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Question

If x,y,z are in A.P. with common difference d and the rank of the matrix ∣∣ ∣∣45x56y6kz∣∣ ∣∣ is 2 then the value of d and k are

A
x4; arbitrary number
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B
arbitrary number, 7
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C
x,5
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D
x2,6
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Solution

The correct option is B arbitrary number, 7
Let A=∣ ∣45x56y6kz∣ ∣
Since x,y,z are in AP with common difference d we get

A=∣ ∣45x56x+d6kx+2d∣ ∣

Apply R3R3+R12R2 to get

A=∣ ∣45x56x+d0k70∣ ∣

Rank of A is given to be 2. So rank will be 2 when k7=0
So k=7 and d is any arbitrary number.

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