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Three Dimensional Geometry — NDA Mathematics PYQs

176 previous year questions from Three Dimensional Geometry with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

176 questionsMathematicsSolutions on every page
1

Passage: The equation of the sphere S is x^2+y^2+z^2-4x-6y-12z+k=0 . Question: If the radius of the sphere S is 8 units, what is the value of k ?

NDA 2026 Solution View → 2

Passage: A line L passing through the point (-1,2,-3) is perpendicular to the plane P given by 2x+3y+z+5=0 . Question: What is the equation of the line L ?

NDA 2026 Solution View → 3

Passage: P(-2,-3,5) , Q(4,-1,5) , R(6,-4,8) and S(2,-6,10) are four points. Question: Consider the following statements : I. PQ is parallel to RS . II. PR is perpendicular to QS .

NDA 2026 Solution View → 4

Passage: A line L passing through the point (-1,2,-3) is perpendicular to the plane P given by 2x+3y+z+5=0 . Question: What are the direction ratios of a line M parallel to the pla

NDA 2026 Solution View → 5

Passage: P(-2,-3,5) , Q(4,-1,5) , R(6,-4,8) and S(2,-6,10) are four points. Question: What is (PQ^2+2QS^2-2PR^2) equal to ?

NDA 2026 Solution View → 6

Passage: The equation of the sphere S is x^2+y^2+z^2-4x-6y-12z+k=0 . Question: What is the radius of the sphere passing through origin and concentric with the sphere S ?

NDA 2026 Solution View → 7

For the following two (02) items: A plane P is parallel to the line having direction ratios 1, 3, 2 and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y

NDA 2025 Solution View → 8

For the following two (02) items: Suppose S is the sphere with the smallest radius that passes through the points A(1, 0, 0) , B(0, 1, 0) and C(0, 0, 1) . On which one of the follo

NDA 2025 Solution View → 9

A(1, 2, -1) , B(2, 5, -2) and C(4, 4, -3) are three vertices of a rectangle. What is the area of the rectangle?

NDA 2025 Solution View → 10

If a line x+1 p = y-1 q = z-2 r where p = 2q = 3r , makes an angle with the positive direction of y -axis, then what is 2 equal to?

NDA 2025 Solution View → 11

What is the equation of the plane passing through the point (1, 1, 1) and perpendicular to the line whose direction ratios are 3, 2, 1 ?

NDA 2025 Solution View → 12

For the following two (02) items: Let A(1, -1, 0) , B(-2, 1, 8) and C(-1, 2, 7) are three consecutive vertices of a parallelogram ABCD . What is the fourth vertex D ?

NDA 2025 Solution View → 13

For the following two (02) items: Suppose S is the sphere with the smallest radius that passes through the points A(1, 0, 0) , B(0, 1, 0) and C(0, 0, 1) . What is the radius of S ?

NDA 2025 Solution View → 14

If a line in 3 dimensions makes angles , and with the positive directions of the coordinate axes, then what is ( + ) ( - ) equal to?

NDA 2025 Solution View → 15

A line makes angles , and with the positive directions of the coordinate axes. If a = ( ^2 ) i + ( ^2 ) j + ( ^2 ) k and b = i + j + k , then what is a b equal to?

NDA 2025 Solution View → 16

For the following two (02) items: A plane P is parallel to the line having direction ratios 1, 3, 2 and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y

NDA 2025 Solution View → 17

Paragraph: Let 2 x^2+2 y^2+2 z^2+3 x+3 y+3 z-6=0 be a sphere. Question: What is the diameter of the sphere?

NDA 2024 Solution View → 18

Paragraph: Let S be the line of intersection of two planes x+y+z=1 and 2 x+3 y-4 z=8 . Question: Which of the following are the direction ratios of S ?

NDA 2024 Solution View → 19

Paragraph: Let S be the line of intersection of two planes x+y+z=1 and 2 x+3 y-4 z=8 . Question: If (l, m, n) are direction cosines of S, then what is the value of 43 (l^2-m^2-n^2

NDA 2024 Solution View → 20

Paragraph: Let L : x+y+z+4=0=2 x-y-z+8 be a line and P : x+2 y+3 z+1=0 be a plane Question: What is the point of intersection of L and P ?

NDA 2024 Solution View → 21

If (1,-1,2) and (2,1,-1) are the end points of a diameter of a sphere x^2+y^2+z^2+2 u x+2 v y+2 w z-1=0 , then what is u+v+w equal to?

