Question

# x, y, z are in AP with common difference d. ∣∣ ∣∣x+yy1y+zz1z+xx1∣∣ ∣∣ =

A
2 d2
B
-3d2
C
- d2
D
d

Solution

## The correct option is B -3d2You can use the property of the determinant that if the terms of one row is given as a sum of two other terms then the determinant can be written as the sum of two determinants as given below, ∣∣ ∣∣x+yy1y+zz1z+xx1∣∣ ∣∣ = ∣∣ ∣∣xy1yz1zx1∣∣ ∣∣ + ∣∣ ∣∣yy1zz1xx1∣∣ ∣∣ ∣∣ ∣∣xy1yz1zx1∣∣ ∣∣ + 0 R2→R2−R1 R3→R3−R1 = ∣∣ ∣∣xy1y−xz−y0z−xx−y0∣∣ ∣∣ = ∣∣ ∣∣xy1dd02d−d0∣∣ ∣∣ = + 1(−d2−2d2) = - 3d2

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