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JEE • MATHEMATICS • 3D GEOMETRY SYSTEM

3d geometry system practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 121-140 of 140 questions

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121 Easy

121 of 140

Application +4 | -1

Locus of centroid of the triangle whose vertices are

\left( {a\cos t,\;a\sin t} \right),\;\left( {b\sin t,\; - b\cos t} \right)

and (1, 0), where

t

is a parameter, is

View answer and explanation

Correct option: B

{\left( {3x - 1} \right)^2} + {\left( {3y} \right)^2} = {a^2} + {b^2}

Option B (

(3x - 1)^2 + (3y)^2 = a^2 + b^2

) is correct because it represents the locus of the centroid, derived by eliminating the parameter t.

122 Easy

122 of 140

Application +4 | -1

The intercepts of the plane

2 x-3 y+4 z=12

on the coordinate axes are given by

View answer and explanation

Correct option: B

6,-4,3

Option B (

6,-4,3

) is correct because it represents the points where the plane intersects the x, y, and z axes.

123 Easy

123 of 140

Application +4 | -1

The ratio in which the line

x + y = 4

divides the line joining the points

\left( {1,\; - 1} \right)

and (5, 7) is

View answer and explanation

Correct option: A

1:2

Option A (

1:2

) is correct because by using the section formula and substituting the point of intersection into the equation of the line, we find the ratio to be

1:2

124 Easy

124 of 140

Application +4 | -1

The number of lines that can be drawn through the point

(4,-5)

at a distance 12 from the point

(-2,3)

View answer and explanation

Correct option: A

Option A (0) is correct because the distance between the given points is less than the required distance, so no such line can be drawn.

125 Easy

125 of 140

Application +4 | -1

If the vertices

A

and

B

of a triangle

A B C

are given by

(2,5)

and

(4,-11)

and

C

moves along the line

L_1: 9 x+7 y+4=0

, the locus of the centroid of the triangle

A B C

is a straight line parallel to

View answer and explanation

Correct option: D

L_1

Option D (

L_1

) is correct because the locus of the centroid will be a line parallel to the line on which C moves.

126 Medium

126 of 140

Application +4 | -1

A

and

B

are the points

(3,4,5)

and

(-1,3,-7)

respectively, then the equation of set of point

P

such that

P A^2+P B^2=k^2

, where

k

is constant, is

View answer and explanation

Correct option: A

2\left(x^2+y^2+z^2\right)-4 x-14 y+4 z+109-k^2=0

Option A (

2(x^2+y^2+z^2)-4x-14y+4z+109-k^2=0

) is correct because it represents the equation derived from the given condition

PA^2 + PB^2 = k^2

127 Easy

127 of 140

Application +4 | -1

The equation of the sphere whose centre has the position vector

(3 \hat{i}+6 \hat{j}-4 \hat{k})

and which touches the plane

{r} \cdot(2 \hat{i}-2 \hat{j}-\hat{k})=10

View answer and explanation

Correct option: A

|{r}-(3 \hat{i}+6 \hat{j}-4 \hat{k})|=4

Option A (

|{r}-(3 \hat{i}+6 \hat{j}-4 \hat{k})|=4

) is correct because it represents the equation of the sphere with the given center and radius.

128 Medium

128 of 140

Application +4 | -1

G\left( {1,\;4} \right)

is the centroid of triangle

ABC

having its two vertices

A

and

B

\left( {4, - 3} \right)

and

\left( { - 9,\;7} \right)

respectively, then area of the triangle

ABC

in square units, is

View answer and explanation

Correct option: C

\frac{{183}}{2}

Option C (

\frac{{183}}{{2}}

) is correct because the area of triangle ABC is three times the area of triangle GAB, where G is the centroid.

129 Easy

129 of 140

Application +4 | -1

The coordinates of the point of intersection of the line

\frac{x+1}{1}=\frac{y+3}{3}=\frac{z-2}{-2}

with the plane

3 x+4 y+5 z=25

are

View answer and explanation

Correct option: B

(5,15,-10)

Option B (

(5,15,-10)

) is correct because these coordinates satisfy both the line equation and the plane equation.

130 Medium

130 of 140

Application +4 | -1

The equation of the set of point

P

, the sum of whose distances from

A(4,0,0)

and

B(-4,0,0)

is equal to 10, is

View answer and explanation

Correct option: B

9 x^2+25 y^2+25 z^2-225=0

Option B (

9 x^2+25 y^2+25 z^2-225=0

) is correct because it represents the equation of the set of points where the sum of distances from A(4,0,0) and B(-4,0,0) is 10, derived using the distance formula and algebraic manipulation.

