Karnataka 1st PUC Mathematics Introduction to Three Dimensional Geometry MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – I PUC
NAME OF THE CHAPTER: INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

Co-ordinate axes & Co-ordinate plane
Consider the three planes which are intersecting mutually perpendicular to each other
at O(origin).
The intersecting lines x o x1, y o y1, z o z1 are perpendicular to each other these lines are
called x, y, z axes. ox, oy, oz are +ve sides & ox1, oy1, oz1 are –ve sides
The planes xoy, yoz & zox are called xy-plane, yz-plane, zx-plane respectively these three
known as co-ordinate planes.
The three co-ordinate planes divided the space into eight parts are called octants, named
as xoyz, x1oyz, x1oy1z, xoy1z, xoyz1, x1oz1, x1oy1z1 & denoted by I, II, III, IV, V, VI, VII, VIII
octants respectively

Co-ordinates of a point in space
Let p (x, y, z) be a point in a space it is called as ordered triplet.
If Draw PM ┴ XY plane ∴ PM = OQ = z
Draw PL ┴ ZY plane ∴ PL = ON = x
Draw PS ┴ XZ plane ∴ PS = OR =y

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Then x, y, z are called the x, y, z co-ordinates respectively (all are +ve).
 The co-ordinate of a point P on the x-axis, y-axis, z-axis are (x, o, o), (o, y, o), (o, o, z),
respectively
 The co-ordinate of a point P on the xy-plane, yz-plane, zx-plane are
(x, y, o), (o, y, z), (x, o, z) respectively.
 The signs of x, y & z would change accordingly. They to ach point in the space there
corresponds an order triplet of real number
Signs of the Co-ordinates of a point in each octant as follows

Octant→
Co-ordinate
I II III IV V VI VII VIII
↓
X + − − + + − − +
Y + + − − + + − −
Z + + + + − − − −

Formula to remember
 Distance between two points A (x1, y1, z1) & B (x2, y2, z2) is
AB = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 +(𝑧2 − 𝑧1 )2

 Distance between origin & any point P (x, y, z) is OP = √𝑥 2 + 𝑦 2 + 𝑧 2
 Distance of the point (a, b c) from
i) 𝑥-axis is √𝑏 2 + 𝑐 2 ii) y-axis is √𝑎2 + 𝑐 2 iii) 𝑧 − 𝑎𝑥𝑖𝑠 is √𝑎2 + 𝑏 2
 Co-ordinates of centroid of a triangle whose vertices P (x1, y1, z1), Q (x2, y2, z2) &
𝑥 +𝑥2 +𝑥3 𝑦1 +𝑦2 +𝑦3 𝑧1 +𝑧2 +𝑧3
R (x3, y3, z3) is G = ( 1 , , )
3 3 3

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Coordinates of a point in space
1. The point (𝟏, −𝟑, 𝟒) lies in the octant (Easy) (KCET-2020)
(a) Third (b) Fourth (c) Eighth (d) Second (Easy)
2. The octant in which the point (𝟐, −𝟒, −𝟕) lies is (Easy) (KCET-2021)
(a) Eighth (b) Fourth (c) Third (d) Fifth (Easy)
3. Find the octant in which the points (-3, 1, 2) and ( −𝟑, 𝟏, −𝟐 ) lie respectively. (Easy)
(a) second, fourth (b) sixth, second
(c) fifth, sixth (d) second, sixth
4. 𝑴 is the foot of the perpendicular drawn from the point 𝑨(𝟔, 𝟕, 𝟖) on the 𝒚𝒛-plane.
What are the coordinates of point ? (Average)
(a) (6, 0 , 0) (b) (6, 7, 0) (c) (6, 0, 8) (d) (0, 7, 8)
5. Let A, B, C be the feet of the perpendicular segments drawn from a point P(3, 4, 5)
on the 𝒙𝒚, 𝒚𝒛 & 𝒛𝒙 planes respectively. The Co-ordinates of A, B & C (Average)
(a) (3, 4, 0), (0, 4, 4), (3, 0, 5) (b) (3, 0, 4), (4, 5, 0), (3, 5, 0)
(c) (3, 5, 0), (0, 5, 4), (0, 3, 4) (d) (3, 4 , 0), (0, 4, 5), (3, 0, 5)
6. L is the foot of the perpendicular drawn from a point (3, 5, 6) on 𝒙 − 𝒂𝒙𝒊𝒔 the
Co-ordinates of L are. (Average)
(a) (3, 0, 0) (b) (0, 6, 0) (c) (0, 0, 5) (d) (0, 5, 6)
7. If P (2,4,5) point in the space and PF perpendicular to xy-plane, then coordinate of
point F is (Average)
(a) (0,0,5) (b) (0,4,5) (c) (2,0,5) (d) (2,4,0)
8. The equation 𝒙𝒚 = 𝟎 in three dimensional space represents (Difficult) (KCET-2024)
(a) A pair of straight line (b) a plane
(c) a pair of plains at right angles (d) a pair of parallel planes
9. The locus of a point for which y  0, z  0? (Average)
(a) equation of x-axis (b) equation of y-axis
(c) equation of z-axis (d) yz- plane
10. The x  axis is the intersection of two planes (Average)
(a) xy and xz (b) yz and zx (c) xy and yz (d) xy and yx
11. Equation of x-axis is considered as (Easy)
(a)𝑥 = 0, 𝑦 = 0 (b)𝑦 = 0 , 𝑧 = 0 (c)𝑧 = 0, 𝑥 = 0 (d) 𝑥 = 0, 𝑦 = 0, 𝑧 = 0