NDA 2024 Solution View → 22

If l, m, n are the direction cosines of a normal to the plane 2 x-3 y+6 z+4=0 , then what is the value of 49 (7 l^2+m^2-n^2 ) ?

NDA 2024 Solution View → 23

If p is the perpendicular distance from origin to the plane passing through (1,0,0),(0,1,0) and (0,0,1) , then what is 3 p^2 equal to?

NDA 2024 Solution View → 24

A line through (1,-1,2) with direction ratios 3,2,2 meets the plane x+2 y+3 z=18 . What is the point of intersection of line and plane?

NDA 2024 Solution View → 25

Paragraph: Let 2 x^2+2 y^2+2 z^2+3 x+3 y+3 z-6=0 be a sphere. Question: The centre of the sphere lies on the plane

NDA 2024 Solution View → 26

Paragraph: Let L : x+y+z+4=0=2 x-y-z+8 be a line and P : x+2 y+3 z+1=0 be a plane Question: What are the direction ratios of the line?

NDA 2024 Solution View → 27

If a direction cosines l, m, n of a line are connected by relation l+2 m+n=0,2 l-2 m+3 n=0 , then what is the value of l^2+m^2-n^2 ?

NDA 2024 Solution View → 28

If a vector of magnitude 2 units makes an angle 3 with 2 i , 4 with 3 j and an acute angle with 4 k , then what are the components of the vector?

NDA 2024 Solution View → 29

If L is the line with DIRECTIONS ratios 3,-2,6 and passing through (1,-1,1) , then what are the coordinates of the points on L whose distance from (1,-1,1) is 2 units?

NDA 2023 Solution View → 30

Which one of the planes is parallel to the line x-2 3 = y-3 4 = z-4 5 ?

NDA 2023 Solution View → 31

What is the equation of the sphere concentric with the sphere x^2+y^2+z^2-2 x-6 y-8 z-5=0 and which passes through the origin?

NDA 2023 Solution View → 32

Paragraph: Let A(1,-1,2) and B(2,1,-1) be the end points of the diameter of the sphere x^2+y^2+z^2+2 u x+2 v y+2 w z-1=0 Question: What is u+v+w equal to?

NDA 2023 Solution View → 33

Paragraph: Let A(1,-1,2) and B(2,1,-1) be the end points of the diameter of the sphere x^2+y^2+z^2+2 u x+2 v y+2 w z-1=0 Question: If P(x, y, z) is any point on the sphere, then wh

NDA 2023 Solution View → 34

A point P lies on the line joining A(1,2,3) and B(2,10,1) . If z -coordinate of P is 7 , what is the sum of other two coordinates?

NDA 2023 Solution View → 35

Paragraph: Consider two lines whose direction ratios are (2,-1,2) and (k, 3,5) . They are inclined at an angle 4 . Question: What is the value of k ?

NDA 2023 Solution View → 36

Paragraph: Consider two lines whose direction ratios are (2,-1,2) and (k, 3,5) . They are inclined at an angle 4 . Question: What are the direction ratios of a line which is perpen

NDA 2023 Solution View → 37

Paragraph: The position vectors of two points A and B are i - j and j + k respectively. Question: Consider the following points: 1. (-1,-3,1) 2. (-1,3,2) 3. (-2,5,3) Which of the a

NDA 2023 Solution View → 38

Paragraph: Let a vector a =4 i -8 j +k make angles , , with the positive directions of x, y, z axes respectivelv, Question: What is equal to?

NDA 2023 Solution View → 39

What is the angle between the lines 2 x=3 y=-z and 6 x=-y=-4 z ?

NDA 2023 Solution View → 40

Paragraph: Let a vector a =4 i -8 j + k make angles , , with the positive directions of x, y, z axes respectively. Question: What is 2 + 2 equal to?

NDA 2023 Solution View → 41

Consider the equation of a sphere x^2+y^2+z^2-4 x-6 y-8 z-16=0 . Which of the following statements is/are correct? 1. z -axis is tangent to the sphere. 2. The centre of the sphere

NDA 2022 Solution View → 42

A plane cuts intercepts 2,2,1 on the coordinate axes. What are the direction cosines of the normal to the plane?

