2 + 2x - 2y - 4z - 19 = 0  is  cut  by  the plane x + 2y + 2z + 7 = 0 is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 17

Centre of sphere = (–1, 1, 2) Radius of sphere 

Perpendicular distance from centre to the plane

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 18

A tetrahedron has vertices at O(0, 0, 0), A(1, 2, 1) B(2, 1, 3) and C(–1, 1, 2). Then the angle between the faces OAB  and ABC will be

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 18

Vector perpendicular to the face OAB

Vector perpendicular  to the face ABC

Angle between the faces  = angle between their normals

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 19

and vectrs (1, a,a2),(1, b, b2) and (1, c,c2) are non- coplanar, then the product abc equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 19



Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 20

Consider points A , B , C and D with position vectors   and respectively. Then ABCD is a

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 20

∴ None of the options is satisfied

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 21

If  are three non - coplanar vectors, then  equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 21


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 22

Two system of rectangular axes have the same origin. If a plane cuts them at distances a, b, c and a' , b' , c' from the origin then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 22

Eq. of planes be 

(⊥ r distance on plane from origin is same.)

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 23

Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y +4z + 5 = 0 is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 23

The planes ar e 2x + y + 2z - 8=0 . ...(1)
and 4x + 2y + 4z + 5 = 0

          ...(2)

∴ Distance  between (1) and (2)

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 24

A line with direction cosin es proportional to 2, 1, 2 meets each of the lines x = y + a = z an d x + a = 2y = 2z . The co-ordinates of each of the points of intersection are given by

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 24

Let a point on th e lin e x = y + a = z is
(λ, λ - a,λ) and a point on th e lin e x + a = 2 y = 2z

 then direction ratio of the line joining these poin ts are

If it respresents the required line, then

on solving we get λ = 3a, μ = 2a
∴ The required points of intersection are (3a, 3aa,3a) and 

or (3a, 2a, 3a) and (a,a,a)

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 25

If the straight lines

with parameters s and t respectively, are co-planar, then λ equals.

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 25

The given lines are

The lines are coplanar, if


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 26

The intersection of the spheres

x2 + y2 + z2 + 7x - 2y - z = 13 and x2 + y2 + z2 -3x + 3y + 4z = 8

is the same as the intersection of one of the sphere and the plane

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 26

The equation s of spheres are

S1 : x2 + y2 + z2 + 7x - 2y - z -13 = 0 and

S2 : x2 + y2 + z- 3x + 3y + 4z-8 = 0

Their plane of intersection is

S1 - S2 = 0 ⇒ 10x - 5y - 5z - 5 = 0
⇒ 2x- y-z = 1

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 27

 be three non -zero vectors such that no two of these are collinear. If the vector   is collinear with   is collinear with   (λ being some non-zero scalar) then equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 27

Let   wh ere t and s are scalars. Adding, we get


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 28

A particles is acted upon by constant forces  and   which displace it from a point   to  the point  The work done in standard units by the forces is given by

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 28

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 29

If    are non-coplanar vectors and l is a real number, then the vectors    are non coplanar for

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 29


coplanar if 

 ∵ Forces are noncoplanar for all λ, except 

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 30

 be such that   If the projection   is equal to that of    and   are perpendicular to each other then 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 30



Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 31

 be non-zero vectors such that  is the acute angle between the vectors  then sinθ equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 31


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 32

If C is the mid point of AB and P is any point outside AB, then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 32


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 33

If the angel θ between the line  and the plane    is such that sin θ = 1/3 then the value of λ is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 33

If θ is the angle between line and plane then   is the angle between line and normal to plane given by

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 34

The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 34

The given lines are  2x = 3y =- z

       [Dividing by 6]
and 6 x = - y = -4z

  [Dividing by 12]
∴ Angle between two lines is

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 35

If the plane 2ax – 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres x2 + y2 + z2 + 6x - 8y- 2z = 13 and x2 + y2 + z2 - 10 x + 4y - 2z = 8 then a equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 35

Centers of given spheres are (– 3, 4, 1) and (5, – 2, 1).
Mid point of centres is (1, 1, 1).
Satisfying this in the equation of plane, we get 2a - 3a + 4a + 6 = 0 ⇒ a =-2.

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 36

The distance between the line  and the plane 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 36

A point on line is (2, –2, 3) its perpendicular distance from the plane x + 5 y +z – 5 = 0 is

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 37

For any vector  the value of     is equal to

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 37

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 38

If non zero numbers a, b, c are in H.P., then the straight line  always passes through a fixed point. That point is 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 38

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 39

Let a, b and c be distinct non- negative numbers. If the vectors  lie in a plane, then c is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 39

Vector    are coplanar

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 40

If  are non coplanar vectors and λ is a real number then 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 40

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 41

Let  and  depends on

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 41


Hence  is independent of x and y both.

