CBSE Class 12 Mathematics Three Dimensional Geometry MCQs Set 03

Question. A line makes angles α, β, γ and δ with the four diagonals of a cube, then cos²α + cos² β+ cos² y + cos²δ is equal to
(a) 1
(b) 4/3
(c) 3/4
(d) 4/5

Answer : B

Question. A line passes through the points ( 6, 7,-1) and (2,-3,1)) The direction cosines of the line so directed that the angle made by it with the positive direction of x-axis is acute, is
(a) 2/3,-2/3,-1/3
(b) −2/3,2/3.1/3
(c) 2/3,-2/3,1/3
(d) 2/3,2/3,1/3

Answer : A

Question. If the projection of a line segment on x,y and z-axes are 3,1 and √15 respectively then length of line segment is
(a) 5
(b) 4+ √15
(c) 5 +√2
(d) 6

Answer : A

Question. The shortest distance between the Z-axis and the line x + y + 2z – 3 = 0, 2x + 3y + 4z – 4 = 0 is
(a) 2
(b) 1/2
(c) 0
(d) 1

Answer : A

Question. Consider the following statements:
I. Equations ax + by + cz + d = 0, a’x + b’y + c’z + d’ = 0 represent a straight line.
II. Equation of the form x-α/l = y-β/m = z-γ/n represent a straight line passing through the point (α , β , γ) and having direction ratio proportional to l, m, n.
Which of the statements given above is/are correct ?
(a) Only I
(b) Only II
(c) Both I and II
(d) Neither I nor II

Answer : C

Question. The line x-1/1 = y-2/-2 = z-1/3 and the plane x + 2y + z = 6 meet at
(a) no point.
(b) only one point.
(c) infinitely many points.
(d) none of these.

Answer : C

Question. The shortest distance between the lines x = y + 2 = 6z – 6 and x + 1 = 2y = – 12z is
(a) 1/2
(b) 2
(c) 1
(d) 3/2

Answer : B

Question. A plane meets the coordinate axes in points A, B, C and the centroid of the triangle ABC is (α, β, ϒ) . The equation of the plane is
(a) x/α + y/β + z/ϒ = 3
(b) αx +βy + ϒz = 3αβϒ
(c) x/α + y/β + z/ϒ =1/2
(d) None of these

Answer : A

Question. The locus of a point, such that the sum of the squares of its distances from the planes x + y + z = 0, x – z =0 and x – 2y + z = 0 is 9, is
(a) x² + y² + z² = 3
(b) x² + y² + z² = 6
(c) x² + y² + z² = 9
(d) x² + y² + z² =12

Answer : C

Question. The sine of the angle between the straight line x-2/3 = y-3/4 = z-4/5 and the plane 2x – 2y + z = 5 is
(a) 10/6√5
(b) 4/5√2
(c) 2/√3/5
(d) √2/10

Answer : D

Question. The equation of two lines through the origin, which intersect the line x-3/2 = y-3/1 = z/1 at angles of π/3 each, are
(a) x/1 = y/2 = z/1 ; x/1 = y/1 = z/2
(b) x/1 = y/2 = z/-1 ; x/1 = y/1 = z/-2
(c) x/1 = y/2 = z/-1 ; x/1 = y/-1 = z/-2
(d) None of the above

Answer : B

Question. What is the length of the projection of 3î + 4ĵ+5kˆ on the xy-plane ?
(a) 3
(b) 5
(c) 7
(d) 9

Answer : B

Question. If a line makes angles 90°, 35°, 45° and with the positive x,y and z-axes respectively, then its direction cosines are
(a) 0-1/√2 and 1/2√
(b) 0,1/2 and -1/2
(c) 0,-1/2 and 1/√2
(d) 1,-1/2 and 1/√2

Answer : A

Question. If a line makes an angle of π/4 with each of y and z axes, then the angle which it makes with x-axis is
(a) π/2
(b) π/3
(c) π/4
(d) π/6

Answer : A

Question. The points P(4, 5, 10), Q(2, 3, 4) and R(1, 2, 1) − are three vertices of a parallelogram PQRS. The coordinates of S are
(a) (3, 4, 5)
(b) (4, 6, 5)
(c) (1, 1, 5)
(d) (1, 3, 5)

