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Three Dimensional Geometry Set-2 MCQ Online Practice Test | Class 12 | Quiz Tweek

Three Dimensional Geometry Set-2

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

Three Dimensional Geometry is crucial in CBSE and competitive exams because it builds the core ideas of lines, planes, and distances in 3D space. Concepts like skew lines, shortest distances, intersection lines, and perpendicularity directly appear in board exams and form high-weight questions in JEE/NEET, where students must handle vector and scalar triple-product based reasoning quickly and accurately.

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Q1. A line $L$ passes through the point $A(1,2,0)$ and has direction vector $\mathbf{d}=(1,1,-1)$ . A plane $\Pi$ has equation $2x-y+z-3=0$ . The shortest distance between $L$ and $\Pi$ is

(A)

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

(B)

$1$

(C)

$\dfrac{\sqrt{6}}{2}$

(D)

$\dfrac{1}{\sqrt{2}}$

Q2. Two lines $L_1$ and $L_2$ are given by

\begin{aligned} L_1 &: \; x=1+2t,\; y=-t,\; z=-1+3t \quad (t\in\mathbb{R})<br> L_2 &: \; x=s,\; y=1+2s,\; z=2-s \quad (s\in\mathbb{R}). \end{aligned}

The shortest distance between $L_1$ and $L_2$ equals

(A)

$\dfrac{5\sqrt{3}}{3}$

(B)

$\dfrac{5}{3}$

(C)

$5\sqrt{3}$

(D)

$\dfrac{5}{\sqrt{6}}$

Q3. Let $L$ be the line of intersection of the planes

\begin{aligned} x+y+z-1&=0<br> 2x-y+3z-4&=0. \end{aligned}

The perpendicular distance from the point $P(2,1,-1)$ to the line $L$ is

(A)

$\sqrt{\dfrac{47}{39}}$

(B)

$\sqrt{\dfrac{47}{26}}$

(C)

$\dfrac{\sqrt{47}}{13}$

(D)

$\sqrt{\dfrac{47}{13}}$

Q4. Find the plane passing through the line of intersection of the planes $P_1: x+2y-z+1=0$ and $P_2: 2x-y+3z-5=0$ which is perpendicular to the plane $P_3: x-y+2z-4=0$ . The required plane is

(A)

$5x-5y-2=0$

(B)

$5x+5y-2=0$

(C)

$x-y+2z-4=0$

(D)

$5x+5y+2z-2=0$

Q5. Two skew lines are given by

\begin{aligned} L_1 &: \; x=1+t,\; y=2t,\; z=1+t \quad (t\in\mathbb{R})<br> L_2 &: \; x=2s,\; y=1+s,\; z=2-s \quad (s\in\mathbb{R}). \end{aligned}

Points $P$ on $L_1$ and $Q$ on $L_2$ such that the segment $PQ$ is the shortest line joining the two lines are

(A)

$P\!\left(\tfrac{5}{3},\tfrac{4}{3},\tfrac{5}{3}\right)\;$ on $L_1$ and $\;Q\!\left(\tfrac{4}{3},\tfrac{5}{3},\tfrac{4}{3}\right)\;$ on $L_2$

(B)

$P\!\left(\tfrac{4}{3},\tfrac{5}{3},\tfrac{4}{3}\right)\;$ on $L_1$ and $\;Q\!\left(\tfrac{5}{3},\tfrac{4}{3},\tfrac{5}{3}\right)\;$ on $L_2$

(C)

$P\!\left(\tfrac{5}{3},\tfrac{4}{3},\tfrac{5}{3}\right)\;$ on $L_1$ and $\;Q\!\left(\tfrac{5}{3},\tfrac{4}{3},\tfrac{5}{3}\right)\;$ on $L_2$

(D)

$P\!\left(\tfrac{4}{3},\tfrac{2}{3},\tfrac{4}{3}\right)\;$ on $L_1$ and $\;Q\!\left(\tfrac{2}{3},\tfrac{4}{3},\tfrac{2}{3}\right)\;$ on $L_2$

Q6. Find the perpendicular distance of the point $P(1,2,3)$ from the plane $3x-4y+12z-26=0$ .

(A)

$\dfrac{13}{5}$

(B)

$\dfrac{1}{13}$

(C)

$\dfrac{5}{13}$

(D)

$\dfrac{5}{\sqrt{41}}$

Q7. Find the shortest distance between the skew lines $L_1:\;(x,y,z)=(1,0,2)+t(2,-1,3)$ and $L_2:\;(x,y,z)=(2,1,-1)+s(1,2,-1)$ .

(A)

$\sqrt{3}$

(B)

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

(C)

$3$

(D)

$\sqrt{6}$

Q8. Find the equation of the plane passing through the point $(2,1,1)$ and containing the line of intersection of the planes $x+2y-z=4$ and $3x-y+2z=5$ .

(A)

$5x+3y+13=0$

(B)

$5x-3y-13=0$

(C)

$3x+5y-13=0$

(D)

$5x+3y-13=0$

Q9. The sphere $x^2+y^2+z^2-4x-2y+6z-11=0$ meets the plane $2x-y+2z=7$ in a circle. The radius of that circle is

(A)

$\dfrac{5}{3}$

(B)

$\dfrac{5\sqrt{5}}{3}$

(C)

$5\sqrt{5}$

(D)

$\dfrac{10}{3}$

Q10. Let $\alpha: x+y+z=1$ and $\beta: 2x-y+z=3$ be two planes. If $L$ is their line of intersection, a sphere centered at the origin tangent to $L$ has radius

(A)

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

(B)

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

(C)

$\sqrt{3}$

(D)

$1$

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