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Match each item in List-I with the correct value from List-II. List-I: (P) If |a| = |b| = |c|, the angle between each pair of vectors is pi/3, and |a + b + c| = sqrt(6), then 2|a| equals (Q) If vector a is perpendicular to (b + c), vector b is perpendicular to (c + a), vector c is perpendicular to (a + b), where |a| = 2, |b| = 3, |c| = 6, then |a + b + c| - 2 equals (R) Given a = 2i + 3j - k, b = -i + 2j - 4k, c = i + j + k, d = 3i + 2j + k, then (1/7)(a x b). (c x d) equals (S) If |a| = |b| = |c| = 2 and a.b = b.c = c.a = 2, then [a b c] * cos(45 deg) equals List-II: (1) 3 (2) 2 (3) 4 (4) 5

1. 3
2. 2
3. 4
4. 5

Correct answer: 2

Solution

P: |a+b+c|² = 3|a|² + 2*(sum of dot products) = 3|a|² + 2*(3*|a|²*cos(60 deg)) = 3|a|² + 3|a|² = 6|a|² = 6 => |a| = 1 => 2|a| = 2. Match (2). Q: a.b + b.c + c.a = 0. |a+b+c|² = 4+9+36 = 49. |a+b+c| = 7. 7-2 = 5. Match (4). R: (axb).(cxd) = (a.c)(b.d) - (a.d)(b.c). a.c = 2-1+3-1 =... compute directly. (1/7)*result = 3. Match (1). S: a.b=2, |a|=|b|=|c|=2. [abc]² = |a|²(|b|²*|c|² - (b.c)²) - a.b*(a.b*|c|² - b.c*a.c) + a.c*(a.b*b.c - |b|²*a.c) = compute to get [abc]*cos45 = 4. Match (3). P->2, Q->4, R->1, S->3.

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