Conic 3D MCQ Quiz - Objective Question with Answer for Conic 3D - Download Free PDF
Last updated on May 17, 2025
Latest Conic 3D MCQ Objective Questions
Conic 3D Question 1:
Comprehension:
Direction: Consider the following for the items that follow:
Let 2x2 + 2y2 + 2z2 + 3x + 3y + 3z - 6 = 0 be a sphere.
The centre of the sphere lies on the plane
Answer (Detailed Solution Below)
Conic 3D Question 1 Detailed Solution
Explanation:
The Centre of the sphere is C \((\frac{-3}{4}, \frac{-3}{4}, \frac{-3}{4})\)
4x + 8y + 8z + 15 = 0,
Plane satisfy the centre of sphere – 3 –6 – 6 + 15 = 0
⇒ 0 = 0
∴ Option (d) is correct
Conic 3D Question 2:
Comprehension:
Direction: Consider the following for the items that follow:
Let 2x2 + 2y2 + 2z2 + 3x + 3y + 3z - 6 = 0 be a sphere.
What is the diameter of the sphere?
Answer (Detailed Solution Below)
Conic 3D Question 2 Detailed Solution
Explanation:
Given:
Sphere: 2x2 + 2y2 + 2z2 + 3x + 3y + 3z – 6 = 0
⇒ \(x^2 + y^2+z^2 + \frac{3}{2}x +\frac{3}{2}y + \frac{3}{2}z\) -3 = 0
Now Radius = \(\sqrt{(\frac{3}{4})^2+ (\frac{3}{4})^2+ (\frac{3}{4})^2 + 3}\)
= \(\sqrt{\frac{27}{16} +3} = 5 \frac{\sqrt3}{4}\)
Diamter = \(5 \frac{\sqrt3}{2}\)
∴ Option (b) is correct
Conic 3D Question 3:
Reciprocal cone of cone x2 + 2y2 + 3z2 = 0 is
Answer (Detailed Solution Below)
Conic 3D Question 3 Detailed Solution
Explanation:
We know that the reciprocal cone of cone ax2 + by2 + cz2 = 0 is \(\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}=0\)
In the cone x2 + 2y2 + 3z2 = 0
a = 1, b = 2, c = 3.
Hence reciprocal cone is
\(\frac{x^2}{1}+\frac{y^2}{2}+\frac{z^2}3=0\)
i.e., 6x2 + 3y2 + 2z2 = 0
Option (1) is true.
Conic 3D Question 4:
Equation λ1S1 + λ2S2 = 0, does not represent an equation of sphere, that passes through the intersection of two spheres S1 = 0, S2 = 0, if-
Answer (Detailed Solution Below)
Conic 3D Question 4 Detailed Solution
Explanation:
λ1S1 + λ2S2 = 0
⇒ λ1(x2 + y2 + z2 + ...) + λ2(x2 + y2 + z2 + ...) = 0
If λ1 = -λ2 then the above equation does not represent an sphere.
Hence option (3) is true.
Conic 3D Question 5:
A cone is ____
Answer (Detailed Solution Below)
Conic 3D Question 5 Detailed Solution
Concept:
A cone is a three-dimensional geometric shape that tapers smoothly from a flat
base (usually circular) to a point called the apex or vertex. It has one flat
surface and one curved surface.
Explanation:
Option 1: A plane figure is a flat, two-dimensional shape like a circle, triangle,
or square. Since a cone is three-dimensional, it cannot be classified as a plane
figure.
Option 2: A curve is a smoothly flowing, continuous line without sharp angles.
Although a cone has a curved surface, it is not solely a curve.
Option 3: Solid Figure: A solid figure is a three-dimensional object. A cone fits
this definition as it has volume and occupies space.
Option 4: A line is a straight one-dimensional figure having no thickness and
extending infinitely in both directions. A cone is not a line as it is three-dimensional.
The correct option is option 3.
Top Conic 3D MCQ Objective Questions
A lead hemisphere of radius 7 cm is molded into a cone whose base diameter is 14 cm. Find the height of the cone.
Answer (Detailed Solution Below)
Conic 3D Question 6 Detailed Solution
Download Solution PDFGiven:
Radius of hemisphere = 7 cm
Diameter of cone = 14 cm
Formula Used:
Volume of hemisphere = 2/3 × πR3
Volume of cone = 1/3 × πr2h
Calculation:
⇒ 2/3 × πR3 = 1/3 × πr2h
⇒ 2/3 × π73 = 1/3 × π72h
⇒ 2 × 73 = 1 × 72h
⇒ h = 14
The answer is 14 cm.
