MCQ Mathematics Chapter 11: INTRODUCTION TO THREE DIMENSIONAL GEOMETRY, HS 1st year

 

---

MCQs

1. What are the three coordinate planes in three-dimensional geometry?
   a) X-axis, Y-axis, Z-axis
   b) XY-plane, YZ-plane, ZX-plane
   c) XO-plane, YO-plane, ZO-plane
   d) XOY-plane, ZOY-plane, XZO-plane

   Answer: b) XY-plane, YZ-plane, ZX-plane

---

2. Which of the following is true about the coordinates of a point in the XY-plane?
   a) x-coordinate is zero
   b) y-coordinate is zero
   c) z-coordinate is zero
   d) All coordinates are zero

   Answer: c) z-coordinate is zero

---

3. What is the distance formula between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$?
   a) $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
   b) $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
   c) $(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2$
   d) $\sqrt{x_2^2 + y_2^2 + z_2^2}$

   Answer: b) $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

---

4. In how many octants does the three-dimensional coordinate system divide space?
   a) 4
   b) 6
   c) 8
   d) 12

   Answer: c) 8

---

5. If the coordinates of a point are $(3, 0, 0)$, where does the point lie?
   a) On the Y-axis
   b) On the Z-axis
   c) On the X-axis
   d) On the XY-plane

   Answer: c) On the X-axis

---

6. What are the coordinates of the origin in three-dimensional geometry?
   a) $(1, 1, 1)$
   b) $(0, 0, 0)$
   c) $(0, 1, 0)$
   d) $(1, 0, 1)$

   Answer: b) $(0, 0, 0)$

---

7. Which mathematician is credited with introducing the three-coordinate system used in three-dimensional geometry today?
   a) Rene Descartes
   b) Leonhard Euler
   c) Antoine Parent
   d) J. Bernoulli

   Answer: b) Leonhard Euler

---

8. The coordinates of a point in the second octant will have signs:
   a) $(+, +, +)$
   b) $(-, +, +)$
   c) $(+, -, +)$
   d) $(-, +, -)$

   Answer: b) $(-, +, +)$

9. What is the equation of the set of points equidistant from the points $A(3, 4, -5)$ and $B(-2, 1, 4)$?
   a) $x + 2y - z = 5$
   b) $10x + 6y - 18z - 29 = 0$
   c) $2x - 3y + z = 10$
   d) $5x + y + z = 12$

   Answer: b) $10x + 6y - 18z - 29 = 0$

---

10. If $P(2, 3, 5)$ and $Q(4, 3, 1)$, what is the distance between them?
    a) $\sqrt{12}$
    b) $6$
    c) $4$
    d) $2\sqrt{5}$

    Answer: d) $2\sqrt{5}$

---

11. In three-dimensional geometry, what determines the octant in which a point lies?
    a) Distance from the origin
    b) Signs of its coordinates
    c) Sum of the coordinates
    d) Projection on the XY-plane

    Answer: b) Signs of its coordinates

---

12. If a point is in the YZ-plane, what can you say about its x-coordinate?
    a) It is positive.
    b) It is zero.
    c) It is negative.
    d) It is undefined.

    Answer: b) It is zero.

---

13. The coordinates of the centroid of a triangle $ABC$ are given by:
    a) $\frac{x_1 + x_2 + x_3}{2}, \frac{y_1 + y_2 + y_3}{2}, \frac{z_1 + z_2 + z_3}{2}$
    b) $\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}$
    c) $x_1 + x_2 + x_3, y_1 + y_2 + y_3, z_1 + z_2 + z_3$
    d) None of these

    Answer: b) $\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}$

---

14. Which octant does the point $(-2, -3, 5)$ lie in?
    a) First
    b) Fifth
    c) Fourth
    d) Seventh

    Answer: b) Fifth

---

15. Which of the following points lies in the XY-plane?
    a) $(1, 2, 3)$
    b) $(2, 0, 3)$
    c) $(3, 4, 0)$
    d) $(0, 0, 0)$

    Answer: c) $(3, 4, 0)$

---

16. What is the equation of the plane formed by the x-axis and the y-axis?
    a) XY-plane
    b) ZX-plane
    c) Z-axis
    d) YZ-plane

    Answer: a) XY-plane

---

17. Who systematically developed three-dimensional coordinate geometry for the first time?
    a) Antoine Parent
    b) Rene Descartes
    c) Leonhard Euler
    d) Pierre Fermat

