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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-11-PLN-01
The Plane — Assignment
Chapter: Three-Dimensional Geometry
Topic: The Plane
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
The normal to the plane $2x+3y-z=5$ is:
  • A.$(2,3,-1)$
  • B.$(5,5,5)$
  • C.$(1,1,1)$
  • D.$(-5,0,0)$
2.
A first-degree equation $Ax+By+Cz+D=0$ (with $A,B,C$ not all zero) represents:
  • A.a line
  • B.a plane
  • C.a sphere
  • D.a point
3.
The $x$-intercept of the plane $4x+y+2z=8$ is:
  • A.$8$
  • B.$2$
  • C.$4$
  • D.$\tfrac12$
4.
Two planes are perpendicular when their normals satisfy:
  • A.$\vec n_1\times\vec n_2=\vec0$
  • B.$\vec n_1\cdot\vec n_2=0$
  • C.$\vec n_1=\vec n_2$
  • D.$|\vec n_1|=|\vec n_2|$
5.
The family of planes through the intersection of $P_1=0$ and $P_2=0$ is:
  • A.$P_1\cdot P_2=0$
  • B.$P_1+\lambda P_2=0$
  • C.$P_1\times P_2=0$
  • D.$P_1/P_2=0$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Find the equation of the plane through $(1,0,-1)$ with normal $\hat i+2\hat j+3\hat k$.
7.
Find the distance of the origin from $2x-3y+6z=14$.
8.
Find the intercepts of the plane $2x+3y-4z=12$.
9.
Are the planes $x+2y-2z=1$ and $2x+4y-4z=7$ parallel?
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Find the angle between the planes $2x+y-2z=5$ and $3x-6y-2z=7$.
11.
Find the distance of $(2,3,-5)$ from the plane $x+2y-2z=9$.
12.
Find the angle between the line of direction $(1,-1,1)$ and the plane $2x-y+z=4$.
13.
Find the equation of the plane through $A(1,1,0)$, $B(1,2,1)$, $C(-2,2,-1)$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Find the equation of the plane through the intersection of $x+y+z=1$ and $2x+3y+4z=5$ that is perpendicular to $x-y+z=0$.
15.
Find the foot of the perpendicular and the image of $P(1,2,3)$ in the plane $2x+y-z=2$.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $(2,3,-1)$
  2. (B) a plane
  3. (B) $2$
  4. (B) $\vec n_1\cdot\vec n_2=0$
  5. (B) $P_1+\lambda P_2=0$
Section B — Short Answer (2 marks)
  1. $x+2y+3z+2=0$.
  2. $2$ units.
  3. $a=6,\;b=4,\;c=-3$.
  4. Yes (normals are proportional).
Section C — Short Answer (3 marks)
  1. $\cos^{-1}\!\left(\tfrac{4}{21}\right)$.
  2. $3$ units.
  3. $\sin^{-1}\!\left(\tfrac{2\sqrt2}{3}\right)$.
  4. $2x+3y-3z=5$.
Section D — Long Answer (5 marks)
  1. $x-z+2=0$.
  2. Foot $\left(\tfrac43,\tfrac{13}{6},\tfrac{17}{6}\right)$; image $\left(\tfrac53,\tfrac73,\tfrac83\right)$.
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