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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.
Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.
- a)Both A and R are true and R is the correct explanation of A
- b)Both A and R are true but R is NOT the correct explanation of A
- c)A is true but R is false
- d)A is false and R is True
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Directions : In the following questions, A statement of Assertion (A)...
Let radius of cone be x and its height be h.
∴ OD = (h – r)
Volume of cone
∴ at h = 4r/3, Volume is maximum
Maximum volume
Hence both A and R are true.
R is not the correct explanation of A.
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Directions : In the following questions, A statement of Assertion (A)...
Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.
Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.
The correct answer is option B, i.e., both A and R are true, but R is not the correct explanation of A.
Explanation:
To understand the assertion and reason given in the question, let's first consider the properties of a cone inscribed in a sphere.
Properties of a cone inscribed in a sphere:
1. The base of the cone lies on the surface of the sphere.
2. The apex of the cone coincides with the center of the sphere.
3. The lateral surface of the cone is tangent to the sphere.
Now, let's analyze the assertion and reason:
Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.
To maximize the volume of the cone, we need to maximize its height. The height of the cone is equal to the altitude of the cone, which is the distance from the apex to the base.
Let's assume the height of the cone is h.
As per the properties of a cone inscribed in a sphere, the base of the cone is a circle with radius r, which is the radius of the sphere.
Therefore, the base of the cone has a circumference of 2πr.
The height of the cone divides the radius of the base into two parts: r - h and h.
Using the Pythagorean theorem, we can write the equation:
(r - h)^2 + h^2 = r^2
Simplifying the equation, we get:
r^2 - 2rh + 2h^2 = r^2
2h^2 - 2rh = 0
h(2h - 2r) = 0
h = 0 or h = r
Since the height cannot be zero, the maximum height of the cone is r.
Therefore, the altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is r.
Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.
The volume of a cone can be calculated using the formula:
Volume of cone = (1/3) * base area * height
For the cone inscribed in a sphere, the base area is given by:
Base area = πr^2
Substituting the values, we get:
Volume of cone = (1/3) * πr^2 * r = (π/3) * r^3
The volume of the sphere is given by:
Volume of sphere = (4/3) * πr^3
Therefore, the ratio of the volume of the cone to the volume of the sphere is:
(π/3) * r^3 / ((4/3) * πr^3) = 1/4
Hence, the maximum volume of the cone is 1/4 of the volume of the sphere, not 8/27.
Therefore, both assertion (A) and reason (R) are true, but reason (
Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.
The correct answer is option B, i.e., both A and R are true, but R is not the correct explanation of A.
Explanation:
To understand the assertion and reason given in the question, let's first consider the properties of a cone inscribed in a sphere.
Properties of a cone inscribed in a sphere:
1. The base of the cone lies on the surface of the sphere.
2. The apex of the cone coincides with the center of the sphere.
3. The lateral surface of the cone is tangent to the sphere.
Now, let's analyze the assertion and reason:
Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.
To maximize the volume of the cone, we need to maximize its height. The height of the cone is equal to the altitude of the cone, which is the distance from the apex to the base.
Let's assume the height of the cone is h.
As per the properties of a cone inscribed in a sphere, the base of the cone is a circle with radius r, which is the radius of the sphere.
Therefore, the base of the cone has a circumference of 2πr.
The height of the cone divides the radius of the base into two parts: r - h and h.
Using the Pythagorean theorem, we can write the equation:
(r - h)^2 + h^2 = r^2
Simplifying the equation, we get:
r^2 - 2rh + 2h^2 = r^2
2h^2 - 2rh = 0
h(2h - 2r) = 0
h = 0 or h = r
Since the height cannot be zero, the maximum height of the cone is r.
Therefore, the altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is r.
Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.
The volume of a cone can be calculated using the formula:
Volume of cone = (1/3) * base area * height
For the cone inscribed in a sphere, the base area is given by:
Base area = πr^2
Substituting the values, we get:
Volume of cone = (1/3) * πr^2 * r = (π/3) * r^3
The volume of the sphere is given by:
Volume of sphere = (4/3) * πr^3
Therefore, the ratio of the volume of the cone to the volume of the sphere is:
(π/3) * r^3 / ((4/3) * πr^3) = 1/4
Hence, the maximum volume of the cone is 1/4 of the volume of the sphere, not 8/27.
Therefore, both assertion (A) and reason (R) are true, but reason (
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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?
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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?.
Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?.
Solutions for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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ample number of questions to practice Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The altitude of the cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.Reason (R): The maximum volume of the cone is 8/27 of the volume of the sphere.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice JEE tests.
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