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IIT JEE Advanced 2010 Paper 1

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Let p be an odd prime number and

T_p

be the following set of 2 × 2 matrices:

T_p

\{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a , b , c \in \{0 , 1 , \dots , p - 1\} \}

Section: Mathematics

Question : 72 of 84

Marks: +1, -0

The number of A in

T_p

Solution:

We must have

a^2 - b^2

= kp. Therefore,

(a + b)(a - b) = kp.

Hence, either a − b = 0 or a + b is a multiple of p

When a = b, the number of matrices is p.

When a + b = multiple of p⇒ a, b has p – 1.

Hence, the total number of matrices = p + p – 1 = 2p −1.

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