NDA 2022 Solution View → 43

Consider the following statements : 1. The direction ratios of y -axis can be 0,4,0 2. The direction ratios of a line perpendicular to z -axis can be 5,6,0 Which of the statements

NDA 2022 Solution View → 44

The plane 6 x+k y+3 z-12=0 where meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x^2+y^2+z^2-2 x-3 y-4 z=

NDA 2022 Solution View → 45

The plane 6 x+k y+3 z-12=0 where meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x^2+y^2+z^2-2 x-3 y-4 z=

NDA 2022 Solution View → 46

Let the plane 2 x k + 2 y 3 + z 3 =2 pass through the point (2,3,-6) . What are the direction ratios of a normal to the plane?

NDA 2022 Solution View → 47

Let the plane 2 x k + 2 y 3 + z 3 =2 pass through the point (2,3,-6) . If p, q and r are the intercepts made by the plane on the coordinate axes respectively, then what is (p+q+r)

NDA 2022 Solution View → 48

Consider the points A(2,4,6), B(-2,-4,-2), C(4,6,4) and D(8,14,12) . Which of the following statements is/ are correct? 1. The points are the vertices of a rectangle A B C D . 2. T

NDA 2022 Solution View → 49

The plane 6 x+k y+3 z-12=0 where meets the coordinate axes at A, B and C respectively. The equation of the sphere passing through the origin and A, B, C is x^2+y^2+z^2-2 x-3 y-4 z=

NDA 2022 Solution View → 50

The locus of a point P(x, y, z) which moves in such a way that z=7 is a

NDA 2021 Solution View → 51

Consider the following statements : 1. A line in space can have infinitely many direction ratios. 2. It is possible for certain line that the sum of the squares of direction cosine

NDA 2021 Solution View → 52

The x y -plane divides the line segment joining the points (-1,3,4) and (2,-5,6)

NDA 2021 Solution View → 53

A B C D E F G H is a cuboid with base A B C D . Let A(0,0,0), B(12,0,0), C(12,6,0) and G(12,6,4) be the vertices. If is the angle between A B and A G ; is the angle between A C and

NDA 2021 Solution View → 54

The number of spheres of radius r touching the coordinate axes is

NDA 2021 Solution View → 55

What is the angle between the two lines having direction ratios (6,3,6) and (3,3,0) ?

NDA 2021 Solution View → 56

If l, m, n are the direction cosines of the line x-1=2(y+3)=1-z , then what is l^4+m^4+n^4 equal to?

NDA 2021 Solution View → 57

What is the projection of the line segment joining A(1,7,-5) and B(-3,4,-2) on y -axis?

NDA 2021 Solution View → 58

What is the number of possible values of k for which the line joining the points (k, 1,3) and (1,-2, k+1) also passes through the point (15,2,-4) ?

NDA 2021 Solution View → 59

The foot of the perpendicular drawn from the origin to the plane x+y+z=3 is

NDA 2021 Solution View → 60

If a line has direction ratios , then what is the sum of the squares of its direction cosines?

NDA 2020 Solution View → 61

Into how many compartments do the coordinate planes divide the space?

NDA 2020 Solution View → 62

Let A be a point in space such that | O A |=12 , where O is the origin. If O A is inclined at angles 45^ and 60^ with x axis and y -axis respectively, then what is O A equal to ?

NDA 2020 Solution View → 63

What is the equation of the plane which cuts an intercept 5 units on the z -axis and is parallel to x y -plane?

NDA 2020 Solution View → 64

What is the length of the diameter of the sphere whose centre is at (1,-2,3) and which touches the plane 6 x-3 y+2 z-4=0 ?

NDA 2020 Solution View → 65

What is the perpendicular distance from the point (2,3,4) to the line x-0 1 = y-0 0 = z-0 0 ?

NDA 2020 Solution View → 66

A point on a line has coordinates (p+1, p-3, 2 p) where p is any real number. What are the direction cosines of the line?

NDA 2019 Solution View → 67

A straight line passes through the point (1,1,1) makes an angle 60^ with the positive direction of z -axis, and the cosine of the angles made by it with the positive directions of

NDA 2019 Solution View → 68

If the line x-4 1 = y-2 1 = z-k 2 lies on the plane 2 x-4 y+z=7 , then what is the value of k ?