    Answer: a) Antoine Parent

---

18. If $A(1, 2, 3)$, $B(4, 5, 6)$, and $C(7, 8, 9)$ are collinear, what is the relationship between $AB$ and $BC$?
    a) $AB = BC$
    b) $AB + BC = AC$
    c) $AB - BC = AC$
    d) None of these

    Answer: b) $AB + BC = AC$

---

19. The distance of the point $(x, y, z)$ from the origin is:
    a) $x + y + z$
    b) $\sqrt{x^2 + y^2 + z^2}$
    c) $(x^2 + y^2 + z^2)$
    d) $\sqrt{(x + y + z)}$

    Answer: b) $\sqrt{x^2 + y^2 + z^2}$

---

20. If the coordinates of two points are $(0, 0, 6)$ and $(6, 0, 0)$, what is the distance between them?
    a) $6$
    b) $\sqrt{72}$
    c) $12$
    d) $\sqrt{36}$

    Answer: b) $\sqrt{72}$

21. What is the distance between the points $(–3, 7, 2)$ and $(2, 4, –1)$?
    a) $\sqrt{50}$
    b) $3\sqrt{5}$
    c) $\sqrt{83}$
    d) $2\sqrt{10}$

    Answer: c) $\sqrt{83}$

---

22. If the coordinates of a point are $(x, y, 0)$, in which plane does it lie?
    a) XY-plane
    b) YZ-plane
    c) ZX-plane
    d) None of these

    Answer: a) XY-plane

---

23. Which octant does the point $(4, -2, -5)$ lie in?
    a) IV
    b) VI
    c) III
    d) VIII

    Answer: d) VIII

---

24. Which of the following points lie on the x-axis?
    a) $(0, y, z)$
    b) $(x, 0, 0)$
    c) $(0, 0, z)$
    d) $(x, y, 0)$

    Answer: b) $(x, 0, 0)$

---

25. The three mutually perpendicular lines in a three-dimensional coordinate system are called:
    a) Coordinate planes
    b) Coordinate axes
    c) Octants
    d) None of these

    Answer: b) Coordinate axes

---

26. What is the relationship between points $P(3, 6, 9)$, $Q(10, 20, 30)$, and $R(25, -41, 5)$ if they form a triangle?
    a) Right-angled triangle
    b) Equilateral triangle
    c) Not a right-angled triangle
    d) Isosceles triangle

    Answer: c) Not a right-angled triangle

---

27. Which coordinate is used to measure the distance from the XY-plane?
    a) x
    b) y
    c) z
    d) Origin

    Answer: c) z

---

28. The coordinate planes divide the space into how many parts?
    a) 4
    b) 6
    c) 8
    d) 12

    Answer: c) 8

---

29. Find the distance between the origin $(0, 0, 0)$ and the point $(2, -3, 6)$:
    a) $\sqrt{49}$
    b) $7$
    c) $2\sqrt{7}$
    d) $\sqrt{36}$

    Answer: b) $7$

---

30. If the coordinates of a triangle’s centroid are $(2, 4, 6)$, the coordinates of one vertex are $(1, 2, 3)$, and another vertex is $(3, 6, 9)$, what are the coordinates of the third vertex?
    a) $(2, 4, 6)$
    b) $(2, 4, 6)$
    c) $(2, 4, 3)$
    d) $(2, 4, 12)$

    Answer: d) $(2, 4, 12)$

---

31. In a 3D coordinate system, the equation $z = 0$ represents:
    a) XY-plane
    b) YZ-plane
    c) ZX-plane
    d) None of these

    Answer: a) XY-plane

---

32. What is the coordinate system used in 3D geometry called?
    a) Polar coordinate system
    b) Cylindrical coordinate system
    c) Rectangular Cartesian coordinate system
    d) None of these

    Answer: c) Rectangular Cartesian coordinate system

---

33. What are the coordinates of the midpoint of the line segment joining $A(1, 2, 3)$ and $B(5, 6, 7)$?
    a) $(3, 4, 5)$
    b) $(4, 5, 6)$
    c) $(3, 6, 9)$
    d) $(1, 1, 1)$

    Answer: a) $(3, 4, 5)$

---

34. The sign of the coordinates $(-, +, -)$ indicates that the point is in which octant?
    a) V
    b) VI
    c) VII
    d) VIII

    Answer: b) VI

---

35. Who introduced the idea of coordinate planes in a letter to Leibnitz?
    a) Euler
    b) Descartes
    c) J. Bernoulli
    d) Fermat

    Answer: c) J. Bernoulli

---

36. What is the value of $x$ if the centroid of a triangle is at $(3, 4, 5)$ and one vertex is $(5, 2, 3)$?
    a) $6$
    b) $8$
    c) $3$
    d) $4$

    Answer: d) $4$

---

37. Which property is true for diagonals in a parallelogram?
    a) They are equal.
    b) They bisect each other.
    c) They are perpendicular.
    d) None of the above.

