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Three Dimensional Geometry Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Three Dimensional Geometry Set-4

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

Three Dimensional Geometry is crucial for Class 12 and competitive exams because it builds your core ability to work with vectors, lines, planes, shortest distances, projections, angles, and transformations in 3D. Many JEE/NEET-style questions directly test these ideas through distance-to-line/plane, skew lines, coplanarity, and geometric loci, so mastering this chapter improves both speed and accuracy.

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Q1. Given points $A(1,2,3)$ and $B(4,-1,2)$ , find the angle between the line $AB$ and the line through the origin with direction vector $(1,-1,0)$ .

(A)

$11.6^\circ$

(B)

$13.1^\circ$

(C)

$16.5^\circ$

(D)

$76.9^\circ$

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

(A)

$\dfrac{6}{\sqrt{35}}$

(B)

$2$

(C)

$\dfrac{12}{\sqrt{31}}$

(D)

$\dfrac{12}{\sqrt{35}}$

Q3. Let $P(3,-1,2)$ and let $L$ be the line $\dfrac{x-1}{2}=\dfrac{y+2}{-1}=\dfrac{z-3}{1}$ . Find the foot of the perpendicular from $P$ to $L$ .

(A)

$\left(\dfrac{5}{3},-\dfrac{7}{3},\dfrac{10}{3}\right)$

(B)

$\left(2,-\dfrac{5}{2},\dfrac{7}{2}\right)$

(C)

$\left(\dfrac{4}{3},-\dfrac{5}{3},\dfrac{8}{3}\right)$

(D)

$\left(\dfrac{5}{3},-\dfrac{5}{3},\dfrac{10}{3}\right)$

Q4. Find the locus of a point $P(x,y,z)$ whose distance from the line $L:\;x=y=0$ equals its distance from the plane $\pi:\;z=1$ .

(A)

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

(B)

$x^2+y^2=|z-1|$

(C)

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

(D)

$x^2+y^2+(z-1)=0$

Q5. Let the family of planes through the intersection of $P_1:2x-y+z-3=0$ and $P_2:x+y-z-2=0$ be $P_1+\lambda P_2=0$ . How many members of this family are at distance $3$ from the origin?

(A)

$1$

(B)

$0$

(C)

$2$

(D)

$3$

Q6. Find the perpendicular distance of the point $P(2,-1,3)$ from the plane $3x+4y-12z+7=0$ .

(A)

$\dfrac{3}{13}$

(B)

$\dfrac{27}{13}$

(C)

$\dfrac{29}{13}$

(D)

$\dfrac{27}{14}$

Q7. The skew lines $L_1$ and $L_2$ are given by $L_1:(x,y,z)=(1,0,2)+t(1,2,-1)$ and $L_2:(x,y,z)=(2,1,1)+s(2,-1,1)$ . The shortest distance between $L_1$ and $L_2$ is

(A)

$\dfrac{\sqrt{35}}{3}$

(B)

$\dfrac{3}{35}$

(C)

$\dfrac{1}{\sqrt{35}}$

(D)

$\dfrac{3}{\sqrt{35}}$

Q8. Let $\pi_1: x-y+z=2$ and $\pi_2: 2x+y-3z=1$ . The line of intersection $\pi_1\cap\pi_2$ meets the perpendicular from $P(3,-1,2)$ at the foot $F$ . The coordinates of $F$ are

(A)

$\left(\dfrac{29}{19},\dfrac{6}{19},\dfrac{15}{19}\right)$

(B)

$\left(\dfrac{29}{19},-\dfrac{6}{19},\dfrac{15}{19}\right)$

(C)

$\left(\dfrac{9}{19},\dfrac{6}{19},\dfrac{15}{19}\right)$

(D)

$\left(\dfrac{29}{19},\dfrac{6}{19},-\dfrac{15}{19}\right)$

Q9. Two skew lines are $L_1:(x,y,z)=(1,2,3)+t(1,-1,2)$ and $L_2:(x,y,z)=(3,0,1)+s(2,1,-1)$ . The endpoints of the common perpendicular (closest points) on $L_1$ and $L_2$ , and the shortest distance between the lines, are

