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Consider a parallelogram constructed on the vectors A → = 5 p → + 2 q → and B → = p → - 3 q → . If p → = 2 , q → = 5 , the angle between p → and q → is π 3 and the length of the smallest diagonal of the parallelogram is k units, then the value of k 2 is equal to

Correct answer

109

Step-by-step solution

Diagonals of the parallelogram are A → + B → ,   A → - B → = 6 p → - q → , 4 p → + 5 q → A → + B → = 36 p → 2 + q → 2 - 12 p → ⋅ q → = 144 + 25 - 12 × 2 × 5 × c o s π 3 = 109 A → - B → = 16 p → 2 + 25 q → 2 + 40 p → ⋅ q → = 16 × 4 + 25 × 25 + 40 × 2 × 5 × c o s π 3 = 889 So, length of the smaller diagonal is 109 units

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