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Class 12 Three Dimensional Geometry worksheet
CBSE Mathematics. Direction cosines, lines in vector and Cartesian form, angles, shortest distance, foot of a perpendicular. Every sheet has 20 questions in three sections and an answer key, and the numbers change every time.
Sample questions
Real questions from this chapter, as the app sets them. Tap Answer to see the answer and working.
Section A · multiple choice
- 1.
The direction cosines of a line with direction ratios 3, 2, −6 are:
- 3/7, 2/7, −6/7
- 3/49, 2/49, −6/49
- 3/11, 2/11, −6/11
- 2/7, 3/7, −6/7
Answer
(a) 3/7, 2/7, −6/7
divide each by √(3² + 2² + (−6)²) = 7
- 2.
The direction ratios of the line through (−2, −5, 4) and (−4, −4, −1) are:
- −2, −5, 4
- 1, −2, −5
- −6, −9, 3
- −2, 1, −5
Answer
(d) −2, 1, −5
subtract the coordinates
- 3.
The Cartesian equation of the line through (5, 2, 5) with direction ratios −3, −3, 3 is:
- (x − 6)/(−3) = (y − 2)/(−3) = (z − 5)/3
- (x + 5)/(−3) = (y + 2)/(−3) = (z + 5)/3
- (x + 3)/5 = (y + 3)/2 = (z − 3)/5
- (x − 5)/(−3) = (y − 2)/(−3) = (z − 5)/3
Answer
(d) (x − 5)/(−3) = (y − 2)/(−3) = (z − 5)/3
(x − x₁)/a = (y − y₁)/b = (z − z₁)/c
Section B · short answer
- 4.
Find the vector and Cartesian equations of the line through (−2, −4, 4) and (−5, 3, −1).
Answer
r = (−2î − 4ĵ + 4k̂) + λ(−3î + 7ĵ − 5k̂); (x + 2)/(−3) = (y + 4)/7 = (z − 4)/(−5)
direction (−3, 7, −5)
- 5.
Find the angle between the lines (x − 3)/(−12) = (y + 1)/12 = (z + 4)/1 and (x − 1)/(−11) = (y − 5)/(−2) = (z + 3)/10.
Answer
θ = cos⁻¹(118/255)
cos θ = |118|/(17 × 15)
Section C · long answer
- 6.
Find the shortest distance between the lines r = (−3ĵ − k̂) + λ(4î + 4ĵ − 6k̂) and r = (−î − 4ĵ − 3k̂) + μ(−î − 2ĵ + 6k̂).
Answer
7/11
a₂ − a₁ = −î − ĵ − 2k̂; b₁ × b₂ = 12î − 18ĵ − 4k̂; distance = |(a₂ − a₁) · (b₁ × b₂)|/|b₁ × b₂| = 14/√484
Fresh sheets for your child
In the app, every Three Dimensional Geometry sheet is new: the app remembers the questions your child has already had and leaves them out. Print the sheet as a PDF, or use notebook mode and write the answers in a notebook.
Coming soon for Android and iPhone.