Three Dimensional Geometry

CBSE Class 12 Mathematics · 131 board questions from this chapter; 129 listed below, 2022–2026

• Direction cosines of line (1-x)/0=y=z are2026
• The length of perpendicular drawn from the point (3, 4, 2) on the line x/0=y/0=z/1 is2026
• Direction ratios of lines l_1 and l_2 respectively are <1,0,0> and <0,-1,0>. The direction ratios of the line perpendicular to…2026
• Direction cosines of line x=y=1-z are2026
• The length of perpendicular drawn from the point (1,2,3) on line x/0=y/1=z/0 is2026
• Direction ratios of lines l_1 and l_2 respectively are <1,-2,3> and <-2, p,-6>. The value of p for which l_1 ∥ l_2, is:2026
• Direction cosines of the line given by equations: (2 x-1)/4=(1-y)/3=(-z)/6 are2026
• If l_1, m_1, n_1 and l_2, m_2, n_2 are direction cosines of lines L_1 and L_2 respectively and θ is the acute angle between them,…2026
• The length of perpendicular drawn from point (2,5,7) on line x/1=y/0=z/0 is2026
• Three points A(0, 1, 1), B(2, 0, -1) and C(1, 0, 3) form Δ ABC. The ar( ABC) is:2026
• Direction ratios of lines l_1 and l_2 are <12, -3, 9> and <4, q, -p> respectively. The values of p and q for which l_1 and l_2…2026
• Assertion: Lines given by x=py+q, z=ry+s and x=p^′ y+q^′, z=r^′ y+s^′ are perpendicular to each other when pp^′+rr^′=1. Reason…2026
• Assertion: A line can have direction cosines < 1, 1, 1 > Reason (R): cos θ=1 is possible for θ=0.2026
• A vector a of magnitude 14 has direction ratios <2,3,-6>. Find the projection of the vector a on i.2026
• Find the co-ordinates of the point on line x=(y-1)/2=(z-2)/3 whose y co- ordinate is 3 times the x co-ordinate.2026
• Find the co-ordinates of foot of perpendicular drawn from (0,0,0) to line x/1=(y+1)/-1=(z-3)/-2.2026
• If the lines (x-3)/1=(1-y)/1=(z+2)/p and (2-x)/3=(y+1)/5=(z+56)/(2 p) are perpendicular to each other, then find the value(s) of…2026
• Find the co-ordinates of the point on the line r=- j+3 k+λ(2 i-2 j+ k) such that the sum of co-ordinates is 3.2026
• Find the angle between the following pair of lines: (x-2)/3=(y+5)/2=(1-z)/-6 and (x-7)/1=y/2=(6-z)/-22026
• Find a point on the line (x-2)/3=(1-y)/2=(z-3)/2 at a distance of √2 units from the point (1, 2, 3). OR Find the shortest…2026
• Show that line AB passing through points A(0,4,1), B(2,3,-1) and the line CD passing through points C(4,5,0), D(2,6,2) are…2026
• Prove that the line through points A(0,-1,-1) and B(4,5,1) intersects the line through points C(3, 9, 4) and D(-4, 4, 4). Hence,…2026
• Find the foot of the perpendicular from the point (0,2,3) on the line (-x-3)/-5=(1-y)/-2=(3 z+12)/9 and hence find the length of…2026
• Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines…2026
• Find the vector and cartesian equations of the line passing through the point of intersection of the lines r=( i+ j- k)+λ(3 i- j)…2026
• Check whether the lines given by (x-1)/2=(y-2)/3=(z-3)/4 and (x-4)/5=(y-1)/2=z are parallel or not. If parallel, find the…2026
• Find the length of the perpendicular drawn from the point P(1, 2, 3) to the line (x-6)/3=(y-7)/2=(z-7)/-2. Also, find the…2026
• Represent the equations of lines l_1 and l_2 in vector form and check whether they are intersecting or not. l_1:…2026
• A line passing through the points A(1, 2, 3) and B(5, 8, 11) intersects the line r=4 i+ j+λ(5 i+2 j+ k). Find the co-ordinates of…2026
• If P is a point on the line segment joining ( 3,6,-1 ) and ( 6,2,-2 ) and y -coordinate of P is 4, then its z -coordinate is:2025
• The coordinates of the foot of the perpendicular drawn from the point A(-2,3,5) on the y -axis is:2025
• If the direction cosines of a line are λ, λ, λ, then λ is equal to:2025
• The equation of a line parallel to the vector 3 i+ j+2 k and passing through the point (4,-3,7) is:2025
