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In triangle ABC, the points D, E and F lie on BC, CA and AB respectively such that BD:DC = 1:4, CE:EA = 3:2 and AF:FB = 3:7. Also, K is a point on AB with AK:KB = 1:3. Then the ratio (AD+BE+CF): CK is

1. 1: 1
2. 2: 5
3. 5: 2
4. None of these

Correct answer: 2: 5

Solution

Using position vectors, AD+BE+CF = (3A+B-4C)/10 while CK = (3A+B-4C)/4. They are parallel, so the magnitude ratio is (1/10):(1/4) = 4:10 = 2:5.

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