JEE Advanced 2013 · Paper 1 · Mathematics Q60

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MathematicsInteger+4 / -1Question 60 of 60

Consider the set of eight vectors $V=\left\{a\hat i+b\hat j+c\hat k:a,b,c\in\{-1,1\}\right\}$ . Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is

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Answer and worked solution

Official answer: 5

Structure. The 8 vectors are the 4 body diagonals of a cube, each in both directions.

Non-coplanar triples. Two opposite vectors are collinear, so a triple must use 3 different diagonals; any 3 diagonals are non-coplanar (e.g. $\det[(1,1,1),(1,1,-1),(1,-1,1)]=-4\ne0$ ). Choose 3 of the 4 diagonals and a sign for each: $4\times2^3=32=2^5$ .

Answer: 5

Topics:Scalar and vector triple products Combinations

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JEE Advanced 2013 · Paper 1, Mathematics question 60