    Answer: b) They bisect each other.

---

38. If $A(3, 2, -5)$ and $B(-1, 3, -7)$, what is the equation of the set of points $P$ such that $PA^2 + PB^2 = k^2$?
    a) $2x^2 + 2y^2 + 2z^2 - 4x + 14y - 4z = 2k^2$
    b) $2x^2 + 2y^2 + 2z^2 + 4x - 14y + 4z = 2k^2 - 109$
    c) $2x^2 + 4x - 14y + 4z = 109$
    d) None of these

    Answer: b) $2x^2 + 2y^2 + 2z^2 + 4x - 14y + 4z = 2k^2 - 109$

39. What is the equation of the plane formed by the y-axis and z-axis?
    a) XY-plane
    b) YZ-plane
    c) ZX-plane
    d) None of these

    Answer: b) YZ-plane

---

40. If $A(1, 2, 3)$, $B(-1, -2, -1)$, $C(2, 3, 2)$, and $D(4, 7, 6)$, what is the shape of the quadrilateral formed by these points?
    a) Parallelogram but not a rectangle
    b) Rectangle
    c) Square
    d) None of these

    Answer: a) Parallelogram but not a rectangle

---

41. If the coordinates of the centroid of a triangle are $(1, 1, 1)$ and two vertices are $(3, -5, 7)$ and $(-1, 7, -6)$, what are the coordinates of the third vertex?
    a) $(1, 1, 1)$
    b) $(1, 1, 2)$
    c) $(1, 1, -1)$
    d) $(1, 1, 0)$

    Answer: b) $(1, 1, 2)$

---

42. What are the coordinates of the fourth vertex $D$ of the parallelogram with vertices $A(3, -1, 2)$, $B(1, 2, -4)$, and $C(-1, 1, 2)$?
    a) $(1, 2, -4)$
    b) $(-3, 2, -4)$
    c) $(1, -4, 3)$
    d) $(5, -2, 0)$

    Answer: b) $(-3, 2, -4)$

---

43. If $P(-2, 3, 5)$, $Q(1, 2, 3)$, and $R(7, 0, -1)$, the points are:
    a) Non-collinear
    b) Collinear
    c) Form a right-angled triangle
    d) None of these

    Answer: b) Collinear

---

44. The coordinates of any point on the z-axis will be of the form:
    a) $(x, y, 0)$
    b) $(0, y, z)$
    c) $(0, 0, z)$
    d) $(x, 0, z)$

    Answer: c) $(0, 0, z)$

---

45. Which of the following points is equidistant from the points $(3, 4, 5)$ and $(-1, 3, -7)$?
    a) $(1, 3.5, 0)$
    b) $(0, 0, 0)$
    c) $(1, 1, 1)$
    d) None of these

    Answer: d) None of these

---

46. The sign of the coordinates $(+, -, +)$ corresponds to which octant?
    a) III
    b) IV
    c) VII
    d) VIII

    Answer: b) IV

---

47. Which of the following statements is true?
    a) Distance between any two points in space is always positive.
    b) Distance can be negative in some cases.
    c) Distance is zero only if the points are the same.
    d) Both a and c

    Answer: d) Both a and c

---

48. If $P(x, y, z)$ is a point in space, what does $x$ represent?
    a) Distance from the YZ-plane
    b) Distance from the XY-plane
    c) Distance from the ZX-plane
    d) None of these

    Answer: a) Distance from the YZ-plane

---

49. The coordinates of a point lying on the YZ-plane will be:
    a) $(0, y, z)$
    b) $(x, 0, z)$
    c) $(x, y, 0)$
    d) None of these

    Answer: a) $(0, y, z)$

---

50. If two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ lie on the same axis, what can be said about their coordinates?
    a) Two coordinates will be equal.
    b) One coordinate will differ, while others will be zero.
    c) All coordinates will differ.
    d) None of these

    Answer: b) One coordinate will differ, while others will be zero.