(A)

Points $\left(\dfrac{39}{35},\dfrac{66}{35},\dfrac{113}{35}\right)$ and $\left(\dfrac{57}{35},-\dfrac{24}{35},\dfrac{59}{35}\right)$ ; distance $=\dfrac{3}{\sqrt{35}}$

(B)

Points $\left(\dfrac{41}{35},\dfrac{66}{35},\dfrac{113}{35}\right)$ and $\left(\dfrac{57}{35},-\dfrac{22}{35},\dfrac{59}{35}\right)$ ; distance $=\dfrac{18}{\sqrt{35}}$

(C)

Points $\left(\dfrac{39}{35},\dfrac{66}{35},\dfrac{113}{35}\right)$ and $\left(\dfrac{57}{35},-\dfrac{24}{35},\dfrac{59}{35}\right)$ ; distance $=\dfrac{18}{\sqrt{35}}$

(D)

Points $\left(\dfrac{39}{35},-\dfrac{66}{35},\dfrac{113}{35}\right)$ and $\left(\dfrac{57}{35},\dfrac{24}{35},\dfrac{59}{35}\right)$ ; distance $=\dfrac{18\sqrt{35}}{35}$

Q10. For which value of $k$ is the line $L:(x,y,z)=(1+t,t,2+t)$ tangent to the sphere $x^2+y^2+z^2-2x-4y-6z+k=0$ ? Also find the point of tangency.

(A)

$k=10$ , point of tangency $\,(1,0,2)$

(B)

$k=12$ , point of tangency $\,(2,1,3)$

(C)

$k=12$ , point of tangency $\,(1,2,3)$

(D)

$k=13$ , point of tangency $\,(2,1,3)$

Q11. Find the perpendicular distance from the point $P(2,1,1)$ to the line $L$ given by $\mathbf r=(1,0,0)+t(1,-1,1)$ .

(A)

$\sqrt{2}$

(B)

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

(C)

$\dfrac{2\sqrt{6}}{3}$

(D)

$\sqrt{3}$

Q12. The shortest distance between the skew lines $L_1:\mathbf r=(1,2,3)+s(1,-1,2)$ and $L_2:\mathbf r=(2,0,1)+t(3,1,-1)$ is

(A)

$\dfrac{23}{\sqrt{66}}$

(B)

$\dfrac{\sqrt{66}}{23}$

(C)

$\dfrac{23\sqrt{2}}{66}$

(D)

$\sqrt{\dfrac{23}{66}}$

Q13. Let planes $\Pi_1: x+2y+3z=6$ and $\Pi_2: 2x-y+z=3$ intersect in a line $L$ . The equation of the plane through $L$ which is perpendicular to the plane $3x+y-z=4$ is

(A)

$x+y+z=3$

(B)

$x+5y+5z=9$

(C)

$5x+5y+5z=9$

(D)

$5y+5z=9$

Q14. The reflection of the point $P(3,-1,4)$ in the plane $2x-y+2z=1$ has coordinates

(A)

$\left(-\frac{1}{9},\,\frac{5}{9},\,\frac{8}{9}\right)$

(B)

$\left(-\frac{29}{9},\,\frac{19}{9},\,-\frac{20}{9}\right)$

(C)

$\left(\frac{29}{9},\,-\frac{19}{9},\,\frac{20}{9}\right)$

(D)

$\left(-\frac{11}{9},\,\frac{23}{9},\,-\frac{4}{9}\right)$

Q15. Find the plane(s) passing through the point $P(0,0,1)$ that make equal angles with the planes $\Pi_1: x+y=0$ and $\Pi_2: x-y=0$ .

(A)

$x+z=1$

(B)

$y+z=1$

(C)

$x+z=1\ \text{and}\ y+z=1$

(D)

$x+y+z=1$

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