CUET Mathematics MCQs Three dimensional Geometry

Question : The d.r. of normal to the plane through (1, 0, 0), (0, 1, 0) which makes an angle p /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 : If (2, 3, 5) is one end of a diameter of the sphere x² + y² + z² – 6x – 12y – 2z + 20 = 0, then the cooordinates of the other end of the diameter are 
(a) (4, 3, 5)
(b) (4, 3, – 3)
(c) (4, 9, – 3)
(d) (4, –3, 3).
Answer : C

Question : If a line makes angles 90°, 60° and θ with x, y and z axes respectively, where θ is acute then the value of θ is
a) 30⁰
b) 60⁰
c) 90⁰
d) 45⁰
Answer : A

Question : If the distance between planes, 4x – 2y – 4 + 1 = 0 and 4x – 2y – 4 + d = 0 is 7, then d is:
(a) 41 or – 42
(b) 42 or – 43
(c) – 41 or 43
(d) – 42 or 44
Answer : C

Question : Let Q be the foot of perpendicular from the origin to the plane 4x – 3y + z + 13 = 0 and R be a point (– 1, – 6) on the plane. Then length QR is :
(a) √14
(b) √19/2
(c) 3√7/2
(d) 3/√2
Answer : C

Question : The intersection of the spheres x² + y² + z² + 7x - 2y - z = 13 and x² + y² + z² - 3x + 3y + 4z = 8 is the same as the intersection of one of the sphere and the plane 
(a) 2x - y - z = 1
(b) x - 2y - z = 1
(c) x - y - 2z = 1
(d) x - y - z = 1
Answer : A

Question : A line with positive direction cosines passes through the point P (2, – 1, 2) and makes equal angles with the coordinate axes. If the line meets the plane 2x + y + z = 9 at point Q, then the length PQ equals 
(a) √2
(b) 2
(c) √3
(d) 1
Answer : C

Question : Statement -1 : The point A(3, 1, 6) is the mirror image of the point B(1, 3, 4) in the plane x – y + z = 5.
Statement -2: The plane x – y + z = 5 bisects the line segment oining A(3, 1, 6) and B(1, 3, 4). 
(a) Statement -1 is true, Statement -2 is true ; Statement - 2 is not a correct explanation for Statement -1.
(b) Statement -1 is true, Statement -2 is false.
(c) Statement -1 is false, Statement -2 is true .
(d) Statement - 1 is true, Statement 2 is true ; Statement -2 is a correct explanation for Statement -1.
Answer : A

Question : The line passing through the points (5, 1, a) and (3, b, 1) crosses the y -plane at the point (0, 17/2, –13/2) Then
(a) a = 2, b = 8
(b) a = 4, b = 6
(c) a = 6, b = 4
(d) a = 8, b = 2
Answer : C

Question : If the angle q between the line x + 1/1 = y - 1/2 = z - 2/2 and the plane 2x – y + √λ z + 4 = 0 is such that sin θ = 3/1 then the value of λ is
(a) 5/3
(b) -3/5
(c) 3/4
(d) -4/3
Answer : A

Question : The distance between the planes 2x + 2y – z + 2 = 0 and 4x + 4y – 2z + s = 0 is
a) 1/6
b) 1
c) 1/4
d) 1/2
Answer : A

Question :  If the lines x-1/2 = y+1/3 = z-1/4  and x-3/1 = y-k/2 = z/1  intersect at a point , then the value of k is
a) 9/2
b) 2/9
c) 2
d) 3/2
Answer : A

Question : The plane containing the line x - 1/1 = y - 2/2 = z - 3/3 and parallel to the line x/1 = y/1 = z/4 passes through the point:
(a) (1, – 2, 5)
(b) (1, 0, 5)
(c) (0, 3, –5)
(d) (– 1, – 3, 0)
Answer : B

Question : If the angle between the line 2(x + 1) = y = + 4 and the plane 2x – y + √λ + 4 = 0 is π/6, then the value of l is:
(a) 135/7
(b) 45/11
(c) 45/7
(d) 135/11
Answer : C

Question : Let the line x–2/3 = y – 1/-5 = z + 2/2 lie in the plane x + 3y – a + β = 0. Then (a, β) equals 
(a) (–6, 7)
(b) (5, –15)
(c) (–5, 5)
(d) (6, –17)
Answer : A

Question : The distance of the point (1, – 5, 9) from the plane x – y + z = 5 measured along a straight x = y = z is 
(a) 10√3
(b) 5√3
(c) 3√10
(d) 3√5
Answer : A

Question : If the three planes x = 5, 2x – 5ay + 3z – 2 = 0 and 3bx + y – 3z = 0 contain a common line, then(a, b) is equal to 
(a) (8/15, 1/5)
(b) (1/5, - 8/15)
(c) (- 8/15, 1/5)
(d) (- 1/5, 8/15)
Answer : B

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 a with the positive x-axis, then cos a equals 
(a) 1
(b) 1/√2
(c) 1/3
(d) 1/√2
Answer : C

Question : A plane which passes through the point (3, 2, 0) and the line x - 4/1 = y - 7/5 = z - 4/4 is
(a) x – y + z =1
(b) x + y + z = 5
(c) x + 2y – z = 1
(d) 2x – y + z = 5
Answer : A