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Distance Between Two points
12. The perpendicular distance from the point P( x, y, z) to xy-plane is (Easy)
(a) x (b) y (c) z (d) √𝑥 2 + 𝑦 2
13. The perpendicular distance from the point P( 6, 7, 8) to zx- plane is (Easy)
(a) 8 (b) 7 (c) 6 (d) 10
14. From which of the following the distance of the point (1,2,3) is 10 (Average)
(a) Origin (b) x-axis (c) y-axis (d) z-axis
15. The perpendicular distance of the point 𝑷(𝟔, 𝟕, 𝟖) from 𝑿𝒀-plane is
(Easy) (KCET-2017)
(a) 7 (b) 6 (c) 8 (d) 5
16. The distance of the point 𝑷(𝒂, 𝒃, 𝒄) from the 𝒙-axis is (Average) (KCET-2020)
(a) √𝑎2 + 𝑏 2 (b) √𝑏 2 + 𝑐 2 (c) 𝑎 (d) √𝑎2 + 𝑐 2
17. The distance of the point 𝑷(−𝟑 , 𝟒 , 𝟓 ) from YZ-plane is (Average) (KCET-2025)
(a) -3 units (b) 3 units (c) 4 units (d) 5 units
18. The point on 𝒙 − 𝒂𝒙𝒊𝒔 which is equidistant from the point A(3, 2, 2) & B(5, 5, 4)
(Average)
5 49
(a) (16, 0, 0) (b) (4 , 0, 0) (c) (9, 0, 0) (d) ( 4 , 0, 0)

19. Let (𝟑, 𝟒, −𝟏)& (−𝟏, 𝟐, 𝟑) be the end points of a diameter of a sphere.
Then the radius of the sphere is equal to (Average)
(a) 2 units (b) 3 units (c) 6 units (d) 7 units

20. The coordinates of a point on y-axis which are at a distance of 5 2
from the point P(3,-2,5) is (Average)
(a) (0,6,0) or (0,-2,0) (b) (0,-6,0) or (0,2,0)
(c) (0,4,0) or (0,-4,0) (d) (0,5,0) or (0,-5,0)
21. If the sum of the squares of the distances of a point from the three coordinate axes
be 36, then its distance from the origin is (Average)

(a) 6 (b) 3 2 (c) 2 3 (d) √6
22. If the points (–1, 3, 2), (–4, 2, –2) and (5, 5, ) are collinear, then   (Easy)
(a)–10 (b)5 (c)–5 (d)10
23. What is the length of foot of perpendicular drawn from the point 𝑷(𝟑, 𝟒, 𝟓) on 𝒚-axis?
(Average)
(a) √41 units (b) √34 units (c) 5 units (d) 4 units

