Maths · Vectors and 3-D geometry · chapter 11 of 13

Class 12 Maths Chapter 11: Three Dimensional Geometry Assertion-reason questions with answers

2 assertion-reason questions on Three Dimensional Geometry, each with its answer and the reason, written from the NCERT chapter. For the CBSE board and JEE Main, which are set from the same NCERT chapter. Open a question's solution to check your working; then take the free test to find your next chapter. Each question gives an Assertion (A) and a Reason (R); choose how they stand to each other.

Question 1
Assertion (A): If a line makes angles α, β, γ with the positive directions of the coordinate axes, then sin2 α + sin2 β + sin2 γ = 2. Reason (R): For any line, cos2 α + cos2 β + cos2 γ = 1.
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Answer: (a) Both A and R are true, and R is the correct explanation of A.

sin2 α + sin2 β + sin2 γ = 3 − (cos2 α + cos2 β + cos2 γ) = 3 − 1 = 2. A follows directly from R.

Question 2
Assertion (A): 3, −6, 2 and −6, 12, −4 are direction ratios of the same line. Reason (R): The direction cosines of a line with direction ratios 3, −6, 2 can be taken as 37, −67, 27.
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Answer: (b) Both A and R are true, but R is not the correct explanation of A.

−6, 12, −4 = −2(3, −6, 2): proportional sets are direction ratios of the same line, so A is true. √(9 + 36 + 4) = 7, so R is true. But A rests on proportionality, not on the values of the direction cosines; R does not explain A.

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