Question : The shortest distance from the plane 12x + 4y +3z = 327 to the sphere 4 2 6 155 x² + y² + z² + x - y - z = is
(a) 39
(b) 26
(c) 11(13/4)
11 (d) 13.
Answer : D

Question : If the lines x-2/1 =y-2/1 =z-4/k and x-1/k = y-4/2 = z-5/1 are coplanar, then k can have
a) Exactly two values
b) Exactly three values
c) Exactly one value
d) Any value
Answer : A

Question : A vector n is inclined to x-axis at 45 , to y-axis at 60 and at an acute angle to z-axis. If n is a normal to a plane passing through the point (√2,-1,1) then the equation of the plane is :
(a) 4√2x + 7y + z - 2
(b) 2x + y + 2z = 2√2 +1
(c) 3√2x - 4y -3z = 7
(d) √2x - y - z = 2
Answer : B

Question : The equation of a plane containing the line x + 1 /-3 = y - 3/2 = z + 2 /1 and the point (0, 7, – 7) is
(a) x + y + z = 0 
(b) x + 2y + z = 21
(c) 3x – 2y + 5z + 35 = 0
(d) 3x + 2y + 5z + 21 = 0
Answer : A

Question : Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
(a) 3/2
(b) 5/2
(c) 7/2
(d) 9/2
Answer : C

Question : The image of the point (–1, 3, 4) in the plane x - 2y = 0 is
(a) (- 17/3, - 19/3, 4)
(b) (15,11, 4)
(c) (- 17/3, - 19/3, 1)
(d) None of these
Answer : D

Question : If a line makes angles α, β, γ with the axes then cos2α + cos2β+cos2γ is equal to
a) - 1
b) 1
c) 2
d) - 2
Answer : A

Question : A equation of a plane parallel to the plane x – 2y + 2z –5 = 0 and at a unit distance from the origin is :
(a) x – 2y + 2z – 3 = 0
(b) x – 2y + 2z + 1 = 0
(c) x – 2y + 2z – 1 = 0
(d) x – 2y + 2z + 5 = 0
Answer : A

Question : Distance between two parallel planes 2x + y + 2 = 8 and 4x + 2y +4 + 5 = 0 is 
(a) 2/9
(b) 2/5
(c) 2/7
(d) 2/3
Answer : C

Question : Equation of the plane which passes through the point of intersection of lines x - 1/3 = y - 2/1 = z - 3/2 and has the largest distance from the origin is:
(a) 7x + 2y + 4 = 54
(b) 3x + 4y + 5 = 49
(c) 4x + 3y + 5 = 50
(d) 5x + 4y + 3 = 57
Answer : C

Question : If the plane 2ax – 3ay + 4a + 6 = 0 passes through the midpoint of the line oining the centres of the spheres x² + y² + z² + 6x - 8y - 2z = 13 and x² + y² + z² -10x + 4y - 2z = 8 then a equals
(a) – 1
(b) 1
(c) – 2
(d) 2
Answer : C

Question : The acute angle between the planes 2x – y+z = 6 and x+y+2z = 3 is
a) 60⁰
b) 30⁰
c) 75⁰
d) 45⁰
Answer : A

Question : The direction cosines of the normal to the plane x + 2y – 3z – 4 = 0 are
a) 1/√14, 2/√14, 3/√14,
b) -1/√14, 2/√14, 3/√14,
c) -1/√14, -2/√14, 3/√14,
d) -1/√14, -2/√14, -3/√14,
Answer : A

Question : The plane x + 2y – z = 4 cuts the sphere x² + y² + z² – x + z – 2 = 0 in a circle of radius
(a) 3
(b) 1
(c) 2
(d) √2
Answer : B

Question : The values of a for which the two points (1, a, 1) and (– 3, 0, a) lie on the opposite sides of the plane 3x + 4y – 12z + 13 = 0, satisfy 
(a) 0 < a < 1/3
(b) – 1 < a < 0
(c) a < – 1 or a < 1/3
(d) a = 0
Answer : D

Question : The equation of a plane through the line of intersection of the planes x + 2y = 3, y –2 + 1= 0, and perpendicular to the first plane is :
(a) 2x – y – 10z = 9
(b) 2x – y + 7z = 11
(c) 2x – y + 10z = 11
(d) 2x – y – 9z = 10
Answer : C

Question : The radius of the circle in which the sphere x² + y² + z² + 2x - 2y - 4z - 19 = 0 is cut by the plane x + 2y + 2z + 7 = 0 is 
(a) 4
(b) 1
(c) 2
(d) 3
Answer : D

Question : Consider the following planes
P : x + y – 2z + 7 = 0
Q : x + y + 2z + 2 = 0
R : 3x + 3y – 6z – 11 = 0 
(a) P and R are perpendicular
(b) Q and R are perpendicular
(c) P and Q are parallel
(d) P and R are parallel
Answer : D

Question : The equation of the plane which cuts equal intercepts of unit length on the coordinate axes is
a) x + y + z = 1
b) x + y + z = 0
c) x + y - z = 1
d) x + y + z = 2
Answer : A

Question : If the angle between the line x = y - 1/2 = z - 3/ λ and the plane x + 2y + 3z = 4 is cos^–1 (√5/14) then λ equals
(a) 3/2
(b) 2/5
(c) 5/3
(d) 2/3
Answer : D