JEE Main Vectors & 3D Geometry: Three Dimensional Geometry – Chapter Test

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Difficulty: 10 medium · 10 hard

Three Dimensional Geometry MCQ questions with answers

The first 3 of 20 questions. Unlock the series for the rest, each with a solution.

1. Q1. The perpendicular distance of the point (1, 2, 3) from the line x = y = z is:

   बिंदु (1, 2, 3) की रेखा x = y = z से लंबवत दूरी है:

   • A. √6
   • B. √2
   • C. √3
   • D. 2√2
   Solution

   Taking A as the origin on the line, AP = (1,2,3) and d = (1,1,1); AP×d = (-1, 2, -1) with magnitude √6, so the distance is √6/√3 = √2.

   रेखा पर A को मूल बिंदु लेने पर, AP = (1,2,3) तथा d = (1,1,1); AP×d = (-1, 2, -1) जिसका परिमाण √6 है, अतः दूरी √6/√3 = √2 है।

2. Q2. Which of the following is NOT a valid set of direction cosines of a line?

   निम्नलिखित में से कौन-सा किसी रेखा की दिक्-कोज्याओं का एक मान्य समुच्चय नहीं है?

   • A. 1/2, 1/2, 1/2
   • B. 1/√2, 1/√2, 0
   • C. 2/3, 2/3, 1/3
   • D. 1/2, 1/2, 1/√2
   Solution

   Direction cosines must satisfy l²+m²+n² = 1. For (1/2, 1/2, 1/2) the sum is 3/4 ≠ 1, so it is invalid; each of the other sets sums to 1.

   दिक्-कोज्याओं को l²+m²+n² = 1 संतुष्ट करना चाहिए। (1/2, 1/2, 1/2) के लिए योग 3/4 ≠ 1 है, अतः यह अमान्य है; अन्य प्रत्येक समुच्चय का योग 1 है।

3. Q3. The direction cosines of the line passing through the points (1, 2, 3) and (3, 5, 9) are:

   बिंदुओं (1, 2, 3) तथा (3, 5, 9) से गुजरने वाली रेखा की दिक्-कोज्याएँ हैं:

   • A. 2/7, 3/7, 5/7
   • B. 1/7, 3/7, 6/7
   • C. 2, 3, 6
   • D. 2/7, 3/7, 6/7
   Solution

   The direction ratios are (2, 3, 6) with magnitude √(4+9+36) = 7, so the direction cosines are 2/7, 3/7, 6/7.

   दिक्-अनुपात (2, 3, 6) हैं जिनका परिमाण √(4+9+36) = 7 है, अतः दिक्-कोज्याएँ 2/7, 3/7, 6/7 हैं।

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