• The line x=1+5 μ, y=-5+μ, z=-6-3 μ passes through which of the following point?2025
• Find the value of λ if the following lines are perpendicular to each other: l_1: (1-x)/-3=(3 y-2)/(2 λ)=(z-3)/3; l_2: (x-1)/(3…2025
• A man needs to hang two lanterns on a straight wire whose end points have coordinates A(4,1,-2) and B(6,2,-3). Find the…2025
• Determine if the lines r=( i+ j- k)+λ(3 i- j) and r=(4 i- k)+μ(2 i+3 k) intersect with each other.2025
• Find the angle at which the given lines are inclined to each other: l_1: (x-5)/2=(y+3)/1=(z-1)/-3; l_2: x/3=(y-1)/2=(z+5)/-12025
• Find the distance of the point P(2,4,-1) from the line (x+5)/1=(y+3)/4=(z-6)/-9. OR Let the position vectors of the points A, B…2025
• Verify that lines given by r=(1-λ) i+(λ-2) j+(3-2 λ) k and r=(μ+1) i+(2 μ-1) j-(2 μ+1) k are skew lines. Hence, find shortest…2025
• Find the distance of the point (-1,-5,-10) from the point of intersection of the lines (x-1)/2=(y-2)/3=(z-3)/4 and…2025
• Find the image A^′ of the point A(1,6,3) in the line x/1=(y-1)/2=(z-2)/3. Also, find the equation of the line joining A and A^′.…2025
• Find the shortest distance between the lines: (x+1)/2=(y-1)/1=(z-9)/-3 and (x-3)/2=(y+15)/-7=(z-9)/5; OR Find the image A^′ of…2025
• Find the foot of the perpendicular drawn from point (2,-1,5) to the line (x-11)/10=(y+2)/-4=(z+8)/-11. Also, find the length of…2025
• Find the foot of the perpendicular drawn from the point ( 1,1,4 ) on the line (x+2)/5=(y+1)/2=(-z+4)/-3. OR Find the point on the…2025
• Show that the line passing through the points A(0,-1,-1) and B(4,5,1) intersects the line joining points C(3,9,4) and D(-4,4,4).2025
• Let the polished side of the mirror be along the line x/1=(1-y)/-2=(2 z-4)/6. A point P(1,6,3), some distance away from the…2025
• Find the point Q on the line (2 x+4)/6=(y+1)/2=(-2 z+6)/-4 at a distance of 3 √2 from the point P(1,2,3). OR Find the image of…2025
• An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of…2025
• If the direction cosines of a line are √3 k, √3 k, √3 k, then the value of k is:2024
• If α, β and γ are the angles which a line makes with positive directions of x, y and z axes respectively, then which of the…2024
• The coordinates of the foot of the perpendicular drawn from the point (0,1,2) on the x -axis are given by:2024
• The vector equation of a line passing through the point ( 1,-1,0 ) and parallel to Y -axis is:2024
• The Cartesian equation of a line passing through the point with position vector a= i- j and parallel to the line r= i+ k+μ(2 i-…2024
• The angle which the line x/1=y/-1=z/0 makes with the positive direction of Y -axis is:2024
• The lines (1-x)/2=(y-1)/3=z/1 and (2 x-3)/(2 p)=y/-1=(z-4)/7 are perpendicular to each other for p equal to:2024
• If a line makes an angle of (π)/4 with the positive directions of both x -axis and z -axis, then the angle which it makes with…2024
• Direction ratios of a vector parallel to line (x-1)/2=-y=(2 z+1)/6 are:2024
• The distance of point P(a, b, c) from y -axis is:2024
• The Cartesian equation of the line passing through the point (1,-3,2) and parallel to the line: r=(2+λ) i+λ j+(2 λ-1) k is2024
• If a line makes an angle of 30^° with the positive direction of x -axis, 120^° with the positive direction of y -axis, then the…2024
• Assertion: A line in space cannot be drawn perpendicular to x, y and z axes simultaneously. Reason ( R ): For any line making…2024
• Find the vector equation of the line passing through the point (2,3,-5) and making equal angles with the co-ordinate axes.2024
• Find the value of p for which the lines r=λ i+(2 λ+1) j+(3 λ+2) k and r= i-3 μ j+(p μ+7) k are perpendicular to each other and…2024
• Find the equation of a line l_2 which is the mirror image of the line l_1 with respect to line l: x/1=(y-1)/2=(z-2)/3, given that…2024
