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Three Dimensional Geometry Set-1 MCQ Online Practice Test | Class 12 | Quiz Tweek

Three Dimensional Geometry Set-1

Practice Important MCQs for Three Dimensional Geometry — Mathematics, Class 12.

Three Dimensional Geometry is crucial in Class 12 and competitive exams because it builds core skills for working with planes, lines, spheres, and their relative positions. Questions often test vector methods (direction ratios, cross/dot products), geometric interpretation (distance, perpendicularity, tangency), and algebraic modeling of spatial objects—exactly the skills used in board exams and JEE/NEET problem-solving.

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Q1. Find the perpendicular distance from the point $P(2,-1,3)$ to the plane $2x-y+2z-3=0$ .

(A)

$\dfrac{4}{3}$

(B)

$\dfrac{8}{3}$

(C)

$8$

(D)

$\dfrac{4\sqrt{2}}{3}$

Q2. The two lines $L_1:\dfrac{x-1}{2}=\dfrac{y+1}{-1}=\dfrac{z-2}{3}$ and $L_2:\dfrac{x+2}{1}=\dfrac{y-2}{2}=\dfrac{z+1}{-1}$ are skew. The shortest distance between $L_1$ and $L_2$ is

(A)

$3$

(B)

$\dfrac{\sqrt{3}}{3}$

(C)

$\dfrac{2}{\sqrt{3}}$

(D)

$\sqrt{3}$

Q3. A plane passes through the line of intersection of the planes $P_1:x+2y-z-1=0$ and $P_2:3x-y+2z+4=0$ and is perpendicular to the plane $P_3:2x+y+z-3=0$ . The equation of that plane is

(A)

$2x-17y+13z+19=0$

(B)

$x+2y-z-1=0$

(C)

$5x-15y+11z+7=0$

(D)

$2x+17y-13z+19=0$

Q4. Let $S$ be the sphere $x^2+y^2+z^2-2x+4y-6z+3=0$ and $\Pi$ the plane $x+y+z-3=0$ . Consider the statements:
P1: $S\cap\Pi$ is a circle of radius $r=\dfrac{4\sqrt{6}}{3}$ .
P2: The centre of that circle is the orthogonal projection of the sphere's centre onto $\Pi$ .
Which of the following is correct?

(A)

Both P1 and P2 are true but P2 does not correctly explain P1.

(B)

P1 is true and P2 is false.

(C)

Both P1 and P2 are true and P2 correctly explains P1.

(D)

P1 is false and P2 is true.

Q5. Let $A(3,4,0)$ and $B(3,0,4)$ . How many planes that contain the line $AB$ are tangent to the sphere $x^2+y^2+z^2=25$ ?

(A)

Exactly two such planes exist.

(B)

No plane containing $AB$ is tangent to the sphere.

(C)

Exactly one such plane exists.

(D)

Infinitely many such planes exist.

Q6. Find the perpendicular distance of the point $P(3,-1,4)$ from the plane $2x-3y+z-6=0$ .

(A)

$\dfrac{3}{\sqrt{14}}$

(B)

$\dfrac{5}{\sqrt{14}}$

(C)

$\dfrac{\sqrt{14}}{2}$

(D)

$1$

Q7. The skew lines are $L_1:\dfrac{x-1}{2}=\dfrac{y+1}{-1}=\dfrac{z-2}{3}$ and $L_2:\dfrac{x}{1}=\dfrac{y-2}{2}=\dfrac{z+1}{-1}$ . The shortest distance between $L_1$ and $L_2$ is

(A)

$\dfrac{\sqrt{3}}{3}$

(B)

$\dfrac{1}{3}$

(C)

$\dfrac{2}{\sqrt{3}}$

(D)

$\dfrac{\sqrt{3}}{6}$

Q8. Planes $\pi_1: x+y+z-3=0$ and $\pi_2: x-y+2z-4=0$ intersect in a line $L$ . The point on $L$ nearest to the origin is

(A)

$\left(\dfrac{4}{7},\dfrac{2}{7},\dfrac{11}{7}\right)$

(B)

$\left(\dfrac{8}{7},-\dfrac{2}{7},\dfrac{11}{7}\right)$

(C)

$\left(\dfrac{8}{7},\dfrac{2}{7},-\dfrac{11}{7}\right)$

(D)

$\left(\dfrac{8}{7},\dfrac{2}{7},\dfrac{11}{7}\right)$

Q9. A sphere passes through $P(1,1,1)$ and is tangent to the plane $x+y+z-6=0$ at the point $Q(2,2,2)$ . The equation of the sphere is

(A)

$(x-1)^2+(y-1)^2+(z-1)^2=1$

(B)

$\left(x-\dfrac{3}{2}\right)^2+\left(y-\dfrac{3}{2}\right)^2+\left(z-\dfrac{3}{2}\right)^2=\dfrac{3}{4}$

(C)

$\left(x-\dfrac{5}{2}\right)^2+\left(y-\dfrac{5}{2}\right)^2+\left(z-\dfrac{5}{2}\right)^2=\dfrac{75}{4}$

(D)

$\left(x-\dfrac{3}{2}\right)^2+\left(y-\dfrac{3}{2}\right)^2+\left(z-\dfrac{3}{2}\right)^2=\dfrac{3}{2}$

Q10. Let $L_1: \mathbf{r}=(1,0,1)+s(2,-1,1)$ and $L_2: \mathbf{r}=(0,1,2)+t(1,2,-1)$ . The pair of points (one on $L_1$ , one on $L_2$ ) at minimum distance from each other is

(A)

$\left(\left(\dfrac{11}{35},\dfrac{12}{35},\dfrac{23}{35}\right),\;\left(\dfrac{2}{35},\dfrac{39}{35},\dfrac{68}{35}\right)\right)$

(B)

$\left(\left(\dfrac{11}{35},-\dfrac{12}{35},\dfrac{23}{35}\right),\;\left(-\dfrac{2}{35},\dfrac{39}{35},\dfrac{68}{35}\right)\right)$

(C)

$\left(\left(\dfrac{22}{35},\dfrac{12}{35},\dfrac{23}{35}\right),\;\left(\dfrac{2}{35},\dfrac{78}{35},\dfrac{68}{35}\right)\right)$

(D)

$\left(\left(\dfrac{11}{7},\dfrac{12}{7},\dfrac{23}{7}\right),\;\left(\dfrac{2}{7},\dfrac{39}{7},\dfrac{68}{7}\right)\right)$

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