Answer : A

Question. If a line lies in the octant OXYZ and it makes equal angles with the axes, then
(a) l=m=n=1/√3
(b) l =m= n = ± 1/√3
(c) l =m=n=-1/√3
(d) l =m= n =± 1/√2

Answer : A

Question. If (1/2,1/3,n) are the direction cosines of a line, then the value of n is
(a) √23/6
(b) 23/6
(c) 2/3
(d) 3/2

Answer : A

Question. The direction cosines l m, and n of two lines are connected by the relations l+ m+ n = 0, lm = 0, then the angles between them is
(a) π /3
(b) π /4
(c) π /2
(d) 0

Answer : A

Question. Consider the following statements
Statement I: The angle between two planes x + 2y + 2z = 3 and –5 x + 3y + 4z = 9 is cos-1(19√2/30) .
Statement II: The angle between the line x-1/2 = y-2/1 = z+3/-2 and the plane x + y + 4 = 0 is 45°.
Choose the correct option.
(a) Statement I is true
(b) Statement II is true
(c) Both statements are true
(d) Both statements are false

Answer : B

Question. The lines x = ay + b, z = cy +d and x = a’y + b’, z = c’y +d’ are perpendicular if
(a) aa’ +bb’ + cc’ + 1 = 0
(b) aa’+bb’+1 = 0
(c) bb’+cc’+1 = 0
(d) aa’ + cc” +1 = 0

Answer : D

Question. Consider the following statements
Statement I : The angle between two planes is twice the angle between their normals.
Statement II : If θ is the angle between two planes, then 180 – θ is also the angle between same planes.
Choose the correct option.
(a) Statement I is true
(b) Statement II is true
(c) Both statements are true
(d) Both statements are false

Answer : B

Question. If a line makes an angle of π / 4 with the positive directions of each of x- axis and y- axis, then the angle that the line makes with the positive direction of the z-axis is
(a) π /4
(b) π /2
(c) π /6
(d) π /3

Answer : B

Question. If the angle θ between the line x+1/1 = y-1/2 = z-2/2 and sin θ = 1/3 then the value of λ is
(a) 5/3
(b) -3/5
(c) 3/4
(d) -4/3

Answer : A

Question. What is the angle between the line 6x = 4y = 3z and the plane 3x + 2y – 3z = 4 ?
(a) 0
(b) π/6
(c) π/3
(d) π/2

Answer : A

Question. A rectangular parallelopiped is formed by drawing planes through the points (–1, 2, 5) and (1, –1, –1) and parallel to the coordinate planes. The length of the diagonal of the parallelopiped is
(a) 2
(b) 3
(c) 6
(d) 7

Answer : D

Question. Four points (0, –1, –1) (–4, 4, 4) (4, 5, 1) and (3, 9, 4) are coplanar. Find the equation of the plane containing them.
(a) 5x + 7y + 11z – 4 = 0
(b) 5x – 7y + 11z + 4 = 0
(c) 5x – 7y –11z – 4 = 0
(d) 5x + 7y – 11z + 4 = 0

Answer : B

Question. The points A(1, 2, 3), B (–1, –2, –3) and C(2, 3, 2) are three vertices of a parallelogram ABCD. The equation of CD is
(a) x/1 = y/2 = z/2
(b) x+2/1 = y+3/2 = z-2/2
(c) x/2 = y/3 = z/2
(d) x-2/1 = y-3/2 = z-2/2

Answer : D

Question. The d.r. of normal to the plane through (1, 0, 0), (0, 1, 0) which makes an angle π/4 with plane x + y = 3 are
(a) 1, √2 ,1
(b) 1, 1, √2
(c) 1, 1, 2
(d) 2 , 1, 1

Answer : B

Question. The coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XY – plane are
(a) (13/5 , 23/5 , 0)
(b) (-13/5 , 23/5 , 0)
(c) (13/5 , -23/5 , 0)
(d) (-13/5 , -23/5 , 0)

Answer : A