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MCQ Class 11 Mathematics Introduction to Three Dimensional Geometry with Answers - Set - 5
Question 1:
The coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ plane is
(a) (0, 17/2, 13/2)
(b) (0, -17/2, -13/2)
(c) (0, 17/2, -13/2)
(d) None of these
Correct Answer – (C)
The line passing through the points (5, 1, 6) and (3, 4, 1) is given as
(x – 5)/(3 – 5) = (y – 1)/(4 – 1) = (z – 6)/(1 – 6)
⇒ (x – 5)/(-2) = (y – 1)/3 = (z – 6)/(-5) = k(say)
⇒ (x – 5)/(-2) = k
⇒ x – 5 = -2k
⇒ x = 5 – 2k
(y – 1)/3 = k
⇒ y – 1 = 3k
⇒ y = 3k + 1
and (z – 6)/(-5) = k
⇒ z – 6 = -5k
⇒ z = 6 – 5k
Now, any point on the line is of the form (5 – 2k, 3k + 1, 6 – 5k)
The equation of YZ-plane is x = 0
Since the line passes through YZ-plane
So, 5 – 2k = 0
⇒ k = 5/2
Now, 3k + 1 = 3 × 5/2 + 1 = 15/2 + 1 = 17/2
and 6 – 5k = 6 – 5 × 5/2 = 6 – 25/2 = -13/2
Hence, the required point is (0, 17/2, -13/2)
Question 2 :
The equation of the plane containing the line 2x – 5y + z = 3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z = 1 is
(a) x + 3y + 6z + 7 = 0
(b) x + 3y – 6z – 7 = 0
(c) x – 3y + 6z – 7 = 0
(d) x + 3y + 6z – 7 = 0
Correct Answer – (D)
Let the equation of the plane is
(2x – 5y + z – 3) + λ(x + y + 4z – 5) = 0
⇒ (2 + λ)x + (λ – 5)y + (4λ + 1)z – (3 + 5λ) = 0
Since the plane is parallel to x + 3y + 6z – 1 = 0
⇒ (2 + λ)/1 = (λ – 5)/3 = (1 + 4λ)/6
⇒ 6 + 3λ = λ – 5
⇒ 2λ = -11
⇒ λ = -11/2
Again,
6λ – 30 = 3 + 12λ
⇒ -6λ = -33
⇒ λ = -33/6
⇒ λ = -11/2
So, the required equation of plane is
(2x – 5y + z – 3) + (-11/2) × (x + y + 4z – 5) = 0
⇒ 2(2x – 5y + z – 3) + (-11) × (x + y + 4z – 5) = 0
⇒ 4x – 10y + 2z – 6 – 11x – 11y – 44z + 55 = 0
⇒ -7x – 21y – 42z + 49 = 0
⇒ x + 3y + 6z – 7 = 0
Question 3 :
For every point P(x, y, z) on the xy-plane
(a) x = 0
(b) y = 0
(c) z = 0
(d) None of these
Correct Answer – (C)
Question 4 :
The projections of a directed line segment on the coordinate axes are 12, 4, 3. The DCS of the line are
(a) 12/13, -4/13, 3/13
(b) -12/13, -4/13, 3/13
(c) 12/13, 4/13, 3/13
(d) None of these
Correct Answer – (C)
Let AB be the given line and the DCs of AB be l, m, n. Then
Projection on x-axis = AB . l = 12 (Given)
Projection on y-axis = AB . m = 4 (Given)
Projection on z-axis = AB . n = 3 (Given)
⇒ (AB²) (l² + m² + n²) = 144 + 16 + 9
⇒ (AB²) = 169 {since l² + m² + n² = 1}
⇒ AB = 13
Hence, DCs of AB are 12/13, 4/13, 3/13
Question 5 :
What is the ratio in which the line joining the points (2,4, 5) and (3, 5, – 4) is internally divided by the xy-plane?
(a) 5 : 4
(b) 3 : 4
(c) 1 : 2
(d) 7 : 5
Correct Answer – (A)
MCQ Class 11 Mathematics Introduction to Three Dimensional Geometry with Answers
Question 6 :
The coordinate of foot of perpendicular drawn from the point A(1, 0, 3) to the join of the point B(4, 7, 1) and C(3, 5, 3) are
(a) (5/3, 7/3, 17/3)
(b) (5, 7, 17)
(c) (5/3, -7/3, 17/3)
(d) (5/7, -7/3, -17/3)
Correct Answer – (A)
Let D be the foot of perpendicular and let it divide BC in the ration m : 1
Then the coordinates of D are {(3m + 4)/(m + 1), (5m + 7)/(m + 1), (3m + 1)/(m + 1)}
Now, AD ⊥ BC
⇒ AD . BC = 0
⇒ -(2m + 3) – 2(5m + 7) – 4 = 0
⇒ m = -7/4
So, the coordinate of D are (5/3, 7/3, 17/3)
Question 7 :
The locus of a point P(x, y, z) which moves in such a way that x = a and y = b, is a
(a) Plane parallel to xy-plane
(b) Line parallel to x-axis
(c) Line parallel to y-axis
(d) Line parallel to z-axis
Correct Answer – (B)
Question 8 :
The angle between the planes r . n1 = d1 and r . n1 = d2 is
(a) cos θ ={|n1| × |n2|}/ (n1. n2)
(b) cos θ = (n1 . n2)/{|n1| × |n2|}²
(c) cos θ = (n1 . n2)/{|n1| × |n2|}
(d) cos θ = (n1 . n2)² /{|n1| × |n2|}
Correct Answer – (C)
The angle between the planes r . n1 = d1 and r . n2 = d2 is defined as
cos θ = (n1 . n2)/{|n1| × |n2|}
Question 9 :
L is the foot of the perpendicular drawn from a point P(6, 7, 8) on the xy-plane. What are the coordinates of point L?
(a) (6, 0, 0)
(b) (6, 7, 0)
(c) (6, 0, 8)
(d) none of these
Correct Answer – (A)
Question 10 :
In three dimensional space the path of a point whose distance from the x-axis is 3 times its distance from the yz-plane is:
(a) y2 + z2 = 9x2
(b) x2 + y2 =3z2
(c) x2 + z2 = 3y2
(d) y2 – z2 = 9x2
Correct Answer – (A)
- NCERT Solutions Class 11 Mathematics Introduction to Three Dimensional Geometry with Answers : Exercise 12.1
- NCERT Solutions Class 11 Mathematics Introduction to Three Dimensional Geometry with Answers : Exercise 12.2
- NCERT Solutions Class 11 Mathematics Introduction to Three Dimensional Geometry with Answers : Exercise 12.3
- NCERT Solutions Class 11 Mathematics Introduction to Three Dimensional Geometry with Answers : Exercise 12 Misc
- NCERT Solutions Class 11 Mathematics Textbook download
