68 practice questions on Introduction to Three Dimensional Geometry, sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Introduction to Three Dimensional Geometry notes.
📐 Mathematics · Class 11 · JEE
Introduction to Three Dimensional Geometry - Practice Questions with Answers
68 free MCQs on Introduction to Three Dimensional Geometry, each with its own worked answer and explanation. Coordinate axes and planes in space, octants, distance between two points, and the section formula
Take the timed Introduction to Three Dimensional Geometry chapterwise test →Easy - 20 questions
Q1.
The three coordinate planes divide space into how many octants?
- A 8
- B 12
- C 4
- D 6
Q2.
The coordinates of the origin in three dimensional space are:
- A (1, 0, 0)
- B (0, 0, 0)
- C (1, 1, 1)
- D (0, 0, 1)
Q3.
A point lying in the XY-plane has which coordinate equal to zero?
- A x
- B y
- C z
- D none of them
Q4.
A point lying on the x-axis has coordinates of the form:
- A (0, y, 0)
- B (0, 0, z)
- C (x, y, 0)
- D (x, 0, 0)
Q5.
The distance of the point P(x, y, z) from the origin is:
- A √(x² + y² + z²)
- B x² + y² + z²
- C √(x + y + z)
- D x + y + z
Q6.
The distance between the points (1, 2, 3) and (1, 2, 5) is:
- A 1
- B 2
- C 3
- D 5
Q7.
The YZ-plane is characterised by the equation:
- A z = 0
- B x = y
- C x = 0
- D y = 0
Q8.
The point (2, 3, 4) lies in which octant?
- A Second
- B Fourth
- C Eighth
- D First
Q9.
The midpoint of the segment joining (2, 4, 6) and (4, 8, 10) is:
- A (3, 6, 8)
- B (6, 12, 16)
- C (2, 4, 4)
- D (1, 2, 2)
Q10.
How many coordinate planes are there in three dimensional geometry?
- A 2
- B 3
- C 4
- D 8
Q11.
The ZX-plane is described by the equation:
- A x = z
- B x = 0
- C y = 0
- D z = 0
Q12.
The distance between the origin and the point (3, 4, 12) is:
- A 19
- B √19
- C 12
- D 13
Q13.
A point on the z-axis has coordinates of the form:
- A (0, 0, z)
- B (z, 0, 0)
- C (0, z, 0)
- D (z, z, 0)
Q14.
In the coordinate triple (x, y, z), the value z is measured along the:
- A y-axis
- B z-axis
- C line y = x
- D x-axis
Q16.
The three coordinate axes in space are:
- A Inclined at 60° to one another
- B Coincident at every point
- C Parallel to one another
- D Mutually perpendicular
Q17.
The point (0, 5, 0) lies on the:
- A y-axis
- B z-axis
- C XY-plane only
- D x-axis
Q18.
The section formula for internal division of PQ in ratio m : n gives the x-coordinate as:
- A (x₁ + x₂)/2
- B (mx₂ + nx₁)/(m + n)
- C (mx₂ − nx₁)/(m − n)
- D (mx₁ + nx₂)/(m + n)
Q19.
The midpoint formula is the section formula applied with ratio:
- A 2 : 1
- B 0 : 1
- C 1 : 1
- D 1 : 2
Q20.
The distance between the points (1, 0, 0) and (0, 1, 0) is:
- A 2
- B 0
- C 1
- D √2
Medium - 20 questions
Q21.
The distance between the points A(2, 3, 5) and B(4, 3, 1) is:
- A 2√5
- B 4
- C √20 units measured along the y-axis only
- D 6
Q22.
The point that divides the join of (1, −2, 3) and (3, 4, −5) internally in the ratio 1 : 3 is:
- A (1/2, −5/2, 4)
- B (3/2, −1/2, 1)
- C (2, 1, −1)
- D (5/2, 5/2, −3)
Q23.
The centroid of the triangle with vertices (1, 2, 3), (4, 5, 6) and (7, 8, 9) is:
- A (12, 15, 18)
- B (2, 3, 4)
- C (4, 5, 6)
- D (3, 4, 5)
Q24.
In what ratio does the XY-plane divide the line joining A(1, 2, 3) and B(2, 4, −6)?
- A 1 : 3
- B 2 : 1
- C 3 : 1
- D 1 : 2
Q25.
The point (−3, 1, 2) lies in the octant where the signs of (x, y, z) are:
- A (−, +, +)
- B (+, +, +)
- C (−, −, +)
- D (+, −, −)
Q26.
If the distance between (2, 3, a) and (2, 3, 1) is 5 units, then a can be:
- A 6 only
- B 6 or −4
- C 4 or −6
- D 5 or −5
Q27.