NDA 2019 Solution View → 69

The equation of the plane passing through the intersection of the planes 2 x+y+2 z=9,4 x-5 y-4 z=1 and the point (3,2,1) is

NDA 2019 Solution View → 70

If the points (x, y,-3),(2,0,-1) and (4,2,3) lie on a straight line, then what are the values of x and y respectively?

NDA 2019 Solution View → 71

The distance between the parallel planes 4 x-2 y+4 z+9=0 and 8 x-4 y+8 z+21=0 is

NDA 2019 Solution View → 72

What are the direction cosines of z -axis?

NDA 2019 Solution View → 73

If the position vectors of points A and B are 3 i -2 j + k and 2 i +4 j -3 k respectively, then what is the length of A B ?

NDA 2019 Solution View → 74

A point on the line x-1 1 = y-3 2 = z+2 7 has coordinates

NDA 2019 Solution View → 75

What is the radius of the sphere x^2+y^2+z^2-6 x+8 y-10 z+1=0 ?

NDA 2019 Solution View → 76

Coordinates of the points O , P , Q and R are respectively (0,0,0),(4,6,2 m),(2,0,2 n) and (2,4,6) . Let L, M, N and K be points on the sides O R, O P, P Q and Q R respectively suc

NDA 2018 Solution View → 77

Consider the following statements: 1. The angle between the planes 2 x-y+z=1 and x+y+2 z=3 is 3 2. The distance between the planes 6 x-3 y+6 z+2=0 and 2 x-y+2 z+4=0 is 10 9 Which o

NDA 2018 Solution View → 78

Let the coordinates of the points A , B , C be (1,8,4) , (0,-11,4) and (2,-3,1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A

NDA 2018 Solution View → 79

What is the equation of the plane passing through the points (-2,6,-6),(-3,10,-9) and (-5,0,-6) ?

NDA 2018 Solution View → 80

A sphere of constant radius r through the origin intersects the coordinate axes in A , B and C . What is the locus of the centroid of the triangle ABC ?

NDA 2018 Solution View → 81

What is the distance of the point (2,3,4) from the plane 3 x -6 y +2 z +11=?

NDA 2018 Solution View → 82

The line x-1 2 - y-2 3 = z-3 3 is given by

NDA 2018 Solution View → 83

What is the equation to the sphere whose centre is at (-2,3,4) and radius is 6 units?

NDA 2018 Solution View → 84

The length of the normal from origin to the plane x+2 y-2 z=9 is equal to

NDA 2017 Solution View → 85

If , and are the angles which the vector OP (O being the origin) makes with positive direction of the coordinate axes, then which of the following are correct? 1. ^2 + ^2 = ^2 2. ^

NDA 2017 Solution View → 86

The point of intersection of the line joining the points (-3,4,-8) and (5,-6,4) with the XY-plane is

NDA 2017 Solution View → 87

If the angle between the lines whose direction ratios are (2,-1,2) and (x, 3,5) is 4 , then the smaller value of x is

NDA 2017 Solution View → 88

A straight line with direction cosines (0,1,0) is

NDA 2017 Solution View → 89

A variable plane passes through a fixed point (a, b, c) and cuts the axes in A , B and C respectively. The locus of the centre of the sphere OABC , O being the origin, is

NDA 2017 Solution View → 90

The line passing through the points (1,2,-1) and (3,-1,2) meets the yz-plane at which one of the following points?

NDA 2017 Solution View → 91

(0,0,0),( a , 0,0),(0, ~b , 0) and (0,0, c ) are four distinct points. What are the coordinates of the point which is equidistant from the four points?

NDA 2017 Solution View → 92

Under which one of the following conditions are the lines x = ay + b ; z = cy + d and x = ey +f ; z = gy + h perpendicular?

NDA 2017 Solution View → 93

The equation of the plane passing through the line of intersection of the planes x+y+z=1,2 x+3 y+4 z=7 , and perpendicular to the plane x-5 y+3 z=5 is given by

NDA 2017 Solution View → 94

The points P (3,2,4), Q (4,5,2), R (5,8,0) and S (2,-1,6) are

NDA 2017 Solution View → 95

Paragraph: Let Q be the image of the point P (-2,1,-5) in the plane 3 x-2 y+2 z+1=0 . Question: Consider the following: 1. The direction ratios of the line segment PQ are 3,-2,2 .