131 Medium

131 of 140

Application +4 | -1

A plane passing through

(1,1,1)

intersects the positive direction at

A, B

and

C

, then the volume of the tetrahedron

O A B C

satisfies (V=Volume)

View answer and explanation

Correct option: B

V \geq \frac{9}{2}

Option B (V ≥ 9/2) is correct because using the equation of a plane and applying AM-GM inequality, the volume of the tetrahedron is greater than or equal to 9/2.

132 Medium

132 of 140

Application +4 | -1

If the axes rectangular, the distance from the point

(3,4,5)

to the point where the line

\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}

meets the plane

x+y+z=17

View answer and explanation

Correct option: C

Option C (3) is correct because it represents the distance from the point to the intersection of the line and the plane.

133 Medium

133 of 140

Application +4 | -1

A sphere of constant radius

2 k

passes through the origin and meets the axes in

A, B, C

. The locus of the centroid of the tetrahedron

{OABC}

View answer and explanation

Correct option: C

x^2+y^2+z^2=k^2

Option A (

x^2+y^2+z^2=k^2

) is correct because it represents the locus of the centroid of the tetrahedron OABC, derived from the sphere's equation and the centroid's coordinates.

134 Medium

134 of 140

Application +4 | -1

If the distance between the points

P\left( {a\cos 48^\circ ,0} \right)

and

Q\left( {0,a\cos 12^\circ } \right)

d,

then

{d^2} - {a^2} =

View answer and explanation

Correct option: D

\frac{{{a^2}}}{8}\left( {\sqrt 5 + 1} \right)

The correct answer is D

\frac{{{a^2}}}{8}\left( {\sqrt 5 + 1} \right)

because it is obtained by using the distance formula and trigonometric identities to simplify the expression.

135 Easy

135 of 140

Application +4 | -1

If a straight line makes an angle of

\frac{\pi}{4}

with each of

x

axis and

y

-axis, then the angle made by the line with

z

-axis is

View answer and explanation

Correct option: C

\frac{\pi}{2}

Option C (

\frac{{\pi}}{{2}}

) is correct because the sum of squares of direction cosines must equal 1.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

136 Easy

136 of 140

Application +4 | -1

The projection of the line segment joining the points

(-1,0,3)

and

(2,5,1)

on the line whose direction ratios are

6,2,3

, is

View answer and explanation

Correct option: A

\frac{22}{7}

Option A (

\frac{{22}}{{7}}

) is correct because it is the projection of the line segment on the line with given direction ratios.

137 Medium

137 of 140

Application +4 | -1

P\left( {3,7} \right)

is a point on the line joining

A\left( {1,1} \right)

and

B\left( {6,16} \right),

then the harmonic conjugate

Q

of point

P

has the coordinates

View answer and explanation

Correct option: D

\left( { - 9, - 29} \right)

Option D ((-9, -29)) is correct because it represents the harmonic conjugate of point P, calculated using the section formula externally.

138 Easy

138 of 140

Analyzing +4 | -1

Points

(-2,4,7),(3,-6,-8)

and

(1,-2,-2)

are

View answer and explanation

Correct option: A

Collinear

Option A (Collinear) is correct because the ratios of the differences in coordinates are equal, indicating that the points lie on the same line.

139 Medium

139 of 140

Application +4 | -1

A plane passes through the point

(1,1,1)

cuts the coordinate axes at

\mathrm{A}, \mathrm{B} \& \mathrm{C}

respectively such that

\mathrm{OA}=\mathrm{OB}=\mathrm{OC}

, where

\mathrm{O}

is the origin and volume of tetrahedron

\mathrm{OABC}

\lambda

then value of

2 \lambda

is equal to

View answer and explanation

Correct option: A

Option A (9) is correct because the volume of the tetrahedron is calculated using the intercepts and then doubled.

140 Medium

140 of 140

Application +4 | -1

The vertex

A

of the triangle

A B C

is on the line

\vec{r}=\hat{i}+\hat{j}+\lambda \hat{k}

and the vertices

B

and

C

have respective position vectors

\hat{i}

and

\hat{j}

. Let

\Delta

be the area of the triangle and

\Delta \in\left[\frac{3}{2}, \frac{\sqrt{33}}{2}\right]

, then the range of values

\lambda

corresponding to '

A

' is

View answer and explanation

Correct option: D

[-4,-2] \cup[2,4]

Option D,

[-4,-2] \cup[2,4]

, is correct because it represents the range of

\lambda

values that satisfy the given area condition for the triangle.

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Review 3D GEOMETRY SYSTEM in your textbook or class notes, then use this set to check your understanding.

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