Verified Solution • 2025–26 Session moderate

QJEEM10Ma6dc2: JEE Main 2010 Mathematics - Class 12 JEEM10Ma6dc2 Solution: A line in three-dimensional space makes angles and with the…

A line ABAB in three-dimensional space makes angles 45∘{45^ \circ } and 120∘{120^ \circ } with the positive xx -axis and the positive yy -axis respectively. If ABAB makes an acute angle θ\theta with the positive zz -axis, then θ\theta equals :

Multiple Choice Options

  • A 45∘{45^ \circ }
  • B 60∘{60^ \circ }
  • C 75∘{75^ \circ }
  • D 30∘{30^ \circ }

Step-by-Step Verified Solution

Direction cosines of the line :

ℓ=cos⁡45∘=12,m=cos⁡120∘=−12,π=cos⁡θ\ell = \cos {45^ \circ } = {1 \over {\sqrt 2 }},m = \cos {120^ \circ } = {{ - 1} \over 2},\pi = \cos \theta

where θ\theta is the angle, which line makes with positive zz -axis.

Now ℓ2+m2+n2=1{\ell ^2} + {m^2} + {n^2} = 1

⇒12+14+cos⁡2θ=1,  cos⁡2θ=14\Rightarrow {1 \over 2} + {1 \over 4} + {\cos ^2}\theta = 1,\,\,{\cos ^2}\theta = {1 \over 4}

⇒cos⁡θ=12    (θ\Rightarrow \cos \theta = {1 \over 2}\,\,\,\,\left( \theta \right. being acute)

⇒0=π3\Rightarrow 0 = {\pi \over 3}

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