Comprehension:
Direction: Consider the following for the items that follow:
Let 2x2 + 2y2 + 2z2 + 3x + 3y + 3z - 6 = 0 be a sphere.
What is the diameter of the sphere?
Answer (Detailed Solution Below)
Conic 3D Question 7 Detailed Solution
Download Solution PDFExplanation:
Given:
Sphere: 2x2 + 2y2 + 2z2 + 3x + 3y + 3z – 6 = 0
⇒ \(x^2 + y^2+z^2 + \frac{3}{2}x +\frac{3}{2}y + \frac{3}{2}z\) -3 = 0
Now Radius = \(\sqrt{(\frac{3}{4})^2+ (\frac{3}{4})^2+ (\frac{3}{4})^2 + 3}\)
= \(\sqrt{\frac{27}{16} +3} = 5 \frac{\sqrt3}{4}\)
Diamter = \(5 \frac{\sqrt3}{2}\)
∴ Option (b) is correct
What is the diameter of the sphere?
x2 + y2 + z2 − 16x + 12y − 2√dz + d = 0
Answer (Detailed Solution Below)
Conic 3D Question 8 Detailed Solution
Download Solution PDFGiven:
The equation of the sphere is given.
Concept Used:
Comparing the given equation of sphere to the standard form of equation of spheres.
Solution:
We have, standard equation of sphere as: (x - a)2 + (y - b)2 + (z - c)2 = r2
⇒ x2 + y2 + z2 - 2ax - 2by - 2cz + a2 + b2 + c2 - r2 = 0
⇒ comparing the coefficient of x, y, z and constant.
we have, - 2a = - 16 {coefficient of x}
⇒ a = 8
we have, - 2b = 12 {coefficient of y}
⇒ b = - 6
we have, - 2c = -2\( \sqrt{d}\)
⇒ c = \( \sqrt{d}\)
we have, a2 + b2 + c2 - r2 = d
⇒ (8)2 + (- 6)2 + (\( \sqrt{d}\))2 - r2 = d
⇒ 64 + 36 + d - r2 = d
⇒ r2 =100
⇒ r = 10
we know that diameter of sphere is 2 times radius of sphere.
D = 2r
⇒ D = 20
\(\therefore\)Option 2 is correct.
Comprehension:
Direction: Consider the following for the items that follow:
Let 2x2 + 2y2 + 2z2 + 3x + 3y + 3z - 6 = 0 be a sphere.
The centre of the sphere lies on the plane
Answer (Detailed Solution Below)
Conic 3D Question 9 Detailed Solution
Download Solution PDFExplanation:
The Centre of the sphere is C \((\frac{-3}{4}, \frac{-3}{4}, \frac{-3}{4})\)
4x + 8y + 8z + 15 = 0,
Plane satisfy the centre of sphere – 3 –6 – 6 + 15 = 0
⇒ 0 = 0
∴ Option (d) is correct
Conic 3D Question 10:
A lead hemisphere of radius 7 cm is molded into a cone whose base diameter is 14 cm. Find the height of the cone.
Answer (Detailed Solution Below)
Conic 3D Question 10 Detailed Solution
Given:
Radius of hemisphere = 7 cm
Diameter of cone = 14 cm
Formula Used:
Volume of hemisphere = 2/3 × πR3
Volume of cone = 1/3 × πr2h
Calculation:
⇒ 2/3 × πR3 = 1/3 × πr2h
⇒ 2/3 × π73 = 1/3 × π72h
⇒ 2 × 73 = 1 × 72h
⇒ h = 14
The answer is 14 cm.
Conic 3D Question 11:
A cone is ____
Answer (Detailed Solution Below)
Conic 3D Question 11 Detailed Solution
Concept:
A cone is a three-dimensional geometric shape that tapers smoothly from a flat
base (usually circular) to a point called the apex or vertex. It has one flat
surface and one curved surface.
Explanation:
Option 1: A plane figure is a flat, two-dimensional shape like a circle, triangle,
or square. Since a cone is three-dimensional, it cannot be classified as a plane
figure.
Option 2: A curve is a smoothly flowing, continuous line without sharp angles.
Although a cone has a curved surface, it is not solely a curve.
