Vidaara.orgClass 12 · Mathematics
CodeVID-M12-11-DCR-01
Direction Cosines & Ratios — Assignment
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.
Direction cosines satisfy:
- A.$l+m+n=1$
- B.$l^2+m^2+n^2=1$
- C.$lmn=1$
- D.$l^2+m^2+n^2=0$
2.
Direction cosines of a line with DRs $1,2,2$ are:
- A.$1,2,2$
- B.$\tfrac13,\tfrac23,\tfrac23$
- C.$\tfrac12,\tfrac12,\tfrac12$
- D.$\tfrac15,\tfrac25,\tfrac25$
3.
DRs of the line joining $(1,2,3),(4,5,6)$ are:
- A.$(3,3,3)$
- B.$(5,7,9)$
- C.$(1,2,3)$
- D.$(4,5,6)$
4.
A line equally inclined to all axes has each direction cosine:
- A.$\tfrac13$
- B.$\tfrac{1}{\sqrt3}$
- C.$1$
- D.$\tfrac{1}{\sqrt2}$
5.
Direction ratios of a line are:
- A.unique
- B.not unique
- C.always cosines
- D.always $1$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the direction cosines of a line with DRs $2,3,6$.
7.
Find the direction ratios of the line joining $(2,3,4)$ and $(5,6,7)$.
8.
Can $\tfrac12,\tfrac12,\tfrac12$ be the direction cosines of a line?
9.
A line makes angles $90^\circ,60^\circ,30^\circ$ with the axes. Find its direction cosines.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find the direction cosines of the line joining $(1,0,0)$ and $(0,1,1)$.
11.
A line makes equal angles with the three axes. Find that angle.
12.
Find the direction cosines for direction ratios $-2,1,2$.
13.
If the direction cosines are proportional to $1,-2,2$, find them.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Show that the points $A(2,3,-4),\ B(1,-2,3),\ C(3,8,-11)$ are collinear.
15.
A line makes angles $90^\circ,135^\circ,45^\circ$ with the $x,y,z$ axes. Find its direction cosines.
Answer Key
Section A — Multiple Choice Questions
- (B) $l^2+m^2+n^2=1$
- (B) $\tfrac13,\tfrac23,\tfrac23$
- (A) $(3,3,3)$
- (B) $\tfrac{1}{\sqrt3}$
- (B) not unique
Section B — Short Answer (2 marks)
- $\tfrac27,\tfrac37,\tfrac67$.
- $(3,3,3)$ or $(1,1,1)$.
- No.
- $0,\ \tfrac12,\ \tfrac{\sqrt3}{2}$.
Section C — Short Answer (3 marks)
- $\left(-\tfrac{1}{\sqrt3},\tfrac{1}{\sqrt3},\tfrac{1}{\sqrt3}\right)$.
- $\cos^{-1}\!\left(\tfrac{1}{\sqrt3}\right)$.
- $-\tfrac23,\tfrac13,\tfrac23$.
- $\tfrac13,-\tfrac23,\tfrac23$.
Section D — Long Answer (5 marks)
- Collinear.
- $\left(0,-\tfrac{1}{\sqrt2},\tfrac{1}{\sqrt2}\right)$.
Generated by Vidaara.org · Assignment VID-M12-11-DCR-01 · vidaara.org