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Test: Three Dimensional Geometry- Case Based Type Questions- 2 - JEE MCQ


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15 Questions MCQ Test Mathematics (Maths) Class 12 - Test: Three Dimensional Geometry- Case Based Type Questions- 2

Test: Three Dimensional Geometry- Case Based Type Questions- 2 for JEE 2024 is part of Mathematics (Maths) Class 12 preparation. The Test: Three Dimensional Geometry- Case Based Type Questions- 2 questions and answers have been prepared according to the JEE exam syllabus.The Test: Three Dimensional Geometry- Case Based Type Questions- 2 MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Three Dimensional Geometry- Case Based Type Questions- 2 below.
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Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 1
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Direction: Read the following text and answer the following questions on the basis of the same:

P1 : x + 3y – z = 0 and P2 : y + 2z = 0 are two intersecting planes. P3 is a plane passing through the point (2, 1, –1) and through the line of intersection of P1 and P2.

Q. ______ is a point on P3.

Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 1
The equation of P3 is x + 9y + 11z = 0.

∴ (–6, –3, 3) is a point on P3.

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 2
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Direction: Read the following text and answer the following questions on the basis of the same:

P1 : x + 3y – z = 0 and P2 : y + 2z = 0 are two intersecting planes. P3 is a plane passing through the point (2, 1, –1) and through the line of intersection of P1 and P2.

Q. Equation of P3 is _______.

Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 2
Let the equation of P3 be P1 + λP2 = 0

i.e., (x + 3y – z) + λ(y + 2z) = 0

P3 passes through the point (2, 1, –1)

∴ (2 + 3 + 1) + λ(1 – 2) = 0

⇒ λ = 6

∴ The equation of P3 is x + 9y + 11z = 0

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Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 3
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Direction: Read the following text and answer the following questions on the basis of the same:

P1 : x + 3y – z = 0 and P2 : y + 2z = 0 are two intersecting planes. P3 is a plane passing through the point (2, 1, –1) and through the line of intersection of P1 and P2.

Q. The angle between P1 and P2 is _______.

Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 3
M

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 4
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Direction: Read the following text and answer the following questions on the basis of the same:

P1 : x + 3y – z = 0 and P2 : y + 2z = 0 are two intersecting planes. P3 is a plane passing through the point (2, 1, –1) and through the line of intersection of P1 and P2.

Q. Equation of plane parallel to P3 and passing through (1, 2, 3) is _______.
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 4
The required equation is (x – 1) +9(y – 2) + 11(z – 3) = 0

i.e., x + 9y + 11z – 52 = 0

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 5
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Direction: Read the following text and answer the following questions on the basis of the same:

P1 : x + 3y – z = 0 and P2 : y + 2z = 0 are two intersecting planes. P3 is a plane passing through the point (2, 1, –1) and through the line of intersection of P1 and P2.

Q. Distance of P3 from origin is _______ units.
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 5
(0, 0, 0) is a point on P3.

∴ Distance of P3 from origin = 0

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 6
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Direction: Read the following text and answer the following questions on the basis of the same:

A mobile tower stands at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with 3 cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at the point (2, 3, 1) as shown in the figure.

Q. The coordinates of the foot of the perpendicular drawn from the top of the tower to the ground are
Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 7
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Direction: Read the following text and answer the following questions on the basis of the same:

A mobile tower stands at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with 3 cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at the point (2, 3, 1) as shown in the figure.

Q. The height of the tower from the ground is
Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 8
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Direction: Read the following text and answer the following questions on the basis of the same:

A mobile tower stands at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with 3 cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at the point (2, 3, 1) as shown in the figure.

Q. The equation of the plane passing through the points A, B and C is
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 8
A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1)

(x - 1)(1 + 2) -y (-2, 0) + (z - 2) (4 - 0)

3x - 3 + 2y + 4z - 8 = 0

3x + 2y + 4z = 11

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 9
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Direction: Read the following text and answer the following questions on the basis of the same:

A mobile tower stands at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with 3 cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at the point (2, 3, 1) as shown in the figure.

Q. The equation of the perpendicular line drawn from the top of the tower to the ground is
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 9
a = 3, b = 2, c = 4 and

(x1, y1, z1) ≡ (2, 3, 1)

∴ (x-2)/3 = (y-3)/2

= (z-1)/4

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 10
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Direction: Read the following text and answer the following questions on the basis of the same:

A mobile tower stands at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with 3 cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at the point (2, 3, 1) as shown in the figure.

Q. The area of ΔABC is
Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 11
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Direction: Read the following text and answer the following questions on the basis of the same:

Consider the plane π1 : 2x – 3y + 4z + 9 = 0 and the point P(1, –2, 3). π1 is a plane parallel to π1 and containing the point P.

Q. The co-ordinates of A are _______.
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 11
Generally A = (2λ + 1, –3λ – 2, 4λ + 3).

A is a point on π1.

⇒ 2(2λ +1) –3 (–3λ – 2) + 4 (4λ + 3) + 9 = 0

⇒ 29λ + 29 = 0

⇒ λ = –1

∴ A = (–1, 1, –1)

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 12
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Direction: Read the following text and answer the following questions on the basis of the same:

Consider the plane π1 : 2x – 3y + 4z + 9 = 0 and the point P(1, –2, 3). π1 is a plane parallel to π1 and containing the point P.

Q. Distance between π1 and π2 is _______ units.
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 12
Distance between π1 and π2

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 13
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Direction: Read the following text and answer the following questions on the basis of the same:

Consider the plane π1 : 2x – 3y + 4z + 9 = 0 and the point P(1, –2, 3). π1 is a plane parallel to π1 and containing the point P.

Q. Equation of π2 is _______.
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 13
Equation of π2 is 2(x – 1) –3(y + 2) +4(z – 3) = 0

i.e., 2x – 3y + 4z – 20 = 0

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 14
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Direction: Read the following text and answer the following questions on the basis of the same:

Consider the plane π1 : 2x – 3y + 4z + 9 = 0 and the point P(1, –2, 3). π1 is a plane parallel to π1 and containing the point P.

Q. A is the foot of perpendicular from π to π1. Equation of PA is _______.
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 14
Here

Equation of PA is

Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 15
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Direction: Read the following text and answer the following questions on the basis of the same:

Consider the plane π1 : 2x – 3y + 4z + 9 = 0 and the point P(1, –2, 3). π1 is a plane parallel to π1 and containing the point P.

Q. The image of π on π1 is _______.
Detailed Solution for Test: Three Dimensional Geometry- Case Based Type Questions- 2 - Question 15
Let P′(α, β, γ) be the image A is mid-point of PP′.

⇒ α = –3, β = 4, γ = –5

∴ P′ = (–3, 4, –5)

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