• Find the equation of the line which bisects the line segment joining points A(2,3,4) and B(4,5,8) and is perpendicular to the…2024
• Equations of sides of a parallelogram ABCD are as follows: AB: (x+1)/1=(y-2)/-2=(z-1)/2 BC: (x-1)/3=(y+2)/-5=(z-5)/3 CD:…2024
• Find the co-ordinates of the foot of the perpendicular drawn from the point (2,3,-8) to the line (4-x)/2=y/6=(1-z)/3. Also, find…2024
• The vertices of Δ ABC are A(1,1,0), B(1,2,1) and C(-2,2,-1). Find the equations of the medians through A and B. Use the equations…2024
• Find the distance between the line x/2=(2 y-6)/4=(1-z)/-1 and another line parallel to it passing through the point (4,0,-5). OR…2024
• The image of point P(x, y, z) with respect to line x/1=(y-1)/2=(z-2)/3 is P^′(1,0,7). Find the coordinates of point P.2024
• Find the equation of the line passing through the point of intersection of the lines x/1=(y-1)/2=(z-2)/3 and (x-1)/0=y/-3=(z-7)/2…2024
• The direction ratios of a line parallel to z -axis are:2023
• The equation of a line passing through point (2,-1,0) and parallel to the line x/1=(y-1)/2=(2-z)/2 is:2023
• The direction cosines of vector BA, where coordinates of A and B are (1,2,-1) and ( 3,4,0 ) respectively, are:2023
• Equation of line passing through origin and making 30^°, 60^° and 90^° with x, y, z axes respectively is2023
• If the point P(a, b, 0) lies on the line (x+1)/2=(y+2)/3=(z+3)/4, then (a, b) is:2023
• If a vector makes an angle of (π)/4 with the positive directions of both x -axis and y -axis, then the angle which it makes with…2023
• Direction cosines of the line (x-1)/2=(1-y)/3=(2 z-1)/12 are:2023
• Equation of a line passing through point (1,1,1) and parallel to z -axis is2023
• The value of λ for which the angle between the lines r= i+ j+ k+p(2 i+ j+2 k) and r=(1+q) i+(1+q λ) j+(1+q) k is (π)/2 is:2023
• Distance of the point (p, q, r) from y -axis is:2023
• If a line makes angles of 90^°, 135^° and 45^° with the x, y and z axes respectively, then its direction cosines are2023
• The angle between the lines 2 x=3 y=-z and 6 x=-y=-4 z is2023
• The point (x, y, 0) on the x y -plane divides the line segment joining the points (1,2,3) and (3,2,1) in the ratio:2023
• If the direction cosines of a line are (1/a, 1/a, 1/a), then:2023
• Equation of a line passing through point ( 1,2,3 ) and equally inclined to the coordinate axis, is2023
• Assertion: If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin^2 α+ sin^2 β+ sin^2 γ=2.…2023
• Assertion ( A ): The lines r= a_1+λ b_1 and r= a_2+μ b_2 are perpendicular, when b_1 · b_2=0. Reason (R): The angle θ between the…2023
• Assertion: A line through the points (4, 7, 8) and (2, 3, 4) is parallel to a line through the points (-1,-2,1) and (1,2,5).…2023
• Assertion: Equation of a line passing through the points ( 1,2,3 ) and (3,-1,3) is (x-3)/2=(y+1)/3=(z-3)/0 Reason (R): Equation…2023
• If the angle between the lines (x-5)/(α)=(y+2)/-5=(z+24/5)/(β) and x/1=y/0=z/1 is (π)/4, find the relation between α and β.2023
• Find the direction cosines of the line whose Cartesian equations are 5 x-3=15 y+7=3-10 z.2023
• If the equation of a line is x=ay+b, z=cy+d, then find the direction ratios of the line and a point on the line.2023
• Find the coordinates of points on line x/1=(y-1)/2=(z+1)/2 which are at a distance of √11 units from origin.2023
• Find the vector and the cartesian equations of a line that passes through the point A(1,2,-1) and parallel to the line 5…2023
• Find the angle between the following two lines: r=2 i-5 j+ k+λ(3 i+2 j+6 k); r=7 i-6 k+μ( i+2 j+2 k)2023
• Find the value of p, so that lines (x-1)/-2=(y-4)/(3 p)=(z-3)/4 and (x-2)/(4 p)=(y-5)/2=(1-z)/7 are perpendicular to each other.2023
• Find the vector equation of the line passing through the point (2,1,3) and perpendicular to both the lines…2023
