68 free MCQs on Three Dimensional Geometry, each with its own worked answer and explanation. Lines and planes in 3D space, direction cosines, distances, and angles
Take the timed Three Dimensional Geometry chapterwise test →68 practice questions on 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 Three Dimensional Geometry notes.
Three mutually perpendicular axes x, y, z meeting at the origin O, with a point P located by its (x, y, z) coordinates.
Q1.
Direction cosines of a line are the cosines of angles made with the:
Q2.
For direction cosines l, m, n of a line:
Q3.
The direction ratios of a line are proportional to its:
Q5.
The distance from the origin to point P(a, b, c) is:
Q6.
The vector equation of a line through point A (position vector a) with direction vector b is:
Q7.
The Cartesian equation of a line through (x<sub>1</sub>,y<sub>1</sub>,z<sub>1</sub>) with direction ratios a,b,c is:
Q8.
The general equation of a plane is of the form:
Q9.
Two lines in 3D that do not intersect and are not parallel are called:
Q10.
The equation of the x-axis in 3D is:
Q11.
If two lines have direction ratios (a<sub>1</sub>,b<sub>1</sub>,c<sub>1</sub>) and (a<sub>2</sub>,b<sub>2</sub>,c<sub>2</sub>), they are parallel if:
Q12.
The normal vector to the plane 2x + 3y - z = 5 is:
Q13.
Direction ratios of the line joining A(1,2,3) and B(4,6,3) are:
Q14.
The distance from origin to the plane ax + by + cz = d is:
Q15.
How many planes pass through a given straight line?
Q16.
The angle between two planes equals the angle between their:
Q17.
A line is perpendicular to a plane if its direction is:
Q18.
The intercept form of a plane is x/a + y/b + z/c = 1. Here a, b, c are:
Q19.
Three points A, B, C are collinear if the vectors AB and AC satisfy:
Q20.
The direction cosines of the positive x-axis are:
Q21.
Find the direction cosines of a line with direction ratios (2, -1, 2).
Q22.
The angle between lines with direction ratios (1,1,0) and (0,1,1) satisfies:
Q23.
Two lines are perpendicular if their direction ratios (a<sub>1</sub>,b<sub>1</sub>,c<sub>1</sub>) and (a<sub>2</sub>,b<sub>2</sub>,c<sub>2</sub>) satisfy:
Q24.
The shortest distance between two parallel lines r = a + lambda*b and r = c + mu*b is:
Q25.
The equation of the plane through origin with normal vector (1, -2, 3) is:
Q26.
The line (x-1)/2 = (y+2)/3 = (z-1)/4 passes through the point:
Q27.
The shortest distance between skew lines r = a + lambda*b and r = c + mu*d is:
Q28.
The angle between planes 2x+3y-z=5 and x-y+2z=3 satisfies cos(theta) =
Q29.
The condition for two lines to be coplanar is:
Q30.
The equation of a plane passing through three non-collinear points is found by:
Q31.
Two planes are parallel if their normal vectors are:
Q32.
The perpendicular distance between the parallel planes 2x+y-2z=3 and 4x+2y-4z=12 is:
Q33.
The line through (1,2,3) perpendicular to plane 2x-y+3z=5 has direction ratios:
Q34.
Four points are coplanar if the vectors from one point to the other three satisfy:
Q35.
The foot of the perpendicular from the origin to the plane 3x+4y+5z=25 is:
Q36.
The plane through (1,0,0), (0,1,0) and (0,0,2) has equation:
Q37.
The line x/1 = y/2 = z/3 makes an angle with the y-axis such that:
Q38.
The line through A(2,1,3) and B(4,3,9) meets x+y+z=6 at the point:
Q39.
The distance from point (1,2,3) to the plane 2x+y+2z=12 is:
Q40.
The ratio in which the plane x+y+z=6 divides the segment joining (1,1,1) and (4,4,4):
Q41.
The shortest distance between skew lines x=y=z and (x+1)/1=y/2=z/3 is:
Q42.
The line through A(1,2,3) and B(4,5,6) meets the plane x+y+z=9 at:
Q43.
The equation of the plane containing line r = i + lambda*(2i+j-k) and parallel to i+j is:
Q44.
The distance between parallel planes 2x-2y+z+3=0 and 4x-4y+2z+5=0 is:
Q45.
The locus of a point equidistant from A(1,0,0) and B(0,1,0) is:
Q46.
The plane through line of intersection of x+y+z=1 and 2x+3y+4z=5 and parallel to x-axis has equation:
Q47.
Two lines coplanar in space share a common plane. The normal to this plane is perpendicular to:
Q48.
The sphere centred at (1,-1,2) passing through the origin has equation:
Q49.
The ratio in which plane x+y+z=9 divides segment from A(2,1,3) to B(5,4,6):
Q50.
The plane through A(2,2,1), B(3,0,1), C(4,-1,0) has equation:
Q51.
The direction cosines of a line making equal angles with all coordinate axes are:
Q52.
The image of the origin in the plane 2x-y+2z=9 is:
Q53.
Two planes x+2y+3z=4 and 2x+4y+6z=5 are:
Q54.
If a line makes angles alpha, beta, gamma with x, y, z axes respectively, then sin<sup>2</sup>(alpha)+sin<sup>2</sup>(beta)+sin<sup>2</sup>(gamma) =
Q55.
The angle bisector planes of two intersecting planes bisect the:
Q56.
The shortest distance between the lines x=1,y=2 and y=3,z=4 (lines parallel to z-axis and x-axis respectively) is:
Q57.
The plane passing through (0,0,0), (1,0,0) and (0,1,1) has normal:
Q58.
The equation of the plane passing through (1,0,-1), (3,2,2), (-1,-1,-2) and passing through all three is:
Q59.
The distance between the points (0, 0, 0) and (2, 3, 6) is:
Q60.
The direction cosines of the x-axis are:
Q61.
The distance of the point (1, 2, 3) from the plane 2x − 3y + 6z − 11 = 0 is:
Q62.
The angle between the planes x + y + z = 1 and x − y + z = 1 is:
Q63.
The direction cosines of a line equally inclined to the three coordinate axes are:
Q64.
The foot of the perpendicular from the origin to the plane 2x + 3y + 6z = 49 is:
Q65.
The equation of the plane through (1,0,0), (0,1,0) and (0,0,1) is:
Q66.
The line with direction (1, 2, 2) and the plane 2x − 2y + z = 5 are:
Q67.
The shortest distance of the point (2, 3, 4) from the x-axis is:
Q68.
The lines (x−1)/2 = (y−2)/3 = (z−3)/4 and (x−2)/3 = (y−3)/4 = (z−4)/5 are:
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