The point equidistant from the three coordinate axes among the following is:
- A (0, 1, 2)
- B (2, 2, 0)
- C (1, 1, 1)
- D (1, 2, 3)
Q28.
The projection (foot of perpendicular) of the point (3, 4, 5) on the XY-plane is:
- A (0, 0, 5)
- B (3, 0, 5)
- C (0, 4, 5)
- D (3, 4, 0)
Q29.
The point which divides the join of (2, 1, 4) and (4, 3, 2) externally in the ratio 1 : 2 is:
- A (0, −1, 6)
- B (0, 1, 6)
- C (6, 5, 0)
- D (3, 2, 3)
Q30.
The distance of the point (1, 2, 3) from the x-axis is:
- A √14
- B √13
- C 1
- D √5
Q31.
If the point (x, 0, 0) is equidistant from (1, 2, 3) and (3, 2, 1), then x equals:
- A 2
- B 3
- C 0
- D 1
Q32.
The points A(1, 2, 3), B(2, 3, 4) and C(3, 4, 5) are:
- A Vertices of an equilateral triangle
- B Vertices of a right triangle
- C Coincident
- D Collinear
Q34.
If A(3, 2, 0), B(5, 3, 2) and C(−9, 6, −3) are the vertices of a triangle, the midpoint of BC is:
- A (7, 9/2, 1)
- B (−2, 9/2, −1/2)
- C (−2, 9, −1)
- D (−4, 9/2, −1/2)
Q35.
In what ratio does the YZ-plane divide the line joining (−2, 4, 7) and (3, −5, 8)?
- A 1 : 2
- B 2 : 1
- C 2 : 3
- D 3 : 2
Q36.
The number of points in space whose each coordinate is either 1 or −1 is:
- A 6
- B 4
- C 3
- D 8
Q37.
If the midpoint of the segment joining (a, 2, 3) and (5, b, 7) is (3, 4, 5), then a + b equals:
- A 7
- B 9
- C 11
- D 5
Q38.
The perpendicular distance of the point (2, −3, 4) from the XY-plane is:
- A √29
- B 4
- C 2
- D 3
Q39.
The triangle with vertices (0, 0, 0), (3, 0, 0) and (0, 4, 0) is:
- A Obtuse angled
- B Not a valid triangle
- C Right angled at the origin
- D Equilateral
Q40.
If P(2, 3, 4) and Q(4, 5, 6), the point dividing PQ in ratio 3 : 1 internally is:
- A (5/2, 7/2, 9/2)
- B (3, 4, 5)
- C (10/3, 13/3, 16/3)
- D (7/2, 9/2, 11/2)
Hard - 28 questions
Q41.
Why can the 2D test for collinearity using slopes not be carried over directly to three dimensions?
- A A line in space has no single slope, since it makes three separate angles with the axes
- B Slopes only exist for lines passing through the origin in any dimension
- C Three points in space can never be collinear in the majority of documented cases
- D Slope is undefined whenever any coordinate is negative under usual circumstances
Q42.
When solving the section formula for an unknown ratio, obtaining a negative value of k means that the point:
- A Lies at the origin of the coordinate system
- B Divides the segment externally rather than internally
- C Does not lie on the line through the two given points
- D Coincides with the midpoint of the segment
Q43.
The locus of a point whose distance from the z-axis is a constant a is:
- A A circle of radius a lying in the XY-plane only
- B A pair of planes parallel to the XY-plane
- C A cylinder of radius a with the z-axis as its axis
- D A sphere of radius a centred at the origin
Q44.
A point equidistant from all four vertices of a tetrahedron is found by:
- A Averaging the four vertices, which always gives the required point
- B Taking the midpoint of the longest edge of the tetrahedron
- C Projecting the centroid onto the XY-plane in every case
- D Solving the three equations obtained by equating squared distances pairwise
Q45.
The points A(1, 2, 3), B(−1, −2, −1), C(2, 3, 2) and D(4, 7, 6) form:
- A A parallelogram, since both pairs of opposite sides are equal and the diagonals bisect each other
- B An equilateral triangle with one repeated vertex according to standard results
- C A regular tetrahedron in most documented configurations
- D Four collinear points under typical conditions
Q46.
If a point P divides AB in the ratio k : 1 and lies on the XY-plane, then k is given by:
- A z₁ + z₂
- B −z₁/z₂
- C z₁/z₂
- D −z₂/z₁
Q47.
The distance formula in three dimensions follows from applying the Pythagorean theorem:
- A Three times, one for each coordinate plane involved
- B Only when all coordinates are positive as generally observed
- C Twice - once in the base plane and once in the vertical direction
- D Once, exactly as in two dimensions with an extra term appended
Q48.
The point on the y-axis equidistant from A(3, 1, 2) and B(5, 5, 2) is:
- A (0, 3, 0)
- B (0, 0, 5)
- C (5, 0, 0)
- D (0, 5, 0)
Q49.