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 42

The plane x + 2y – z = 4 cuts the sphere x2 + y+ z2 – x + z – 2 = 0 in a circle of radius

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 42

Perpendicular distance of centre   from x + 2y – 2 = 4  is given by


radius of sphere



Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 43

 where   are any three vectors such that  are

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 43

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 44

The values of a, for which points A, B, C with position vectors  respectively are the vertices of a right angled  triangle with C = π/2 are

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 44

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 45

The two lines x = ay+b , z = cy+d ; and x = a' y+b' , z = c' y+d' are perpendicular to each other if

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 45

Equation of lines  

Line are perpendicular ⇒ aa '+ 1 + cc '= 0

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 46

The image of the point (–1, 3, 4) in the plane x - 2y = 0 is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 46




N is mid point of PP'

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 47

If a line makes an angle of π /4 with the positive directions of each of x- axis and y- axis, then the angle that the line makes with the positive direction of the z-axis is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 47

Let the angle of line makes with the positive direction of z-axis is a direction cosines of line with the +ve directions of x-axis, y-axis, and z-axis is l, m, n respectively.


Hence, angle with positive direction of the 

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 48

If are unit vectors and θ is the acute angle between them, then  is a unit vector for

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 48

Given   is acute angle between  and


Hence, there is exactly one value of θ for which  is a unit vector..

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 49

If (2, 3, 5) is one end of a diameter of the sphere x2 + y2 + z2 – 6x – 12y – 2z + 20 = 0, then the cooordinates of the other end of the diameter are

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 49

For given sphere centre is (3, 6, 1) Coordinates of one end of diameter of the sphere are (2, 3, 5). Let the coordinates of the other end of diameter are (α, β,γ)

∴ Coordinate of other end of diameter are (4, 9, –3)

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 50

Let  If the vectors   in the plane of  then x equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 50


 lies in the plane of 

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 51

Let L be the line of intersection of the planes 2x + 3y + z = landx + 3y + 2z= 2. IfL makes an angle α with the positive x-axis, then cos α equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 51

Let the direction cosines of line L be l, m, n, then 2l + 3m + n = 0 ....(i)
and l + 3m + 2n = 0 ....(ii)
on solving equation (i) and (ii), we get

Line L, makes an angle α with +ve x-axis

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 52

The vector   lies in the plane of the vectors  and bisects the angle between   Then  which one of the following gives possible values of α and β?

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 52

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 53

The non-zero vectors are   are related by  and   Then the angle between 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 53

 and are opposite in direction 
∴ Angle between

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 54

The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point  Then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 54

Equation of line through (5, 1, a) and

∴ Any point on this line is a

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 55

If the straight lines    and  intersect at a point, then the integer k is equal to

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 55

The two lines intersect if shortest distance between them is  zero i.e.


∵ k is an integer, therefore k = –5

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 56

Let the line  lie in the plane x + 3y – αz + β = 0. Then (α, β) equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 56

x + 3y – αz + β  = 0
∴ Pt (2, 1, – 2) lies on the plane
i.e. 2 + 3 + 2α + β = 0  or 2α + β + 5  = 0 ....(i)
Also normal to plane will be perpendicular to line,
∴ 3 × 1 – 5 × 3 + 2 × (– α) = 0  ⇒ α = – 6
From equation (i) then, b = 7
∴ (α, β) = (– 6, 7)

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 57

The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 57

Let P (x1, y1, z1) and Q (x2, y2, z2) be the initial and final points of the vector whose projections on the three coordiante axes are 6, – 3, 2 then


So that direction ratios of 
∴ Direction cosines of  are

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 58

If   are non-coplanar vectors and p, q are real numbers, then the equality   holds for :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 58

∴  Exactly one value of (p, q)

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 59

Let  Then the vector satisfying 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 59



 which is not possible

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 60

If the vectors   and  are mutually orthogonal, then (λ, μ)=

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 60

Since   are mutually orthogonal 

On solving (i) and (ii), we get λ = -3, μ = 2

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 61

Statement -1 : The point A(3,  1,  6) is the mirror image of the point B(1, 3, 4) in the plane x – y + z = 5.
Statement -2: The plane x – y + z = 5 bisects the line segment joining A(3, 1, 6) and B(1, 3, 4).

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 61

A(3, 1, 6); B = (1, 3, 4) Mid-point of AB = (2, 2, 5) lies on the plane. and d.r’s of AB = (2, –2, 2)
d.r’s of normal to plane = (1, –1, 1).
Direction ratio of AB and normal to the plane are proportional therefore, AB is perpendicular to the plane
∴ A is image of B
Statement-2 is correct but it is not correct explanation.

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 62

A line AB in three-dimensional space makes angles 45° and 120° with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle θ with the positive z-axis, then θ equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 62

Direction cosines of the line :

where θ is the angle, which line makes with positive z-axis.

