If the points A, B, C and D have position vectors $a, 2 a + b, 4 a + 2 b$ and $5 a + 4 b$ respectively. Then the three collinear points are?

$A, B$ and $C$
$A, C$ and $D$
$A, B$ and $D$
$B, C$ and $D$

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Solution

The correct option is C
$A, B$ and $D$

Apply condition for collinearity,
We know that a set of three points are said to be collinear if and only if they lie in the same line in the same plane.
Given, points $A, B, C$ and $D$ have position vectors $a, 2 a + b, 4 a + 2 b$ and $5 a + 4 b$ respectively.
$A B = (2 a + b) - a = a + b$
$A C = (4 a + 2 b) - a = 3 a + 2 b$
$A D = (5 a + 4 b) - a = 4 (a + b)$
Since, $A D = 4 A B$
So, $A D$ and $A B$ lie in the same line.
Therefore $A, B$ and $D$ are collinear points.
Hence, option C is the correct option.

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