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11th Grade > Mathematics

INTRODUCTION TO THREE DIMENSIONAL GEOMETRY MCQs

Introduction To Three Dimensional Geometry

Total Questions : 49 | Page 1 of 5 pages
Question 1. The equation of the set of points which are equidistant from (1,2,3) and (3,2,-1).
  1.    x = 2z
  2.    x+2y+3z=0
  3.    x+2z=0
  4.    2y-3z=0
 Discuss Question
Answer: Option A. -> x = 2z
:
A
Let P(x,y,z) represent the set of points whichequidistant from A(1,2,3) and B(3,2,-1).
Equating AP=BP using the distance formula, we have
(x1)2+(y2)2+(z3)2=(x3)2+(y2)2+(z+1)22x4y6z=6x4y+2z4x8z=0x=2z
Question 2. The coordinates of the point P which divides the line segment joining A(1,-2,3) and B(3,4,-5) internally in the ratio 2:3 is
  1.    (9,2,-1)
  2.    P=(95,25,−15)
  3.    (-3,-14,19)
  4.    P=(35,145,−195)
 Discuss Question
Answer: Option B. -> P=(95,25,−15)
:
B
Let P(x,y,z) divide the line segment joining A(1,-2,3) and B(3,4,-5) internally in the ratio 2:3. Therefore,
x=2×3+3×12+3=95y=2×4+3×(2)2+3=25z=2×(5)+3×32+3=15
P=(95,25,15)
Question 3. Which of the following signs of x,y and z coordinates respectively of a point corresponds to octant VI?
  1.    - ,+, -
  2.    -, -, +
  3.    +, -, -
  4.    +, -, +
 Discuss Question
Answer: Option A. -> - ,+, -
:
A
For a point in octantVI, the x and z coordinates are negative and the y coordinate is positive.
Question 4. The points (4,-2,3), (5,1,2) and (7,7,0) lie on a straight line. State true or false.
  1.    True
  2.    False
  3.    7 units
  4.    7.5 units
 Discuss Question
Answer: Option A. -> True
:
A
Let A=(4,-2,3), B=(5,1,2) and C=(7,7,0).
AB=(54)2+(1(2))2+(23)2=12+32+12=11
BC=(75)2+(71)2+(02)2=22+62+22=44=211
AC=(74)2+(7(2))2+(03)2=32+92+32=99=311
AC=AB+BC. Hence, A, B and C are collinear. The statement is true.
Question 5. The centroid of a triangle whose vertices are (0,3,5), (1,4,-2) and (2,-1,3) is ___.
  1.    (1,2,2)
  2.    (0,4,3)
  3.    (2,1,3)
  4.    (3,6,6)
 Discuss Question
Answer: Option A. -> (1,2,2)
:
A
Let C(x,y,z) be the centroid of the triangle whose vertices are (0,3,5), (1,4,-2) and (2,-1,3)
x=0+1+23=33=1y=3+413=63=2z=52+32=63=2
The centroid of the triangle is (1,2,2)
Question 6. What is the distance between the points (3,0,4) and (-1,3,4)?
  1.    4 units 
  2.    5 units
  3.    6 units
  4.    7 units
 Discuss Question
Answer: Option B. -> 5 units
:
B
Using the distance formula, the distance between the points (3,0,4) and (-1,3,4) is
d=(13)2+(30)2+(44)2=42+52+02=25=5units
Question 7. Given that P(3,2,-4), Q(5,4,-6) and R(9,8,-10) are collinear, find the ratio in which Q divides PR.
  1.    1:2
  2.    1:3
  3.    1:4
  4.    2:3
 Discuss Question
Answer: Option A. -> 1:2
:
A
Let Qdivide PR in the ratio k:1. Equate the x-coordinateof the point of division to 5.
3+9kk+1=53+9k=5k+54k=2k=12
Since it is given that P, Q and Rare collinear, we need to calculate k onlyfor any one of thecoordinates.
Question 8. In which of the following octants is the x-coordinate of a point negative?
  1.    II, IV, VI, VIII
  2.    III, V, VI, VII
  3.    II, III, VII, VIII
  4.    II, III, VI, VII
 Discuss Question
Answer: Option D. -> II, III, VI, VII
:
D
The x-coordinate of a point is negative in the octants II, III, VI, VII
Question 9. The point (2,0,3) lies in which octant?
  1.    Octant I
  2.    Octant IV
  3.    Both A and B
  4.    None
 Discuss Question
Answer: Option D. -> None
:
D
The point (2,0,3) lies on the x-z plane as its y-coordinate is 0. Hence it is part of no octant.
Question 10. The distance between the points (-3,7,2) and (2,4,-1) is ___ .
  1.    5 units
  2.    √41 units
  3.    √43 units
  4.    7 units
 Discuss Question
Answer: Option C. -> √43 units
:
C
The distance between the points (-3,7,2) and (2,4,-1),
d=(32)2+(74)2+(12)2d=43

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