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24. If the distance between the points (𝒂, 𝟎, 𝟏) and (𝟎, 𝟏, 𝟐) is √𝟐𝟕, then the value of 𝒂 is
(Average)
(a) 5 (b) ±5 (c) -5 (d) 0
25. Reflection of the point (𝜶, 𝜷, 𝜸) in 𝑿𝒀 plane is (Difficult) (KCET-2017)
(a) (0,0, 𝛾) (b) (𝛼, 𝛽, −𝛾) (c) (−𝛼, −𝛽, 𝛾) (d) (𝛼, 𝛽, −𝛾)
26. Find the equation of the set of the points 𝑷 such that its distances from the points
𝑨(𝟑, 𝟒, −𝟓) and 𝑩(−𝟐, 𝟏, 𝟒) are equal. (Average)
(a) 10𝑥 + 6𝑦 − 18𝑧 − 29 = 0 (b) 10𝑥 + 18𝑦 − 6𝑧 − 29 = 0
(c) 5𝑥 + 3𝑦 − 9𝑧 − 29 = 0 (d) 10𝑥 + 6𝑦 − 18𝑧 − 45 = 0
27. Find the coordinates of a point which is equidistant from the four points
𝑶(𝟎, 𝟎, 𝟎), 𝑨(𝒍, 𝟎, 𝟎), 𝑩(𝟎, 𝒎, 𝟎) and 𝑪(𝟎, 𝟎, 𝒏). (Difficult)
𝑙 𝑚 𝑛 𝑙 𝑛
(a) (𝑙, 𝑚, 𝑛) (b) ( , , ) (c) ( , 𝑚, ) (d) (2𝑙, 2𝑚, 2𝑛)
2 2 2 2 2

28. The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B
are (3, –5, 7) and (–1, 7, – 6), respectively, find the coordinates of the point C.
(Average)
(a) (1,1,2) (b) (3,1,2) (c) (1,2,2) (d) (1,-1,2)
29. A point moves so that the sum of its distances from the points (4,0,0) and (-4,0,0)
remains 10. The locus of the point is (Difficult)

(a) 9 x 2  25 y 2  25 z  225 (b) 9 x 2  25 y 2  25 z  225
2 2

(c) 9 x 2  25 y 2  25 z  225 (d) 9 x 2  25 y 2  25 z  225  0
2 2

30. Statement 1: The perpendicular distance from the point P(6, 7, 8) to zx - plane is 7
Statement 2: The shortest distance of the point (a, b, c) from the x-axis is √𝑏 2 + 𝑐 2
(Difficult)
A) Statement 1 is true and Statement 2 is false.
B) Statement 1 is false and Statement 2 is false.
C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation
for Statement 1
D) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for
Statement 1

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31. Statement 1: A point is in XY plane. Its distance measured parallel to X axis is 10 units
and that of parallel to Y axis is 20 units. Then its coordinates is (10,20,0).
Statement 2: The coordinates of any point in the XZ-plane is (0, b, c). (Difficult)
a) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of the
STATEMENT 1
b) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct explanation
of the STATEMENT 1
c) STATEMENT 1 is true and STATEMENT 2 is false.
d) STATEMENT 1 is false and STATEMENT 1 is true.
32. For the figure given below, consider the following statements 1 and 2 (Difficult)

Statement 1: The coordinates of point D is (1, -2, 8)
Statement 2: The length of diagonal is AB= 20
a) Statement 1 is true and Statement 2 is false
b) Statement 1 is false and Statement 2 is true
c) Both Statement 1and 2 are true
d) Both Statement 1 and 2 are false.
33. The mid points of the sides of a triangle are (𝟏, 𝟓, −𝟏), (𝟎, 𝟒, −𝟐) and (𝟐, 𝟑, 𝟒) then
centroid of the triangle (Average)
1 1
(a) (1, 4, 3) (b) (1,4, 3) (c) (-1, 4, 3) (d) (3 , 2,4)

INTRODUCTION TO THREE DIMENSIONAL GEOMETRY
1 2 3 4 5 6 7 8 9 10
b a d d d a d c a a
11 12 13 14 15 16 17 18 19 20
b c b c c b b d b b
21 22 23 24 25 26 27 28 29 30
b d b b d a b d c c
31 32 33
c a b

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