---

51. The length of the diagonal of a cube with side length $a$ in 3D space is:
    a) $\sqrt{2}a$
    b) $\sqrt{3}a$
    c) $2a$
    d) $3a$

    Answer: b) $\sqrt{3}a$

---

52. The equation $x^2 + y^2 + z^2 = r^2$ represents:
    a) A circle in 3D space
    b) A sphere
    c) A paraboloid
    d) None of these

    Answer: b) A sphere

---

53. Which mathematician extended geometry to more than three dimensions?
    a) Leonhard Euler
    b) Rene Descartes
    c) Einstein
    d) Antoine Parent

    Answer: c) Einstein

---

54. What does the triplet $(x, y, z)$ represent in 3D geometry?
    a) A vector
    b) A scalar
    c) A point in space
    d) A line equation

    Answer: c) A point in space

---

55. What is the distance between $P(2, -1, 3)$ and $Q(-2, 1, 3)$?
    a) $2\sqrt{5}$
    b) $2\sqrt{2}$
    c) $4\sqrt{2}$
    d) $\sqrt{5}$

    Answer: a) $2\sqrt{5}$

---

56. The octants are separated by:
    a) Coordinate axes only
    b) Coordinate planes only
    c) Both coordinate axes and planes
    d) None of these

    Answer: b) Coordinate planes only

---

57. The midpoint formula for two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is:
    a) $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})$
    b) $(\frac{x_1 - x_2}{2}, \frac{y_1 - y_2}{2}, \frac{z_1 - z_2}{2})$
    c) $(x_1x_2, y_1y_2, z_1z_2)$
    d) None of these

    Answer: a) $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})$

---

58. A line in 3D space passing through the origin can be represented as:
    a) $\vec{r} = t\vec{a}$
    b) $\vec{r} = \vec{a} + t\vec{b}$
    c) $\vec{r} = t(\vec{a} + \vec{b})$
    d) $\vec{r} = \vec{a}$

    Answer: a) $\vec{r} = t\vec{a}$

---

59. If $A(1, 2, 3)$, $B(4, 5, 6)$, and $C(7, 8, 9)$, what type of triangle is formed?
    a) Isosceles
    b) Right-angled
    c) Collinear points (No triangle)
    d) Scalene

    Answer: c) Collinear points (No triangle)

---

60. A point lies on the origin if its coordinates are:
    a) $(0, 0, 0)$
    b) $(1, 1, 1)$
    c) $(x, y, z)$
    d) None of these

    Answer: a) $(0, 0, 0)$

MCQs for NEET, JEE, IIT, NIT, CUET, CTET, and SSC Entrance Exams: Your Ultimate Preparation Guide

Are you preparing for competitive exams like NEET, JEE, IIT, NIT, CUET, CTET, or SSC? Multiple Choice Questions (MCQs) are a proven way to enhance your preparation strategy. These MCQs are designed to strengthen your concepts, boost problem-solving skills, and improve time management—key elements to ace any entrance exam.

Why MCQs Are Essential for Competitive Exams?

1. Concept Clarity: MCQs cover important topics, ensuring a deep understanding of fundamental concepts.
2. Time Management: Practicing MCQs improves speed and accuracy, essential for high-stakes exams.
3. Exam Pattern Familiarity: MCQs simulate the real exam pattern, helping you feel confident on the big day.

Benefits of Using These MCQs for NEET, JEE, IIT, and More

• Targeted Preparation: Specially curated MCQs focus on the specific syllabus of exams like NEET, JEE, and CUET, making your study focused and efficient.
• Wide Subject Coverage: Includes questions from Physics, Chemistry, Mathematics, Biology, and more.
• Boosts Competitive Edge: High-quality MCQs ensure you're ready to tackle even the trickiest questions.

Exams These MCQs Help With:

• NEET: For aspiring medical students.
• JEE & IIT: For engineering and technical aspirants.
• NIT Admissions: A stepping stone to prestigious National Institutes of Technology.
• CUET: Common University Entrance Test for undergraduate programs.
• CTET: A mandatory test for teachers.
• SSC: Government job aspirants preparing for SSC exams.

Pro Tips to Use MCQs Effectively:

1. Regular Practice: Dedicate time daily to solve topic-wise MCQs.
2. Review and Revise: Go through solutions and explanations to understand your mistakes.
3. Mock Tests: Attempt full-length MCQ tests to build exam stamina.

Get Started Now!

Our extensive MCQ collections are tailored to help you succeed in NEET, JEE, IIT, NIT, CUET, CTET, and SSC. Start practicing today and take a step closer to your dream career. Bookmark this page and share it with fellow aspirants.

Tags:

MCQs for NEET, JEE MCQs, IIT preparation, CUET questions, CTET practice, SSC MCQs, competitive exam tips, entrance exam success.

Make your preparation smarter, not harder. Let these MCQs guide your way to success!