If the origin is the centroid of a triangle with vertices (2a, 2, 6), (−4, 3b, −10) and (8, 14, 2c), then a, b and c are:
- A a = −2, b = −16/3, c = 2
- B a = 2, b = 16/3, c = −2
- C a = −2, b = 16/3, c = −2
- D a = 4, b = −5, c = 1
Q50.
Three points are collinear in space if and only if:
- A They lie in the same octant of the coordinate system
- B The largest of the three pairwise distances equals the sum of the other two
- C All three pairwise distances are equal to one another
- D Their coordinates sum to zero in each of the three components
Q51.
The locus of a point equidistant from the points A and B in space is:
- A A sphere with AB as its diameter in most cases
- B The single midpoint of AB and no other point
- C The plane perpendicular to AB through its midpoint
- D The straight line AB extended indefinitely
Q52.
A point P lies on the x-axis and is at distance 5 from the point (0, 3, 4). The coordinates of P are:
- A (5, 0, 0) or (−5, 0, 0)
- B (3, 0, 0) or (−3, 0, 0)
- C No such point exists
- D (0, 0, 0) only
Q53.
If A(1, 2, 3) and the midpoint of AB is (2, 3, 4), then the coordinates of B are:
- A (3, 4, 5)
- B (1, 1, 1)
- C (4, 6, 8)
- D (3/2, 5/2, 7/2)
Q54.
The ratio in which the line joining (2, 4, 5) and (3, 5, −4) is divided by the XY-plane is:
- A 5 : 1
- B 5 : 4
- C 4 : 5
- D 1 : 1
Q55.
Why does a point's distance from a coordinate axis use only two of its three coordinates?
- A Distances in space are always computed pairwise between two coordinates only
- B The third coordinate is always zero for points measured from an axis
- C The perpendicular from the point meets the axis at the foot sharing that axis coordinate, so that coordinate contributes nothing
- D Every axis lies in the XY-plane, making the third coordinate irrelevant
Q56.
The vertices A(0, 7, 10), B(−1, 6, 6) and C(−4, 9, 6) form a triangle that is:
- A Equilateral
- B Scalene with no right angle
- C Degenerate, since the points are collinear
- D Right angled and isosceles
Q57.
For a point dividing a segment externally, the denominator in the section formula is:
- A m − n, which is why m = n gives no valid point
- B m + n, exactly as in internal division
- C Always 2, independent of the ratio chosen
- D mn, the product of the two ratio terms
Q58.
The point (1, 2, −3) lies in the octant characterised by:
- A (+, +, +)
- B (+, +, −)
- C (+, −, −)
- D (−, +, −)
Q59.
If a parallelepiped is formed with edges along the axes and one vertex at the origin and the opposite vertex at (a, b, c), its main diagonal has length:
- A abc
- B √(ab + bc + ca)
- C √(a² + b² + c²)
- D a + b + c
Q60.
The centroid of a tetrahedron with vertices A, B, C and D is found by:
- A Averaging only three vertices and ignoring the fourth
- B Taking the midpoint of the longest edge in every case
- C Projecting the triangle centroid of ABC onto the fourth vertex
- D Averaging all four position vectors, dividing each coordinate sum by 4
Q61.
The distance between the points (1, 2, 3) and (4, 6, 3) is:
- A 5
- B √29
- C 6
- D 4
Q62.
The midpoint of the segment joining (2, 4, 6) and (4, 8, 10) is:
- A (3, 6, 8)
- B (6, 12, 16)
- C (2, 4, 4)
- D (1, 2, 2)
Q63.
The point dividing the segment from (1, 2, 3) to (4, 5, 6) in the ratio 2:1 internally is:
- A (3, 4, 5)
- B (2, 3, 4)
- C (5, 6, 7)
- D (3, 3, 4)
Q64.
The signs of the coordinates of the point (−1, 2, −3) are:
- A (+, +, +)
- B (−, +, −)
- C (−, −, −)
- D (+, −, +)
Q65.
The centroid of the triangle with vertices (1, 2, 3), (2, 3, 4) and (3, 4, 5) is:
- A (2, 3, 4)
- B (3, 4, 5)
- C (1, 2, 3)
- D (6, 9, 12)
Q66.
The point on the x-axis equidistant from (1, 2, 3) and (3, 2, −1) is:
- A (0, 0, 0)
- B (1, 0, 0)
- C (2, 0, 0)
- D (−1, 0, 0)
Q67.
The distance of the point (3, 4, 5) from the origin is:
- A 5√2
- B 5
- C 10
- D √48
Q68.
The reflection of the point (1, 2, 3) in the xy-plane is:
- A (1, 2, −3)
- B (−1, −2, 3)
- C (1, −2, 3)
- D (−1, 2, 3)
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