NDA 2016 Solution View → 96

Paragraph: A line L passes through the point P (5,-6,7) and is parallel to the planes x + y + z =1 and 2 x - y -2 z =3 . Question: What are the direction ratios of the line of inte

NDA 2016 Solution View → 97

Paragraph: A line L passes through the point P (5,-6,7) and is parallel to the planes x + y + z =1 and 2 x - y -2 z =3 . Question: What is the equation of the line L ?

NDA 2016 Solution View → 98

Paragraph: Let Q be the image of the point P (-2,1,-5) in the plane 3 x-2 y+2 z+1=0 . Question: Consider the following: 1. The coordinates of Q are (4,-3,-1) . 2. PQ is of length m

NDA 2016 Solution View → 99

A plane P passes through the line of intersection of the planes 2 x - y +3 z =2, x + y - z =1 and the point (1,0,1) . If the plane P touches the sphere x^2+y^2+z^2=r^2 , then what

NDA 2016 Solution View → 100

A plane P passes through the line of intersection of the planes 2 x - y +3 z =2, x + y - z =1 and the point (1,0,1) . What is the equation of the plane P?

NDA 2016 Solution View → 101

A plane P passes through the line of intersection of the planes 2 x - y +3 z =2, x + y - z =1 and the point (1,0,1) . What are the direction ratios of the line of intersection of t

NDA 2016 Solution View → 102

The lines 2 x=3 y=-z and 6 x=-y=-4 z

NDA 2015 Solution View → 103

What are the co-ordinates of the foot of the perpendicular drawn from the point (3,5,4) on the plane z=0 ?

NDA 2015 Solution View → 104

The radius of the sphere 3 x^2+3 y^2+3 z^2-8 x+4 y+8 z-15=0 is

NDA 2015 Solution View → 105

The projections of a directed line segment on the coordinate axes are 12, 4, 3 respectively. What are the direction cosines of the line segment?

NDA 2015 Solution View → 106

The projections of a directed line segment on the coordinate axes are 12, 4, 3 respectively. What is the length of the line segment?

NDA 2015 Solution View → 107

From the point P(3,-1,11) , a perpendicular is drawn on the line L given by the equation x 2 = y-2 3 = z-3 4 . Let Q be the foot of the perpendicular. What are the direction ratios

NDA 2015 Solution View → 108

A point P(1,2,3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes. What is the length of one of the

NDA 2015 Solution View → 109

From the point P(3,-1,11) , a perpendicular is drawn on the line L given by the equation x 2 = y-2 3 = z-3 4 . Let Q be the foot of the perpendicular. What is the length of the lin

NDA 2015 Solution View → 110

The lengths of the intercepts on the co-ordinate axes made by the plane 5 x+2 y+z-13=0 are

NDA 2015 Solution View → 111

Let , , be distinct real numbers. The points with position vectors i + j + k , i + j + k and i + j + k

NDA 2015 Solution View → 112

The direction ratios of the line perpendicular to the lines with direction ratios 1,-2,-2 and 0,2,1 are

NDA 2015 Solution View → 113

A triangular plane ABC with centroid (1, 2, 3) cuts the coordinate axes at A, B, C respectively. What is the equation of plane ABC?

NDA 2015 Solution View → 114

A point P(1,2,3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes. What is the equation of the plane

NDA 2015 Solution View → 115

Consider a sphere passing through the origin and the points (2,1,-1),(1,5,-4),(-2,4,-6) . What is the centre of the sphere?

NDA 2014 Solution View → 116

Consider a sphere passing through the origin and the points (2,1,-1),(1,5,-4),(-2,4,-6) . Consider the following statements: 1. The sphere passes through the point (0,4,0) . 2. The

NDA 2014 Solution View → 117

The line joining the points (2,1,3) and (4,-2,5) cuts the plane 2 x+y-z=3 . What is the ratio in which the plane divides the line?

NDA 2014 Solution View → 118

Consider the plane passing through the points A (2,2,1), B (3,4,2) and C (7,0,6) . What are the direction ratios of the normal to the plane?

NDA 2014 Solution View → 119

Paragraph : A straight line passes through (1,-2,3) and perpendicular to the plane 2 x+3 y-z=7 . Question : Where does the line meet the plane?

NDA 2014 Solution View → 120

Paragraph : A straight line passes through (1,-2,3) and perpendicular to the plane 2 x+3 y-z=7 . Question : What are the direction ratios of normal to plane?

NDA 2014 Solution View →
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