Option 3: Solid Figure: A solid figure is a three-dimensional object. A cone fits
this definition as it has volume and occupies space.
Option 4: A line is a straight one-dimensional figure having no thickness and
extending infinitely in both directions. A cone is not a line as it is three-dimensional.
The correct option is option 3.
Conic 3D Question 12:
Equation of the sphere that passes through the points (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c) is -
Answer (Detailed Solution Below)
Conic 3D Question 12 Detailed Solution
Concept:
General equation of sphere is x2 + y2 + z2 + 2ux + 2vy + 2wz + c = 0
Explanation:
Now, equation of sphere is x2 + y2 + z2 + 2ux + 2vy + 2wz + c = 0 ....... (i)
Passes through (0,0,0), (a,0,0), (0,b,0), (0,0,c)
Now, when (i) equation passes through (0,0,0) ⇒ c = 0
Now, when (i) equation passes through (a,0,0) ⇒ u = \(\frac{-a}{2}\)
Similarly, For (0,b,0) we get v = \(\frac{-b}{2}\) and for (0,0,c) we get w = \(\frac{-c}{2}\)
By substituting all value we get
x2 + y2 + z2 - ax - by - cz = 0
Now, By completing the square we get
\((x - \frac{a}{2})^2\) + \((y - \frac{b}{2})^2\) + \((z - \frac{c}{2})^2\) = \(\frac{1}{4}\) (a2 + b2 + c2)
Hence, Option (3) is true
Conic 3D Question 13:
Comprehension:
Direction: Consider the following for the items that follow:
Let 2x2 + 2y2 + 2z2 + 3x + 3y + 3z - 6 = 0 be a sphere.
What is the diameter of the sphere?
Answer (Detailed Solution Below)
Conic 3D Question 13 Detailed Solution
Explanation:
Given:
Sphere: 2x2 + 2y2 + 2z2 + 3x + 3y + 3z – 6 = 0
⇒ \(x^2 + y^2+z^2 + \frac{3}{2}x +\frac{3}{2}y + \frac{3}{2}z\) -3 = 0
Now Radius = \(\sqrt{(\frac{3}{4})^2+ (\frac{3}{4})^2+ (\frac{3}{4})^2 + 3}\)
= \(\sqrt{\frac{27}{16} +3} = 5 \frac{\sqrt3}{4}\)
Diamter = \(5 \frac{\sqrt3}{2}\)
∴ Option (b) is correct
Conic 3D Question 14:
What is the diameter of the sphere?
x2 + y2 + z2 − 16x + 12y − 2√dz + d = 0
Answer (Detailed Solution Below)
Conic 3D Question 14 Detailed Solution
Given:
The equation of the sphere is given.
Concept Used:
Comparing the given equation of sphere to the standard form of equation of spheres.
Solution:
We have, standard equation of sphere as: (x - a)2 + (y - b)2 + (z - c)2 = r2
⇒ x2 + y2 + z2 - 2ax - 2by - 2cz + a2 + b2 + c2 - r2 = 0
⇒ comparing the coefficient of x, y, z and constant.
we have, - 2a = - 16 {coefficient of x}
⇒ a = 8
we have, - 2b = 12 {coefficient of y}
⇒ b = - 6
we have, - 2c = -2\( \sqrt{d}\)
⇒ c = \( \sqrt{d}\)
we have, a2 + b2 + c2 - r2 = d
⇒ (8)2 + (- 6)2 + (\( \sqrt{d}\))2 - r2 = d
⇒ 64 + 36 + d - r2 = d
⇒ r2 =100
⇒ r = 10
we know that diameter of sphere is 2 times radius of sphere.
D = 2r
⇒ D = 20
\(\therefore\)Option 2 is correct.
Conic 3D Question 15:
Comprehension:
Direction: Consider the following for the items that follow:
Let 2x2 + 2y2 + 2z2 + 3x + 3y + 3z - 6 = 0 be a sphere.
The centre of the sphere lies on the plane
Answer (Detailed Solution Below)
Conic 3D Question 15 Detailed Solution
Explanation:
The Centre of the sphere is C \((\frac{-3}{4}, \frac{-3}{4}, \frac{-3}{4})\)
4x + 8y + 8z + 15 = 0,
Plane satisfy the centre of sphere – 3 –6 – 6 + 15 = 0
⇒ 0 = 0
∴ Option (d) is correct