CBSE Class 12 Mathematics Three Dimensional Geometry MCQs Set 02

Question. The three planes x + y = 0, y + z = 0 and x + z = 0
(a) meet in a unique point
(b) meet in a line
(c) meet taken two at a time in parallel lines
(d) None of these

Answer : A

Question. The points (2, 3, 4), (−1, −2, 1) and (5, 8, 7) are
(a) formed an isosceles triangle
(b) formed an equilateral triangle
(c) collinear
(d) None of the above

Answer : C

Question. The direction cosines of the side AB of the ∆ ABC whose vertices are A( 3,5,-4), B (-1,1,2) ) and C (-5,-5,-2) are///3
(a) − 4/√17,-4/√17,6/√17
(b) −/√17,-2/√17,3/√17
(c) − 2/√17,2/√17,-3/√17
(d) None of these

Answer : B

Question. A line makes acute angles of α, β and γ with the coordinate axes such that cos α cos β = cos β=cos y=2/9 and cos y cos α =4/9, then cos α+ cos β+cos y is equal to
(a) 25/9
(b) 5/9
(c) 5/3
(d) 2/3

Answer : C

Question. Lines OA and OB are drawn from O with direction cosines proportional to (1,2, -1) and ( , ,3,-2,3), respectively. The direction ratios of the normal to the pane AOB are
(a) (4, 3, 2)
(b) (4,-3,-2)
(c) (-4,3,-2)
(d) (4,3,-2)

Answer : D

Question. If O is the origin and OP = 3 with direction ratios – 1, 2, – 2, then coordinates of P are
(a) (1, 2, 2)
(b) (– 1, 2, – 2)
(c) (– 3, 6, – 9)
(d) (-1/3, 2/3, 2/3)

Answer : B

Question. The equation of the plane passing through three noncollinear points with position vectors a̅, b̅, c̅ is

Answer : B

Question. The vector equation of the symmetrical form of equation of straight line x-5/3 = y+4/7 = z-6/2 is

Answer : D

Question. The distance between the lines given by

(a) √59/14
(b) √59/7
(c) √118/7
(d) √59/7

Answer : B

Question. If O be the origin and the coordinates of P be (1, 2, –3), then the equation of the plane passing through P and perpendicular to OP is
(a) x + 2y + 3z = –5
(b) x + 2y + 3z = – 14
(c) x + 2y – 3z =14
(d) x + 2y – 3z =5

Answer : C

Question. The lines whose vector equations are

are perpendicular for all values of λ and μ if p =
(a) 1
(b) –1
(c) – 6
(d) 6

Answer : D

Question. A variable plane remains at constant distance p from the origin.If it meets coordinate axes at points A, B, C then the locus of the centroid of ΔABC is
(a) x^-2 + y^-2 + z^-2 = 9p^-2
(b) x-3 + y^-3 + z^-3 =9p^-3
(c) x² + y2 + z² = 9p²
(d) x³ + y³ + z³ = 9p³

Answer : A

Question. Two lines r̅ = a̅₁ + λb̅₁ and r̅ = a̅₂ + μb̅₂ are said to be coplanar, if
(a) (a₂ – a₁) . (b₁ × b₂) = 0
(b) (x₁, y₁, z₁)
(c) Both (a) and (b)
(d) None of the above

Answer : A

Question. The lines in a space which are neither intersecting nor parallel, are called
(a) concurrent lines
(b) intersecting lines
(c) skew lines
(d) parallel lines

Answer : C

Question. If a line makes angles α β γ , and with the positive directions of the coordinate axes, then the value of sin²α+sin² β+ sin² y is
(a) 0
(b) 1
(c) 2
(d) −1

Answer : C

Question. If a line makes angles α, β γ and with the axes, then cos2α+ cos2β+ cos2y is equal to
(a) −2
(b) −1
(c) 1
(d) 2

Answer : B

Question. O is the origin. OP makes an angle of 45° and 60° with the positive direction of x and y-axes respectively. OP = 12 units. Then, the coordinates of P are
(a) (6, 6, 6)
(b) (6, 6, −6)
(c) (6 √2, 6, ± 6)
(d) (6, ± 6, 6)

Answer : C

Question. The points (5,2,4), (6,-1,2) ) and ( 8,-7,k) are collinear, if k is equal to
(a) − 2
(b) 2
(c) 3
(d) − 1

Answer : A

Question. The vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector

Answer : A

Question. The angle θ between two planes A₁x + B₁y + C₁z + D₁ =0 and A₂x + B₂y + C₂z + D₂ =0 is given by cos θ equal to
(a) A₁A₂ + B₁B₂ + C₁C₂

Answer : C

Question. If r = a1 + λb1 and r = a2 + μb2 are the equations of two lines, then cos θ =

Answer : B

Question.

Codes
      A     B    C
(a) 1     3     2
(b) 2     3     1
(c) 3     1     2
(d) 2     1     3

Answer : C

Question. The distance of the point (–5, –5, –10) from the point of intersection of the line r . = 2î – ĵ + 2k̂ + λ (3î + 4ĵ+ 2k̂ )and the plane r· (î – ĵ + k̂)= 5 is
(a) 13
(b) 12
(c) 4 √15
(d) 10 √2

Answer : A

Question. Which of the following is/are true?
I. The vector equation of the line passing through the point (1, 2, –4) and perpendicular to the two lines x-8/3 = y+19/-16 = z-10/7 and x-15/3 = y-29/8 = z-5/-5 is r = (î + 2ĵ – 4kˆ ) + λ (2î + 3ĵ + 6kˆ )
II. If a plane has intercepts a, b, c and is at a distance of p units from the origin, then p2 = a2 + b2 + c2.
(a) Only I is true
(b) Only II are true
(c) Both I and III are true
(d) Neither I nor II is true

Answer : A

Question. Let L be the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2z = 2. If L makes an angle α with the positive x-axis, then cos α equals
(a) 1
(b) 1/√2
(c) 1/√3
(d) 1/2

Answer : C

Question. The vector equation of the plane which is at a distance of 6/√29 from the origin and its normal vector from the origin is 2î – 3ĵ + k̂, is

(c) Both a and b
(d) None of the above

Answer : B

Question. The distance of a point (2, 5, –3) from the plane r . (6î – 3ĵ + 2k̂) = 4 is
(a) 13
(b) 13/7
(c) 13/5
(d) 37/7

Answer : B

Question. What is the condition for the plane ax + by + cz + d = 0 to be perpendicular to xy-plane ?
(a) a = 0
(b) b = 0
(c) c = 0
(d) a + b +c = 0

Answer : C

Question. Distance between the parallel planes 2x – y + 3z + 4 = 0 and 6x – 3y + 9z – 3 = 0 is:
(a) 5/√3
(b) 4/√6
(c) 5/√14
(d) 3/2√3

Answer : C