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 63

If the angle between the line   and the plane  then λ equals

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 63

If θ be the angle between the given line and plane, then


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 64

then the value 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 64

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 65

The vectors  are not perpendicular and   are two vectors satisfying   Then the vector 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 65

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 66

Statement-1: The point A(1, 0, 7)) is the mirror image of the point B(1, 6, 3) in the line : 

Statement-2: The line bisects the line segment joining A(1, 0, 7) and B(1, 6, 3) .

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 66

The direction ratios of the line segment joining points
A(1, 0, 7) and B(1, 6, 3) are 0, 6, – 4.
The direction ratios of the given line are 1, 2, 3.
Clearly 1 × 0 + 2 × 6 + 3 × (-4) = 0
So, the given line is perpendicular to line AB.
Also , the mid point of A and B is (1, 3, 5) which lies on the given line.
So, the image of B in the given line is A, because the given line is the perpendicular bisector of line segment joining points A and B, But statement-2 is not a correct explanation for statement-1.

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 67

 be two unit vectors . If the vectors  are perpendicular to each other,, then the angle between

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 67


 are perpendicular to each other

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 68

A equation of a plane parallel to the plane x – 2y + 2z – 5 = 0 and at a unit distance from the origin is :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 68

Given equation of a plane is x – 2y + 2z – 5 = 0
So, Equation of parallel plane is given by x – 2y + 2z + d = 0

Now, it is given that distance from origin to the parallel plane is 1.

So equation of required plane x – 2y + 2z ± 3 = 0

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 69

If the line  intersect, then k is equal to:

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 69

Given lines in vector form are

These will intersect if shortest distance between them = 0

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 70

Let ABCD be a parallelogram such that  and ∠BAD be an acute angle.  is the vector that coincide with the altitude directed from the vertex B to the side AD, then 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 70

Let ABCD be a parallelogram such that

be an acute angle.

We have

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 71

Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 71

2x + y + 2z – 8 = 0    …(Plane 1)

              …(Plane 2)
Distance between Plane 1 and 2

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 72

If the lines  are coplanar, then k can have

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 72

Given lines will be coplanar

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 73

If the vectors   are the sides of a triangle ABC, then the length of the median through A is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 73

∵ M is mid point of BC

Length of median AM

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 74

The image of the line   in the plane 2x - y +z + 3 = 0 is the line:

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 74



Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 75

The angle between the lines whose direction cosines satisfy the equations l + m +n= 0 and l2 = m2 + n2 is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 75

Given l + m + n = 0 and l2 = m2 + n2
Now, (–m –n)2 = m2 + n2
⇒ mn = 0   ⇒ m = 0  or n = 0
If m = 0 then l = –n


Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 76

then λ is equal to

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 76

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 77

Let   be three non-zero vectors such that no two of them are collinear and   If θ is the
angle between vectors   then a value of sin θ is :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 77



 are non collinear, the above equation is possible only when

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 78

The equation of the plane containing the line 2x – 5y + z = 3; x + y + 4z = 5, and parallel to the plane, x + 3y + 6z = 1, is :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 78

Equation of the plane containing the lines
2x – 5y + z = 3 and x + y + 4z = 5  is
2x – 5y + z – 3 + λ (x + y + 4z – 5) = 0
⇒ (2 + λ) x + (–5 + λ) y + (1 + 4λ)z + (–3 – 5λ) = 0  ...(i)

Since the plane (i) parallel to the given plane x + 3y + 6z = 1

Hence equation of the required plane is

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 79

The distance of the point (1, 0, 2) from the point of intersection of the line  and the plane

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 79

General point on given line ≡ P(3r + 2, 4r – 1, 12r + 2)
Point P must satisfy equation of plane
(3r + 2) – (4r – 1) + (12r + 2) = 16
11r + 5 = 16
r = 1

P(3 × 1 + 2, 4 × 1 – 1, 12 × 1 + 2) = P(5, 3, 14)
distance between P and (1, 0, 2)

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 80

If the line,  lies in the plane, lx + my – z = 9,then l2 + m2 is equal to :

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 80

Line lies in the plane ⇒ (3, –2, –4) lie in the plane
⇒ 3ℓ – 2m + 4 = 9 or 3ℓ – 2m = 5 ..... (1)
Also, ℓ, m,–1 are dr's of line perpendicular to plane and 2, –1, 3 are dr's of line lying in the plane
⇒ 2ℓ – m – 3 = 0 or 2ℓ – m = 3 .....(2)
Solving (1) and (2) we get ℓ = 1 and m = –1
⇒ ℓ2 + m2 = 2

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 81

 be three unit vectors such that  is not parallel to  then
the angle between  is: 

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 81

On comparing both sides

where θ is the angle between

Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 82

The distance of the point (1, –5, 9) from the plane x – y + z = 5 measured along the line x = y = z is :     

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Vector Algebra and Three Dimensional Geometry - Question 82



Putting these in eqn of plane :-

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