• Find the coordinates of the foot of the perpendicular drawn from point (5,7,3) to the line (x-15)/3=(y-29)/8=(z-5)/-5. OR If a=…2023
• Find the coordinates of the foot of the perpendicular drawn from the point P(0,2,3) to the line (x+3)/5=(y-1)/2=(z+4)/3. OR Three…2023
• Find the distance between the lines: r=( i+2 j-4 k)+λ(2 i+3 j+6 k); r=(3 i+3 j-5 k)+μ(4 i+6 j+12 k)2023
• Find the image of the point (2,-1,5) in the line (x-11)/10=(y+2)/-4=(z+8)/-11; OR Vertices B and C of ABC lie on the line…2023
• Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4,2,-6), Q(5,-3,1), R(12,4,5) and S(11,9,-2).…2023
• Find the vector and the Cartesian equations of a line passing through the point ( 1,2,-4 ) and parallel to the line joining the…2023
• Find the value of b so that the lines (x-1)/2=(y-b)/3=(z-3)/4 and (x-4)/5=(y-1)/2=z are intersecting lines. Also, find the point…2023
• Show that the following lines do not intersect each other: (x-1)/3=(y+1)/2=(z-1)/5; (x+2)/4=(y-1)/3=(z+1)/-2 OR Find the angle…2023
• The Cartesian equation of a line A B is: (2 x-1)/12=(y+2)/2=(z-3)/3 Find the direction cosines of a line parallel to line AB.2022
• Find the vector equation of a line passing through a point with position vector 2 i- j+ k and parallel to the line joining the…2022
• The foot of a perpendicular drawn from the point (-2, -1, -3) on a plane is (1, - 3, 3). Find the equation of the plane.2022
• Find the values of λ, for which the distance of point (2,1, λ) from plane 3 x+5 y+ 4 z=11 is 2 √2 units.2022
• If a line makes an angle α, β, γ with the coordinate axes, then find the value of cos 2 α+ cos 2 β+ cos 2 γ.2022
• If the distance of the point (1,1,1) from the plane x-y+z+λ=0 is 5/(√3), find the value (s) of λ.2022
• Write the cartesian equation of the line PQ passing through points P(2,2,1) and Q(5,1,-2). Hence, find the y-coordinate of the…2022
• Find the shortest distance between the following lines: r=3 i+5 j+7 k+λ( i-2 j+ k) and r=(- i- j- k)+μ(7 i-6 j+ k).2022
• Find the equation of the plane passing through the line of intersection of the planes r ·( i+ j+ k)=10 and r ·(2 i+3 j- k)+4=0…2022
• Find the vector equation of the plane passing through the intersection of the planes r ·(2 i+2 j-3 k)=7 and r ·(2 i+5 j+3 k)=9…2022
• Find the co-ordinates of point where the line (x-3)/-1=(y+4)/1=(z+5)/6 crosses the plane passing through points (7/2,…2022
• Show that the lines: (1-x)/2=(y-3)/4=z/-1 and (x-4)/3=(2 y-2)/-4=z-1 are coplanar.2022
• Find the equation of the plane passing through points (2, 1, 0), (3, -2, -2) and (1, 1, -7). Also, obtain its distance from the…2022
• If a line makes 60° and 45° angles with the positive directions of x-axis and z-axis respectively, then find the angle that it…2022
• Find the co-ordinates of the point on the line (x+2)/3=(y+1)/2=(z-3)/2 which is at a distance of 5 units from the point (1, 3, 3).2022
• Find the distance of the point (2, 3, 4) measured along the line (x-4)/3=(y+5)/6=(z+1)/2 from the plane 3 x+2 y+2 z+5=0. 2 2 3 3…2022
• Find the equations of the planes passing through the line of intersection of the planes r ·( i+3 j)=6 and r ·(3 i- j-4 k)=0,…2022
• Find the distance of the point (-1, -5, -10) from the point of intersection of the line (x-2)/3=(y+1)/4=(z-2)/12 and the plane…2022
• Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the zx-plane. Case-Study Based Question2022
• Find the distance of the point (1, -2, 9) from the point of intersection of the line r=4 i+2 j+7 k+λ(3 i+4 j+2 k) and the plane r…2022
• Electrical transmission wires which are laid down in winters are stretched tightly to accommodate expansion in summers. Two such…2022
• Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines r=λ( i+2 j- k)…2022

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