Bihar Board Class 12 (Commerce) Three Dimensional Geometry — practice questions
104 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice Bihar Board Class 12 (Commerce) Three Dimensional Geometry in the app →If l, m, n are the direction cosines of a line in space, thenThe vector equation of a line passing through point a and parallel to vector b isThe equation of a plane in normal form (with unit normal n̂ and perpendicular distance d from origin) isA plane meets the axes at (a,0,0), (0,b,0), (0,0,c). Its equation isThe direction cosines of the line joining (1, 2, 3) and (4, 5, 6) areThe Cartesian equation (x-1)/2 = (y-2)/3 = (z-3)/4 representsThe angle θ between two lines with direction vectors b₁ and b₂ is given byThe distance of the point (x₀, y₀, z₀) from the plane ax + by + cz + d = 0 isThe shortest distance between skew lines r = a₁ + λb₁ and r = a₂ + μb₂ isTwo lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar if and only ifThe angle between two planes with normals n₁ and n₂ is given byThe angle ϕ between a line with direction b and a plane with normal n satisfiesThe equation of a plane passing through point (x₁, y₁, z₁) with normal vector (a, b, c) isTwo lines with direction vectors b₁ and b₂ are parallel if and only ifThe equation of a plane through the intersection of π₁ = 0 and π₂ = 0 has the formThe perpendicular distance from a point P with position vector p to a line r = a + λb isThe angle between the lines with direction ratios (1, 2, 2) and (2, 1, 2) satisfiesThe distance of the point (2, 1, -1) from the plane 2x - y + 2z + 5 = 0 isThe angle between the planes 2x - y + 2z = 5 and x + 2y - 2z = 4 satisfiesThe equation of the plane through the points (1,0,0), (0,1,0), (0,0,1) isThe distance between the parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z = 5 isThe line r = (2, -1, 4) + λ(3, 0, 2), expressed in symmetric Cartesian form, isThe shortest distance between r = (1,2,3) + λ(2,3,4) and r = (2,4,5) + μ(3,4,5) isThe equation of a plane parallel to x + 2y - 3z = 5 and passing through (1, 1, 1) isThe plane through the points (1,1,0), (1,0,1) and (0,1,1) has equationA line has direction ratios (1, 1, 2) and meets the plane x - y + z = 0. The angle ϕ between them satisfiesThe line through the origin with direction ratios (1, 1, 1) meets the plane x + y + z = 6 atThe foot of the perpendicular from origin to the plane 2x + 3y - 6z = 14 isThe image of the point (1, 2, 3) in the plane x + y + z = 0 isA line makes angles 90, 60 and 30 degrees with the x, y, z axes. Its direction cosines are:A line has direction ratios 2, -1, -2. One set of its direction cosines is:Find the direction cosines of the line through (-2, 4, -5) and (1, 2, 3).Reversing the direction of a directed line in space makes its direction cosines:When a directed line is reversed, each direction angle is replaced by its:Two parallel lines drawn in the same sense in space have:How many sets of direction ratios does a line in space have?A line makes equal acute angles with all three axes. Each of its direction cosines is:The points (2, 3, 4), (-1, -2, 1) and (5, 8, 7) are:Direction ratios of the segment from (1, 2, 3) to (4, 6, 15) are:A line through (x1, y1, z1) with direction ratios a, b, c has Cartesian form:The line through (5, 2, -4) parallel to 3i + 2j - 8k has Cartesian form:A line passes through (1, 2, 3) parallel to 3i + 2j - 2k. Which point lies on it?On the line r = 2i - j + 4k + λ(i + 2j - k), the point for λ = 2 is:In r = a + λb, the vector b represents the line's:In the equation r = a + λb of a line, the vector a is the position vector of:A line in space is uniquely fixed when it passes through:A line through (1, 2, 3) and (3, 4, 5) has direction ratios proportional to:Which point lies on the line through the origin parallel to the x-axis?For two lines with direction cosines l1, m1, n1 and l2, m2, n2, cos θ equals:Lines with direction ratios (a1, b1, c1) and (a2, b2, c2) are perpendicular when:The lines with direction vectors i + 2j + 2k and 3i + 2j + 6k have cos θ equal to:Lines with direction ratios 3, 5, 4 and 1, 1, 2 meet at an angle whose cosine is:Two lines have direction ratios 1, 1, 0 and 0, 1, 1. They are inclined at:For which k are lines with ratios (k, 2, 3) and (1, -1, 2) perpendicular?The angle between two lines in space is conventionally taken as:Skew lines in space are lines that are:Which statement is true of every pair of skew lines?The shortest distance between two parallel lines is the:For r = a1 + λb1 and r = a2 + μb2 (skew), the shortest distance is:Shortest distance between r = i + j + λ(2i - j + k) and r = 2i + j - k + μ(3i - 5j + 2k):The parallel lines r = i + 2j - 4k + λ(2i + 3j + 6k) and r = 3i + 3j - 5k + μ(2i + 3j + 6k) are how far apart?How far apart are the parallel lines r = λi and r = 3j + 4k + μi?What is the shortest distance between the x-axis and the line through (0, 0, 4) parallel to the y-axis?Two parallel lines in space are always:If two lines lie in the same plane, they must either cross or be:Direction cosines of a vector are the cosines of its angles with the:Which triple can be the direction cosines of a line?If a line has l = 1/2 and m = 1/2, then n equals:A line has direction cosines 0, 1/2 and n. What is n?A line makes 60 degrees with the x-axis and 60 degrees with the y-axis. It can make which angle with the z-axiDirection ratios of line PQ are 2, -3, 4. Those of QP can be:An undirected line in space has how many sets of direction cosines?To find the direction cosines of a line missing the origin, we draw through the origin a line that is:Direction ratios a, b, c relate to cosines l, m, n by a = λl, where λ is:Which point lies on the line through (1, 2, 3) with direction (3, 1, -4)?The line (x-1)/2 = (y+3)/4 = (z-5)/6 passes through:The line (x-1)/2 = (y+3)/4 = (z-5)/6 has direction cosines:Line through (1, -1, 2) and (3, 4, -2), and line through (0, 3, 2) and (3, 5, 6), are:Lines with ratios a, b, c and b-c, c-a, a-b are inclined at:If (b1 x b2) . (a2 - a1) = 0 for two lines, the lines are:A unit vector along the common perpendicular of lines with directions b1, b2 is:Lines through two disjoint pairs of four non coplanar points are:What is the length of the segment from P(-2, 4, -5) to Q(1, 2, 3)?How many straight lines pass through two distinct points in space?The line r = (2i - j + 4k) + λ(i + 2j - k) has direction ratios:Parametric equations x = 1 + 2λ, y = 3 - λ, z = 4λ give direction ratios:A line with direction ratios (1, 2, 2) is perpendicular to the line with ratios:Which description fits a pair of skew lines in space?What is the shortest distance between two lines that cross at a point?The shortest segment between two skew lines has which property?A line has direction ratios 2, 2, 1. What is the cosine of its angle with the x-axis?Lines with direction ratios 2, 4, 6 and 1, 2, 3 make what angle with each other?A line has the vector equation r = (i + j) + t(3i - 2j + k). Its direction ratios areA line through points (1, 2, 3) and (4, 6, 8) is parallel to which vector?The angle between two skew lines is that between two lines whichTwo parallel lines in space always satisfy which of these conditions?Which is the lowest dimension of space in which a pair of skew lines can exist?Two lines lying in one common plane must behave in which way?Two lines in three-dimensional space are skew exactly when they areFour points define two skew lines only if the points form the vertices of a tetrahedron ofRotating a line about a skew, non-perpendicular line sweeps outIn higher-dimensional geometry, a line is also described as a flat of which dimension?The common perpendicular to two skew lines with direction vectors b and d runs along which vector?Points A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear because the direction ratios of AB and BC are