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JEE Advanced 2010Paper 1
Exam
JEE Advanced
Year
2010
Paper
Paper 1
Subject
Mathematics
Chapter
Matrices

JEE Advanced 2010 · Paper 1 · Mathematics · Matrices

Paragraph: Let p be an odd prime number and T_p be the following...

Paragraph: Let pp be an odd prime number and TpT_p be the following set of 2×22 \times 2 matrices T_p=\left\{A=\left[\begin{array}{ll}a & b \\ c & a \end{\text{ array }}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\} Question: The number of AA in TpT_p such that the trace of AA is not divisible by pp but det (A)(A) is divisible by p is [Note : The trace of a matrix is the sum of its diagonal entries.]

Options

  1. A(p−1)(p2−p+1)(p-1)\left(p^2-p+1\right)
  2. Bp3−(p−1)2p^3-(p-1)^2
  3. C(p−1)2(p-1)^2
  4. D(p−1)(p2−2)(p-1)\left(p^2-2\right)

Correct answer

C. (p−1)2(p-1)^2

Step-by-step solution

Trace of A=2aA=2 a, will be divisible by pp iff a=0a=0. ∣A∣=a2−bc|A|=a^2-b c, for (a2−bc)\left(a^2-b c\right) to be divisible by pp. There are exactly (p−1)(p-1) ordered pairs (b,c)(b, c) for any value of aa. ∴\therefore Required number is (p−1)2(p-1)^2. Hence, (c) is the correct option.
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